Operator algebras, operator theory and applications

Operator Theory: Advances and Applications Vol. 181

Editor: I. Gohberg (GLWRULDO2I½FH School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) H. Widom (Santa Cruz)

Operator Algebras, Operator Theory and Applications Maria Amélia Bastos Israel Gohberg Amarino Brites Lebre Frank-Olme Speck Editors

Birkhäuser Basel · Boston · Berlin

Editors: Maria Amélia Bastos Amarino Brites Lebre Frank-Olme Speck Departamento de Matemática Instituto Superior Técnico, U.T.L. $YHQLGD5RYLVFR3DLV  /LVERD3RUWXJDO  HPDLODEDVWRV#PDWKLVWXWOSW DOHEUH#PDWKLVWXWOSW IVSHFN#PDWKLVWXWOSW

 

Israel Gohberg School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University 5DPDW$YLY,VUDHO HPDLOJRKEHUJ#PDWKWDXDFLO

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Contents Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Participants of WOAT 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Summer School: Lecture Notes S.C. Power Subalgebras of Graph C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

B. Silbermann C ∗ -algebras and Asymptotic Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . .

33

H. Upmeier Toeplitz Operator Algebras and Complex Analysis . . . . . . . . . . . . . . . . . . .

67

Workshop: Contributed Articles F.P. Boca Rotation Algebras and Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 121 G. Bogveradze and L.P. Castro On the Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators with Semi-almost Periodic Symbols . . . . . . . . . . . . . . . . 143 L.P. Castro and D. Kapanadze Diffraction by a Strip and by a Half-plane with Variable Face Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

A.C. Concei¸c˜ ao and V.G. Kravchenko Factorization Algorithm for Some Special Matrix Functions . . . . . . . . . . 173 E. Gots and L. Lyakhov On a Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 R. El Harti Extensions of σ-C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

A.Y. Karlovich Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 207

vi

Contents

Yu.I. Karlovich Nonlocal Singular Integral Operators with Slowly Oscillating Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Y.I. Karlovich and L.V. Pessoa Poly-Bergman Projections and Orthogonal Decompositions of L2 -spaces Over Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 V. Kokilashvili and S. Samko Vekua’s Generalized Singular Integral on Carleson Curves in Weighted Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 V. Manuilov On Homotopical Non-invertibility of C ∗ -extensions . . . . . . . . . . . . . . . . . . . 295 S. Mendes Galois-fixed Points and K-theory for GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . 309 K.M. Mikkola and I.M. Spitkovsky Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory . . . . . . . . . . . . . . . . . 321 K. Nourouzi Compact Linear Operators Between Probabilistic Normed Spaces . . . . 347 V.S. Rabinovich and S. Roch Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schr¨ odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 V.S. Rabinovich, S. Roch and B. Silbermann On Finite Sections of Band-dominated Operators . . . . . . . . . . . . . . . . . . . . 385 H. Rafeiro and S. Samko Characterization of the Range of One-dimensional Fractional Integration in the Space with Variable Exponent . . . . . . . . . . . . . . . . . . . . . 393 C.C. Ramos, N. Martins and P.R. Pinto Orbit Representations and Circle Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 N. Samko and B. Vakulov On Generalized Spherical Fractional Integration Operators in Weighted Generalized H¨older Spaces on the Unit Sphere . . . . . . . . . .

429

Editorial Introduction This volume is devoted to the International Summer School and Workshop on Operator Algebras, Operator Theory and Applications, WOAT 2006, held at Instituto Superior T´ecnico in Lisbon, Portugal on 1–5 September 2006. WOAT 2006 was a satellite conference of the International Congress of Mathematicians 2006 that was held in Madrid, Spain. Operator Algebras and Operator Theory are important areas of Mathematics that play an important role in different mathematics areas and its applications, particularly in Mathematical Physics and Numerical Analysis. The main aim of WOAT 2006 was to bring together researchers in the Operator Algebras and Operator Theory areas. This volume contains three lecture notes of the Summer School courses and nineteen articles, contributions to the workshop of the WOAT 2006. The lecture notes, written by leading experts in the fields, are focused on: • Subalgebras of Graph C ∗ -Algebras (S. Power) A self contained introduction to two novel classes of non self-adjoint operator algebras, namely the generalized analytic Toeplitz algebras associated with the Fock spaces of a directed graph and subalgebras of graph C ∗ algebras, are given. The topics are independent but in both cases the focus is on techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids. • C ∗ -Algebras and Asymptotic Spectral Theory (B. Silbermann) An introduction to asymptotic spectral theory is presented using the elementary theory of C ∗ algebras. Given a bounded sequence of matrices with increasing size the spectra, ε-pseudospectra and the singular values of theses matrices are characterized. Three fundamental notions are discussed: stability, fractality and Fredholm sequences. The theory is applied to finite sections of quasidiagonal operators, Toeplitz operators, and operators with almost periodic diagonals. • Toeplitz Operator Algebras and Complex Analysis (H. Upmeier) Recent investigations are presented concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C ∗ -algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation).

viii

Editorial Introduction

The articles were based contributions to the workshop, the majority of them being centered on the main topics of the workshop: • Crossed product C ∗ -algebras. C ∗ -algebras of operators on Hardy and Bergman spaces. Invertibility theory for non-local C ∗ -algebras. Von Neumann algebras. • Approximate methods in operator algebras. Asymptotic properties of approximation operators. • Toeplitz, Hankel, and convolution type operators and algebras. Symbol calculi. Invertibility and index theory. • Operator theoretical methods in diffraction theory. Factorization theory and integrable systems. Applications to Mathematical Physics. The organizers gratefully acknowledge the support of the WOAT 2006 sponsors: the Portuguese Foundation for Science and Technology, Center for Mathematics and its Applications, Center for Mathematical Analysis, Geometry and Dynamical Systems, Research Project FCT/FEDER/POCTI/MAT/59972/2004, as well as Caixa Geral de Dep´ositos, Cˆamara Municipal de Lisboa, Funda¸c˜ao Calouste Gulbenkian, Embassy of Germany in Portugal, Funda¸c˜ao Luso-Americana, and Reitoria da Universidade T´ecnica de Lisboa. Lisbon, September 2007 The Editorial Board

WOAT 2006 – Program Friday, 1 September 2006 Workshop – Room ACI Registration Opening Session Session W1 Nikolai Nikolski Coffee break David Evans Lewis Coburn Lunch

09:00–09:30 09:30–09:50 09:55–10:45 10:50–11:10 11:10–12:00 12:05–12:55 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Summer School – Room ACI Stephen Power Break Course B/Lecture 1 Konrad Schm¨ udgen Coffee break Course C/Lecture 1 Bernd Silbermann Course A/Lecture 1

Saturday, 2 September 2006

09:00–09:50 09:55–10:45 10:50–11:05

11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Course Course Course Course

Session W3 Room QA1.1 D. Mushtari I. Todorov R. Popescu V. Strauss

Workshop Session W2 – Room QA02.3 Ilya Spitkovsky Vladimir Manuilov Coffee break Session W4 Session W5 Room QA1.2 Room QA1.3 M. Ptak E. Gots H. Kaptanoglu P. Lopes M.C. Cˆ amara A. Montes-Rodr´ıguez A. Karlovich H. Rafeiro Lunch

Summer School – Room ACI Harald Upmeier Break Course A/Lecture 2 Stephen Power Coffee break Course B/Lecture 2 Konrad Schm¨ udgen Course D/Lecture 1

A: Subalgebras of Graph C ∗ -algebras B: C ∗ -algebras – Selected topics C: C ∗ -algebras and Asymptotic Spectral Theory D: Toeplitz Operator Algebras and Multivariable Complex Analysis

WOAT 2006 – Program Monday, 4 September 2006

09:00–09:50 09:55–10:20 10:25–10:50 10:50–11:05

11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Workshop Session W6 – Room QA1.1 Session W7 – Room QA1.2 Florin Boca Yuri Karlovich Session W8 – Room QA1.1 Session W9 – Room QA1.2 A. Katavolos R. Duduchava L. Marcoux L. Castro Coffee break Session W10 Session W11 Session W12 Room QA1.1 Room QA1.2 Room QA1.3 M. Dritschel J. Rodriguez A. Concei¸ca ˜o H. Tandra R. Marreiros C. Diogo E. Lopushanskaya K. Nourouzi T. Malheiro H. Mascarenhas M.C. Martins Lunch Summer School – Room ACI Bernd Silbermann Break Course D/Lecture 2 Harald Upmeier Coffee break Course A/Lecture 3 Stephen Power Course C/Lecture 2

Tuesday, 5 September 2006

09:00–09:50 09:55–10:45 10:50–11:05 11:05–11:30 11:35–12:00 12:05–12:30 12:35–13:00 13:00–14:00 14:00–15:30 15:30–15:45 15:45–17:15 17:15–17:30 17:30–19:00

Workshop Session W13 – Room QA1.1 Session W14 – Room QA1.2 Mikhail Agranovich Steffen Roch Nikolai Rabinovich Stefan Samko Coffee break Session W15 – Room QA1.1 Session W16 – Room QA1.2 C. Fernandes N. Samko R. El Harti L. Pessoa C. Ramos G. Bogveradze S. Mendes A. Nolasco Lunch Summer School – Room ACI Konrad Schm¨ udgen Break Course C/Lecture 3 Bernd Silbermann Coffee break Course D/Lecture 3 Harald Upmeier Course B/Lecture 3

List of Participants of WOAT 2006 Agranovich, Mikhail Moscow Institute of Electronics and Mathematics, Russia Al-Rashed, Maryam Imperial College London, United Kingdom Bastos, M. Am´elia Universidade T´ecnica de Lisboa/IST, Portugal Becher, Florian University of Freiburg, Germany Boca, Florin University of Illinois-Urbana-Champaign, USA Bogveradze, Giorgi Universidade de Aveiro, Portugal Bravo, Ant´ onio Universidade T´ecnica de Lisboa/IST, Portugal Cˆ amara, M. Cristina Universidade T´ecnica de Lisboa/IST, Portugal Campos, Hugo Universidade do Algarve/FCT, Portugal Campos, Lina Universidade do Algarve/FCT, Portugal Carvalho, Catarina Universidade T´ecnica de Lisboa/IST, Portugal Castro, Lu´ıs Universidade de Aveiro, Portugal Coburn, Lewis The State University of New York at Buffalo, USA Concei¸ca ˜o, Ana Universidade do Algarve/FCT, Portugal Diogo, Cristina Instituto Superior de Ciˆencias do Trabalho e da Empresa, Portugal Dritschel, Michael University of Newcastle, United Kingdom Duduchava, Roland A. Razmadze Mathematical Institute, Georgia

xii

List of Participants of WOAT 2006

Efendiev, Messoud GSF/TUM-Munich, Germany El Harti, Rachid University Hassan I, Morocco Erkursun, Nazife Middle East Technical University, Turkey Esteves, Ana Universidade de Aveiro, Portugal Evans, David Cardiff University, United Kingdom Fernandes, Cl´ audio Universidade Nova de Lisboa/FCT, Portugal Ferreira dos Santos, Ant´ onio Universidade T´ecnica de Lisboa/IST, Portugal Gots, Ekaterina Voronezh State Technological Academy, Russia Habgood, Joe Queens University Belfast, United Kingdom Kaptanoglu, H. Turgay Bilkent University, Turkey Karlovich, Yuri Universidad Aut´ onoma del Estado de Morelos, Mexico Karlovych, Oleksiy Universidade do Minho, Portugal Katavolos, Aristides University of Athens, Greece Kravchenko, Viktor Universidade do Algarve/FCT, Portugal Lazar, Aldo Tel Aviv University, Israel Lebre, Amarino Universidade T´ecnica de Lisboa/IST, Portugal Lopes, Paulo Universidade T´ecnica de Lisboa/IST, Portugal Lopushanskaya, Ekaterina Voronezh State University, Russia Malheiro, Teresa Universidade do Minho, Portugal Manuilov, Vladimir Moscow State University, Russia Marcoux, Laurent W. University of Waterloo, Canada Marreiros, Rui Universidade do Algarve/FCT, Portugal

List of Participants of WOAT 2006 Martins, Maria do Carmo Universidade dos A¸cores, Portugal Mascarenhas, Helena Universidade T´ecnica de Lisboa/IST, Portugal Mendes, S´ergio University of Manchester, United Kingdom Montes-Rodr´ıguez, Alfonso Universidad de Sevilla, Spain Moura Santos, Ana Universidade T´ecnica de Lisboa/IST, Portugal Mushtari, Daniar Kazan State University, Russia Nikolski, Nikolai Universit´e Bordeaux, France and Steklov Institute of Mathematics, Russia Nolasco, Ana Universidade de Aveiro, Portugal Nourouzi, Kourosh K.N.Toosi University of Technology, Iran Oliveira, Isabel Universidade T´ecnica de Lisboa/IST, Portugal Oliveira, Lina Universidade T´ecnica de Lisboa/IST, Portugal Pereira, Paulo Universidade de Aveiro, Portugal Pessoa, Lu´ıs Universidade T´ecnica de Lisboa/IST, Portugal Pinto, Paulo Universidade T´ecnica de Lisboa/IST, Portugal Popescu, Radu Universidade T´ecnica de Lisboa/IST, Portugal Power, Stephen Lancaster University, United Kingdom Ptak, Marek University of Agriculture of Krakow, Poland Quint˜ ao Braga, Maria Jo˜ ao Universidade Cat´ olica Portuguesa, Portugal Rabinovich, Vladimir Instituto Politecnico Nacional, ESIME, Mexico Rafeiro, Humberto Universidade do Algarve/FCT, Portugal Ramos, Carlos ´ Universidade de Evora, Portugal Roch, Steffen Technical University Darmstadt, Germany

xiii

xiv

List of Participants of WOAT 2006

Rodrigues, C´ atia Universidade de Aveiro, Portugal Rodr´ıguez, Juan Universidade do Algarve/FCT, Portugal Rodr´ıguez Mart´ınez, Alejandro Universidad de Sevilla, Spain Samko, Natasha Universidade do Algarve/FCT, Portugal Samko, Stefan Universidade do Algarve/FCT, Portugal Sangha, Amandip University of Oslo, Norway Santos, Pedro Universidade T´ecnica de Lisboa/IST, Portugal Schm¨ udgen, Konrad University of Leipzig, Germany Silbermann, Bernd Technical University of Chemnitz, Germany Skill, Thomas Philipps-University Marburg, Germany Speck, Frank-Olme Universidade T´ecnica de Lisboa/IST, Portugal Spitkovsky, Ilya College of William & Mary, USA Strauss, Vladimir Simon Bolivar University, Venezuela Tandra, Haryono Bandung Institute of Technology, Indonesia Teixeira, Francisco Universidade T´ecnica de Lisboa/IST, Portugal Todorov, Ivan Queen’s University Belfast, United Kingdom Upmeier, Harald Marburg University, Germany

Operator Theory: Advances and Applications, Vol. 181, 3–32 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Subalgebras of Graph C*-algebras Stephen C. Power Abstract. I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras LG , associated with the “Fock space” of a graph G, and subalgebras of graph C*algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids. Mathematics Subject Classification (2000). Primary 47L40. Keywords. Operator algebras, directed graphs.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 The Cuntz algebras, intuitively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3 Toeplitz algebras for countable directed graphs . . . . . . . . . . . . . . . . . . . . . . . . .

17

4 Subalgebras of On . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5 Appendix: Digraph algebras and limit algebras . . . . . . . . . . . . . . . . . . . . . . . . .

27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

1. Introduction I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras LG , associated with the “Fock space” of a graph G, and subalgebras of graph C*-algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids.

4

S.C. Power

1.1. Generalities on operator algebras Let us set the scene with a bird’s eye view of how operator algebras “come about” and comment on their morphisms. I shall take the term operator algebra to mean a complex algebra of bounded linear operators on a separable complex Hilbert space. For example, the operator algebra A could be the set of all complex single variable polynomials in a given operator T ; that is A = {p(T ) = a0 I + a1 T + · · · + an T n : p(z) = a0 1 + a1 z + · · · + an z n }. Often, the operator algebras of interest are manufactured by specifying a set of generators (such as the set {I, T } in the example) on a Hilbert space both of which (set and space) arise from a “foundational” mathematical structure, such as a group, or a graph, or a dynamical system. We might call this a spatial setting since the Hilbert space is in place at the outset and the operator algebra is taken to be the algebra generated by the given generators. The term “generated” may mean simply the unclosed complex algebra or it may refer to the closure of this algebra in some topology, usually either the operator norm topology or the weak operator topology. We do not assume that the generated algebra is self-adjoint, that is, closed under the conjugate transpose operation X → X ∗ , although of course that will follow if the set of generators is a self-adjoint set. Operator algebras are also constructed in a Hilbert space-free way, for example, as a particular operator algebra, within some huge general class of operator algebras, satisfying a universal property for (perhaps) a set of generators and relations. Alternatively the algebra A might be constructed in the category of normed algebras with the expectation that A is isometrically isomorphic to an operator algebra by virtue of the fact that the ingredients for A are operator algebras. For example A might be a Banach algebra direct limit of operator algebras, or, again, simply a quotient of operator algebras. We shall focus on spatial viewpoints. However, let us recall that the celebrated Gelfand–Naimark theorem can bring us back to a spatial context from a space free one. In truth, there are usually more ready-to-hand ways of providing a Hilbert spaces on which an indirectly constructed operator algebra can sit. Theorem 1.1. Let A be a C*-algebra (an involutive complete normed algebra with ab ≤ ab and a∗ a = a2 for all a, b ∈ A). Then there is a Hilbert space H (a separable one is possible if A is separable) and an isometric star homomorphism A → B(H). A fundamental question for a class of operator algebras is: When are two operator algebras A1 and A2 isomorphic? The strongest sense of isomorphism is undoubtedly unitary equivalence, that is, A1 = U A2 U ∗ for some isometric onto map U from the Hilbert space of A2 to that of A1 . Here the operator algebras really are the same if only we would identify the Hilbert spaces. A weaker form of isomorphism which also takes account of how the operator algebra sits on the underlying Hilbert space is star-extendible isomorphism. This requires that there is a map φ : A1 → A2 which is the restriction

Subalgebras of Graph C*-algebras

5

of an adjoint respecting algebra isomorphism φ : C ∗ (A1 ) → C ∗ (A2 ) between the generated C*-algebras. Weaker still, and now ignoring how the operator algebras are manifested, an isometric algebra isomorphism is simply an algebra isomorphism which is an isometric linear map. For nonselfadjoint operator algebras this form of isomorphism is usually the essential case to elucidate. In truth, while these forms of isomorphisms certainly are different, in the case of operator algebras constructed from the same spatial scheme, as alluded to above, the resulting forms of isomorphism type are usually the same. By this I mean that if A1 and A2 are isomorphic in one of the three sense above then they are isomorphic in the other senses. Is there a metatheorem here I wonder? Let us also note a companion question to that above, which is generally deeper, concerning the symmetries of an operator algebra A. What is the isometric automorphism group of an operator algebra? Naturally one expects that when two instances of a foundational structure are isomorphic then this entails an isometric isomorphism between the associated operator algebras and indeed this is generally a routine verification. (We might more formally realize the association as a functor.) But how about the converse direction? If A(S1 ) and A(S2 ) are the (norm closed say) operator algebras obtained from the foundation structures S1 and S2 , and if Φ : A1 → A2 is an isometric algebra isomorphism then does this somehow induce an isomorphism between S1 and S2 ? This would provide a satisfyingly definitive answer to the isomorphism question and it is in this connection that there is, as we say, clear blue water between the non-self-adjoint and the self-adjoint theory. Algebras of the former category seem to remember their foundations while the self-adjoint algebras need not.1 As an indication of the general landscape ahead here is a list of the ingredients of several operator algebra contexts concerning a hierarchy of Toeplitz algebras. A: The classical context : The Hardy–Hilbert space H 2 for the unit circle, the unilateral shift operator S, with dim(I − SS ∗ ) = 1, the disc algebra A(D), the function algebra H ∞ (D), realizations of A(D) and H ∞ (D) as “analytic” Toeplitz algebras, and the Toeplitz C*-algebra TZ+ . Let us outline some important classical facts. The Hilbert space 2 (Z+ ) is unitarily equivalent to the Hardy space H 2 of the Hilbert space L2 (T) of square integrable functions on the circle. Here H 2 denotes the closure in L2 (T) of the space of polynomials in z. The basis matching unitary U : 2 (Z+ ) → H 2 which does this (with U en := z n for n ≥ 0) effects a unitary equivalence between the unilateral shift S (with Sen = en+1 , n ≥ 0) and the multiplication operator Tz : f → zf, f ∈ H 2 . That is U SU ∗ = Tz . C*-algebras it is K-theory and associated invariants that often lead to classifying invariants. In general such invariants are insufficient as they are generally determined by the diagonal part A ∩ A∗ of the operator algebra A.

1 For

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S.C. Power

More generally, the Toeplitz operator Tφ with symbol function φ in C(T) or L∞ (T) is given by Tφ : f → PH 2 φf . When φ is analytic, that is, is the boundary function of an analytic function on the disc, then Tφ is simply the restriction of the multiplication operator Mφ to H 2 . In this way we have a representation of the so called disc algebra A(D) as an algebra of operators on H 2 . A good exercise to do at least once is to show that the Toeplitz algebra TZ+ = C ∗ (I, Tz ) (which is equal to U C ∗ (I, S)U ∗ ) contains every compact operator K on H 2 . The idea is to start with the rank one operator I − Tz Tz∗ and “move it around” with the shifts Tz , Tz∗ to obtain every rank one operator of the form f → f, z k z l . For example, show that Tzn (I − Tz Tz∗ ) is the matrix unit operator En,0 : f → f, z 0 z n . Then from these operators and their adjoints create all the matrix units Ei,j . Finally, take linear combinations to approximate any finite rank operator. Once this is done one a little more work leads to the following theorem. Theorem 1.2. (i) For each Toeplitz operator Tφ and compact operator K we have Tφ + K ≥ Tφ . (ii) The Toeplitz algebra TZ+ is equal to the set of operators Tφ +K with φ in C(T) and K compact and the quotient TZ+ /K is naturally isomorphic to C(T). On the other hand the norm closed operator algebra generated by Tz is abelian and isometrically isomorphic to the disc algebra. Indeed, it is the algebra {Tφ : φ ∈ A(D)}. The weak operator topology closed algebra is similarly a copy of H ∞ (D), namely, {Tφ : φ ∈ H ∞ (D)}. On occasion we simply write H ∞ for this operator algebra when the context is clear. Recall that the weak operator topology is the weakest topology for which the spatial linear functionals T → T f, g are continuous. There are a great many ways in which one can move on from the Toeplitz context above and below I discuss some aspects of the following operator algebra directions. B: The (spatial) free semigroup context (Section 3.): The Fock space 2 (F+ n) for the free semigroup on n generators, the freely noncommuting shifts L1 , . . . , Ln with dim(I − (L1 L∗1 + · · · + Ln L∗n )) = 1, the noncommutative disc algebra An and free semigroup algebra Ln , and the Cuntz–Toeplitz C*-algebra on 2 (F+ n ). C: The (spatial) graph context (Section 3.): The Fock space of a directed graph G = (V, E), the freely noncommuting partial isometries Le , e ∈ E, the tensor algebra AG , the free semigroupoid algebras LG , and the Cuntz–Krieger– Toeplitz C*-algebras TG = C ∗ (AG ). D: The (universal) free semigroup context (Sections 2, 4): The freely noncommuting isometries S1 , . . . , Sn with S1 S1∗ + · · · + Sn Sn∗ = I, the Cuntz algebras On = C ∗ (S1 , . . . , Sn ).

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E: The (universal) graph context (Section 4): The (universal) graph C*algebra C*(G) of a countable directed graph G = (V, E) with partial isometry generators Se , for e ∈ E, and relations
Se Se∗ = Px , Se∗ Se = Ps(e) , r(e)=x

where {Px : x ∈ V } is a family of orthogonal projections and e = (r(e), s(e)).

2. The Cuntz algebras, intuitively The Cuntz algebra On is a certain C*-algebra generated by n isometries, S1 , . . . , Sn say, satisfying S1 S1∗ + · · · + Sn Sn∗ = 1. That is, their range projections Si Si∗ are orthogonal and sum to the identity operator. In fact I could have dropped the word “certain” because of the following remarkable uniqueness property. Theorem 2.1. If n ≥ 2 and s1 , . . . , sn is any family of n isometries in a unital C*algebra with s1 s∗1 + · · · + sn s∗n = I, then C ∗ ({s1 , . . . , sn }) is naturally isometrically isomorphic to On . Our main aim is to obtain tools and results that will help in understanding norm closed subalgebras of the Cuntz algebras. In this connection we are prepared to consider operator algebras generated by semigroups of words in the generators and to contemplate quite general subalgebras. Perhaps it is fair to say that a C*-algebraist is largely happy with the state of knowledge of the Cuntz algebras. He/she is more interested in looking for generalisations and wider classes to classify and understand (such as Graph C*-algebras, or C*-algebras allied to dynamical systems). Our view here is quite different – we are intending to linger with On and look inside it with a view to understanding classes of nonselfadjoint operator algebras. Our orientation and motivation comes partly from the existing theory of limit algebras which are found as nonselfadjoint subalgebras of approximately finite C*-algebras. We shall focus on the Cuntz algebras for clarity but the methods we discuss do extend to more general graph C*-algebras. 2.1. Cuntz algebra basics One direct way to define On is to look into the next section, take the freely noncommuting shifts L1 , . . . , Ln on the Fock space for the free semigroup on n generators, take the generated C*-algebra and divide out by the ideal of compact operators. (This should sound familiar if n = 1!) In taking the quotient we lose the Hilbert space and gain equality in place of the one-dimensional defect dim(I − L1 L∗1 + · · · + Ln L∗n ) = 1. The uniqueness allows us to consider two other models for On which will in fact be our viewpoint. I call these models the interval picture and the Cantorised interval picture. The first uses isometric operators S1 , . . . , Sn on L2 [0, 1] whose i ranges are the orthogonal subspaces L2 [ i−1 n , n ]. For definiteness we may define √ i−1+x i Si ( n ) = nf (x), for x ∈ [0, 1], and Si f )(t) = 0 for t not in [ i−1 n , n ].

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Notice that a product Si1 Si2 . . . Sik has range L2 [Ek ] where Ek is an interval with |Ek | = n−k . Write Sµ for this product where µ is the word i1 i2 . . . ik . These k-fold products have distinct ranges and so if the lengths of µ and ν are |µ| = |ν| = k then Sµ∗ Sν is the zero operator when µ and ν differ. (Exercise: Prove this algebraically.) It follows in general that if µ and ν have differing lengths and Sµ∗ Sν is nonzero then either Sµ∗ Sν = Sµ∗
where µ = νµ , or Sµ∗ Sν = Sν
where ν = µν  . On the other hand products with stars on the right are always non-zero. Indeed, from the interval picture we see that for |µ| = |ν| = k the operator Sµ Sν∗ acts as an isometry from L2 [Eν ] to L2 [Eµ ] for certain intervals Eµ , Eν with lengths |n|−k . Moreover, the set of operators {Sµ Sν∗ : |µ| = |ν| = k} satisfies the relations of a matrix unit system. The span of this set, Fkn say, is thus a copy of the matrix algebra Mnk , and we have the matrix algebra tower F1n ⊆ F2n ⊆ F3n · · · . n We see then that the generators of On provide a distinguished subalgebra F∞ = ∞ n ∞ n ∪k=1 Fk which we refer to as a matricial star algebra of type n . Write F for the closure of this subalgebra in On . Notice that the diagonal matrix units are those of the form Sµ Sµ∗ . The closed linear span of these is an abelain algebra, C say, in F n and On which plays a distinguished role.

0

1

1

0 1/4

1/2

Figure 1. Interval picture for the operator S1 S2 in O2 . Theorem 2.1 gives rise immediately to an important family of automorphisms of On , the so called gauge automorphisms γz , for z ∈ T, which satisfy γz (Si ) = zSi , 1 ≤ i ≤ n. (Alternatively, these automorphisms are inherited from easily defined unitarily implemented automorphisms of the Cuntz-Toeplitz C*-algebra.)

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Proposition 2.2. (i) Each operator a in the star algebra generated by S1 , . . . , Sn has a representation N N

(S1∗ )i a−i + a0 + ai S1i a= i=1

i=1

n F∞

where ai ∈ for each i. This representation is unique if for each i ≥ 0 we have ai = ai Pi and a−i = Pi a−i where Pi is the final projection of S1i . (ii) The linear maps Ei , defined by Ei (a) = ai , extend to continuous, contractive, linear maps from On to F n . (iii) The generalised Cesaro sums Σk (a) =

N


1−

k=1

N
 |k|  ∗ k |k|  (S1 ) E−k (a) + Ek (a)sk1 1− N N k=0

converge to a as N → ∞. Proof. Our observations above show that the linear span of the operators Sµ Sν∗ is the algebra generated by the generators. The representation in (i) (with the dilation actions carried by S1 alone) follows from this by means of formulae such as Sµ = (Sµ (S1∗ )k )S1k = aS1k where k = |µ|. The key to uniqueness is to make use of “recovery formulae” such as  2π dθ a0 = γz (a) 2π 0 where the integral is a Riemann integral of a norm continuous function. The representation in (i) can be viewed as a generalised Fourier series representation for the operator a. In fact to any operator a in On one may assign generalised Fourier coefficients ak in F n by means of the maps Ek (.). The operators ak S1k (k > 0) and (S1∗ )k a−k (k < 0) appear as the conventional Fourier series coefficients for the norm continuous operator-valued function fa : z → γz (a). The Cesaro polynomials pn (z) for the continuous function fa converge uniformly in operator norm on |z| = 1 by classical theory. Finally, the specialisation z = 1 gives the desired norm convergence of the generalised Ces´aro polynomials.
Exercises. (i) Show that E0 is faithful, that is, if a ≥ 0, a = 0 then E0 (a) = 0. (ii) Show that   ∗k γz (aS1 )dz S1k , k > 0. ak = T

The Cantorised interval picture for On is a refinement of the interval presentation. The beauty of this perspective is that it provides a context for defining binary relation (and groupoid) invariants for subalgebras. The essence of the idea is captured in Figure 2, shown with 2-fold branching relevant to the n = 2 case.

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0

1 01

00

11

10 101

100

0

x

1

X

Figure 2. Cantorised interval picture. Let X be the set of infinite paths on the tree, starting at vertex 0 or vertex ∞ 1. This set is identifiable with the direct product k=1 {0, 1}, consisting of points x which are zero-one sequence x1 x2 . . . . Each vertex word w in the tree, such as 1001, gives rise to an “interval” Ew of points x whose product expansion starts with w. With the usual product topology the direct product is a Cantor space whose topology has the set of intervals as a base of closed-open sets. For each pair of vertex words w1 , w2 there is a partial homeomorphism αw2 ,w1 from Ew1 to Ew2 defined by matching tails: if x = w1 xp xp+1 . . .

then

αw2 ,w1 (x) = w2 xp xp+1 . . .

Notice that for |w1 | = |w2 | = k the partial homeomorphism has an action on ∗ the set X that bears close analogy with Sw2 Sw and its interval picture, where we 1 have relabeled the generators as S0 and S1 . In fact we can add the natural product probability measure to X and present On on L2 (X) in terms of (newly labeled) generators S0 and S1 induced by the partial homeomorphisms α∅,0 : x → 0x, α∅,1 : x → 1x. That is, for i = 1, 2 we have (Si f )(α∅,0 (x)) =

√ 2f (x),

and (Si f )(y) = 0 if y is not in the range of α∅,i . Exercise. Obtain the partial homeomorphism that is associated with the partial ∗ . (Here we have the 0-1 subscript labeling as opposed to isometry S1 S0∗ + S00 S11 the 1-2 labeling.) 2.2. Normalising partial isometries We now come to an important class of partial isometries associated with the abelian diagonal algebra C.

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Definition 2.3. A partial isometry in F n (or, more generally, in On ) is C-normalising if vCv ∗ ⊆ C and v ∗ Cv ⊆ C. The obvious examples are the matrix units Sµ Sν∗ for |µ| = |ν| = k, and the sums v of these when the initial projections are orthogonal and the final projections are orthogonal. Also we may multiply such a v by a unitary diagonal element d. The resulting C-normalising partial isometry has support which can be indicated pictorially, as shown. The coordinates have been arranged so that the support picture represents v intuitively as a continuous matrix (although the d information is now lost). The picture should be “Cantorised” and viewed as a subset of X × X. 0

1

0

1 ∗ Figure 3. The support the partial isometry S2 S1∗ + S11 S22 .

The next theorem is an extremely useful characterisation. It shows in particular that for subalgebras of F n an isometric isomorphism α : A1 → A2 with α(C) = C preserves the set of normalising partial isometries in the algebras. Theorem 2.4. Let v be an element of F n . Then the following assertions are equivalent: (i) v is a C-normalizing partial isometry. (ii) v is an orthogonal sum of a finite number of partial isometries of the form dSµ Sν∗ , where |µ| = |ν| and d ∈ C. (iii) For all projections p, q ∈ C, the norm qvp is equal to 0 or 1. First we note some general “recovery facts” about F n which show how operators may be approximated in an explicit manner. For b ∈ F n the “diagonal part” ∆(b) ∈ C may be defined as the limit of the block diagonal operators bk := Σi ekii bekii

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where ekii are the diagonal matrix units of Fkn . The map ∆ : F n → C is a projection and moreover is faithful in the sense that ∆(b∗ b) = 0 entails b = 0. Likewise one can use block diagonal maps (via matrix unit projections taken from the commutants n n n of Fm in Fkn , k = m, m + 1, . . . ) to define explicit maps ∆m : F n → F˜m where F˜m ∗ n n m m is the C*-algebra C (C, Fm ) (which in fact is identifiable with Fm ⊗ (e11 Ce11 ) and we have ∆m (b) → b as m → ∞ for all b ∈ F n . In this way (and analogously to Cesaro convergence) we can approximate a general element b in F n by explicitly n . constructed approximants ∆m (b) in F˜m Proof of Theorem 2.4: The directions (ii) =⇒ (i) =⇒ (iii) are elementary so assume that v is an element which satisfies the zero one norm condition. Choose m large so that v = ∆m (v) + v  , v   < 1. Since v satisfies the zero one norm condition this is also true of ∆m (v) and v  . Indeed, this holds for operators in the matrix subalgebras and by approximation holds in general. The implication (iii) implies (ii) is straightforward for elements of F˜kn . Thus it remains to show that if v  has norm less than one and satisfies the zero one norm condition then v  = 0. This too follows from approximation. Exercise. Show that F n has no proper closed two-sided ideals. (Hint: E0 is faithful.) Remark. The map a → ∆(E0 (a)) is a positive faithful contraction onto the diagonal algebras but it is not (as above) defined as a limit of block diagonal parts. (eg consider a = S1 ). (However, as we note in the Notes below, it may be defined in a more subtle algebraic manner.) n 2.3. Subalgebras of F∞ With the two models for On above and generalised Fourier series we are almost ready to contemplate the following vague question. What are the natural subalgebras of On ?

Before turning to this we should of course look first inside the C*-algebra n Mn and the algebras F∞ , F n. The most natural subalgebras of Mn are perhaps those unital subalgebras A which are spanned by a subset of the standard matrix unit system {eij : 1 ≤ i, j ≤ n}. In fact these subalgebras are precisely those that contain the diagonal algebra C spanned by {eii }. (It is this latter property that we essentially use to define infinite-dimensional variants.) In this case the set E = {(i, j) : eij ∈ A} can be viewed as the set of edges of a directed graph G with n vertices. It follows that (i, i) ∈ E for all i and that (i, k) ∈ E if (i, j), (j, k) ∈ E. Conversely, if G is the graph (V, E), with E such a reflexive and transitive relation, then A(G) := span{eij : (i, j) ∈ E} is a complex unital subalgebra containing C. These algebras are the building block algebras of limit algebras. See the Appendix for a fuller discussion.

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It is an elementary but worthwhile exercise to show that the operator algebra A(G) remembers the graph G: Proposition 2.5. (Recovery theorem.) If A(G1 ) and A(G2 ) are isometrically isomorphic algebras then the graphs G1 , G2 are isomorphic. Sketch proof: (i) Projections must map to projections (since projections are the idempotents with norm one), (ii) minimal projections map to minimal projections, (iii) one can reduce (via unitary equivalence) to the case that diagonal projections map to diagonal projections and this gives the needed vertex map. We can think of E both as a “support set” for the algebra (should we view a matrix (aij ) as a function ij → aij ), and also, more usefully, as a binary relation that comes with the algebra. In these terms the proposition says that a digraph algebra has isometric isomorphism type determined by the isomorphism type of its binary relation. In Section 4 I give a generalisation of this fact for a wide class of subalgebras of On . First however, let us look inside the matricial star algebra n F∞ and its closure F n . In fact we may as well consider a more general class of approximately finite algebras. Definition 2.6. (a) A unital matricial star algebra is a complex algebra B for which there exists a spanning set {ekij : 1 ≤ i, j ≤ nk , k = 1, 2, . . . } such that (i) for each k the set {ekij : 1 ≤ i, j ≤ nk } is a matrix unit system for Mnk , (ii) for each k, Mnk ⊆ Mnk+1 and moreover the inclusion map is a C*algebra injection which maps each ekij to a sum of matrix units from {ek+1 : 1 ≤ i, j ≤ nk }. ij (b) A regular matricial algebra is a complex algebra A which is a unital subalgebra of a matricial star algebra containing the diagonal subalgebra C = span{ekij }. It is straightforward to see that the regular matricial algebra A in the matricial star algebra B is the union of the algebras Ak = A ∩ Mnk = span{ekij : ekij ∈ A} and that each Ak is a digraph algebra A(Gk ) relative to the given matrix unit sysn tem. Our subalgebra F∞ is a particular unital matricial star algebra in which each inclusion map has the same multiplicity n. Further examples of such algebras can be obtained with the technology of direct limits: first take a tower of appropriate inclusions maps, A(G1 ) → A(G2 ) → . . . . Algebraic direct limits then provide the algebra A = lim→ A(Gk ) as well as B and C.

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2.4. Subalgebras of unital AF C*-algebras Suppose we have the purely algebraic setting C ⊆ A ⊆ B given in Definition 2.4. The matricial star algebra carries a unique C*-algebra norm and taking operator norm closures gives the triple inclusion C ⊆ A ⊆ B. Here B is a UHF C*-algebra, C is a particularly nice abelian subalgebra in B and A is an instance of a limit algebra A = lim(A(Gk ), φk ) →

where the inclusion maps φk : A(Gk ) → A(Gk+1 ) are particularly nice. (In the terminology of the Appendix, they are star-extendible and regular.) We give three key results for such limit algebras. The first, rather surprisingly perhaps, shows that any norm closed algebra A with C ⊆ A ⊆ B is necessarily the closure of a regular matricial algebra. This is justification for the opinion that the subalgebras of UHF C*-algebras which contain the distinguished masa are the natural generalisations of finite-dimensional digraph algebras. The following “local recovery” theorem gives a key step towards understanding the limit algebras A. Theorem 2.7. (Inductivity principle.) Let B be a unital matricial star algebra with subalgebra chain {Mnk } and diagonal C and let B, C be their norm closures. If A ⊆ B is a norm closed subalgebra containing C then A is the closed union of the digraph algebras Ak = A ∩ Mnk , k = 1, 2, . . . . The next two theorems (and that above) have more general forms for subalgebras of AF C*-algebras but we state them here for subalgebras of the UHF C*-algebra F n . We first need to define the appropriate substitute for the graph that underlies a digraph algebra and for this the Cantorised interval picture provides what we need, both for F n and for On . In fact we are going to define an isometric isomorphism invariant for the algebra A together with its diagonal C which is in the category of topological binary relations. Often (always?!) the binary relation is a complete isometric isomorphism invariant for the algebra alone. ∞ Let X be the Cantor space k=1 {0, 1, . . . , n − 1}. For words µ, ν with the same length k recall that Sµ Sν∗ is a partial isometry on L2 (X) which is induced by the partial homeomorphisms αµ,ν : νxk+1 xk+2 · · · → µxk+1 xk+2 . . . . Define the topological space R to be the set in X × X which is the union of the graphs of these partial homeomorphism: R = {(α(x), x) : x ∈ dom(α), α = αµ,ν , |µ| = |ν| = k, k = 1, 2, . . . } (k)

k for the graph of the partial homeomorphism for the matrix unit eij Write Ei,j n in Fk and one can conceive of these sets as the “support set” of the matrix (k) units. The diagonal matrix units eij provide closed-open sets Eiik in the diagonal ∆ = {(x, x) : x ∈ X} and these give a base for the natural Cantor space topology. k We topologise R by taking the sets Eij as a base for the topology.

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Exercise. Show that R is an equivalence relation. Show that the topology is not the relative product topology. It follows from Theorem 2.7 that if C ⊆ A ⊆ F n then A, and the given subalgebra chain, determines a subset R(A) of R, namely k R(A) := R(A) := ∪{Eij : ekij ∈ Ak }.

With the relative topology, R(A) is known as the topological binary relation of A. In fact the topological binary relation R(A) is determined by the pair (A, C) and serves as the analogue of the graph of a digraph algebra. The following uniqueness theorem also follows from Theorem 2.7. Theorem 2.8. (Spectral theorem for subalgebras.) Let A1 , A2 be norm closed subalgebras of F n which contain the canonical diagonal algebra C. If R(A1 ) = R(A2 ) then A1 = A2 . That R(A) is intrinsic to the pair A, C is also revealed by the following proposition. We identify X with the Gelfand space of C. Proposition 2.9. As a set, R(A) is the set of points (x, y) in X × X for which there exists a ∈ A and δ > 0 such that paq ≥ δ for all projections p, q in C with x(p) = y(q) = 1. We can now state the following classification theorem which, loosely paraphrased, asserts that a triangular subalgebra of F n remembers its topological binary relation. Theorem 2.10. Let A1 and A2 be norm-closed subalgebras of F n with Ai ∩ A∗i = C for i = 1, 2. (Such algebras are said to be triangular.) Then the following statements are equivalent n n and A2 ∩ F∞ are isometrically isomorphic normed algebras. (i) A1 ∩ F∞ (ii) A1 and A2 are isometrically isomorphic operator algebras. (iii) The topological binary relations R(A1 ), R(A2 ) are isomorphic, that is, there is a homeomorphism α : M (C) → M (C) such that α × α : R(A1 ) → R(A2 ) is a homeomorphism.

The key to the proofs of Theorem 2.7, Theorem 2.8, Proposition 2.9 and Theorem 2.10 is the structure of partial isometries given in Theorem 2.4. It is this which which makes the link between operator algebra entities and the underlying topological binary relation. Remarkably, perhaps, there is a close generalisation of Theorem 2.10 to gauge invariant subalgebras of On . (Theorem 4.3.) Sketch proof of Theorem 2.10. Let Φ : A1 → A2 be an isometric isomorphism. By triangularity, the set of projections p in Ai generate Ci . Since the projections are the norm one idempotents it follows that Φ(C1 ) = C2 . By the zero-one characterisation

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in Theorem 2.4 it follows that Φ maps the C1 -normalising partial isometries to C2 normalising partial isometries. Considering the support of a normalising partial isometry as a closed-open set in M (Ci ) × M (Ci ) it follows (from the finiteness of Theorem 2.4 (ii)) that Φ maps a base of closed open sets to a base of closed-open sets. Moreover Φ induces a point bijection α : R(A1 ) → R(A2 ). (Take intersections of neighbourhoods.) The point bijection is a topological homeomorphism since its map on sets extends the original bijection of closed-open sets. Open problems. (See also the Notes below.) 1. For the algebraic context C ⊆ A ⊆ B given at the end of Section 2.3 is C unique in A up to automorphisms of A? If this is the case then R(A) becomes a well-defined invariant for the algebra A. 2. Is R(A) an algebraic isomorphism invariant for a regular matricial algebra A? 2.5. Notes Theorem 1.2 is usually referred to as Coburn’s theorem. For more on this, Cuntz algebras and other C*-algebras, see, for example, Davidson [3] and the references therein. On is usually defined as the universal C*-algebra generated by isometries S1 , . . . , Sn with S1 S1∗ + · · · + Sn Sn∗ = I. The direct sum of irreducible realisations of such relations gives generators for this algebra; the universal property is that to any isometric realisation s1 , . . . , sn of the relations there should exist a canonical homomorphism On → C ∗ ({s1 , . . . , sn }) and this is readily checked for this universal direct sum. The uniqueness theorem, Theorem 2.1, will follow now from the simplicity of On , and this in turn follows readily from an algebraic formulation of the map E0 : On → Fn . (See [3],[1].) For if J is an ideal and a ∈ J is nonzero, then a∗ a is a positive nonzero element in J and so it follows from the algebraic formulation that a0 = E0 (a) is in J. Since Fn is simple the simplicity of On follows. In fact more is true in that one can find elements x, y ∈ On such that xab = I. To do this one uses the algebraic definitions of E0 and ∆ to get, in J, an operator d = pdp = x1 ay2 close to a diagonal matrix unit p. Then one finds the appropriate isometry Sµ to conjugate p to an operator close to the identity. The normalising partial isometry theorem, from which Theorem 2.7, Theorem 2.8 and Theorem 2.10 are easily obtained, are discussed further in [21]. Here one can also find their natural extensions to AF C*-algebras. The open problems are essentially the problems 7.8, 7.9 of [21]. Those problems are stated for closed algebras but because of inductivity the problems really reside in pure algebra and are stated in these terms here. If A is self-adjoint then the masa C is unique up to automorphism. More is true: for any other matrix unit system for A, with subalgebra system {Ak } and masa C  (as in the definition), there is an automorphism A → A which maps C to C  . (See [21].) However in this case the automorphism, which is determined by an intertwining diagram A1 → An1 → Am1 → An2 → · · · , can be constructed in such a way so that there is an intertwining diagram with regular maps in the sense that the normaliser of each diagonal algebra maps into

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17

the normaliser of the diagonal of the next algebra. (See the Appendix.) Part of the difficulty of the general nonselfadjoint problem is that it is known that there are diagonal masas (of the type above) which although automorphically equivalent are not equivalent through an intertwining diagram of regular maps. See [9] for this subtlety, and see [10], [24] for further discussions. The importance of the problem is that all kinds of putative invariants can be associated with classes of regular direct system and one would like these constructs (partial isometry homology groups for example) to be invariants for the algebra rather than the pair (A, C).

3. Toeplitz algebras for countable directed graphs We now take up a different topic and formally define the analytic Toeplitz algebras AG and LG . Let G be a finite or countable directed graph, with edge set E(G) and vertex set V (G). The free semigroupoid F+(G) determined by G is a set with partially defined associative multiplication. The set consists of the vertices, which act as multiplicative units, and all finite directed paths in G. The partially defined product is the natural operation of concatenation of paths, with a vertex considered as a path. Thus a nonunit element of F+(G) is a path (or word) w = e1 e2 . . . en where the initial vertex of each ei , for i < n, is equal to the final vertex of ei+1 . Vertices may appear in a word of edges, redundantly, to indicate information. For example given a path w in F+(G) we have w = ywx when the initial and final vertices of w are, respectively, x and y. Let HG = 2 (F+(G)) be the Hilbert space with orthonormal basis of vectors ξw indexed by elements w of F+(G). For each edge e ∈ E(G) and vertex x ∈ V (G) define partial isometries and projections on HG by the following left-sided actions on basis vectors:  ξew if ew ∈ F+(G) Le ξw = 0 otherwise and  ξxw = ξw if w = xw ∈ F+(G) Lx ξw = 0 otherwise. We also write Px for the projection Lx . If G has a single vertex x then ξx is referred to as the vacuum vector and the operators Lw are isometries. If there are, additionally, only finitely many edges e1 , . . . , en then each path w is a free word in these edges and the semigroupoid of paths is simply the free (unital) semigroup F+ n with n generators. For n = 2 one can visualise the action of the two isometries Le1 and Le2 as downward left and right shifts of basis vectors placed at the vertices of a downward branching tree. Definition 3.1. (i) The free semigroupoid algebra LG is the weak operator topology closed algebra generated by the projections Lx and the (partial) shift operators Le ; LG

= wot–Alg {Le , Lx : e ∈ E(G), x ∈ V (G)}.

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(ii) The algebra AG is the norm closed algebra generated by {Le , Lx : e ∈ E(G), x ∈ V (G)}. This Toeplitz algebra is also referred to as the tensor algebra for G. The algebra LG can also be thought of as being generated by the left “regular” representation λG : F+ G → B(HG ), λG (e) = Le , which faithfully represents the as partial isometries. In the case of a noncomposable elements semigroupoid F+ G w1 and w2 one can check that the corresponding partial isometries have zero product. 3.1. Examples and matrix function algebras It should be apparent that the algebra LG for the graph with a single vertex x and single loop edge e = xex is unitarily equivalent to the analytic Toeplitz algebra TH ∞ ; the Fock space naturally identifies with the Hardy space H 2 , and Le is then unitarily equivalent to the unilateral shift Tz . More generally, the noncommutative analytic Toeplitz algebras Ln , n ≥ 2, also known as the free semigroup algebras, arise from the graphs with a single vertex and n distinct loop edges, while L∞ comes from the single vertex graph with countably many loops. (i) If G is a finite graph with no directed cycles, then the Fock space HG is finite-dimensional and so too is LG . As an example, consider the graph with three vertices and two edges, labelled x1 , x2 , x3 , e, f where e = x2 ex1 , f = x3 f x1 . Then the Fock space is spanned by the vectors {ξx1 , ξx2 , ξx3 , ξe , ξf } and with this basis the general operator X = αLx1 + βLx2 + γLx3 + λLe + µLf in LG is represented by the matrix ⎤ ⎡ α 0 0 0 0 ⎢ 0 β 0 0 0⎥ ⎥ ⎢ ⎥ X⎢ ⎢ 0 0 γ 0 0⎥ . ⎣λ 0 0 β 0 ⎦ µ 0 0 0 γ Here, LG is isometrically isomorphic to (but not unitarily equivalent to) the socalled digraph algebra (see later) in M3 (C) consisting of the matrices ⎡ ⎤ α 0 0 ⎣λ β 0⎦ . µ 0 γ (ii) Consider the graph G with two vertices x, y, a loop edge e = xex and the edge f = yex directed from vertex x to vertex y. The tree graph for Fock space takes the form shown in Figure 4. The semigroupoid algebra LG is generated by {Le , Lf , Px , Py }. If we make the natural identifications HG = Px HG ⊕ Py HG  H 2 ⊕ H 2 , then         Tz 0 0 0 I 0 0 0 , Px  Le  , Lf  , Py  . 0 0 Tz 0 0 0 0 I Thus, LG is seen to be unitarily equivalent to a matrix Toeplitz algebra, which we can also view as (isometrically and weak star – weak star isomorphic to) the

Subalgebras of Graph C*-algebras

x

19

y

e

f

ee

fe

eee

fee

Figure 4. Fock space graph. matrix function algebra

 ∞ H H0∞

0 CI



where H0∞ is the subalgebra of H ∞ formed by functions h with h(0) = 0. Exercise. Add to G a “returning” directed edge g = xgy to obtain a graph G . Show that LG
contains a “copy” of L2 by virtue of the fact that it contains isometries with mutually orthogonal ranges. (iii) Let n ≥ 1 and consider the cycle graph Cn which has n vertices x1 , . . . , xn and n edges en = x1 en xn and ek = xk+1 ek xk for k = 0, . . . , n − 1. Identify Lxi HG with H 2 for each i in the natural way (respecting word length). Then HG = Lx1 HG ⊕ . . . ⊕ Lxn HG  Cn ⊗ H 2 and the operator α1 Le1 + . . . + αn Len is identified with the operator matrix ⎡ ⎤ 0 αn Tz ⎢α1 Tz ⎥ 0 ⎢ ⎥ ⎢ ⎥ α2 Tz 0 ⎢ ⎥. ⎢ ⎥ .. .. ⎣ ⎦ . . αn−1 Tz Hn∞ ∞

∞

0

Write for the subalgebra of H arising from functions of the form h(z n ) with h in H . It follows that the algebra LCn is isomorphic to the matrix function algebra ⎡ ⎤ Hn∞ z n−1 Hn∞ . . . zHn∞ ⎢ .. ⎥ ⎢ zHn∞ Hn∞ . ⎥ ⎢ ⎥. ⎢ ⎥ .. .. ⎣ ⎦ . . ... Hn∞ z n−1 Hn∞

20

S.C. Power

3.2. Fourier series There is a companion algebra to LG coming from “right shifts”. Consider the right regular representation ρG : F+(G) → B(HG ) determined by a directed graph G. This yields partial isometries ρG (w) ≡ Rw
for w ∈ F+(G) acting on HG defined by the equations Rw
ξv = ξvw , where w is the word w in reverse order. The corresponding algebra is RG

= wot–Alg {Re , Rx : e ∈ E(G), x ∈ V (G)}.

Given edges e, f ∈ E(G), observe that Le Rf ξw = ξewf = Rf Le ξw , for all w ∈ F+(G), so that Le Rf = Rf Le , and similarly for the vertex projections. In fact, we have Proposition 3.2. The commutant LG of LG is equal to RG . The proof of this proposition makes use of an important tool, namely the Fourier series expansion of elements of LG . Recall that Px is the projection for the subspace spanned by basis vectors ξw with w = xw. Write Qx for the projection onto the subspace spanned by basis vectors ξw with w = wx. (The projections Qx correspond to the “components” of the Fock space when basis elements are linked by the natural tree structure.) The proposition below takes a simple form when there is a single vertex, i.e., LG = Ln . Proposition 3.3. Let A  ∈ LG , x ∈ V (G), and let aw ∈ C be the coefficients for which Aξx =Qx Aξx = w=wx aw ξw . Then the Cesaro sums associated with the formal sum w∈F+(G),w=wx aw Lw , given by 
|w|  aw L w 1− Σk (Qx A) = k w=wx;|w|≤k

converge in the strong operator topology to Qx A. 3.3. LG remembers the graph G Let us first observe why Ln and Lm are not isomorphic when n = m. This can be seen from the following theorems the first of which introduces another important tool, namely the eigenvectors for L∗G . By an eigenvector for L∗n we really mean a vector ν which is a joint eigenvector for the n-tuple of generators L∗e1 , . . . , L∗en , so that L∗ei ν = αi ν, , 1 ≤ i ≤ n, for some complex numbers αi ∈ C. The notation below is that w(λ) is the complex number obtained on substituting λi for ei in the word w. Theorem 3.4. The eigenvectors for L∗n are complex multiples of the unit vectors
w(λ)ξw , νλ = (1 − λ2 )1/2 w∈F+ n

for λ = (λ1 , . . . , λn ) in the open unit ball Bn ⊆ Cn . Furthermore L∗ei νi = λi νλ , for each i.

Subalgebras of Graph C*-algebras

21

Note that for λ in the open unit ball n

 λi Lei 2 = |λi |2 = λ2 < 1 so that I −

1

n

1 λi Lei is invertible, with inverse  −1  k



I− λi Lei = λi Lei = w(λ)Lw .



e

k≥0

e

w

In particular w w(λ) is convergent, and a similar shows the norm to be (1 − λ2 )−1/2 . Eigenvectors are important since they are allied to characters, i.e., multiplicative linear functionals φ : Ln → C, φ : An → C. Indeed, the map φλ : An → C defined by φλ (A) = Aνλ , νλ  satisfies φλ (p(Le1 , . . . , Len )) = νλ , p(L∗e1 , . . . , L∗en )νλ  = νλ , p(λ1 , . . . , λn )νλ  = p(λ1 , . . . , λn ). It follows that the vector functional actually defines a character. One can go on to show that the character space of An is homeomorphic to closed unit ball. The dimension of the character space serves as a classifying invariant for the algebras An and Ln . More generally one has the following theorem, and an analogous result for the weakly closed free semigroup algebras. Theorem 3.5. Let G, G be directed graphs. Then the following assertions are equivalent. (i) G and G are isomorphic graphs. (ii) AG and AG
are unitarily equivalent. (iii) AG and AG
are isometrically isomorphic. 3.4. Notes There are a number of approaches to the classification theorem above. In Kribs and Power [14], following the free semigroup algebra analysis of Davidson and Pitts [5], [6], wandering vectors are analysed to obtain a Beurling type theorem for invariant subspaces of the algebra. Using this one obtains unitarily implemented automorphisms of LG that act transitively on the set of eigenvectors. Now the eigenvectors are parametrised by the union of unit balls for each vertex with loop edges. In particular isomorphisms can be normalised to the special case where vacuum vectors map to vacuum vectors. Consequently the ideal A0G generated {Le : e ∈ E(G)} is preserved. Theorem 3.5 is straightforward in this case. See also Solel [29] and Katsoulis and Kribs [13]. Other topics that can be found in these papers, and others, are the determination of unitary automorphisms, the structure of partial isometries, the reflexivity and hyper-reflexivity of the algebras LG , and determination of the Jacobson radical and semisimplicity.

22

S.C. Power The Hilbert space Hn is readily identifiable with the Fock space
⊕(Cn )⊗k Hn = C ⊕ k∈Z+

formed by the direct sum of multiple tensor products of Cn . With this formulation the operators Le are conveniently specified by the shift property Le (ξ1 ⊗ · · · ⊗ ξk ) = ξe ⊗ ξ1 ⊗ · · · ⊗ ξk where ξ1 ⊗ · · · ⊗ ξk is an elementary tensor in the k-fold tensor product summand. In general the generating operators Le are partial isometries acting on a natural generalized Fock space Hilbert space, in which not all tensors are admissible. Although we have not needed the tensor formalism this does provide a fundamental construction allowing for further generalisations, most notably for the tensor algebras of correspondences. See, for example, Muhly and Solel [17]. We remark that there is presently a theory of higher rank versions of such non-selfadjoint operator algebras being developed which is associated with higher rank graphs and with higher rank correspondences (product systems). For more on this see Kribs and Power [16], Power [25], Solel [30] and Power and Solel [26]. The following result from [15] gives a graph theoretic condition corresponding, roughly speaking, to the separation of the algebras LG into two classes, those which are “matrix function like” and those that are “free semigroup like”. The following notion parallels somewhat the requirement that a C∗ -algebra contain O2 , or that a discrete group contain a free group. Definition 3.6. A wot-closed algebra A is partly free if there is an inclusion map L2 → A which is the restriction of an injection between the generated von Neumann algebras. If the map can be chosen to be unital, then A is said to be unitally partly free. A directed graph G is said to have the double-cycle property if there are distinct minimal cycles w = xwx, w = xw x over some vertex x in G. Theorem 3.7. The following assertions are equivalent for a countable directed graph G with a finite number of vertices. (i) G has the double-cycle property. (ii) LG is partly free.

4. Subalgebras of On Returning to the themes of Section 2, we are now ready to look inside On . Our context is that of a norm closed subalgebra A ⊆ On which contains the canonical diagonal subalgebra C associated with the given generators of On . Let us first note that there are a number of ways such algebras arise. (i) Generator constraints: (a) If S is a semigroup of operators of the form Sµ Sν∗ which contains all the projections Sµ Sµ∗ then the norm closed linear span is

Subalgebras of Graph C*-algebras

23

a subalgebra of On which contains the canonical diagonal subalgebra. Note that this algebra is left invariant by the gauge automorphisms of On . (b) Let A1 be the norm closed algebra generated by the diagonal algebra C and the single operator S1 . Let A2 be the (nonunital) subalgebra which is the ideal in A1 generated by 1 − S1 . The abelain algebra A1 /com(A1 ) can be naturally identified with the disc algebra A(D), while A2 /com(A2 ) identifies with the ideal of functions h(z) with h(1) = 0. The gauge automorphisms of On rotate the ideals of A(D) and so the algebra A2 is not gauge invariant. (ii) Fourier series constraints: Let A ⊆ F n be a triangular subalgebra with A ∩ A∗ = C. Then A = {a ∈ On : E0 (a) ∈ A, Ek (a) = 0, k < 0} is a triangular subalgebra of On . Once again, A is gauge invariant. (iii) Extrinsic constraints: Let N ⊆ C be a totally ordered family of projections. For example, N could consist of the projections corresponding to the intervals [0, k/2n] in the interval picture of On . To such a nest of projections one can assign the nest subalgebra A = On ∩ AlgN = {a ∈ On : (1 − p)ap = 0, for all p ∈ N }. Once again, masa normalising partial isometries give a key tool for recovering the underlying “coordinates” from the algebra structure. Each partial isometry Sµ Sω∗ is C-normalising, as are finite sums of these when they have orthogonal ranges and orthogonal domains. Also we may multiply these sums by unitary elements of C to obtain further examples. These turn out to be all the normalising partial isometries and they may be characterised in intrinsic terms as in the following theorem. In terms of the interval picture, or the Cantor interval picture one can, once again, indicate pictorially the support of such a partial isometry, as shown. 0

1

0

1

Figure 5. The support a normalising partial isometry in O2 .

24

S.C. Power

Theorem 4.1. Let v be a contraction in On . Then the following assertions are equivalent: (i) v is a C-normalizing partial isometry. (ii) v is an orthogonal sum of a finite number of partial isometries of the form dSµ Sν∗ , where d ∈ C. (iii) For all projections p, q ∈ C, the norm qvp is equal to 0 or 1. Proof. The implications from (ii) to (i) to (iii) are routine and left to the reader. Let v be a contraction with the zero one norm condition. We claim first that E0 (v) is a C-normalizing partial isometry in F n . The intuitive reason for this follows on contemplating the (Cantor space) support supp(v) of v which is defined as the set of points (y, x) in X × X such that for some δ > 0 qvp = 1 for all projections p, q in C with y(q) = x(p) = 1. Considering continuous matrices we can see that the support supp(w) of a finite sum w of products of the generators and their adjoints will be the union of the sets supp(E0 (w)) and supp(w − E0 (w)), the former set consisting of the diagonal segments parallel to the main diagonal. Because of the essential disjointness of these sets it follows from (iii) that E0 (w) satisfies the zero-one condition and so by Theorem 2.10 is a normalising partial isometry. For a rigorous proof we can argue as follows. If E0 (v) were not a normalising partial isometry then by Theorem 2.10 we would be able to find δ > 0 and projections p, q in C so that δ ≤ qE0 (v)p ≤ 1 − δ. Moreover, for each N it is possible to choose the projections in such a way so that qwp = 0 for any standard partial isometry w of the form Sµ Sν∗ with |µ| = |ν| and 0 ≤ |µ|, |ν| < N . Choose v  in the star algebra generated by the generators with v − v   < δ/3. Since v  − E0 (v  ) is a linear combination of Sµ Sν∗ with |µ| = |ν| we may choose p, q as above so that pvq − pv  q = pvq − pE0 (v  )q. It follows that pvq is not zero or one, a contradiction. Now suppose that m > 0. If |ν| = m and |λ| − |µ| = m, then the product Sν∗ Sλ Sµ∗ is either zero or of the form Sλ1 Sµ∗ with |λ1 | = |µ|. It follows that if Φm (v) is the mth term in the series expansion of v (Φm (v) = am S1m for m ≥ 0), then Sν∗ Φm (v) = E0 (Sν∗ v). Since v satisfies the zero one condition, so does Sν∗ v and the argument above shows that Sν∗ Φm (v) is C-normalizing and so has the desired form. This, in turn, implies that Sν Sν∗ Φm (v) is C-normalizing and has the required form for any word ν of length m. Consequently, Φm (v) has the desired form. In a similar fashion, we can show that when m < 0, Φm (v) also has the desired form (consider adjoints, for example). Finally, if w is a partial isometry and ww∗ xw∗ w = 0 then w + ww∗ xw∗ w > 1. From this observation and the Ces´ aro convergence of generalised Fourier series, it follows that the operators Φm (v) are non-zero for only finitely many values of m and that v is the orthogonal sum of these operators. Thus v itself has the desired form.


Subalgebras of Graph C*-algebras

25

Recall the Cantor interval picture and the partial homeomorphisms αµν . The dilation factor of αµν we define to be k = |ν| − |µ|. Previously we focused on the case k = 0 appropriate to F n . We now define the counterpart to R(F n ). The Cuntz groupoid R(On ) is, intuitively speaking, the support of the algebra in the Cantorised interval picture, with record taken of the dilation factors. More formally it is the set R(On ) = {(x, k, y) : x = αµν (y) for some αµν }, together with (i) the totally disconnected topology with (as before) the set of graphs Eµν , for the partial homeomorphisms αµν , as a base, (ii) the natural partially defined multiplication coming from composition of appropriate partial homeomorphisms. It is natural now to seek to obtain for subalgebras of On results analogous to those in Section 2. It turns out that there is a complication in that “synthesis”, as expressed in Theorem 2.7, may fail. However, it is precisely the gauge invariant closed subalgebras containing C that are determined by their groupoid support: Theorem 4.2. Let A be a closed subalgebra of On containing the canonical diagonal masa C. Then A is generated by the partial isometries Sµ Sν∗ belonging to A if and only if A is invariant under the gauge automorphisms γz for |z| = 1. For a gauge invariant algebra as above we define an associated topological semigroupoid R(A). This is the set R(A) = {(x, k, y) : x = αµν (y) for µ, ν such that Sµ Sν∗ ∈ A} with the relative topology and partially defined multiplication. With the characterisation of normalising partial isometries given above it is now possible to prove the following analogue of Theorem 2.10, and in a similar manner to the proof of that theorem. To paraphrase, gauge invariant triangular subalgebras of On remember their semigroupoids and are classified by them. Theorem 4.3. Let A1 and A2 be norm-closed subalgebras of On with Ai ∩ A∗i = C for i = 1, 2. Then the following statements are equivalent (i) A1 and A2 are isometrically isomorphic operator algebras. (ii) The semigroupoids R(A1 ), R(A2 ) are isomorphic, that is, there is a homeomorphism α : R(A1 ) → R(A2 ) which respects the partially defined multiplication. Proof. The direction (i) =⇒ (ii) is similar to that of the proof of Theorem 2.10. The direction (ii) =⇒ (ii) is more straightforward. One lifts the map α to a map on the semigroup of normalising partial isometries generated by the Sµ Sν ∗ and this can be extended to an isometric algebra isomorphism.
Suppose now that G is a countable directed graph (V, E) with range and source maps r, s : E → V . One can generalise the Cuntz relations as we indicated above in context E of the introduction. To each edge e there is a partial isometry

26

S.C. Power

Se and to each vertex x a projection Px . The initial projection of Se is Ps(e) while the range projection is dominated by Pr(e) . Moreover, under a given Py the range projections sum to that projection:
Se Se∗ = Py e:r(e)=y

with weak operator topology convergence if the edge incidence is infinite. We see then that it is simply the graph that encodes partial isometry generators and relations. The graph C*-algebra C ∗ (G) is defined to be the associated universal C*algebra. Much is known about the structure of this diverse class. See for example [27]. Once again, words in the generators and their adjoints have a reduced form Sα Sβ∗ where, in the graph case, α = α1 . . . αn is a directed path in G (directed from left to right in our convention) and there are natural counterparts to methods and results in Section 2.1. Moreover, if G has no source vertices, in the sense that r is onto, then the abelian C*-algebra generated by the projections Sα Sα∗ is a masa. In this setting, with the simplifying assumption of finite incidence at every vertex (so-called row finiteness) one has exact counterparts to all the results of this section. 4.1. Notes Theorems 4.1, 4.2 and 4.3 are taken from Hopenwasser, Peters and Power [11] where one can also find the more general variants for graph C*-algebras. The methods related to the AF classification, Theorem 2.10, has assisted in the analysis of many particular families of subalgebras of AF C*-algebras and much is known of the structure of ideals and representations, for example. On the other hand subalgebras of graph C*-algebras have not received such attention but it may be timely to do so. In these lectures we have been led from operator algebra considerations to specific topological groupoids and semigroupoids, for F n and On . It is an important and natural consideration to complete the circle and construct operator algebras associated with general abstract topological groupoids. For this see Renault [28], Paterson [18] and Raeburn [29]. For further perspectives on non-self-adjoint operator algebras see the recent article of Donsig and Pitts [8] who also comment on variants of the open problems in Section 2.

Subalgebras of Graph C*-algebras

27

5. Appendix: Digraph algebras and limit algebras We give a brief self-contained account of digraph algebras A(G) and some examples of direct limit algebras. 5.1. Digraph Algebras A digraph is a directed graph G = (V, E) with no multiple directed edges, so that E ⊆ V × V and each edge e in E can be written as (x, y) with initial vertex y and final vertex x. If V = {1, . . . , n} then the subspace A(G) ⊆ Mn (C) is defined by A(G) := span{ei,j : (i, j) ∈ E}, where {eij } is the standard matrix unit system for Mn (C). Suppose further that E, when regarded as a binary relation, is reflexive. Thus (v, v) ∈ E for all v ∈ V . Then Dn ⊆ A(G), where Dn = span {eii }. Note that A(G) is a complex algebra if and only if G is transitive, that is, if (i, j) and (j, k) are edges then so is (i, k). We say that A(G) is a standard digraph algebra in this case. Examples. (i) If Km is the complete digraph on {1, . . . , m} then A(Km ) = Mm (C). (ii) Let D2m be a 2m-sided polygon with alternating directions on the edges and loops at each vertex. Then A(D2m ) is identifiable as a rather sparse algebra of matrices. (iii) Let Tn be the subalgebra of upper triangular matrices in Mn (C). Then Tn is a digraph algebra. (iv) Given the digraph G, we can construct G × Km , the relative product by replacing each vertex of G by Km , and replacing each proper edge of G by all the n2 edges between the new vertices. The digraph algebra A(G × Km ) is identifiable with A(G) ⊗ Mm (C) . Definition 5.1. A ⊆ Mn (C) is a digraph algebra if A is a complex algebra and A contains a maximal abelian self-adjoint algebra (masa) D. Proposition 5.2. If D ⊆ Mn (C) is a masa then there is a unitary matrix u ∈ Mn (C) such that uDu∗ = Dn , the standard masa. Proof. D being a masa means that if D properly contains D and D is also a self-adjoint algebra, then D is not abelian. Let {p1 , . . . , pt } ⊆ D be a maximal set of pairwise orthogonal projections. Then rank pi = 1, for all i, by maximality of D. For if not split the projection into a sum of two projections to obtain a larger abelian algebra. It follows, again by maximality, that t = n and so there is a unitary u such that for all i, upi u∗ = eii as required.
Proposition 5.3. If D ⊆ A(G) is a masa then there exist a unitary u ∈ A(G) such that uDu∗ = Dn . Thus, maximal abelian self-adjoint subalgebras of digraph algebras are unique up to inner conjugacy (inner unitary equivalence). Proof. Use the previous proposition, combining unitaries in each block of the block diagonal self-adjoint subalgebra A(G) ∩ A(G)∗ .
Since an algebra in Mn (C) which contain the standard masa is a standard digraph algebra we have the following corollary.

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S.C. Power

Corollary 5.4. Every digraph algebra A ⊆ Mn (C) is inner conjugate to a standard digraph algebra. That is uAu∗ = A(G) for some digraph G and some unitary u in A. If standard digraph algebras are unitarily equivalent then by Proposition 5.2 we can assume that the unitary equivalence maps the standard masa to the standard masa and it follows readily that the graphs are isomorphic. 5.2. Maps between digraph algebras  To begin to understand algebras of the form A = k A(Gk ), where the building block algebras are nested, i.e., A(Gk ) ⊆ A(Gk+1 ), we must consider the nature and m variety of the possible inclusion maps A(Gk ) → A(Gk+1 ). Let (fij )i,j=1 ⊆ Mn (C) be operators with the relations of an m × m matrix unit system, that is, ⎤ fij fjk = fik ∀ ijk ⎦ fij∗ = fji (∗) fij fkl = 0 if j = k Then the map φ : Mm → Mn which is defined to be the linear extension of the correspondences eij → fij , is an injective C ∗ -algebra homomorphism. Conversely, it is straightforward to show that if φ is such a map, then {φ(eij )} satisfy the relations (∗). Definition 5.5. Let A(G1 ) ⊆ Mm , A(G2 ) ⊆ Mn be unital standard digraph algebras with connected graphs. Then an algebra homomorphism φ : A(G1 ) → A(G2 ) is said to be star-extendible if φ is the restriction of a C ∗ -algebra map between the finite-dimensional C*-algebras C ∗ (A(G1 )) and C ∗ (A(G2 )). Example (i) The map φ : T2 → T4 given by ⎛ a 0   ⎜ 0 a a b φ =⎜ ⎝ 0 0 0 c 0 0 is a star-extendible algebra injection. Example (ii) The map φ : T2 → T4 given by ⎛ a   ⎜ a b φ: →⎜ ⎝ c

√ b/√2 b/ 2 c 0

b c

0 0 a

√ −b/√ 2 b/ 2 0 c

⎞ 0 0 ⎟ ⎟ 0 ⎠ 0

is not star-extendible (and is an isometric algebra injection). Example (iii) φ : T2 → T4 given by ⎡ ⎤ a b   ⎢ ⎥ a b a b ⎥ φ: →⎢ ⎣ ⎦ c c c is a star-extendible algebra injection.

⎞ ⎟ ⎟ ⎠

Subalgebras of Graph C*-algebras

29

Remark 1 Star-extendible injective maps between digraph algebras are necessarily isometric: since they are restrictions of C*-algebra maps. This is an elementary fact from spectral theory. Indeed, note first that if p ∈ Mm is a nonzero selfadjoint projection then ||p|| = 1, by the definition of the operator norm. If φ is a star homomorphism then φ(p) is a projection, so φ(p) = 1 = p. We claim that φ(a) = a if a is a self-adjoint operator in Mm . By the spectral theorem a = λ1 p1 + · · · + λm pm , and we can suppose a = |λ1 | ≥ |λk | for k = 2, . . . , m, where p1 , . . . , pm are pairwise orthogonal projections. Let qi = φ(pi ) 1 ≤ i ≤ m. Then φ(a) = λ1 q1 + · · · + λm qm , with q1 . . . qm pairwise orthogonal projections. Thus (exercise) φ(a) = |λ1 | = a. Finally if b ∈ Mm is a general element, then φ(b)2 = φ(b)∗ φ(b) = φ(b∗ )φ(b) = φ(b∗ b) = b∗ b = b2 . Definition 5.6. Let A(G1 ), A(G2 ) be digraph algebras with standard matrix unit systems {ekij : (ij) ∈ E(Gk )}, k = 1, 2, as usual. (i) An algebra injection φ : A(G1 ) → A(G2 ) is a standard regular injection, with respect to {e1ij }, and {e2ij }, if φ is star-extendible and maps each e1ij to a sum of matrix units in {e2kl } (ii) An algebra injection ψ : A(G1 ) → A(G2 ) is a regular (star-extendible) injection if there exists a unitary operator u in A(G2 ) such that ψ(a) = uφ(a)u∗ ∀a ∈ A(G1 ), where φ is a standard regular injection. We say that ψ is inner equivalent (or, less precisely, unitarily equivalent) to φ when such a relationship holds. Remark. The unitary u belongs to A2 ∩ A∗2 . Indeed, by the spectral theorem, u = λ1 p1 + · · · + λr pr where pi is the spectral projection for the eigenspace for λi and each pi lies in the self-adjoint subalgebra. Exercises. (i) Prove that there are uncountably many inner conjugacy classes of embedding φ : T2 → T4 which are star-extendible and unital. (ii) Prove that there are only finitely many inner conjugacy classes of regular embeddings between two digraph algebras. 5.3. Direct limits We leave it to the reader to recall the definition of the direct limit algebra of a direct system and we now give some standard direct systems of triangular matrix algebras. The examples are all determined by regular star extendible inclusion maps. Limits of finite-dimensional operator algebras with respect to isometric irregular embeddings are less well understood and have not received much attention. The interested reader can find some classifications in this direction in [24]. Standard limits. Let n2 = rn1 and let σ : Mn1 → Mn2 be the inclusion map such that (1)

(2)

(2)

(2)

σ(ei,j ) = ei,j + ei+n1 ,j+n1 + · · · + ei+(r−1)n1 ,j+(r−1)n1

30

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or equivalently, identifying Mn2 with Mr ⊗ Mn1 , σ(a) = I ⊗ a, so that, in block matrix terms, ⎡ ⎤ a 0 ... 0 ⎢ 0 a ⎥ ⎥. σ(a) = ⎢ ⎣ ⎦ 0 a Then σ(Tn1 ) ⊆ Tn2 , and so, repeating, we may construct the regular matricial algebra Aσ = lim(Tnk , σ), when nk |nk+1 for all k. Refinement limits. Let n2 = rn1 and let ρ : Mn1 → Mn2 be the inclusion map such that (1)

(2)

(2)

(2)

ρ(ei,j ) = e(i−1)n1 +1,(j−1)n1 +1 + e(i−1)n1 +2,(j−1)n1 +2 + · · · + e(i−1)n1 +r,(j−1)n1 +r or equivalently, identifying Mnk with Mn1 ⊗Mr , ρ(a) = a⊗I, so that, ρ((aij )) is the inflated matrix (aij Ir ). Again, ρ(Tn1 ) ⊆ Tn2 , and for a sequence with nk |nk+1 for all k we can define the regular matricial algebra Aρ = lim(Tnk , ρ). This algebra is not isometrically isomorphic to the Aσ algebra for the same sequence (nk ) despite the fact that their generated C*-algebras are isomorphic. Countable total order limits. The standard embeddings σ and the refinement embeddings ρ can also be alternated in which case one obtains a more general class of algebras, the so called alternation algebras. The three classes described correspond to a Cantor space product coordinate indexing by Z− , Z+ , and Z respectively. One can generalise this further to obtain strange triangular algebras whose background Cantor product is indexed over an arbitrary countable order. Moreover this countable order is an algebra isomorphism invariant. For further detail see [23].

Subalgebras of Graph C*-algebras

31

References [1] J. Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185. [2] K.R. Davidson, Free Semigroup Algebras: a survey. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 209–240, Oper. Theory Adv. Appl. 129, Birkh¨ auser, Basel, 2001. [3] K.R. Davidson, C*-algebras by Example, Fields Institute Monograph Series, vol. 6, American Mathematical Society, 1996. [4] K.R. Davidson, E. Katsoulis, Nest representations of directed graph algebras, Proc. London Math. Soc., to appear. [5] K.R. Davidson, D.R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275–303. [6] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc., 78 (1999), 401–430. [7] K.R. Davidson, E. Katsoulis and J. Peters, Meet-irreducible ideals and representations of limit algebras. J. Funct. Anal. 200 (2003), no. 1, 23–30. [8] A.P. Donsig and D.R. Pitts Coordinate Systems and Bounded Isomorphisms for Triangular Algebras. math.OA/0506627 66 pages. [9] A.P. Donsig and S.C. Power, The failure of approximate inner conjugacy for standard diagonals in regular limit algebras. Canad. Math. Bull. 39 (1996), no. 4, 420–428. [10] P.A. Haworth and S.C. Power, The uniqueness of AF diagonals in regular limit algebras. J. Funct. Anal. 195 (2002), no. 2, 207–229. [11] A. Hopenwasser, J. Peters and S.C. Power, Subalgebras of Graph C*-Algebras, New York J. Math. 11 (2005), 1–36. [12] A. Hopenwasser and S.C. Power,Limits of finite dimensional nest algebras, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 1, 77–108. [13] E. Katsoulis, D.W. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann., 330 (2004), 709–728. [14] D.W. Kribs, S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2004), 75–117. [15] D.W. Kribs, S.C. Power, Partly free algebras, Operator Theory: Advances and Applications, Birkh¨ auser-Verlag Basel/Switzerland, 149 (2004), 381–393. [16] D.W. Kribs and S.C. Power, The H ∞ algebras of higher rank graphs, Math. Proc. of the Royal Irish Acad., 106 (2006), 199–218. [17] P. Muhly, B. Solel, Tensor algebras, induced representations, and the Wold decomposition, Can. J. Math. 51 (4), 1999, 850–880. [18] A. L. Paterson, Groupoids, inverse semigroups and their operator algebras, Progress in Mathematics, Vol. 170, Birkh¨ auser, Boston, 1999. [19] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46. [20] G. Popescu, Noncommuting disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137–2148.

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[21] S.C. Power, Limit algebras: an introduction to subalgebras of C ∗ -algebras, Pitman Research Notes in Mathematics Series, vol. 278, (Longman Scientific & Technical, Harlow, 1992) CRC Press ISBN: 0582087813. [22] S.C. Power, Lexicographic semigroupoids, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 365–377. [23] S.C. Power, Infinite lexicographic products of triangular algebras, Bull. London Math. Soc. 27 (1995), no. 3, 273–277. [24] S.C. Power, Approximately finitely acting operator algebras, J. Funct. Anal. 189 (2002), no. 2, 409–468. [25] S.C. Power, Classifying higher rank analytic Toeplitz algebras, preprint 2006, preprint Archive no., math.OA/0604630. [26] Operator algebras associated with unitary commutation relations, preprint March 2007. [27] I. Raeburn, Graph algebras C.B.M.S lecture notes, vol. 103, Amer. Math. Soc., 2006. [28] J. Renault, A groupoid approach to C ∗ -algebras, Springer, Berlin, 1980. [29] B. Solel, You can see the arrows in a quiver algebra, J. Australian Math. Soc., 77 (2004), 111–122. [30] B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps, J. Funct. Anal.235 (2006), 593–618. Stephen C. Power Department of Mathematics and Statistics Lancaster University England LA1 4YF e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 33–66 c 2008 Birkh¨
auser Verlag Basel/Switzerland

C ∗ -algebras and Asymptotic Spectral Theory Bernd Silbermann Abstract. The presented material is a slightly polished and extended version of lectures given at Lisbon, WOAT 2006. Three basic topics of numerical functional analysis are discussed: stability, fractality, and Fredholmness. It is further shown that these notions are corner stones in order to understand a few topics in asymptotic spectral theory: asymptotic behavior of singular values, ε-pseudospectra, norms. Four important examples are discussed: Finite sections of quasidiagonal operators, Toeplitz operators, band-dominated operators with almost periodic coefficients, and general band-dominated operators. The elementary theory of C ∗ -algebras serves as the natural background of these topics. Mathematics Subject Classification (2000). 47B35. Keywords. C∗-algebras, operator sequences, asymptotics, finite sections.

1. Introduction One goal of functional analysis is to solve equations with “infinitely” many variables, and that of linear algebra to solve equations in finitely many variables. Numerical analysis builds a bridge between these fields. Functional numerical analysis is concerned with the theoretical foundation of numerical analysis. Given a bounded linear operator A acting on some Hilbert space H, that is A ∈ B(H), consider the equation Ag = h ,

(1.1)

where h ∈ H is given and g is to find if this equation is supposed to be uniquely solvable. Even if the operator A is continuously invertible (and this will be assumed in what follows), it is as a rule impossible to compute the solution A−1 h. Then one tries to solve (1) approximately. For, one chooses a sequence (hn ) ⊂ H of elements which approximates the right-hand side h, and a sequence (An ) of operators

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which approximates the operator A, and one replaces (1.1) by the approximation equations An gn = hn , n = 1, 2, . . . (1.2) the solutions gn of which are sought in H (or in certain subspaces Hn of H) again. Approximation of h by hn means that h − hn H → 0 as n → ∞. It is tempting to suppose that the operators An also approximate A in the norm, but this assumption does not work in practice. The point is that usually A acts on an infinite-dimensional space, whereas one will, of course, try to choose the An as acting on spaces of finite dimension, i.e., as finite matrices. But the only operators which can be approximated in norm by finite rank operators, are the compact ones. The kind of approximation which fits much better to the purpose of numerical analysis is that of pointwise or strong convergence: the sequence (An ) converges strongly to the operator A if Ah − An hH → 0 for every h ∈ H (notation: s-lim An = A). We write s∗ -lim An = A, if s-lim An = A and s-lim A∗n = A∗ . Basic question: suppose that An is invertible for all n ≥ n0 . Does the sequence (gn ) of solutions of (1.2) converge to the solution g of (1.1)? The answer is NO! Let l2 := {(x0 , . . . , xn , . . . ) :




|xk |2 < ∞},

k∈Z+

˜l2 := {(. . . , x−n , . . . , x0 , . . . , xn , . . . ) :




|xk |2 < ∞}.

k∈Z

Example 1. Let ε = (εn ) be a sequence with εn > 0 and lim εn = 0 and Aε an infinite matrix given by ⎞ ⎛ ε0 1 ⎟ ⎜ 1 ε0 0 ⎟ ⎜ ⎟ ⎜ 0 ε 1 1 ⎟ ⎜ ⎟ ⎜ 1 ε 0 1 ⎟ ⎜ (with respect to the standard basis of l2 ). ⎟ ⎜ 1 0 ε 2 ⎟ ⎜ ⎜ .. ⎟ ⎜ . ⎟ 1 ε 2 ⎠ ⎝ .. .. . . Consider Pn Aε Pn , Pn : l2 → l2 , (x0 , . . . , xn , xn+1 , . . . ) → (x0 , . . . , xn , 0, 0, . . . ). Then:  ≤2 if n is odd (Pn Aε Pn )−1 Pn  = ≥ ε−1 if n is even. n Thus, if n is even, then (Pn Aε Pn )−1 Pn  → ∞, that is there is an h ∈ H such that gn = (Pn Aε Pn )−1 Pn h
A−1 h = g by the Steinhaus-Banach Theorem.

C ∗ -algebras and Asymptotic Spectral Theory

35

Let us turn back to the general situation: Suppose that (Pn ) is a sequence of orthogonal projections such that s-lim Pn = I, and (An ) a bounded sequence of operators An : imPn → imPn with the property that there is an n0 such that An is invertible for n ≥ n0 and sup A−1 n Pn  < ∞. If s-lim An Pn = A, then n≥n0

−1 A−1 x → 0 for every x ∈ H : n Pn x − A −1 −1 A−1 x ≤ A−1 x + Pn A−1 x − A−1 x n Pn x − A n Pn x − Pn A −1 x + Pn A−1 x − A−1 x → 0 . ≤ A−1 n Pn  x − An Pn A ∗ Remark. Suppose again that sup A−1 n Pn  < ∞ and s -lim An Pn = A. Then A n≥n0

is invertible. Indeed, we have An Pn x ≥ CPn x

(n ≥ n0 , C > 0) .

Passing to limits gives Ax ≥ Cx . Thus im A = im A and ker A = {0}. Using A∗n Pn x ≥ CPn x we get in the same manner A∗ x ≥ Cx . Thus A∗ = im A∗ and ker A∗ = {0}. Definition 1. A sequence of operators An ∈ B(imPn ) is called stable if there exists a number n0 such that the operators An are invertible for every n ≥ n0 and if the norms of their inverses are uniformly bounded: sup A−1 n Pn  < ∞ .

n≥n0

The above discussion shows the crucial role of stability in analysis. How to prove stability? There is no general idea. In most cases it is very complicated. Easy cases: • A = B + iS, B positive and S selfadjoint, • A = I + T , T compact, and An = Pn APn , where (Pn ) is a sequence of orthogonal projections with slim Pn = I. Exercise: prove that in both cases the sequence (An ) is stable.

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We will show that the stability problem can frequently be tackled by the help of C ∗ -algebra techniques. Recall that a complex Banach algebra is called C ∗ algebra if there is an involution a → a∗ such that aa∗  = a2 . Given two C ∗ algebras A and B, a ∗-homomorphism ϕ : A → B is a continuous homomorphism such that ϕ(a∗ ) = ϕ(a)∗ for all a ∈ A.

2. Algebraization of stability Let H be a (separable) Hilbert space and (Ln ) be a sequence of orthoprojections on H with s-lim Ln = I. Definition 2. Let F be the set of all sequences (An )∞ n=0 of operators An ∈ B (im Ln ) which are uniformly bounded: sup An Ln  < ∞ . n≥0

The natural operations (An ) + (Bn ) := (An + Bn ), (An )(Bn ) := (An Bn ), λ(An ) := (λAn ) , (An )∗ := (A∗n ) make F to an algebra with involution. Proposition 1. F is a C ∗ -algebra (prove it). We are mainly interested in the asymptotic behavior of the sequences belonging to F . This means that sequences which differ in a finite number of entries only will have the same asymptotic behavior, and therefore can be identified. For this goal we introduce the set G of all sequences (Gn ) in F with lim Gn Ln  = 0. n→∞

Proposition 2. G is a closed ideal in F (prove it). The following theorem reveals a perfect frame to study stability problems in an algebraic way. Theorem 1. (A. Kozak) A sequence (An ) ∈ F is stable ⇔ the coset (An ) + G is invertible in the quotient algebra F /G. Proof. ⇒: If (An ) is stable, then (A−1 n )n≥n0 is bounded for some sufficiently large n0 by definition. We make (A−1 ) n n≥n0 to a bounded sequence −1 , A (B0 , B1 , . . . , Bn0 −1 , A−1 n0 n0 +1 , . . . ) in F by freely choosing operators Bi ∈ B(imLi ). It is evident that this sequence is an inverse of (An ) modulo G. ⇐ Let conversely, (An )+ G be invertible in F /G. Then there are sequences (Bn ) ∈ F as well as (Gn ) and (Hn ) in G such that An Bn = Ln +Gn , Bn An = Ln +Hn . If n is large enough, then Gn  < 12 , Hn  < 12 , and a Neumann series argument yields the invertibility of Ln + Gn and Ln + Hn as well as the uniform boundedness of their inverses by 2. Hence, Bn (Ln +Gn )−1 , (Ln +Hn )−1 Bn are uniformly bounded. Thus, the operators An are invertible for all sufficiently large n, and their inverses are uniformly bounded.


C ∗ -algebras and Asymptotic Spectral Theory

37

Proposition 3. For all (An ) ∈ F, (An + G)F /G = lim sup An Ln  n→∞

(2.1)

(where lim sup stands for lim superior).


Proof. Exercise.

Formula (2.1) gives raise to ask if there are interesting sequences in F for which lim sup in (2.1) can be replaced by lim. This question is important in order to prove that the condition numbers of a stable sequence converge. Recall, that the condition number (cond A) for an invertible matrix (operator) A is defined by cond A := A A−1  (for computational purposes: cond A should be small). The right tool to study this and related questions is another fundamental notion of numerical analysis – that of a fractal sequence, which we are now going to discuss. It is not important in this place that the elements of the sequences under consideration are operators. So we will use slightly generalized definitions of the C ∗ -algebras F and G, namely, given unital C ∗ -algebras Cn , n = 0, 1, 2, . . . , with identity elements en , let F stand for the set of all bounded sequences (c0 , c1 , . . . ) with cn ∈ Cn , and let G refer to the set of all sequences (c0 , c1 , . . . ) in F with cn  → 0 as n → ∞. Defining elementwise algebraic operations and an elementwise involution, and taking the supremum norm, we make F to a C ∗ -algebra and G to a closed ideal of F . Thus, F is the product of the C ∗ -algebras Cn , and G-their restricted product. Given a strongly monotonically increasing sequence η : Z+ → Z+ , let Fη and Gη denote the product and the restricted product of the C ∗ -algebras Cη(0) , Cη(1) , . . . , respectively, and let Rη stand for the restriction mapping Rη : F → Fη , (an ) → (aη(n) ). The mapping Rη is a ∗-homomorphism from F onto Fη . Further, given a C ∗ -subalgebra A of F , let Aη refer to the image of A under Rη . By the first isomorphy theorem for C ∗ -algebras ([7], Theorem 1.45), Aη actually is a C ∗ -algebra. Definition 3. Let A be a C ∗ -subalgebra of F . (a) A ∗-homomorphism W : A → B of A into a C ∗ -algebra B is fractal if for every strongly monotonically increasing sequence η, there is a ∗-homomorphism Wη : Aη → B such that W = Wη Rη . (b) The algebra A is fractal, if the canonical homomorphism π : A → A/A ∩ G is fractal. (c) A sequence (an ) ∈ F is fractal, if the smallest C ∗ -subalgebra of F containing (an ), is fractal. Roughly spoken: given a subsequence (aη(n) ) of a sequence (an ) which belongs to a fractal algebra A, it is possible to reconstruct the original sequence (an ) from its subsequence modulo sequences in A ∩ G.

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Consequences: • (aη(n) ) ∈ Gη ⇒ (an ) ∈ G ([7], Theorem 1.66),

(2.2)

• (aη(n) ) stable ⇒ (an ) stable

(2.3)

(see Theorem 4 below). ∗

Theorem 2. ([7], Theorem 1.7.1) Let A be a fractal C -subalgebra of F . If (an ) ∈ A, then the limit lim an  exists and is equal to (an ) + G.  |an |2 < ∞} and the bounded Example 2. Consider again l2 := {(an )n∈Z+ : n∈Z+

linear operators Pn , Rn : l2 → l2 given by (ak ) → (a0 , a1 , . . . , an , 0, 0, 0, . . . ) , (ak ) → (an , an−1 , . . . a0 , 0, 0, 0, · · · ) , respectively. Let Cn = B(imPn ), and let F W refer to the set of all sequences (An ) ∈ F ˜ (An ) := s-lim Rn An Rn for which the strong limits W (An ) := s-lim An Pn and W ∗ ∗ ˜ as well as the strong limits W (An ) and W (An ) exist. The set F W actually forms a C ∗ -subalgebra of F (prove it or compare the proof of Theorem 1.18 (a) in [7]). ˜ : F W → B(l2 ) turn out to be fractal: given a strongly The ∗-homomorphism W, W ˜ η : Fη → B(l2 ) via monotonically increasing sequence η, we can define Wη , W Wη (Aη(n) ) := s- lim Aη(n) Pη(n) and ˜ η (Aη(n) ) := s- lim Rη(n) Aη(n) Rη(n) . W ˜ =W ˜ η Rη . The algebra F W is not fractal: consider Then, obviously, W = Wη Rη , W the sequence (An ) ∈ F, where A2n+1 = 0 and A2n = diag (0, . . . , 0, 1, 0, . . . , 0), where the 1 stands in the center of this diagonal matrix. It is easily seen, that ˜ (An ) = 0, but (An ) ∈ (An ) ∈ F W , W (An ) = 0, W / G(⊂ F W ). For the special choice η(n) = 2n + 1 one obtains Rη (An ) = (A2n+1 ) ∈ Gη . By (2.2) F W cannot be fractal. One the other hand, F W contains interesting fractal subalgebras as we will see later on.

3. Asymptotic behavior Given a sequence (An ) ∈ F one can ask how the spectra (ε-pseudospectra) of the entries develop. We need some definitions. Let (Mn )∞ n=1 be a set sequence with values in the set of all subsets of the complex plane. For instance, if (An ) ∈ F, then the mapping n → sp An is a set sequence in this sense. Definition 4. Let (Mn )∞ n=1 be a set sequence. The partial limiting set or limes superior lim sup Mn (resp. the uniform limiting set or limes inferior lim inf Mn ) of the sequence (Mn ) consists of all points m ∈ C which are a partial limit (resp. limit) of a sequence (mn ) of points mn ∈ Mn (partial limit of a sequence (mn ) is by definition a limit of some subsequence of (mn )).

C ∗ -algebras and Asymptotic Spectral Theory

39

Observe that the partial limiting set  lim sup Mn is non-empty if infinitely many of the Mn are non-empty and if Mn is bounded, whereas the uniform n

limiting set can be empty even under these restrictions as the trivial example Mn = {(−1)n } shows. Let CC denote the set of all non-empty and compact subsets of C. The Hausdorff distance of two elements A and B of CC is defined by   h(A, B) := max max dist (a, B), max dist (b, A) , a∈A

b∈B

where dist (a, B) = min |a − b|. The function h is actually a metric on CC . We b∈B

denote limits with respect to this metric by h-lim. Proposition 4. ([7], Proposition 3.6) Let (Mn ) be a set sequence taking values in CC . Then lim sup Mn and lim inf Mn coincide if and only if the sequence (Mn ) is h-convergent. In that case lim sup Mn = lim inf Mn = h- lim Mn . Example 3. Let V : l2 → l2 be the shift operator acting by (a0 , a1 , a2 , . . . ) → (0, a0 , a1 , a2 , . . . ) and consider (Pn V Pn ). It is easy to see that the matrix representation of Pn V Pn with respect to the standard basis of im Pn equals ⎞ ⎛ 0 ⎟ ⎜ 1 0 ⎟ ⎜ ⎟ ⎜ 1 0 0 ⎟. ⎜ ⎟ ⎜ 0 1 0 ⎟ ⎜ ⎠ ⎝ 1 0 1 0 Hence, sp Pn V Pn = {0} for all n and lim inf sp Pn V Pn = lim sup sp Pn V Pn = {0}, but sp V = {z ∈ C : z ≤ 1} ⊂ spF /G ((Pn V Pn ) + G). Therefore h-lim Pn V Pn exists but this limits has almost nothing to do with sp V . What is the reason for this unpleasant fact? One can prove that for (an ) ∈ F a point s ∈ C belongs to the partial limiting set lim sup sp an if and only if the sequence (an −sen ) is not spectrally stable (Theorem 3.17 in [7]). (A sequence (an ) is spectrally stable if its entries an are invertible for sufficiently large n and if the spectral radii ρ(a−1 n ) of their inverses are uniformly bounded.) Spectral stability is a very involved notion and not much is known. We accomplish this discussion with Theorem 3. ([7], Theorem 3.19) Let Cn = Cn×n and (An ) ∈ F. Then  lim sup sp (An + Cn ) = spF /G ((An ) + G). (Cn )∈G

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One conclusion: Spectral stability is very sensitive with respect to perturbations from G (contrary to stability). These difficulties disappear if we restrict our attention to sequences for which stability and spectral stability coincide. Corollary 1. ([7], Corollary 3.18) If (an ) ∈ F is a sequence of normal elements, then lim sup sp an = spF /G ((an ) + G). For fractal algebras we get refinements. Theorem 4. ([7], Theorem 3.20) Let A be a fractal C ∗ -subalgebra of F which contains the identity. (a) A sequence (an ) ∈ A is stable if and only if it possesses a stable (infinite) subsequence. (b) If (an ) ∈ A is normal, then lim sup sp an = lim inf sp an = h-lim sp an . (c) If (an ) ∈ A is normal, then the limit lim ρ(an ) exists and is equal to ρ((an ) + G) (ρ-spectral radius). Let us shortly discuss limiting sets of singular values (because of their importance in numerical analysis). Let B be a unital C ∗ -algebra and a ∈ B. The set  (a) of the singular values of a is defined to be {λ ∈ R+ : λ2 ∈ sp (a∗ a)}. Since the determination of the singular values is equivalent to the determination of the spectrum of a self-adjoint element, the previous results have the following evident analogues for singular value sets.   Theorem 5. If (an ) ∈ F, then lim sup (an ) = ((an ) + G). Theorem 6. If A is a fractal C ∗ -subalgebra of F containing the identity and if (an ) ∈ A, then


lim sup (an ) = lim inf (an ) = h- lim (an ) . The last topic in this section is ε-pseudospectra. A computer working with finite accuracy cannot distinguish between a noninvertible matrix and an invertible matrix the inverse of which has a very large norm. This suggests the following definition reflecting finite accuracy. Definition 5. Let B be a C ∗ -algebra with identity e and let ε be a positive constant. An element a ∈ B is ε-invertible if it is invertible and a−1  < 1ε . The ε-pseudospectrum spε (a) of a consists of all λ ∈ C for which a − λe is not εinvertible. It is easily seen that ε-invertible elements of a C ∗ -algebra form an open set, and that ε-pseudospectra are compact and non-empty subsets of C. The following theorem provides an equivalent description of the ε-pseudospectrum which offers a way for numerical computations at least for (finite) matrices.

C ∗ -algebras and Asymptotic Spectral Theory

41

Theorem 7. ([7], Theorem 3.27) Let B be a unital C ∗ -algebra and ε > 0. Then, for every a ∈ B, the ε-pseudospectrum is equal to  spε (a) = sp (a + p). p∈B p≤ε

Let us still remark that (unital) C ∗ -algebras are also inverse closed with respect to ε-invertibility. What about limiting sets of ε-pseudospectra? Theorem 8. ([7], Theorem 3.31) Let (an ) ∈ F and ε > 0. Then /G ((an ) + G). lim sup spCε n (an ) = spF ε

The proof of Theorem 8 is based on the following result. Proposition 5. (Daniluk) ([3], Theorem 3.14) Let B be a C ∗ -algebra with identity e, let a ∈ B, and suppose a − λe is invertible for all λ in some open subset U of the complex plane. If (a − λe)−1  ≤ C for all λ ∈ U, then (a − λe)−1  < C for all λ ∈ U. In other words: the analytic function U → B, λ → (a − λe)−1 satisfies the maximum principle. This is a surprising fact since – in contrast to complex-valued analytic functions – the maximum principlefails in general for operator-valued λ 0 analytic functions (consider C → C2×2 , λ → ). 0 1 It is an open question for which Banach algebras Daniluk’s result is true (one particular answer is in [3], Theorem 7.15). In case A is a fractal C ∗ -subalgebra of F we have the following refinement of Theorem 8: /G ((an ) + G) . h- lim spCε n (an ) = spF ε

4. First applications I. Quasidiagonal operators and their finite sections Recall that a bounded linear operator T on a separable (complex) Hilbert space is said to be quasidiagonal if there exists a sequence (Pn )n∈N of finite rank orthogonal projections such that s-lim Pn = I and which asymptotically commute with T , that is [T, Pn ] := T Pn − Pn T  → 0 as n → ∞ . In particular, every selfadjoint or even normal operator is quasidiagonal as well as their perturbations by compact operators. However it is by no means trivial to single out a related sequence (Pn ). For instance, for multiplication operators in periodic Sobolev spaces H λ related sequences can explicitly be given: these are orthogonal projections on some spline spaces. Let T be quasidiagonal with respect to (Pn ) = (Pn )n∈N . Consider the C ∗-subalgebra F l of F , the last one defined by help of (Pn ), consisting of all sequences of

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F for which s∗ - lim An Pn exist. It is not hard to prove that J := {(Pn KPn )+(Cn ): K-compact, (Cn ) ∈ G} forms a two-sided closed ideal in F l (but not in F !) Let C(Pn ) (T ) denote the smallest C ∗ -subalgebra of F l containing the sequences (Pn T Pn ), (Pn ), and the ideal J. Proposition 6. The quotient algebra C(Pn ) (T )/G is isometrically isomorphic to the smallest C ∗ -subalgebra C(T ) of B(H) containing T, I, and all compact operators. This isomorphism is given by the quotient map induced via s-lim An Pn ((An ) ∈ C(Pn ) (T )). Sketch of the proof. Suppose s-lim An Pn =: A is invertible. Then A−1 ∈ C(T ), and since every element in C(T ) is quasidiagonal, A−1 also owns this property, and Pn − Pn APn A−1 Pn  = Pn AA−1 Pn − Pn APn A−1 Pn  = Pn (Pn A − APn )A−1 Pn  → 0 . Hence, (Pn APn ) is stable. This means that (Pn APn ) + G is invertible if and only if s-lim Pn APn is invertible. Now it is sufficient to prove that Pn APn − An ∈ G. For, it is sufficient to show this for the special case An = Pn B1 Pn B2 Pn . We have Pn B1 B2 Pn − Pn B1 Pn B2 Pn  = Pn (Pn B1 − B1 Pn )B2 Pn  → 0 as n → ∞.
Corollary 2. A sequence (An ) ∈ C(Pn ) (T ) is stable if and only if s-lim An is invertible. Moreover, C(Pn ) (T ) is fractal. Now it is evident that the theory of Section 3 applies. Proposition 7. Let (An ) ∈ C(Pn ) (T ) and A = s-lim An . Then (a) lim An  = s-lim An Pn . (b) lim inf spε An = lim sup spε An = spε (s-lim An ) (ε > 0). (c) If (An ) is normal, then (d) lim inf



lim inf sp An = lim sup sp An = sp (s- lim An ) .   (An ) = lim sup (An ) = (s-lim An ).

In the papers [5], [6] Nathaniel Brown proposed further refinements into two directions: speed of convergence and how to choose the sequence (Pn ) of orthoprojections in some special cases such as quasidiagonal unilateral band operators, bilateral band operators or operators in irrational rotation algebras. Remark. If (an ) ∈ F l is stable and s∗ -lim an = A , A+K invertible and K compact, then (an + Pn KPn ) is stable (this sequence equals (an )(Pn + Pn a−1 n Pn KPn ) for n large enough). II. Toeplitz operators and their finite sections Let a ∈ L∞ (T) and denote by ak the kth Fourier coefficient of a: ak =

1 2π

2π 0

a(eiθ )e−ikθ dθ , k ∈ Z ,

where

T := {z ∈ C : |z| = 1}.

C ∗ -algebras and Asymptotic Spectral Theory

43

Then the Laurent operator L(a) on ˜ l2 , the Toeplitz operator T (a) on l2 , and 2 the Hankel operator H(a) on l are given via their matrix representation with respect to the standard bases of ˜l2 and l2 by L(a) = T (a) =

(ak−j )∞ k,j=−∞ , ∞ (ak−j )∞ k,j=0 , H(a) = (aj+k+1 )k,j=0 .

Here is a list of elementary properties of these operators (see any textbook on Toeplitz operators). • If a ∈ L∞ (T), then the Laurent operator L(a) is bounded on ˜l2 . • (Brown/Halmos) If a ∈ L∞ (T), then the Toeplitz operator T (a) is bounded on l2 , and T (a) = a∞ . • (Nehari) If a ∈ L∞ (T), then H(a) is bounded on l2 , and ∞

H(a) = distL∞ (T) (a, H ). • T (ab) = T (a)T (b) + H(a)H(˜b), where ˜b(t) := b( 1t ). • T (a)∗ = T (a). • (Coburn) Let a ∈ L∞ (T) \ {0}. Then at least one of the spaces ker T (a) and l2 /imT (a) consists of the zero element only. Proposition 8. ([3], Chapter 1) (i) Let a ∈ C(T). The Toeplitz operator T (a) is Fredholm on l2 if and only if 0 ∈ / a(T). In this case, ind T (a) = −wind a, where wind a refers to the winding number of the curve a(T), provided with the orientation inherited by the usual counter-clockwise orientation of the unit circle, around the origin. / a(T) (ii) Let a ∈ C(T). The Toeplitz operator is invertible on l2 if and only if 0 ∈ and wind a = 0. (iii) Let a ∈ C(T). Then H(a) is compact on l2 . (iv) The smallest C ∗ -subalgebra T (C) of B(l2 ) containing all Toeplitz operators with continuous generating functions, decomposes as 2 ˙ T (C) = {T (a) : a ∈ C(T)}+K(l ),

where K(l2 ) stands for the (closed) ideal of all compact operators. Now let us turn to the finite section method for Toeplitz operators (with continuous generating function). The first question is about the stability of the sequence (Pn T (a)Pn ), where Pn : l2 → l2 is the projection defined by (a0 , a1 , . . . , an , an+1 , . . . ) → (a0 , . . . , an , 0, 0, . . . ) . This problem was investigated by many people. G. Baxter, 63 : (Pn T (a)Pn ) stable in l1 if and only if T (a) is invertible (a ∈ W, 0 ∈ / a(T), wind a = 0, where W stands for the Wiener algebra). I. Gohberg, I. Feldmann, 65 : (Pn T (a)Pn ) stable in l2 if and only if T (a) is invertible.

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Later on related results for classes of discontinuous generating functions were achieved: QC, C +H ∞ , P C, P QC. For instance QC stands for the class of all quasi continuous functions, P QC for the smallest closed subalgebra of L∞ (T) containing the algebra P C of all piecewise continuous functions and QC. Treil, 87 : There are generating functions a with only one point of discontinuity such that T (a) is invertible but (Pn T (a)Pn ) is not stable. Recall the definition of the algebra F W (Example 2): A sequence (An ) ∈ F belongs to F W , if and only if the strong limits W (An ) := s˜ (An ) = s-lim Rn An Rn as well as W (A∗n ) , W ˜ (A∗n ) exist. Because of lim An Pn , W ∞ Rn T (a)Rn = Pn T (˜ a)Pn (a ∈ L (T)) it is easy to see that (Pn T (a)Pn ) ∈ F W . Moreover, Rn KRn tends strongly to zero for every compact operator due to the weak convergence of (Rn ) to zero. Hence, the smallest C ∗ -subalgebra S(C) in F containing all sequences (Pn T (a)Pn ), a ∈ C(T), is actually contained in F W . Our next goal is to describe the structure of the algebra S(C). For, we need Widom’s identity Pn T (ab)Pn = Pn T (a)Pn T (b)Pn + Pn H(a)H(˜b)Pn + Rn H(˜ a)H(b)Rn (prove it). The collection JW := {(Pn KPn + Rn LRn + Cn ) : K, L compact, (Cn ) ∈ G} forms a closed two-sided ideal in F W . Theorem 9. ([7], Theorem 1.5.3) JW ⊂ S(C). Moreover, each element (An ) ∈ S(C) can uniquely be written as An = Pn T (a)Pn + Pn KPn + Rn LRn + Cn , where K, L are compact operators, (Cn ) ∈ G. Using this representation, it is evident that ˜ (An ) = T (˜ a) + L , W (An ) = T (a) + K , W ˜ = G. and ker W ∩ ker W ˜ : S(C) → B(l2 ) and glue them toNow take the ∗-homomorphisms W, W 0 gether to obtain a ∗-homomorphism smb : S(C) → B(l2 ) × B(l2 ), ˜ (An )) . (An ) → (W (An ), W Furthermore, it is clear that ker smb0 = G. Thus, the quotient homomorphism smb : S(C)/G → B(l2 ) × B(l2 ) is correctly defined and is injective. Notice that an injective ∗-homomorphism is isometric.

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45

Theorem 10. (i) The map smb is a ∗-isomorphism from S(C)/G onto the C ∗ -subalgebra of ˜ (An )) with (An ) running B(l2 ) × B(l2 ) which consists of all pairs (W (An ), W through S(C). ˜ (An ) are invertible oper(ii) (An ) ∈ S(C) is stable if and only if W (An ) and W ators. (iii) S(C) is fractal. Now it is clear that the theory of Section 3 applies. Theorem 11. ˜ (An )}. (a) lim An  = max{W (An ) , W ˜ (An ). (b) lim inf spε An = lim sup spε An = spε W (An ) ∪ spε W ˜ (An ). a) = spε W If (An ) = (Pn T (a)Pn ), then spε W (An ) = spε T (a) = spε T (˜ ˜ (An ). (c) If (An ) is normal, then lim inf spAn = lim sup sp An = sp W (An )∪sp W    ˜  (d) lim inf (An ) = lim sup (An ) = (W (An )) ∪ (W (An )).

5. Fredholm sequences Now we are going to introduce a third fundamental notion, namely that one of Fredholm sequences. First we introduce Fredholm sequences in some restricted form and finally in full generality. We introduce C ∗ -subalgebras of F which are generalizations of the algebras l F and F W and which give raise to consider Fredholm sequences. Let H be an infinite-dimensional Hilbert space and (Ln ) be a sequence of orthogonal projections such that Ln → I strongly as n → ∞. The related C ∗ -algebra of all bounded sequences is again denoted by F . We shall assume that all projections Ln are finite rank operators. Let T be a (possibly infinite) index set and suppose that, for every t ∈ T , we are given an infinite-dimensional Hilbert space H t with identity operator I t as well as a sequence (Ent ) of partial isometries Ent : H t → H such that • the initial projections Ltn of Ent converge strongly to I t as n → ∞, • the range projection of Ent is Ln , • the separation condition ∗

(Ens ) Ent → 0 weakly as n → ∞

(5.1)

holds for every s, t ∈ T with s = t. (Recall that an operator E : H  → H is a partial isometry if EE ∗ E = E and that E ∗ E and EE ∗ are orthogonal projections which are called the initial t and the range projections of E, respectively). For brevity, write E−n instead t ∗ t t of (En ) , and set Hn := im Ln and Hn := im Ln . Let F T stand for the set of all sequences (An ) ∈ F for which the strong limits t t s-limn→∞ E−n An Ent and s- lim(E−n An Ent )∗

46

B. Silbermann

exist for every t ∈ T , and define mappings W t : F T → B(H t ) by W t (An ) := t s- limn→∞ E−n An Ent . It is easily seen that F T is a C ∗ -subalgebra of F which contains the identity, and that the W t are ∗-homomorphisms. The separation condition (5.1) ensures that, for every t ∈ T and every comt ) belongs to the algebra F T , pact operator K t ∈ K(H t ), the sequence (Ent K t E−n and that for all s ∈ T  t if s = t K t W s (Ent K t E−n )= . (5.2) 0 if s = t Conversely, (5.2) implies (5.1). Moreover, the ideal G belongs to F T . So we t can introduce the smallest closed ideal J T which contains all sequences (Ent K t E−n ) t t with t ∈ T and K ∈ K(H ) as well as all sequences (Gn ) ∈ G. Remark. The algebra F W provides an example of this type. Indeed T consists only ˜ , and of two points, say 1 and 2. Then W 1 = W, W 2 = W J1 J2

= {(Pn KPn + Cn ) : K compact, (Cn ) ∈ G} , = {(Rn LRn + Cn ) : L compact, (Cn ) ∈ G} .

The ideal JT is exactly the earlier introduced ideal JW . The separation condition (5.1) is obviously fulfilled (recall that Rn tends weakly to zero). There are examples which show that indeed infinite index sets T are needed ([7], 4.5.1–4.5.2, for instance). Theorem 12. ([7], Theorem 6.1) (a) A sequence (An ) ∈ F T is stable if and only if the operators W t (An ) are invertible in B(H t ) for every t ∈ T and if the coset (An ) + J T is invertible in the quotient algebra F T /J T . (b) If (An ) ∈ F T is a sequence with invertible coset (An ) + J T , then all operators W T (An ) are Fredholm on H t , and the number of the non-invertible operators among the W t (An ) is finite. Notice that this theorem can be used to give a different proof of Theorem 10, (ii). Definition 6. (a) A sequence (An ) ∈ F T is called Fredholm if the coset (An ) + J T is invertible. (b) If the sequence (An ) ∈ F T is Fredholm, then its nullity α(An ), deficiency β(An ) and index ind (An ) are defined by

α(An ) := dim ker W t (An ), β(An ) := dim coker W t (An ), t∈T

t∈T

and ind (An ) := α(An ) − β(An ). It is a triviality to carry over well-known properties of Fredholm operators to Fredholm sequences.

C ∗ -algebras and Asymptotic Spectral Theory

47

Remark. As we will see later on, this notion of Fredholm sequence depends on the underlying algebra F T . We shall also see that Fredholmness of a sequence in the sense of Definition 6 implies its Fredholmness in a general sense which has still to be defined. Let (An ) ∈ F be arbitrary. We order the singular values of An as follows (ln = rank Ln ): 0 ≤ s1 (An ) ≤ · · · ≤ sln (An )(= An ) . For the sake of convenience let us also put s0 (An ) = 0. Recall that usually the singular values are ordered in the reverse manner. Definition 7. We say that (An ) ∈ F has the k-splitting property if there is a k ∈ Z+ such that lim sk (An ) = 0 , n→∞

while the remaining ln − k singular values stay away from zero, that is sk+1 (An ) ≥ δ > 0 for n large enough. The number k is also called the splitting number. Notice if (An ) has the 0-splitting property then (An ) is stable (hint: if s1 (An ) = 0 then An is invertible and A−1  = s1 (An )−1 ). Theorem 13. (a) Let (An ) ∈ F T be Fredholm. Then (An ) is subject to the k-splitting property  t Ht t En Pker W t (An ) E−n ). with k = α(An ) and sα(An ) (An ) ≤ An ( t∈T

(b) If for (An ) ∈ F T there is at least one t1 ∈ T such that W t1 (An ) is not Fredholm, then lim sl (An ) = 0 for all l ∈ Z+ . n→∞

Assertions (a) and (b) can be proved slightly modifying the idea of the proof of Theorem 6.11 and using Theorem 6.67 in [7]. A complete proof of Theorem 13 is contained in [17]. We present here the proof of Theorem 13, (a), for the special case F l (we use here (Pn ) instead of (Ln ) in accordance with the notations of 4.I.). Proof. We shall make use of the following alternative description of the singular values (as approximation numbers): sj (An ) :=

min

B∈Flln

An − B ,

n−j

ln where Fm denotes the collection of all ln × ln -matrices of rank at most m. Let Rn be the orthoprojection onto im (Pn Pker A Pn ), where A = s-lim An Pn . It is easy to check that im Rn = im Pn Pker A Pn ,

rank Rn = rank Pn Pker A Pn = rank Pker A = dim ker A =: k for n large enough, and Rn − Pn Pker A Pn  → 0 as n → ∞ .

48

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Consequently, ||An Rn || → 0 as n → ∞, and (An Rn ) ∈ G. Consider the sequence (Bn ) ∈ F l , Bn := A∗n An (Pn −Rn )+Pn Pker A Pn . Obviously, this sequence is also Fredholm and s-lim Bn Pn = A∗ A + Pker A is invertible. Then (Bn ) is stable by Theorem 12, (a). Since rank (Pn − Rn ) = ln − k for n large enough we get for those n   sk (An ) ≤ || An − An A∗n An (Pn − Rn )Bn−1 Pn || ≤ || (An Bn − An A∗n An (Pn − Rn )) Pn || ||Bn−1 Pn || ≤ ||Bn−1 Pn || ||An Pn Pker A Pn || . Since (Bn ) is stable, there exists for n large enough a constant C with ||Bn−1 Pn || ≤ C. Thus we have sk (An ) ≤ C||An Pn Pker A Pn || → 0 as n → ∞ . Now consider sk+1 (An ). By using the well-known inequality sk+1 (A∗n An ) ≤ ≤ sk+1 (An )||A∗n || and that ||A∗n || is bounded (recall that A∗n Pn converges strongly to A∗ = 0) it has to be shown that sk+1 (A∗ An ) is bounded away from zero (n large enough). We have sk+1 (A∗n An ) = = ≥ =

min

|| (A∗n An − B) Pn ||

min

|| ((A∗n An + Pn Pker A Pn ) − B − Pn Pker A Pn ) Pn ||

n B∈Flln −k−1

n B∈Flln −k−1

min

n B∈Flln −1

|| ((A∗n An + Pn Pker A Pn ) − B) Pn ||

s1 (A∗n An + Pn Pker A Pn ) ≥ δ > 0

for n large enough since (A∗n An + Pn Pker A Pn ) is stable, and we are done.




Corollary 3. If (An ) ∈ F T is Fredholm, then ind (An ) = 0 . Proof. One has only to use that the matrices A∗n An and An A∗n are unitarily equivalent. This shows that the splitting numbers of (An ) and (A∗n ) coincide.
Example 4. The sequence (Pn V Pn ) belongs to both algebras F l and F W . This sequence is Fredholm in F W but not in F l . If it would be Fredholm in F l then ind V = 0; but ind V = −1. Theorem 13 has remarkable applications. Let us mention some simple results: • If T (a)(a ∈ C(T)) is Fredholm, then the Moore-Penrose inverses (Pn T (a)Pn )+ converge strongly to T (a)+ if and only if dim ker Pn T (a)Pn = α(Pn T (a)Pn ) for n large enough.

C ∗ -algebras and Asymptotic Spectral Theory

49

A deeper study of this problem is presented in [3], Chapter 4. • Let T (a) (a ∈ C(T)) be Fredholm and K be compact. Since (Pn (T (a) + K)Pn ) is subject to the splitting property with splitting number α(Pn (T (a)+ a) and dim ker T (˜ a) is known by K)Pn ) = dim ker(T (a) + K) + dim ker T (˜ Coburns Theorem, dim ker(T (a)+K) can be found numerically (in principle). • If T (a) (a ∈ C(T)) is Fredholm and a is smooth then sα (Pn T (a)P  n )βtends fast |k| |ak | < ∞ to zero. For instance, if the Fourier coefficients ak of a fulfill k∈Z

for some β > 0 then sα (Pn T (a)Pn ) = O(1/nβ ). • It was mentioned before that multiplication operators Ma with continuous functions a are quasidiagonal in L2 (T). The corresponding sequence of finitedimensional projections can be taken as orthogonal projections on some spline spaces. To be more precise let T := {|z| = 1} be parametrized be ϕ : [0, 1] → T , ϕ(t) = e2πit . A sequence of partitions (∆k )k∈N , ∆k := {σ0k , . . . , σnk k } , 0 = k −σj ) → σ0k < σ1k < · · · < σnk k = 1, is said to be admissible if h∆k := max(σj+1 δ 0 as k tends to infinity. We denote by S˜ (∆k ) the space of all ψ ∈ C(T) such that ψ ◦ ϕ is (δ − 1) times continuously differentiable and the restriction of k ) is a polynomial of degree ≤ δ (smoothest ψ ◦ ϕ to each interval (σjk , σj+1 splines). Let P∆k denote the orthogonal projections of L2 (T) onto S˜δ (∆k ). Then (see [8], Section 2.14) (I − P∆k )f P∆k  → 0 ,

P∆k f (I − P∆k ) → 0

as k → ∞, where –  ·  stands for the operator norm in L2 (T), – f is continuous, – (∆k ) is admissible. Consider the singular integral operator A with continuous coefficients:  1 g(τ ) dτ Ag = ag + bSg , where (Sg)(t) := πi τ −t T

(Recall that this integral exists for g ∈ L (T) almost everywhere in the sense of Cauchy’s principal value). A singular integral operator A is called strongly (locally) elliptic, if there is a continuous function c on T, a linear operator T with T  < 1 and a compact operator K such that 2

A = c(I + T ) + K , c(t) = 0 for all t ∈ T . It follows that A is Fredholm with index 0 (even invertible). It is well known that A is strongly elliptic if and only if a(t) + λb(t) = 0 ∀t ∈ T und ∀λ ∈ [−1, 1] . If A is strongly elliptic and (∆k ) admissible then (P∆k AP∆k ) is stable

50

B. Silbermann (use P∆k AP∆k = P∆k cIP∆k (I + T )P∆k + P∆k KP∆k + C∆k , C∆k  → 0). If a and b are merely continuous N × N -matrix functions, then A is strongly elliptic if and only if det(a(t) + λb(t)) = 0 for ∀t ∈ T and ∀λ ∈ [−1, 1]

(see [8], Section 13.31). In this case A is Fredholm of index 0, but might be not invertible. In any case, (P∆k AP∆k ) is Fredholm and α(P∆k AP∆k ) = dim ker A where (∆k ) is admissible. Now we turn to general Fredholm sequences. Definition 8. Let B be a unital C ∗ -algebra. An element k ∈ B is of central rank one if, for every b ∈ B, there is an element µ(b) belonging to the center of B such that kbk = µ(b)k. An element of B is of finite central rank if it is the sum of a finite number of elements of central rank one, and it is centrally compact if it lies in the closure of the set of all elements of finite central rank. We denote the set of all centrally compact elements in B by J (B). It is easy to check that J (B) forms a closed two-sided ideal in B. Proposition 9. ([7], Proposition 6.33) A sequence (An ) ∈ F is centrally compact if and only if, for every ε > 0, there is a sequence (Kn ) ∈ F such that sup An − Kn  < ε and sup dim im Kn < ∞ . n

Definition 9. A sequence (An ) ∈ F is a Fredholm sequence if it is invertible modulo the ideal J(F ) of the centrally compact sequences. Theorem 14. ([7], Theorem 6.35) A sequence (An ) ∈ F is Fredholm if and only if there is an l ∈ Z+ such that lim inf sl+1 (An ) > 0 .

n→∞

Conclusion. If (An ) ∈ F T is Fredholm then it is also Fredholm in the sense of Definition 9.

6. Applications continued: Around finite sections of operators with almost periodic diagonals This material is based on [10]. 6.1. Example.  Let ˜l2 := {(xn )n∈Z : |xn |2 < ∞}. n∈Z

The Almost Mathieu Operator is the operator Hα,λ,θ : ˜l2 → ˜l2 , which acts on a sequence x = (xn )n∈Z ∈ ˜l2 by (Hα,λ,θ x)n := xn+1 + xn−1 + λxn cos 2π(nα + θ).

C ∗ -algebras and Asymptotic Spectral Theory

51

Only recently the long-standing Ten Martini problem was solved, see [1], [9], and for a introduction to the topic [2]. The problem consists in describing the spectrum of Hα,λ,θ . The result says (in a somewhat incomplete form) that • If α is rational, α = pq and p, q relatively prime with q > 0, then the spectrum of Hα,λ,θ is the union of exactly q closed and pairwise disjoint intervals, θ πp ∈ / Z. • If α ∈ [0, 1) is irrational, then the spectrum is a Cantor type set (it means: nowhere dense, closed, and does not contain isolated points). This result is a qualitative one! It does not allow to say that a given number µ belongs to the spectrum (or not). There is (at least in present time) only one way to tackle this problem, namely the use of approximation methods. For, introduce the projection operators Pn and P˜n : P˜n x = {. . . , 0, x−n , . . . , xn , 0, . . . } Pn x = {. . . , 0, x0 , . . . , xn , 0, . . . } . First idea: consider the operators (matrices) (P˜n Hα,λ,θ P˜n ) (restricted to im P˜n and with respect to the standard basis) and compute the eigenvalues using Matlab or something else. Then the question arises, is this spectrum some how related to the spectrum of the Almost Mathieu Operator Hα,λ,θ ? The following is devoted to some theory around this problem. However, we will merely make use of the projections Pn . The operator Hα,λ,θ is an example of a band operator with almost periodic diagonals. 6.2. Band-dominated operators with almost periodic diagonals and related Toeplitz-like operators Recall that a general band-dominated operator A : ˜l2 → ˜l2 is the norm limit of band operators, that is of operators of the kind k


al U l ,

−k

where Um : ˜l2 → ˜l2 is the shift operator given by (Um x)(n) = x(n − m) and al I : l → l are multiplication operators with al ∈ l∞ (Z). If we replace the elements al ∈ l∞ (Z) by almost periodic ones, that is the set {Um a}m∈Z is relatively compact in the norm of l∞ (Z), then we obtain the class of band and band-dominated operators with almost periodic diagonals. It is easy to see that the collection AP (Z) ⊂ l∞ (Z), of all almost periodic sequences as well as the collection AAP of all band-dominated operators with almost periodic diagonals form C ∗ -algebras. Simple example: The Laurent operator L(a) with continuous generating function a ∈ C(T) is obviously an element of AAP (Z). ˜2

˜2

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B. Silbermann

Let us introduce the following class of Toeplitz-like operators. Clearly, l2 can be thought of as a subspace of ˜l2 . Let P denote the orthogonal projection onto l2 and Q := I − P . Consider T (A) : l2 → l2 , A ∈ AAP (Z), T (A) := P AP |imP . If A = L(a), a ∈ C(T), then T (L(a)) is denoted simply by T (a), and this is a familiar Toeplitz operator. Introduce J : ˜ l2 → ˜l2 (flip operator) by (xn ) → ˜ (x−n−1 )) and H(A) := P AQJ , A := JAJ. Then one has ˜ (A, B ∈ AAP (Z)) , T (AB) = T (A)T (B) + H(A)H(B) which reminds the basic identity relating Toeplitz and Hankel operators and it is this identity for A = L(a), B = L(b), a, b ∈ C(T). ˜ are compact operators! Notice: H(A), H(B) Let AAP (Z+ ) denote the smallest C ∗ -subalgebra of B(l2 ) containing all operators T (A), A ∈ AAP (Z). Then • T (A) = A (6.1) 2 ˙ ) • AAP (Z+ ) = {T (A) : A ∈ AAP (Z)}+K(l The first identity is based on a remarkable fact which plays an important role in what follows. Let us have a closer look. Let H refer to the set of all sequences h : Z+ → Z which tend to +∞ or −∞. Definition 10. An operator Ah ∈ B(˜l2 ) is called a norm limit operator of the operator A ∈ B(˜l2 ) with respect to the sequence h ∈ H if U−h(k) AUh(k) → Ah as k → ∞ in norm. The set of all norm limit operators is called the norm operator spectrum σop (A). Theorem 15. A ∈ AAP (Z+ ) is Fredholm if and only if each Ah ∈ σop (A) is invertible. Proposition 10. If A ∈ AAP (Z) then A ∈ σop (A). This result is in force since so-called distinguished sequences exist for A ∈ AAP (Z). Definition 11. A monotonically increasing sequence h : Z+ → Z+ is called distinguished if Ah exists and equals A. The first assertion in (6.1) is now easy to prove: Consider U−h(k) P AP Uh(k) = U−h(k) P Uh(k) U−h(k) AUh(k) U−h(k) P Uh(k)  ! "  ! "  ! " ↓ strongly ↓ in norm ↓ strongly I A I and apply the Banach-Steinhaus Theorem.

C ∗ -algebras and Asymptotic Spectral Theory

53

Further conclusion: ess sp A = spA for A ∈ AAP (Z) and T (A) is Fredholm if and only if A is invertible (ess sp A = sp(A + K(˜l2 )). Example 5. Almost Mathieu operators. We have ak (n) = =

U−k Hα,λ,θ Uk = U−1 + U1 + ak I , a(n + k) = λ cos 2π((n + k)α + θ) λ(cos 2π(nα + θ) cos 2π(kα) − sin 2π(nα + θ) sin 2π(kα)) .

Let α ∈ (0, 1) be irrational. We write α as a continued fraction with nth approximant pqnn such that 1

α = lim

n→∞

1

b1 + b2 +

1 ..

. bn−1 +

1 bn

with uniquely determined positive integers. Write this continued fraction as pn /qn with positive and relatively prime integers pn , qn . These integers satisfy the recursions pn = bn pn−1 + pn−2 , qn = bn qn−1 + qn−2 with p0 = 0, p1 = 1, q0 = 1 and q1 = b1 , and one has for all n ≥ 1 # # # # #α − pqnn # < q12 , n ⇒ |αqn − pn | ≤ q1n → 0 . Now it is not hard to see that (qn ) is a distinguished sequence for Hα,λ,θ (note: (qn ) is independent on λ and θ). 6.3. Distinguished sequences and finite sections In what follows we fix a strongly monotonically increasing sequence h : Z+ → Z+ and define AAP,h (Z) := {A ∈ AAP (Z) : Ah exists and Ah = A} . It is easy to check that AAP,h (Z) is a C ∗ -subalgebra of B(˜l2) which is moreover shift invariant, i.e., U−k AUk again lies in AAP,h (Z+ ) whenever A does. Let AAP,h (Z+ ) refer to the smallest closed subalgebra of B(l2 ) which contains all operators T (A) with A ∈ AAP,h (Z). For instance, all Toeplitz operators with continuous generating functions lie in this algebra. Thus • K(l2 ) ⊂ AAP,h (Z+ ) and 2 ˙ • AAP,h (Z+ ) = {T (A) : A ∈ AAP,h (Z)}+K(l ).

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Let us turn over to finite sections. For, let Fh denote the set of all bounded sequences (An ) of matrices An ∈ Ch(n)×h(n) . Provided with pointwise defined operations and the supremum norm, Fh becomes a C ∗ -algebra (An  – norm of the operator defined by An on im Ph(n) ). Finally, we let SAP,h (Z+ ) denote the smallest closed subalgebra of Fh which contains all sequences (Ph(n) T (A)Ph(n) ) with operators A ∈ AAP,h (Z). Define Rn : l2 → l2 , (xn )n≥0 → (xn , xn−1 , . . . , x0 , 0, 0, . . . ). Theorem 16. The C ∗ -algebra SAP,h (Z+ ) consists exactly of all sequences of the form   Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Rh(n) LRh(n) + Ch(n) (6.2) with A ∈ AAP,h (Z) , K, L ∈ K(l2 ) , Ch(n)  → 0 as n → ∞, and each sequence in SAP,h (Z+ ) can be written in the form (6.2) in a unique way. ˜ : SAP,h (Z+ ) → AAP,h (Z+ ) by Define mappings W, W W (An ) = ˜ (An ) = W

s- lim Ph(n) An Ph(n) , s- lim Rh(n) An Rh(n) .

These mappings are ∗-homomorphisms. Their importance is given by the following stability theorem. Theorem 17. A sequence (An ) ∈ SAP,h (Z+ ) is stable if and only if the operators ˜ (An ) are invertible, that is, if W (An ), W An = Ph(n) T (A)Ph(n) + Ph(n) KPh(n) + Rh(n) LRh(n) + Ch(n) ˜ +L with K, L, Ch(n) as above, then (An ) is stable if and only if T (A) + K , T (A) are invertible. Moreover, SAP,h (Z+ ) is fractal. The proof is basically the same as in the Toeplitz case. 6.4. Spectral approximations The last theorem in Section 6.3 is one of the keys to study spectral approximations. Theorem 18. Let A := (An ) ∈ SAP,h (Z+ ) be a self-adjoint sequence. Then the ˜ (A). spectra sp An converges in the Hausdorff metric to sp W (A) ∪ sp W Theorem 19. Let A := (An ) ∈ SAP,h (Z+ ). Then the set of the singular values    ˜ (A)). (An ) converges in the Hausdorff metric to (W (A)) ∪ (W Theorem 20. A sequence A = (An ) ∈ SAP,h (Z+ ) is Fredholm if and only if its ˜ (A) is a Fredholm operstrong limit W (A) is a Fredholm operator. In this case W ator too, and ˜ α(A) = dim ker W (A) + dim ker W(A) ; moreover, lim sα (An ) = 0. n→∞

These theorems can be completed by results concerning ε-pseudospectra and the so-called Arveson’s dichotomy (the last for self-adjoint sequences).

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Arveson’s dichotomy: Given a self-adjoint sequence A := (An ) ∈ SAP,h (Z+ ) and an open interval U ⊂ R, let Nn (U ) refer to the number of eigenvalues of An in U , counted with respect to their multiplicity. A point λ ∈ R is called essential for A, if for every open interval U containing λ, lim Nn (U ) = ∞ ,

n→∞

and λ ∈ R is called a transient point for A is there is an open interval U containing λ such that sup Nn (U ) < ∞ . n

Theorem 21. Let A := (An ) ∈ SAP,h (Z+ ) be self adjoint, s-lim An = T (A) + K. (s-lim An is necessarily of this form because of Theorem 16). Then every point λ ∈ sp A is essential, and every point λ ∈ R\sp A is transient for A. Moreover, for every point λ ∈ R\sp A, the sequence A−λP where (P := (Ph(n) ) ∈ SAP,h (Z+ )) is Fredholm and there is an open interval U ⊂ R containing λ such that sup Nn (U ) = α(A − λP).

n

Theorem 21 is a consequence of Theorem 7.12 in [7] and of the fact that $ = ess sp A. $ ess sp A = sp A = sp A The first assertion of Theorem 21 implies in particular that each real number is either essential or transient for A. This property is usually referred to as the Arveson’s dichotomy of that sequence. Remark. If A ∈ AAP (Z) is selfadjoint and h is a distinguished sequence, then Theorem 21 offers the possibility to determine the spectrum of A numerically. Test calculations will be provided in the next section. Let us mention that we could also use the projection P$n for this aim. The related theory (see Section 7) is not simpler as that for the projections Pn , but gives essentially the same results. A deep study of the finite sections for general band-dominated operators in ˜l2 is carried out in [15]. Let us mention also the recent book [4], where spectral properties of banded Toeplitz matrices are studied. 6.5. Test calculations Here we shall demonstrate how the theory can be used to determine numerically the spectrum of the Almost Mathieu operator for some choices of the parameters α, λ and θ (using Matlab). For each of the triples  √        √ 2 1 1 1 2 5−1 2 2 , 2, 0 , , 2, , , 2, 0 , , 2, , 2, , , 5 5 2 7 2 2 2 2

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in place of (α, λ, θ), we choose a distinguished sequence of the corresponding Almost Mathieu operator which depends only on & % √ √ 2 5−1 2 2 αj ∈ , , , , 5 7 2 2 namely α1 α2 α3

= = =

α4

=

2 5 : h1 (k) = 5k , 2 7 : h2 (k) = 7k , √ √ k √ k 2 1 2) , 5 : h3 (k) = 2 (1 +  2) +(1 − √ √ √  √ k √ k 5−1 5+ 5 1+ 5 5− 5 1− 5 : h (k) = + 4 2 10 2 10 2

.

For irrational αk , this choice has been done via continued fractions. Notice that the sequences h3 and h4 are rapidly growing. For instance, h3 (13) = 47321 , h4 (23) = 46368. The results are plotted in Pictures 1–7. The results for α4 , λ = 2 , θ = 0, 5 and h5 (k) := 2k (non-distinguished!) are plotted in Picture 8. The computations clearly indicate the advantage of distinguished sequences over non-distinguished (compare Pictures 6 and 8). For irrational α the Cantorlike structure of the spectrum is also somehow reflected in the computations (see Pictures 4 and 5). The computations also show that the speed of converges is very high. There is only a guess why it should be, but not a proof.

Picture 1: Eigenvalues of Ph1 (k)Hα,λ,θ Ph1 (k) with α = 2/5, λ = 2, θ = 1/2.

C ∗ -algebras and Asymptotic Spectral Theory

Picture 2: Eigenvalues of Ph2 (k)Hα,λ,θ Ph2 (k) with α = 2/7, λ = 2, θ = 0.

Picture 3: Eigenvalues of Ph2 (k)Hα,λ,θ Ph2 (k) with α = 2/7, λ = 2, θ = 1/2.

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Picture 4: Eigenvalues of Ph3 (k)Hα,λ,θ Ph3 (k) with α =

√ 2/2, λ = 2, θ = 0.

Picture 5: The eigenvalues of Ph3 (k)Hα,λ,θ Ph3 (k) with α = which lie in the interval (−2.4, −2.8).

√ 2/2, λ = 2, θ = 0

C ∗ -algebras and Asymptotic Spectral Theory

59

√ Picture 6: Eigenvalues of Ph4 (k)Hα,λ,θ Ph4 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5.

√ Picture 7: The eigenvalues of Ph4 (k)Hα,λ,θ Ph4 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5, which lie in the interval (1.8, 2.6).

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√ Picture 8: Eigenvalues of Ph5 (k)Hα,λ,θ Ph5 (k) with α = ( 5 − 1)/2, λ = 2, θ = 0.5.

7. Applications continued: Band-dominated operators and their finite sections We reproduce in this section some recent results obtained in [12], [13], however with different proofs. We give a complete study of the stability problem for finite sections of band-dominated operators and use these results to get an index formula for Fredholm operators belonging to that class. Recall that a general band-dominated operator A : ˜l2 → ˜l2 is the norm limit of band operators, that is of operators of the kind k


al U l ,

−k

where Ul are the earlier defined shift operators, and al I : ˜l2 → ˜l2 are the multiplication operators (al x)m := al (m)xm with al = (al (m)) ∈ l∞ (Z). Let P, Q be the orthoprojections in ˜l2 , introduced in Section 6.2. Denote by H the set of all sequences h : N → Z which tend to −∞ or +∞. An operator Ah ∈ B(˜l2 ) is called the limit operator of A ∈ B(˜l2 ) with respect to the sequence h ∈ H if U−h(n) AUh(n) tends ∗-strongly to Ah as n → ∞. The set σ op (A) of all limit operators of a given operator A ∈ B(˜l2 ) is called the operator spectrum of A. It is not hard to see that every limit operator of a Fredholm operator is invertible. It is a remarkable fact that for band-dominated operators the reverse

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61

implication is true. More precisely, the following theorem is in force (see [11], Chapter 2): Theorem 23. Let A ∈ B(˜l2 ) be a band-dominated operator. Then (a) every sequence h ∈ H possesses a subsequence g such that the limit operator Ag exists, (b) the operator A is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded, (c) for each compact operator K one has σop (A + K) = σ op (A). Moreover, (A + K)g = Ag if one of the operators A (A + K)g , Ag with respect to the sequence g ∈ H exists. Notice that the collection of all band-dominated operators actually forms a C ∗ subalgebra S of B(˜l2 ) which contains all compact operators. Since P AQ, QAP are compact operators for all band-dominated operators, the study of the Fredholm properties of A ∈ S can be reduced to those of P AP + QBQ. ˜ n : ˜l2 → ˜l2 by Let P˜n be defined as in Section 6.1 and define R (xn )n∈Z → (. . . , 0, 0, x−1 , x−2 , . . . , x−n , xn , xn−1 , . . . , x0 , 0, 0, . . . ) . ˜ n play an important role in the study of the stability of (P˜n AP˜n ), The operators R A ∈ S. It is not hard to see that the following version of Widom’s formula holds true (A, B ∈ S): ˜ n LR ˜ n + Cn (7.1) P˜n AB P˜n = P˜n AP˜n B P˜n + R with some compact operator L, and (Cn ) ∈ G (exercise; hint: consider expressions of the type P˜n P ABP P˜n − P˜n P AP P˜n P BP P˜n and those where P is replaced by Q). The ideas of 4.II are again not directly applicable. The point is that the strong ˜ n AR ˜n does not exist for A ∈ S in general. The situation changes when limit s-lim R we consider suitable subsequences. Theorem 24. (a) Let A ∈ S and h ∈ H be a sequence tending to +∞ for which the limit operator Ah of A with respect to the sequence h exists. Then there is a subsequence g of h such that the limit operator A−g of A exists. Moreover, ˜ g(n) AR ˜ g(n) = P JAh JP + QJA−g JQ . s∗ - lim R (b) An analogous result is true for sequences tending to −∞. ˜ g(n) ) tends weakly Proof. (a ) Since A− P AP − QAQ is a compact operator and (R to zero, it needs only to consider P AP + QBQ. Obviously, ˜ g(n) = P JU−g(n) P AP Ug(n) JP + QJUg(n) QAQU−g(n) JQ , ˜ g(n) (P AP + QAQ)R R where g is a subsequence of h for which the limit operator A−g of A exists (Theorem 23, (a)). A straightforward computation shows the claim.


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Let h ∈ H be a sequence tending to +∞ and introduce the algebra Fh with ˜ respect to the sequence (Ph(n) ) and let FhW stand for the C ∗ -subalgebra of Fh constituted by all sequences of Fh for which the limits ˜ h(n) Ah(n) R ˜ h(n) s∗ - lim P˜h(n) Ah(n) P˜h(n) and s∗ - lim R exist. Again, the set  ( ' ˜ ˜ h(n) LR ˜ h(n) + Ch(n) : K, L − compact, (Ch(n) ) ∈ Gh JhW := P˜h(n) K P˜h(n) + R ˜

actually forms a closed two-sided ideal in FhW . (Gh stands here of the closed twosided ideal of Fh consisting of all sequences tending to zero in norm). The algebra ˜ FhW is of the type described in Section 5. Thus, the general theory mentioned in Section 5 applies. ˜

Corollary 4. A sequence (Ah(n) ) ∈ FhW is stable if and only if ˜ h(n) Ah(n) R ˜ h(n) are invertible, (a) the operators s-lim P˜h(n) Ah(n) Ph(n) and s-lim R ˜ ˜ ˜ W W W (b) the coset (Ah(n) ) + Jh is invertible in Fh /Jh . Corollary 5. Let h ∈ H be a sequence tending to +∞ and let A ∈ S be an operator for which ˜ h(n) AR ˜ h(n) s∗ - lim R ˜ exists. Then (P˜h(n) AP˜h(n) ) belongs to FhW .

Proposition 11. Let h ∈ H be a sequence tending +∞ and let A ∈ S be Fredholm. Then there exists a subsequence g of h such that (Pg(n) APg(n) ) is a Fredholm ˜ sequence in FgW . Proof. It is easy to see that A has a regularizer B in S, that is AB = I +M1 , BA = I + M2 , where M1 , M2 are compact operators. Using (7.1) we get ˜ n LR ˜ n + Cn , K, L-compact and (Cn ) ∈ G . P˜n AP˜n B P˜n = P˜n + P˜n K P˜n + R By Theorem 24 and Corollary 5 there is a subsequence h1 of h such that ˜ h (n) AR ˜h (n) exists. By the same reasoning there is a subsequence g1 of h1 s∗ -lim R 1 1 ˜ g (n) B R ˜g (n) exists. Hence, (Pg (n) APg (n) ), (Pg (n) BPg (n) ) ∈ such that s∗ -lim R 1 1 1 1 1 1 ˜

)

˜

˜

FgW1 , and (pg1 (n) APg1 (n) ) + JgW1 is invertible from the right in FgW1 /JgW1 . Again choosing a suitable subsequence g of g1 one obtains in the same fashion as above ˜ that (Pg(n) APg(n) ), (Pg(n) BPg(n) ) belong to FgW and ˜

˜

˜

(Pg(n) APg(n) ) + JgW is already invertible in FgW /JgW .




op op (A), σ− (A) denote the sets of all limit operators of A ∈ S Theorem 25. Let σ+ with respect to sequences tending to +∞ and −∞, respectively. Then the sequence (P˜n AP$n ) is stable if and only if the operator A, and all operators op (A) , QAh Q + P with Ah ∈ σ+

C ∗ -algebras and Asymptotic Spectral Theory

63

and all operators op P Ah P + Q with Ah ∈ σ− (A)

are invertible on ˜l2 . Proof. Sufficiency: Suppose that (P˜n AP˜n ) is not stable. Then there is a sequence h tending to +∞ such that lim (P˜h(n) AP˜h(n) )−1 P˜h(n)  equals +∞ (if P˜h(n) AP˜h(n) is not invertible on im P˜h(n) , then we set (P˜h(n) AP˜h(n) )−1 Ph(n)  = ∞). Then there is by Proposition 10 a subsequence g of h such that (Pg(n) APg(n) ) is a Fredholm ˜ sequence in FgW . By the computations in the proof of Theorem 24 we may assume that ˜ g(n) AR ˜ g(n) s∗ - lim R

= P JAg JP + QJA−g JQ = (P JAg JP + Q)(P + QJA−g JQ) .

Since JP J = Q and JQJ = P , we have J(P JAg JP + Q)J = QAg Q + P . Hence, P JAg JP + Q is invertible. By the same reasoning it follows that also P + QJA−g JQ is invertible. Using Corollary 4 we get that (Pg(n) APg(n) ) is stable. But this is a contradiction. The proof of the necessity is much more easier and is left to the reader.
Our next topic is the index of a Fredholm band-dominated operator A ∈ S. For A ∈ S we put A+ = P AP + Q and A− = P + QAQ . It is easy to see that op op (A) ∪ {I} and σ op (A− ) = σ− (A) ∪ {I} . σ op (A+ ) = σ+

Further, the operator A ∈ S is Fredholm if and only if A+ and A− are Fredholm operators. Moreover, the equality P AP + QAQ = A+ A− = A− A+ shows that, for a Fredholm operator A ∈ S, ind A = ind A+ + ind A− . In addition, we shall use the notation ind+ A := ind A+ and ind− A := ind A− and call ind+ A and ind− A the plus- and the minus-index, respectively. Theorem 26. op (A) have the same plus1. Let A ∈ S be Fredholm. Then all operators in σ+ index, and this number coincides with the plus-index of A. Analogously, all op (A) have the same minus-index, and this number coincides operators in σ− with the minus-index of A.

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2. If A ∈ S is Fredholm, then ind A = ind+ B + ind− C , op op where B ∈ σ+ (A) and C ∈ σ− (A) are arbitrary taken. op (A) choose a sequence h tending Proof. It needs only to prove 1: Given B ∈ σ+ to + infinity and for which B is the limit operator for A. Then B ∈ σop (A+ ). Notice that the limit operator of A+ with respect to −h exists and equals I. Then ˜ (Ph(n) A+ Ph(n) ) ∈ FhW . The Fredholmness of A+ implies that (Ph(n) A+ Ph(n) ) is a Fredholm sequence in Fh(n) and, by Corollary 3, ind A+ + ind (QBQ + P ) = 0. Since B is invertible it follows that ind (QBQ + P ) = −ind (P BP + Q) (use that ind (P BP + QBQ) = 0). Hence, ind A+ = ind+ B. Analogously, ind A− = ind− C, op where C ∈ σ− (A) is arbitrarily chosen.


Finally, let us mention an application of Theorems 25 and 26 to finite sections of band-dominated operators with slowly oscillating coefficients. A function a ∈ l∞ (Z) is slowly oscillating if lim (a(x + k) − a(x)) = 0 for all k ∈ Z .

x→±∞

It is one of the remarkable features of band-dominated operators with slowly oscillating coefficients that their limit operators are Laurent operators with continuous generating functions defined on the unit circle in the complex plane (see [11], Chapter 2). Theorem 27. Let A ∈ S be a band-dominated operator with slowly oscillating coefficients. Then the sequence (P˜n AP˜n ) is stable if and only if A is invertible and ind A+ = 0. Proof. Let A be invertible and ind A+ = 0. Then we have also ind A− = 0. By op op (A) (σ− (A)) have the same plus-index (minusTheorem 26 all operators in σ+ index) and this number coincides with the plus-index (minus-index) of A. Thus, all these indices are equal to zero. By a theorem of Coburg (usually formulated op for Toeplitz operators) the operators P BP + Q(P + QCQ) with B ∈ B+ (A) op (C ∈ σ− (A)) are invertible (see also part 4.II). Now it remains to apply Theorem 26. The reverse statement is obvious.
Theorem 25 can be found in [11] under the additional assumption that all op op operators QAh Q + P, Ah ∈ σ+ (A) and all operators P Ah P + Q, Ah ∈ σ− (A), are uniformly invertible. Without this assumption it occurs in [12]. Theorem 26 was originally proved in [14] by help of K-theory and reproved recently in [13] using asymptotic spectral theory. Theorem 27 is Theorem 6.2.4 in [11], however for band operators with slowly oscillating coefficients. In the general setting Theorem 27 was proved by S. Roch in [16] and is now simply a corollary to Theorems 25 and 26.

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The approach pointed out in Section 7 can be also used to get relatively simple proofs of some further results obtained by S. Roch in [15]. Let me mention only two: • computation of the α-number of a Fredholm sequence (P˜n AP˜n ), A ∈ S. • Any sequence (P˜n AP˜n ), A ∈ S and A = A∗ , has the Arveson dichotomy.

References [1] A. Avila and S. Jitomirskaja: The Ten Martini problem. -arXiv:math. DS/ 0503363. [2] F.-P. Boca: Rotation C ∗ -algebras and Almost Mathieu Operators. Theta Series in Advanced Mathematics 1, The Theta Foundation, Bucharest 2001. ¨ ttcher and B. Silbermann: Introduction to Large Truncated Toeplitz Ma[3] A. Bo trices. Springer-Verlag, New York 1999. ¨ ttcher and S.M. Grudsky: Spectral Properties of Bounded Toeplitz Ma[4] A. Bo trices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 2005. [5] N.P. Brown: Quasidiagonality and the finite section method. Math. Comp. 76 (2007), no. 257, 339–360 (electronic). [6] N.P. Brown: AF Embeddings and the Numerical Computation of Spectra in Irrational Rotation Algebras. Num. Funct. Anal. Optim. 27 (2006), no. 5-6, 517–528. [7] R. Hagen, S. Roch, and B. Silbermann: C ∗ -Algebras and Numerical Analysis. Marcel Dekker, Inc., New York, Basel 2001. ¨ ssdorf and B. Silbermann: Numerical Analysis for Integral and related [8] S. Pro Operator Equations. Akademie Verlag, Berlin 1991. [9] J. Puig: Cantor spectrum for the almost Mathieu Operator. Comm. Math. Phys. 244 (2004), 2, 297–309. [10] V. Rabinovich, S. Roch, and B. Silbermann: Finite sections of band-dominated operators with almost periodic coefficients. Operator Theory: Advances and Applications 170, Birkh¨ auser Verlag, Basel, Boston, Berlin 2006. [11] V. Rabinovich, S. Roch, and B. Silbermann: Limit Operators and Their Applications. Operator Theory: Advances and Applications 150, Birkh¨ auser Verlag, Basel, Boston, Berlin 2004. [12] V. Rabinovich, S. Roch, and B. Silbermann: On finite sections of banddominated operators. Preprint 2486 (2006), Fachbereich Mathematik, TU Darmstadt. [13] V. Rabinovich, S. Roch, and B. Silbermann: The finite section approach to the index formula for band-dominated operators. Preprint 2488 (2006), Fachbereich Mathematik, TU Darmstadt. [14] V. Rabinovich, S. Roch, and J. Roe: Fredholm indices of band-dominated operators. Integral Equations and Operator Theory 49 (2004), 2, 221–238. [15] S. Roch: Finite sections of band-dominated operators. Preprint Nr. 2355 (2004), Fachbereich Mathematik, TU Darmstadt.

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[16] R. Roch: Band-dominated operators on lp -spaces: Fredholm indices and finite sections. Acta Sci. Math (Szeged) 70 (2004), 783–797. [17] A. Rogozhin and B. Silbermann: Banach algebras of operator sequences: Approximation Numbers. J. Operator Theory, in print. Bernd Silbermann Technische Universit¨ at Chemnitz Fakult¨ at f¨ ur Mathematik D-09107 Chemnitz, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 67–118 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Toeplitz Operator Algebras and Complex Analysis Harald Upmeier

1. Introduction The aim of this survey article is to present the recent work concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C ∗ -algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation). Acknowledgment A substantial part of the results presented here is joint work with J. Arazy (University of Haifa). The financial support of the German-Israeli foundation (GIF No. 696-17.6/2001) is gratefully acknowledged. The author would like to thank the referee for valuable comments.

2. Complex geometry of bounded symmetric domains In the following consider a complex vector space Z = Cd , endowed with a norm  · , and let D = {z ∈ Z : z < 1} be its unit ball. In general, the boundary ∂D = {z ∈ Z : z = 1} will not be smooth. Consider the set S = ∂ex D ⊂ ∂D of all extreme points and let K = U (Z) = {g ∈ GL(Z) : gz = z}

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be the isometry group. Then K ⊂ GL(Z) is a closed subgroup, therefore K is a compact Lie group. Clearly, the natural K-action satisfies K · D = D, K · ∂D = ∂D, K · S = S. Example. For Z = Cd , define the scalar product
zi w i (z|w) = zw∗ = i

and let z = (z|z)

1/2

be the Hilbert norm. Then D = {z ∈ Cd : (z|z) < 1}

is the Hilbert unit ball. In this case K = U (d) is the unitary group and S = ∂D = S 2d−1 is the odd sphere. Example. More generally, consider now the matrix space Z = Cr×(r+b) , endowed with the operator norm z = sup σ(zz ∗ )1/2 , and let D = {z ∈ Cr×(r+b) : z < 1} = {z ∈ Cr×(r+b) : I − zz ∗ > 0} be the corresponding matrix unit ball, with closure D = {z ∈ Cr×(r+b) : I − zz ∗ ≥ 0}. In this case we have K = U (r) × U (r + b)  (u, v), via the action z → uzv ∗ , and S = {z ∈ Cr×(r+b) : zz ∗ = I} consists of all isometries. In the special case b = 0 we obtain S = U (r). In this case the domain D is said to be of tube type. We now introduce holomorphic functions. By definition, a (possibly vectorvalued) function f : D → Cm is holomorphic, if it has a power series expansion
f (z) = cα z1α1 · · · znαn α∈Nn

(compact convergence) with coefficients cα ∈ Cm . By collecting monomials of equal (total) degree, we obtain the expansion


cα z1α1 · · · znαn = pk (z) f (z) = k≥0 |α|=k

k≥0

into a series of k-homogeneous polynomials. Let Pk (Z, Cm ) be the space of all k-homogeneous polynomials pk : Z → Cm satisfying pk (λz) = λk pk (z) ∀ λ ∈ C. The space
Pk (Z, Cm ) P(Z, Cm ) = k≥0

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of all polynomials is dense in the space of all holomorphic functions O(D, Cm ) under compact convergence. Definition 2.1. Let G = Aut (D) = {g : D → D biholomorphic} denote the holomorphic automorphism group. It is a real Lie group (by a deep theorem of H. Cartan). Moreover, K = {g ∈ G : g(0) = 0}  G is a compact subgroup. Definition 2.2. The domain D is called symmetric iff G acts transitively on D, i.e., ∀ z ∈ D ∃ g ∈ G, z = g(0)). Clearly, this is equivalent to D = G/K. For arbitrary domains, the condition of symmetry requires that at each point there exists a globally defined geodesic reflection. In general, this is a stronger condition than homogeneity, but in our case the unit ball D is circular and has the obvious reflection s0 (z) = −z at the origin. Example. If Z = Cr×(r+b) and D is the matrix unit ball, we obtain the pseudounitary group      ∗ ∗    ab ab 1 0 a c 1 0 G = SU (r, r+b) = ∈ GL (2r + b) : = cd b d 0 − 1 b∗ d∗ 0 −1 acting on D via Moebius transformations   ab (z) = (az + b)(cz + d)−1 . cd Its maximal compact subgroup is    a0 K= : a ∈ U (r), d ∈ U (r + b) 0d Example. If Z = Cd = C1×d and D is the Hilbert unit ball, we obtain as a special case G = SU (1, d). Example. For dimension d = 1, we have Z = C and D is the unit disk. In this case, we may identify G =

SU (1, 1) ≈ SL(2, R),

K

U (1)

=

It is a fundamental fact [K], [K1], [LO2], [U6] that hermitian symmetric domains have an algebraic description in terms of the so-called Jordan algebras and Jordan triples. In order to explain this connection, consider the Lie algebra exp g −→ G of the real Lie group G = Aut (D), which can be realized as follows:

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Consider a 1-parameter group t → gt ∈ G, and define, for f ∈ O(D, Cm ), the infinitesimal generator   ∂ ∂f (gt (z)) ##  f (z). = (Xf )(z) = h(z) f (z) = h(z) t=0 ∂t ∂z Here h : D → Z = Cd is a holomorphic vector field, with commutator bracket     ∂ ∂k ∂h ∂ ∂ , k = h −k . h ∂z ∂z ∂z ∂z ∂z Let s0 (z) = −z be the symmetry at the origin 0 ∈ D. For the adjoint action of G on g, defined via   ∂ ∂ = h(g(z)) g  (z) , Ad (g) h ∂z ∂z we obtain the Cartan decomposition g = k ⊕ ℘, where k = {X ∈ g : Ad (s0 ) X = X} is the 1-eigenspace and ℘ = {X ∈ g : Ad (s0 ) X = −X} is the (−1)-eigenspace. By definition, we obtain a Lie triple system [k, k] ⊂ k ⊃ [℘, ℘], [k, ℘] ⊂ ℘ ⊃ [℘, k]. The crucial step towards Jordan triples is the following Theorem 2.3. Let D ⊂ Z be a bounded symmetric domain. Then there exists a Jordan triple product Z × Z × Z → Z, denoted by u, v, w → {uv ∗ w} = {wv ∗ u}, such that   ∂ ℘ = (v − {zv ∗ z}) : v∈Z ∂z ∂ k = {h(z) : h ∈ gl(Z) linear , h{zv ∗ z } = {h(z ) h(v )∗ h(z )}}. ∂z In order to define the Jordan triple property, let Z × Z → End (Z) be the mapping u, v → u
v ∗ , given by (u
v ∗ ) z = {uv ∗ z}. Since [[℘, ℘] ℘] ⊂ [k, ℘] ⊂ ℘, we obtain the Jordan triple identity [u
v ∗ , x
y ∗ ] = {uv ∗ x}
y ∗ − x
{yu∗v}∗ .

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Example. In the matrix case Z = Cr×(r+b) , let   at b t (z) = (at z + bt )(ct z + dt )−1 ∈ SU (r, r + b) ct dt be a 1-parameter group of Moebius transformations. Its differential is computed as d ## ˙ + b˙ − z cz ˙ − z d˙ # (at z + bt )(ct z + dt )−1 = az dt t=0  ˙ with ac˙˙ db˙ ∈ su (r, r + b). The eigenspace decomposition is given by   ∂ ˙ ˙ : a˙ ∈ u(r), d ∈ u(r + b) k = (az ˙ − z d) ∂z and



 ∂ ∗ r×(r+b) ˙ ˙ ˙ : b∈C ℘ = (b − z b z) . ∂z Therefore, according to Theorem 2.3 we obtain {zv ∗ z} {uv ∗ w} and u
v∗ =

= =

zv ∗ z, 1 ∗ ∗ 2 (uv w + wv u)

1 (Luv∗ + Rv∗ u ). 2

Any bounded symmetric domain D ⊂ Z has an important K¨ ahler geometric structure called the Bergman metric. To define this metric, consider the Bergman operators Z × Z → End (Z), denoted by u, v → B(u, v), and defined as B(u, v)z = z − 2{uv ∗ z} + {u{vz ∗ v}∗ u}. Example. For the matrix case Z = Cr×(r+b) , we have B(u, v)z

=

z − uv ∗ z − zv ∗ u + u(vz ∗ v)∗ u

= =

z − uv ∗ z − zv ∗ u + uv ∗ zv ∗ u (1 − uv ∗ ) z(1 − v ∗ u).

Therefore B(u, v) = L1−uv∗ R1−v∗ u . Definition 2.4. Define a G-invariant Hermitian metric on D as follows: ∀ z ∈ D ∀ u, v ∈ Tz D ≈ Z we put hz (u, v) = (u|v)z = (B(z, z)−1 u|v), using the normalized K-invariant inner product (u|v) on T0 D = Z given by (u|v) = where p is the “genus” of D.

2 trZ u
v ∗ , p

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Example. For matrices Z = Cr×(r+b) we have 2r + b Luv∗ + Rv∗ u = tr uv ∗ 2 2 and hence the Bergman metric is given by trZ u
v ∗ = trZ

hz (u, v) = tr (1 − zz ∗ )−1 u(1 − z ∗ z)−1 v ∗ . In terms of the Bergman operator, the Bergman kernel function K : D × D → C of D can be expressed as K(z, w) = det B(z, w)−1 . It is a fundamental fact that there exists a sesqui-polynomial called the Jordan triple determinant N : Z × Z → C such that the Bergman kernel function has the form K(z, w) = det B(z, w)−1 = N (z, w)−p for all z, w ∈ D. Example. In the matrix case Z = Cr×(r+b) we have det B(z, w) = detZ [L1−zw∗ R1−w∗ z ] = detZ L1−zw∗ detZ R1−w∗ z = detCr (1r − zw∗ )r detCr+b (1r+b − w∗ z)r+b = det (1r − zw∗ )2r+b . It follows that N (z, w) = det (1 − zw∗ ). Jordan triples are closely related to Jordan algebras which were originally introduced in quantum mechanics. Definition 2.5. A real vector space X ≈ Rd is called a real Jordan algebra iff X has ◦ a commutative, non-associative product X × X −→ X, denoted by x, y → x ◦ y = y ◦ x, which satisfies the Jordan algebra identity x2 ◦ (x ◦ y) = x ◦ (x2 ◦ y). If, in addition, we require x2 + y 2 = 0 =⇒ x = y = 0, the Jordan algebra X is called formally-real or Euclidean since, in this case it has a strictly positive inner product which agree with (u|v) on the complexification Z = X C . We have the following classification [JNW]: Every (irreducible) Euclidean Jordan algebra has a realization X ≈ Hr (K) = {self-adjoint r × r-matrices x = (xij ) ∈ Kr×r , x∗ij = xji } for the anti-commutator product 1 (xy + yx). 2 More precisely, we have the following possibilities, depending on the “rank” r x◦y =

if r ≥ 4, then K = R, C, H = quaternions if r = 3, then K = R, C, H, O = octonions.

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If r = 2, then K = (0) is a real Hilbert space, i.e., we obtain “formal” 2 × 2 matrices   α b H2 (K)  ∗ b δ with α, δ ∈ R, b ∈ K. It follows from the classification that X is uniquely characterized by only two invariants: r = rank X and the characteristic multiplicity a = dimR K. Every Euclidean Jordan algebra X has a positive cone Ω = {x2 : x ∈ X \ {0}}, corresponding to the positive definite matrices Hr+ (K) and a Jordan algebra determinant N : X → R, which is a degree r polynomial defined via Cramer’s rule ∇x N . x−1 = N (x) Example. For the rank 2 case, N

  α b = αδ − (b|b) b∗ δ

is the Lorentz metric on R1, 1+a . If (X, ◦) is a real Jordan algebra, the complexification Z := X C becomes a complex Jordan ∗-algebra, and the associated Jordan triple (of tube type) on Z is defined as follows: {uv ∗ w} = (u ◦ v ∗ ) ◦ w + (w ◦ v ∗ ) ◦ u − v ∗ ◦ (u ◦ w). For the fine structure of Jordan triples and the corresponding symmetric domains, we need the notion of tripotent (= triple idempotent) and associated Peirce decomposition. This is analogous to the well-known root decomposition of (semi-simple) Lie algebras which underlies the Lie theoretic approach to symmetric spaces. In any Jordan triple Z, an element c ∈ Z is called a tripotent iff {cc∗ c} = c. In this case we have the Peirce decomposition Z = Z1 (c) ⊕ Z1/2 (c) ⊕ Z0 (c), where Zα (c) = {z ∈ Z : {cc∗ z} = αz} is the α-eigenspace for α = 0, 1/2, 1. Example. For matrices Z = Cr×(r+b) a tripotent c = cc∗ c is a partial isometry. In case k r−k b   0 0 r 1k , c= r−k 0 0 0

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the Peirce decomposition is given by  Z1/2 (c) Z1 (c) Cr×(r+b) = Z1/2 (c) Z0 (c)

Z1/2 (c) Z0 (c)

 .

The Peirce decomposition can be constructed more generally for any maximal frame e1 , . . . , er of orthogonal tripotents, where r = rank Z. Here orthogonality means ei
e∗j = 0 for i = j, a stronger condition than orthogonality for the scalar product of Z. In this case
⊕ Zij , Z= 0≤i≤j≤r

where

1 i (δ + δkj ) z}. 2 k Proposition 2.6. If D is an irreducible symmetric domain, the joint Peirce decomposition yields the characteristic multiplicities ⎫ Z00 = (0) ⎪ ⎪ ⎬ Zii = C ei 1 ≤ i ≤ r a 1≤i<j≤r ⎪ dimC Zij = ⎪ ⎭ dimC Z0j = b 1≤j≤r Zij = {z ∈ Z : {ek e∗k z} =

In particular, b = 0 iff D is of tube type. Example. In the matrix case Z = Cr×(r+b) let e1 , . . . , er be the diagonal matrix units. The associated Peirce decomposition looks as follows ⎛ ⎞ Z11 Z12 Z1r Z01 ⎜ Z12 Z22 Z02 ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . Z1r

Zrr

Z0r

and we have a = 2 (“complex case”).

3. Weighted Bergman spaces of holomorphic functions We will now introduce an important class of Hilbert spaces belonging to the holomorphic discrete series. Let Z be an irreducible Jordan triple of rank r, with characteristic multiplicities a, b. The dimension is then given by d = r + a r(r−1) + rb 2 or, equivalently, a d = 1 + (r − 1) + b. r 2 The precise meaning of the “second” multiplicity b will be given in Proposition 2.6 below. Define the genus p = 2 + a(r − 1) + b. The tube type case corresponds to b = 0 or, equivalently, to p2 = dr . Fix a parameter λ > p − 1 = 1 + a(r − 1) + b

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and consider the probability measure on D given by dµλ (z) =

ΓΩ (λ) N (z, z)λ−p dz, π d ΓΩ (λ − d/r)

where ΓΩ denotes the Koecher-Gindikin Γ-function of the symmetric cone Ω discussed in more detail below. The weighted Bergman space of holomorphic functions on D is defined as  2 Hλ (D) = {ψ ∈ O(D, C) : dµλ (z) |ψ(z)|2 < ∞}. D

For λ = p, dµp (z) is a multiple of the Lebesgue measure dz and  Hp2 (D) = {ψ ∈ O(D, C) : dz |ψ(z)|2 < ∞} D

is the standard (unweighted) Bergman space. The inner product (conjugate-linear in the first variable) is given by   ΓΩ (λ) dz N (z, z)λ−p ψ1 (z) ψ2 (z). (ψ1 |ψ2 )λ = dµλ (z) ψ1 (z) ψ2 (z) = d π ΓΩ (λ − dr ) D

D

Here we follow the convention of mathematical physics, where Hilbert state spaces, such as Hλ2 (D), have a scalar-product which is conjugate-linear in the first variable, whereas the underlying phase space Cn (in its K¨ahler polarization) has the K¨ahler metric conjugate-linear in the second variable. Proposition 3.1. Hλ2 (D) has the reproducing kernel Kλ (z, w) = N (z, w)−λ = det B(z, w)−λ/p . It follows that the orthogonal projection Pλ : L2 (D, dµλ ) → Hλ2 (D) has the form  (Pλ ψ)(z) = dµλ (z) N (z, w)−λ ψ(w). D

Define an irreducible projective unitary representation Uλ : G → U (Hλ2 (D)) of G by putting (Uλ (g −1 ) ψ)(z) = det ψ  (z)λ/p ψ(g(z)). Viewed as a quantum deformation, λp = h1 corresponds to the inverse Planck constant. The representations Uλ constitute the scalar holomorphic discrete series of G. Using the Jordan triple determinant, we may describe the analytic continuation of the scale of weighted Bergman spaces [L3], [FK1], [VR], [WA]: Define theWallach set W (D) = {λ ∈ C : (N (zi , zj )−λ )1≤i, j≤n

0 ∀ z1 , . . . , zn ∈ D}.

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Theorem 3.2. The Wallach set is a disjoint union ' a ( ' ( a W (D) =  : 0 ≤  < r ∪ λ > (r − 1) 2 2 consisting of r discrete points and a continuous part. More explicitly, the Wallach parameters are given as follows:

Figure 1. The scale of Wallach parameters z

a (r − 1) 2 }|

{z a (r − 1) 2

0 a  2

1+b }|

{z d r Hardy

a (r − 1) 2 }|

{z p−1

d a + r 2

weighted Bergman }|

{

p Bergman

Example. In the matrix case Z = Cr×(r+b) we have a = 2, and hence dr = r+b, p = 2r + b. The Shilov boundary S is a homogeneous space under the group K, therefore S = K/L where L := { ∈ K : e = e} for some e ∈ S. In case D ⊂ X C is of tube type and e is the unit element of the associated Euclidean Jordan algebra X, L = Aut (X) agrees with the automorphism group of X. The Shilov boundary S = K/L, endowed with the K-invariant probability measure, corresponds to the Wallach parameter d a λ = = 1 + (r − 1) + b r 2 giving rise to the Hardy space H 2 (S) = {ψ ∈ L2 (S) : ψ holomorphic on B}. The associated Szeg˝ o projection P : L2 (S) → H 2 (S) has the form  (P ψ)(z) = ds N (z, s)−d/r ψ(s). S

The proof of the fundamental Theorem 3.2 depends on harmonic analysis (PeterWeyl decomposition) for the compact Lie group K. Consider the algebra P(Z) of all polynomials p : Z → C and the natural algebra action K × P(Z) → P(Z) defined by (Uk−1 p)(z) := p(kz). A deep result [S] asserts that
Pm1 ≥...≥mr (Z) P(Z) = m1 ≥...≥mr ≥0

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is a direct sum of irreducible K-modules, labelled by all partitions m = (m1 , . . . , mr ) ∈ Nr+ . Moreover, the highest weight vector has the form Nm (z) = N1 (z)m1 −m2 N2 (z)m2 −m3 · · · Nr (z)mr , where Nk is the Jordan determinant of


Zij

1≤i≤j≤k

(generalized minor). This is the main result of [U3]. Example. For the simplest partition m = (1, 0, . . . , 0), P10···0 (Z) = Z  is the dual space of linear forms on Z. In this case N10···0 (z) = (e1 |z), where we fix a frame (e1 , . . . , er ). As a Hilbert space completion, we obtain the Segal-Bargmann-Fock space  1 dz e−(z|z) |ψ(z)|2 < ∞} H 2 (Z) = {ψ ∈ O(Z) : d π Z

of entire functions, with reproducing kernel expanded in a series
e(z|w) = Km (z, w), m1 ≥···≥mr ≥0

where Km (z, w) denotes the reproducing kernel of Pm (Z) ⊂ H 2 (Z). The FarautKoranyi binomial formula [FK1] is
N (z, w)−λ = (λ)m Km (z, w), m

where (λ)m =

r . Γ(λ + mj − (j − 1) a2 ) ΓΩ (λ + m) = ΓΩ (λ) Γ(λ − (j − 1) a2 ) j=1

denotes the Pochhammer symbol. By checking positivity of the coefficients, one obtains the desired conclusion a a W (D) = {λ ∈ C : (λ)m ≥ 0 ∀ m ∈ Nr+ } = { : 0 ≤  < r} ∪ {λ > (r − 1)}. 2 2

4. Toeplitz operators on symmetric domains A deep relationship between analysis on symmetric domains and operator theory concerns C ∗ -algebras generated by Toeplitz operators in the multi-variable setting. In general, for any complex Hilbert space H, let L(H) be the C ∗ -algebra of all bounded linear operators on H. For f ∈ L∞ (D), define the Bergman-Toeplitz operators Tfλ ∈ L(Hλ2 (D)) for parameter λ > p − 1 by Tfλ ψ = Pλ (f ψ) ∀ ψ ∈ Hλ2 (D).

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More precisely, (Tfλ ψ)(z)

 =

dµλ (z) Kλ (z, w) f (w) ψ(w) D

=

ΓΩ (λ) d π ΓΩ (λ − d/r)



dz N (z, z)λ−p N (z, w)−λ f (w) ψ(w).

D

∞

Similarly, for f ∈ L (S), define Hardy-Toeplitz operators Tf ∈ L(H 2 (S)) for parameter λ = dr by Tf ψ = P (f ψ) ∀ ψ ∈ H 2 (S). More precisely,

 (Tfλ ψ)(z) =

ds N (z, s)−d/r f (s) ψ(s).

S

In both cases we consider the Toeplitz C ∗ -algebras Tλ (D) =

C ∗ (Tfλ : f ∈ C(D)) ⊂ L(Hλ2 (D))

T (S) =

C ∗ (Tf : f ∈ C(S)) ⊂ L(H 2 (S))

In principle this could be defined for any Wallach parameter ·

λ ∈ W (D) = Wdisc (D) ∪ Wcont (D), since Hλ is a Hilbert space with reproducing kernel N (z, w)−λ (z, w ∈ D). However, there is a difference between the discrete and continuous part: Proposition 4.1. Hλ is a Hilbert-module iff λ > a2 (r − 1). It follows that only in this case the Toeplitz operators Tz1 , . . . , Tzd form a d-tuple of bounded operators. For a deeper analysis of Toeplitz C ∗ -algebras, we need some geometric preparations concerning the boundary of symmetric domains. Proposition 4.2. The Shilov boundary has the Jordan theoretic characterization S = ∂ex D = {e ∈ Z : {ee∗ e} = e tripotent, e
e∗ invertible, Z0 (e) = (0)}. In the “tube type” case, we obtain the simpler characterization S = {e ∈ Z tripotent : e
e∗ = idZ , Z = Z1 (e)}. The boundary structure of a symmetric domain D can now be described in algebraic terms [LO2]. Let D ⊂ Z be the unit ball of a Jordan triple and consider a tripotent c = {cc∗ c}, with Peirce decomposition Z = Z1 (c) ⊕ Z1/2 (c) ⊕ Z0 (c). Then Z1 (c) is a Jordan ∗-algebra with unit element c such that k = rank Z1 (c). On the other hand, Z0 (c) is a Jordan subtriple and the norm satisfies z1 + z0  = max (z1 , z0 ) for all z1 ∈ Z1 (c), z0 ∈ Z0 (c). The unit ball D0 (c) = {z0 ∈ Z0 (c) : z0  < 1}

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79

is a symmetric domain of lower rank r − k and c + D0 (c) ⊂ ∂D, since c + z0  = max ( c , z0 ) = 1.  !"  !" =1

<1

Let S0 (c) = ∂ex D0 (c) denote the Shilov boundary of D0 (c). Proposition 4.3. We have a disjoint union ·  c + D0 (c) ∂D = 0 =c={cc∗ c}

into boundary components, equivalently, the closures c + D0 (c) constitute all the faces of D. For the Shilov boundary there is a similar, nondisjoint, union  S= c + S0 (c). c=0 tripotent

Example. For the matrix space Z = Cr×(r+b) , with elements written as z = (u, v), consider the rank k tripotent ⎞ ⎛ 1 ⎟ ⎜ .. 0 ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 1 ⎟ ∈ Z, ⎜ c=⎜ ⎟ 0 ⎟ ⎜ ⎟ ⎜ . . 0 ⎠ ⎝ . 0 whose Peirce decomposition looks as follows

Z=

k r−k



k r−k Z1 (c) Z1/2 (c) Z1/2 (c) Z0 (c)

b Z1/2 (c) Z0 (c)



 

u v3

v1 w1

 v2 . w2

In this case, we obtain the boundary component     ∗ w 1 0 0 c + D0 (c) = : (w1 , w2 ) 1∗ < I 0 w1 w2 w2 and its Shilov boundary c + S0 (c) =



1 0 0 w1

0 w2



  ∗ w : (w1 , w2 ) 1∗ = I . w2

Our first main result concerns the structure and representations of the HardyToeplitz C ∗ -algebra [U1], [U2].

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Definition 4.4. If T ⊂ L(H) is a C ∗ -algebra, a representation of T is a ∗-homomorphism π : T → L(Hπ ). Define the spectrum T  = Spec T = {π : T → L(Hπ ) irreducible ∗-representation} modulo unitary equivalence. There exists a canonical “Zariski-type” topology on Spec T which in the non-commutative case is in general not Hausdorff. Example. If S is a compact space, consider the commutative C ∗ -algebra C(S) = {f : S → C continuous}. Then C(S) = S, since for all s ∈ S : χs (f ) := f (s) defines a character. Now let S = ∂ex D be the Shilov boundary of a bounded symmetric domain and let T (S) be the Toeplitz C ∗ -algebra on H 2 (S). It acts irreducibly on H 2 (S), yielding the “natural” representation. Theorem 4.5. (i) For any tripotent c = {cc∗ c} there exists a unique irreducible representation σc : T (S) → T (S0 (c)) acting on H 2 (S)

acting on H 2 (S0 (c))

such that for all f ∈ C(S) we have S (c)

σc (TfS ) = Tfc0

,

where fc ∈ C(S0 (c)) is defined by fc (s) := f (c + s) for all s ∈ S0 (c) (ii) the representations σc are mutually inequivalent (iii) these are all irreducible ∗-representations of T (S), i.e., we may identify Spec T (S) = {c ∈ Z : {cc∗ c} = c tripotent}, realized as a “stratified” non-Hausdorff space. A similar structure theorem holds for Bergman-Toeplitz operators [U7]. Let D ⊂ Z be an irreducible symmetric domain of rank r, with characteristic multiplicities a, b. Consider parameters a λ > (r − 1) 2 in the continuous Wallach set (Hilbert modules), or the subclass λ > 1 + a(r − 1) + b corresponding to the weighted Bergman spaces (discrete series). Let c = {cc∗ c} be a tripotent of rank k. Then D0 (c) ⊂ Z0 (c) is an irreducible symmetric domain of rank r − k, with characteristic multiplicities a, b. Moreover, the shifted parameter a a λ − k > (r − k − 1) 2 2

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81

belongs to the “little” continuous Wallach set. Similarly, in the Bergman case a λ − k > 1 + a(r − k − 1) + b 2 belongs to the “little” discrete series. Theorem 4.6. Let Tλ (D) be the Toeplitz C ∗ -algebra acting on Hλ2 (D) (weighted Bergman space). Then (i) For any tripotent c = {cc∗ c}, there exists a unique irreducible ∗-representation σc : Tλ (D) → Tλ−k a/2 (D0 (c)) such that D (c)

0 σc (TλD (f )) = Tλ−k a/2 (fc )

for all f ∈ C(D), with restriction fc ∈ C(D0 (c)) defined by fc (w) := f (c + w) for all w ∈ D0 (c). (ii) The representations σc are mutually inequivalent (iii) each irreducible ∗-representation of Tλ (D) has this form, i.e., similar to the Hardy case, we may identify Spec Tλ (D) with the set of all tripotents c = {cc∗ c} of Z. Note that the structure and representations of these Toeplitz C ∗ -algebras remain “rigid” along the continuous part of the Wallach set, since we are dealing with a “non-commutative” topological structure. The discrete points, however, show a different behavior.

5. Convolution C ∗ -algebras on non-commutative Hardy spaces It turns out that the structure theorems for Toeplitz C ∗ -algebras belong to a much more general theory concerning convolution operators on compact symmetric spaces. Thus we may consider situations where only the Shilov boundary S, but not the domain D itself, is a symmetric space. In this section we use “left” quotient spaces which are more convenient for convolution algebras. (Accordingly the domain D, which in this section plays only a minor role, should be realized as D = K \ G.) Let S =L\K be a homogeneous compact manifold, with stabilizer group L = {k ∈ K : e · k = e}. Here e ∈ S is the “unit element”. L2 (K) carries the left-regular K-action (t f )(s) := f (t−1 s) ∀ s, t ∈ K, f ∈ L2 (K). Consider the group C ∗ -algebra C∗ (K) = C ∗ (u : u ∈ L1 (K))

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generated by all left-convolution operators  (u f )(s) := du(t)(t f )(s). K

Its weak closure W ∗ (K) = W ∗ (t : t ∈ K) is the group W ∗ -algebra. Identify L2 (S) ≈ {f ∈ L2 (K) : f (k) = f (k) ∀  ∈ L}  L2 (K) closed

with all left L-invariant functions, and consider the orthogonal projection π : L2 (K) → L2 (S) defined via averaging over L:

 dt f (k).

(πf )(ek) = L

Note that in this setting we write S as a left quotient space, endowed with a right K-action. Proposition 5.1. For symmetric domains, the Szeg˝ o distribution dE(s) = N (s, e)−d/r on S defines via convolution the orthogonal projector E : L2 (K) → H 2 (S). π

P

Proof. Consider the orthogonal projections L2 (K) −→ L2 (S) −→ H 2 (S). The substitutions t = e · τ and k = στ −1 yield    (P ◦ π) f (e · σ) = dt N (e · σ, t)−d/r (πf )(t) = dτ N (e · σ, e · τ )−d/r d f (τ ) S

K



−d/r

dτ N (e · σ, e · τ )

= =

dτ N (e · στ

f (τ ) =

K



L



−1

−d/r

, e)

f (τ )

K

dk N (e · k, e)−d/r f (k −1 σ) = (N (−,e)−d/r f )(σ).

K

In this special case, the binomial formula yields the expansion  
d Km (s, e) N (s, e)−d/r = r m m1 ≥···≥mr ≥0

into K-spherical functions.




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We will now consider “non-commutative” Hardy spaces more generally. If K is a compact Lie group with involution σ : K → K and K σ = {k ∈ K : σ(k) = k} denotes the fixed point subgroup, then S = K σ \ K = Kσ is a compact symmetric space, and we have the Peter-Weyl decompositions
Kα ⊗ Kα , L2 (K) = ˆ α∈K

ˆ is the unitary dual of K (irreducible representations), and where K
L2 (S) = Kα ⊗ φα , ˆσ α∈K

where ˆ σ = {α ∈ K ˆ : Kα contains a non-zero K σ -invariant vector} Sˆ = K is the subset of all spherical representations. Here Kα denotes the (finite-dimenˆ In terms of the Lie algebra decomposition sional) representation space of α ∈ K. k = kσ ⊕ kσ , consider a maximal torus t = tσ ⊕ tσ , where kσ ⊃ tσ = i a is a maximal abelian “Cartan subspace”. Define r := dim tσ to be the rank of S. Then there exist natural embeddings as discrete subsets ˆ → i t = L(t, i R) (highest weight) K and Sˆ → a (restricted highest weight). Now let V ⊂ a be an arbitrary open polyhedral cone. The basic result for Fourier analysis on H 2 (S) is the following [L1], [L2]. Theorem 5.2. Let V  ⊂ a denote the (closed) dual cone. Then the Hardy space corresponding to the (open) cone V has a Peter-Weyl decomposition
HV2 (S) = Kα ⊗ φα .  ˆ α∈S∩V

Moreover, the Cauchy-Szeg˝ o distribution can be expanded into a series
EV (s) = φα (s)  ˆ α∈S∩V

of spherical functions, and HV2 (S) can be realized via boundary values of holomorphic functions on the K-invariant domain DV = K exp (V )(e) ⊂ S C .

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Here the complexification S C = (K σ )C \ K C exp

is a complex manifold and, choosing a “Weyl chamber” t+ σ ⊂ tσ ⊂ k −→ K, we obtain a polar decomposition S C = K exp (i tσ ) and may define the complex phase space (“non-commutative tube domain”) by DV := K exp (V ) where V ⊂ t+ σ is an open polyhedral convex cone. Letting Λ = V  ⊂ i tσ be the polar cone, the “non-commutative” Hardy space
HV2 (S) := Kα ⊗ φα ˆ σ ∩Λ α∈K

plays the role of “Hilbert state space” in this setting. Theorem 5.3. If DV is pseudoconvex (i.e., a domain of holomorphy), then one may identify HV2 (S) = {h ∈ L2 (S) : h holomorphic on DV }. Using Proposition 5.1, we show next that the Hardy space of hermitian symmetric domains (at least of tube type) fits into the more general framework. In fact, let K be the compact Lie group of Jordan (triple) automorphisms, with involution subgroup K σ consisting of all Jordan algebra automorphisms. In this case, on the Lie algebra level, kσ corresponds to the Jordan multiplication operators and the Cartan subspace i tσ is obtained by diagonal multiplication operators, isomorphic to Rr via spectral theory. The associated cone V = Rr> is precisely the positive octant, a very special case of a polyhedral cone. The Hardy space decomposition
⊕ Pm1 ,...,mr H 2 (S) = m1 ≥···≥mr ≥0

into irreducible K-modules labelled by integer partitions, with Pm1 ,...,mr , generated by the highest weight vector N1 (z)m1 −m2 N2 (z)m2 −m3 · · · Nr (z)mr , where N1 , . . . , Nr are the Jordan theoretic minors ⎞ ⎛ N1 ⎟ N2 ⎜ ⎟ ⎜ ⎟, ⎜ N3 ⎝ ⎠ .. .. . .

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is now understood by realizing the partitions within the dual cone of V . Moreover, the Cauchy-Szeg˝ o kernel N (z, w)−d/r = det (I − zw∗ )−d/r will now be interpreted as a convolution kernel on the Shilov boundary. In principle, the same could be done for the Bergman kernel N (z, w)λ = det (I − zw∗ )−λ , for parameter λ > p − 1. Thus also the Berezin quantization could be realized on the Shilov boundary. Returning to the general case of a compact symmetric space S = L \ K, let P : L2 (S) → HV2 (S) be the orthogonal projection, realized as a convolution operator  (PV h)(s) := EV (st−1 ) h(t) dt, K

where EV (s) =




φα (s)

ˆ σ ∩V  α∈K

is the Cauchy-Szeg˝o distribution. We may regard KV (s, t) := EV (st−1 ) as a generalized “reproducing kernel”. If f ∈ C(S) is a continuous symbol function, let TS (f ) = PV f PV be the associated Toeplitz operator (depending on V ), given by TS (f ) h := PV (f h). These operators are in general non-commuting, but satisfy TS (f )∗ = TS (f ). The Toeplitz C ∗ -algebra TV (S) := C ∗ (TS (f ) : f ∈ C(S)) is the uniform closure of the algebra generated by all operators TS (f ) with continuous symbol f . In order to describe the generalization of Theorem 4.6 to the general case of compact Lie groups K, let again V be a polyhedral cone, with polar cone V  realized in the (dual) Cartan subspace. The basic geometric idea, generalizing the boundary stratification of symmetric domains, is as follows: For any face A of V there exists a subgroup KA (not necessarily closed) with Cartan subspace A spanned by the face. Consider the associated foliation K/KA of Kronecker type, which is Hausdorff iff KA is closed. + Theorem 5.4. There exist C ∗ -ideals IKA  IK  TV (S) in the Toeplitz C ∗ -algebra, A + with subquotient IKA /IKA ≈ C ∗ (K/KA ) isomorphic to the foliation C ∗ -algebra.

The proof is based on C ∗ -duality theory [LPRS], [NT]. As a first step, TV (S) is identified with a corner of the co-crossed product Cˆ ∗ (K) ⊗ C 0 (K) δ

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for a suitable co-action δ on a “completed” C ∗ -algebra Cˆ ∗ (K). On the other hand, the foliation C ∗ -algebra C ∗ (K/KA ) = C0 (K) ⊗ C ∗ (KA ) can be realized as a crossed product [CO]. In view of the well-known Katayama duality K(L2 (K)) = C ∗ (K) ⊗ C 0 (K) δ

one obtains C ∗ -representations of TV (S) on L2 (KA ). This is of course the crucial step, and we give a sketch of the construction. Let T (A) be the tangent space of ˜ such that T (A) ⊂ m ˜ is maximal A. Then there exists a Lie subalgebra ˜k = ˜l ⊕ i m, abelian. Now consider the (non-closed) analytic subgroup KA := exp ˜k = Kc × Ke , which can be factored into a compact times Euclidean part. o distribution relative to the face A. Then there Theorem 5.5. Let EA be the Szeg˝ exists a C ∗ -representation σ

A TV (S) −→ L(L2 (KA )),

which maps the Toeplitz operator T (f ) to TA (fA ), where fA := f |KA is the restriction and TA is the Toeplitz operator defined relative to EA . Proof. Let Cˆ ∗ (K) be the C ∗ -algebra generated by all left convolution operators λgE with g ∈ A(K), the Fourier algebra of K. Then there exist representations A(K) f

−→ L(L2 (KA )) → fA

and

Cˆ ∗ (K) −→ L(L2 (KA )) → λgA EA . λgE The latter representation is best described via the Fourier decomposition W (K) ∼ = ∗

(∞)


L∞ (Kα ),

ˆ α∈K



with λu → u ˆα =

u(s)∗ sα ds

K

denoting the Fourier coefficient. Here sα denotes the action of s ∈ K for the representation α. Now let g(s) = (φ|sγ ψ) be a matrix coefficient function, with φ, ψ ∈ Hγ belonging to a not necessarily irreducible K-representation. Then an / ˆ ˆ (γ ⊗ α) ψ. Now let β ∈ K ˆ A , and let α ∈ K easy computation shows gE(α) = φ∗ E ˜ β ⊂ Kα as a KA -invariant be a sequence such that α|KA  β. Realizing K / subspace, one shows that gE(α| ˜ β )  g A EA (β). In this way the representation K is constructed via a limiting argument. For more details, cf. [U5], [W].


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Once the representations σA are constructed, Theorem 5.4 can be expressed as follows: Defining 0 + IA = Ker σA  IA = Ker σB B
A

it follows that

+ IA /IA ∼ = C ∗ (FA ),

where FA = K/KA is the Kronecker foliation associated with the non-closed subgroup KA . This C ∗ algebra is of type I iff KA is closed. The “local version” can be reformulated in the following global version: There exists a C ∗ -filtration K = I1  I2  · · ·  Ir  TV (S), such that for every k ⊕
∼ Ik+1 /Ik = C ∗ (FA ). codim A=k

In the special case of symmetric domains, the composition factors Ik+1 /Ik ∼ = C(Sk ) ⊗ K of the filtration are even realized as fibre bundles over the compact manifold Sk of all tripotents (partial isometries) of rank k. Hence in the case of symmetric domains we obtain a fibration of boundary faces instead of a foliation in the more general setting. A few remarks about the associated index theory. In full generality the analytic index is given by a mapping 
⊕  Ind K1 C ∗ (FB ) −→A K0 (C ∗ (FA )), B
A

and should be related to a longitudinally elliptic operator DA along the foliation FA . The details of this construction have only been worked out for a basic class of non-symmetric domains, the so-called Reinhardt domains. In this case K = S = Tn is the n-torus, whereas L = {1}. The complexification S C = (C \ {0})n ⊂ Cn is open and dense. A domain D ⊂ Cn is called Reinhardt iff (z1 , . . . , zn ) ∈ D =⇒ (ei t1 z1 , . . . , ei tn zn ) ∈ D. Let |D| = {(|z1 |, . . . , |zn |) : z ∈ D} ⊂ Rn+ be the associated “absolute” domain and define V = log |D| ⊂ Rn− ,

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H. Upmeier

assuming that D is contained in the unit polydisk. Important examples, for n = 2, are the L-shaped domains and the so-called Hartogs’ wedge, shown in Figure 2. Figure 2. L-shaped domains and the Hartogs’ wedge |z2 |

|z2 |

T2

2 (1, 1) T

T2 log |z2 |

|D| log |z1 |

V = log |D|

|D| log |z2 |

|z1 |

|z1 |

slope − θ1

V = log |D|

log |z1 |

Via groupoid methods, Toeplitz operators on Reinhardt domains have been studied in [CM], [SSU]. In the special case of Hartogs’ wedge, one obtains the following Theorem 5.6. [SSU]: Let V = VΘ1 , Θ2 be the convex cone with slopes Θ1 and Θ2 , coming from Hartogs’ wedge. Then the Toeplitz C ∗ -algebra has a 2-step filtration K  I2  T (T2 ). Here T (T2 )/I2 = C(T2 ) is the commutator ideal, and I2 /K is stably isomorphic to AΘ1 ⊕ AΘ2 , where AΘ = C ∗ (u, v : uv = e2πiΘ vu)

unitary:

denotes the irrational rotation algebra (non-commutative torus) induced by the Kronecker foliation T2 /RΘ (cf. Figure 3). Figure 3. The Kronecker foliation T2 /RΘ

Θ

Based on this C ∗ -algebra filtration, one obtains a real-valued index theorem: tr

Z2 ≈ K 1 (T2 ) −→ K0 (AΘ ) −→ R, ≈

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89

corresponding to the dense embedding (m, n) → m + n Θ. These results have also applications to complex analysis, for example the existence of smooth (Reinhardt) domains with non-compact ∂-Neumann operator, and a characterization of proper holomorphic maps between Reinhardt domains in terms of their associated Toeplitz C ∗ -algebras. It is interesting to note that Reinhardt domains are closely related to multi-variable Wiener-Hopf operators [MR], [U12], by embedding higher-dimensional cones in compact spaces via the Cayley transform.

6. Quantization of Hermitian symmetric domains A different area of interaction between analysis on symmetric domains and Hilbert space operator theory concerns quantization methods in the sense of mathematical physics. Let D = G/K be a hermitian symmetric space of non-compact type, realized as the open unit ball of a Jordan triple Z = Cd . The group G = {biholomorphic automorphisms of D} = Aut(D) is a semi-simple Lie group, with maximal compact subgroup K = {Jordan triple automorphisms} = Aut(Z) ⊂ GL(Z). Example. In case D = {z ∈ Cp×q : z < 1} is the matrix ball for the operator norm, Z = Cp×q has the Jordan triple product uv ∗ w + wv ∗ u , 2 and we obtain the Moebius transformations   ab G = SU (p, q)  (z) = (az + b)(cz + d)−1 , cd whereas K = S(U (p) × U (q)) consists of the linear isometries. Here r = min (p, q) is the rank of D. {uv ∗ w} =

Example. In case D = {z ∈ Cr×r : z < 1, z t = z} consists of symmetric matrices, the tangent space Z = {z ∈ Cr×r : z t = z} becomes a Jordan sub-triple of Cr×r , and G = Sp (2r, R) ⊃ K = U (r). In the following we will also consider the flat case D = Cn ≈ T ∗ (Rn ), identified with the cotangent bundle. Here G = U (n)  Cn is a semi-direct product.

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The quantization Hilbert spaces are the weighted Bergman spaces described in more detail in Section 3. Consider the G-invariant measure dµ0 (z) = N (z, z)−p dz, where N (z, w) denotes the “Jordan triple determinant” and p is the genus. Example. For the matrix ball, N (z, w) = Det (I − zw∗ ) and the genus is p + q. Define the scalar holomorphic discrete series as follows. For ν > p − 1, ΓΩ (ν) N (z, z)ν−p dz π d ΓΩ (ν − d/r) is a probability measure and the associated weighted Bergman space dµν (z) =

Hν2 (D) := {ψ ∈ L2 (D, µν ) : ψ holomorphic} has the projective unitary representation (Uν (g −1 )ψ)(z) = (Det g  (z))ν/p ψ(g(z)) and the reproducing kernel ν Kν (z, w) = N (z, w)−ν = Kw (z).

Accordingly, we have



ψ(z) =

dµν (w) Kν (z, w) ψ(w)

∀ ψ ∈ Hν2 (D).

D

For the flat case,

 ν d

e−ν(z|z) dz π denotes the Gauss measure, with corresponding Segal-Bargmann-Fock space dµ (z) =

H2 (Cn ) := {ψ ∈ L2 (Cn , dµ ) : ψ holomorphic} and its unitary representation (π (tb ) ψ)(z) = eν(z|b)−ν(b|b)/2 ψ(z − b) for the translations tb induced by b ∈ Cn . Here  = ν1 plays the role of Planck’s constant. By definition [AU4] a covariant functional calculus (or “quantization”) is determined by a densely defined ∗-linear mapping L2 (D, dµ0 )

A

−→

∗−linear

L2 (Hν2 (D)) (Hilbert-Schmidt operators),

associating to f the operator Af with active symbol f . The covariance condition can be expressed as Af ◦g−1 = Uν (g) Af Uν (g)−1 .

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The adjoint mapping A∗ : L2 (Hν2 (D)) −→ L2 (D, dµ0 ), associating to an operator T the passive symbol A∗T of T , is formally defined by duality  dµ0 (w) A∗T (w) f (w) = tr T ∗ Af . D

Composing both mappings, we obtain the link transform (or generalized Berezin transform) B = A∗ A : L2 (D, dµ0 ) → L2 (D, dµ0 ), which is a G-invariant pseudo-differential operator. It is often convenient to express the quantization by an integral representation  Af = dµ0 (w) f (w) Aw , D

where w → Aw is a covariant field of (bounded) operators, satisfying the covariance condition Ag(w) = Uν (g) Aw Uν (g)−1 . For any (non-compact) Hermitian symmetric space, we have the Plancherel decomposition  dλ 2 Gλ L (G/K) = |c(λ)|2 Rr

into a direct integral of principal series representations πλ, where c(λ) is HarishChandra’s c-function [HE]. Here, by definition, the representation space Gλ contains a spherical vector φλ, explicitly given by  φλ = dk πλ(k) eλ. K

For symmetric domains of tube type, eλ(z) = N (z + z ∗ )λ+ρ denotes the conical functions for λ = (λ1 , . . . , λr ), where as before we put N (x)λ = N1 (x)λ1 −λ2 N2 (x)λ2 −λ3 · · · Nr (x)λr and N1 , N2 , . . . , Nr = N are the Jordan algebraic minors. The half-sum ρ of positive roots has the components 1+b a (j − 1) + . 2 2 Now let A be a G-covariant calculus. The Berezin transform ρj =

σ ◦ A = A∗ ◦ A

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H. Upmeier

is G-invariant and hence, by diagonalization, ∗ A(λ) · id  σ ◦ A|Gλ = A Gλ , ∗ A(λ) = σ(A  where A eλ )(0) is the λ-eigenvalue. For the spectral resolution of Berezin-type transforms it suffices to determine these eigenvalues or, equivalently, to express σ ◦ A in terms of some set of generators ∆1 , ∆2 , . . . , ∆r of the algebra of G-invariant differential operators on D, where ∆1 is the Laplace-Beltrami operator. The primary example of a covariant quantization is the Toeplitz-Berezin calculus

T ν : L∞ (D) −→ L(Hν2 (D))

(bounded operators),

associating to f the Toeplitz operator Tfν defined via Tfν ψ = Pν (f ψ), where Pν : L2 (D, dµν ) → Hν2 (D) is the orthogonal projection [B2], [B3]. Its integral representation has the form  ν (Tf ψ)(z) = dµν (w) Kν (z, w) f (w) ψ(w). D

Therefore the adjoint (TT∗ )(w) =

ν ν (Kw |T Kw ) Kν (w, w)

coincides with the Berezin symbol and the defining operator field Twν is given by ν the rank 1 projection onto C · Kw . It follows that the Berezin transform f → Bf = ∗ T T f has the integral representation  Kν (z, w) Kν (w, z) f (w), (Bf )(z) = cν dµ0 (w) Kν (z, z) Kν (w, w) D

where cν is chosen so that B1 = 1. In the flat case of the Segal-Bargmann-Fock space H2 (Cn ), it has been shown in [G] that Bf = e−∆ f coincides with the heat semi-group. In the curved setting, we have instead [BLU], [E]: Theorem 6.1. For any bounded symmetric domain, the Toeplitz quantization satisfies the correspondence principle in the following sense (i) Tfν Tgν − Tfνg  → 0 ν  → 0. (ii) ν[Tfν , Tgν ] − i T{f,g} Here {f, g} denotes the G-invariant Poisson bracket.

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For the explicit calculation of the eigenvalues of the Toeplitz-Berezin transform, we pass to the Siegel domain realization Z = U ⊕ V , via the biholomorphic Cayley transform D −→ {(u, v) ∈ Z : u + u∗ − {ev ∗ v} ∈ Ω} C

≈

onto a symmetric Siegel domain. Here U = X C is the complexification of a Euclidean Jordan algebra, x ◦ y denotes the Jordan product and Ω = {x2 : x ∈ X invertible} is the positive cone of X. In general, V is a complex representation space of X, with V = {0} corresponding to the tube type. The Cayley transform is given by √ C(u, v) = ((e + u) ◦ (e − u)−1 , 2 v ◦ (e − u)−1 ), where e is the unit element of X. By the classification, we may identify X with the algebra Hr (K) of all self-adjoint (r×r)-matrices over K, under the anti-commutator . The associated matrix entries belong to product x ◦ y = xy+yx 2 ⎧ r≥4 ⎨ R, C, H R, C, H, O r = 3 exceptional K= ⎩ Euclidean space r = 2 spin factors, and we call a = dimR K the characteristic multiplicity. Using the conical functions N s (x) = N1 (x)s1 −s2 N2 (x)s2 −s3 · · · Nr (x)sr ,

the Euler integral  r d1 −r . a ΓΩ (s) = dx N (x)−d1 /r N s (x) e−(x|e) = (2π) 2 Γ (sj − (j − 1)) 2 j=1 Ω

is called the Koecher-Gindikin Γ-function and d1 = d − rb is the dimension of X. Using this fundamental concept, the Toeplitz-Berezin eigenvalues have been computed in [UU], yielding the beautiful formula T ∗ T (λ) =

ΓΩ (ρ + ν −

d r

+ λ) ΓΩ (ρ + ν −

ΓΩ (ν −

d r ) ΓΩ

(ν)

d r

− λ)

,

where ρ denotes the half-sum of positive roots. A second example of a covariant calculus is the Weyl quantization W : L2 (D, dµ0 ) −→ L2 (Hν2 (D)), associating to f the Weyl operator Wf defined via its integral representation  Wf = dµν (w) f (w) Uν (sw ) D

[UU1], [UU2]. Here sw ∈ G denotes the geodesic symmetry, uniquely determined by sw (w) = w, sw (w) = −Id.

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H. Upmeier Figure 4. The geodesic reflection

sw (z) w z

By duality, we obtain the Weyl symbol WT∗ (w) = tr T Uν (sw ). In the flat case, we have the formulas sw (z) = 2w − z, s0 = −id, sw = tw ◦ s0 ◦ t−w which induce for the complex wave Hilbert space H2 (Cn ) the representations = e(z|b)−(b|b)/2 ψ(z − b) and

(π(tb )ψ)(z)

= ψ(2w − z) e2(z−w|w).

(π(sw )ψ)(z) This yields the explicit form (WfC ψ)(z) =

 dw f ( Cn

z + w (z−w|z+w)/2 )e ψ(w). 2

Remark 6.2. Both in the setting of bounded symmetric domains and for the flat case, the quantization Hilbert spaces have also a real representation via a socalled Segal-Bargmann transform. For the real wave Hilbert space L2 (Rn ), the representations (π(ty,η ) φ)(x)

=

ei(y−2x)η φ(x − y),

π(sy,η φ)(x)

=

e4i(y−x)η φ(2y − x)



lead to (WfR

ψ)(x) =

dy Rn



 dη f

Rn

x+y ,η 2

 e2i(y−x)η φ(y).

It should be noted that the well-known Moyal ∗-product arises this way. In order to determine the eigenvalues of the link transform for the Weyl calculus, let G = KAN be the Iwasawa decomposition, and for λ ∈ a let eλ denote the conical function of G/K, with  φλ = dk k eλ K

the associated spherical function. Realized as a Siegel domain G/K = {(u, v) : u + u∗ − 2{ev ∗ v} ∈ Ω},

Toeplitz Operator Algebras and Complex Analysis

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the conical functions have the form eλ(u, v) = N λ+ρ (u + u∗ − 2{ev ∗ v}). Taking o = (e, 0) as a base point, we first apply the so called product formula [AU4], [AU6], which asserts that whenever A, B are two covariant operator calculi on Hν2 (D), the eigenvalues of the link transform can be expressed as ∗ B (λ) =  A

(Aeλ Ko |Ko )(Ko |Beλ Ko ) ∗ T (λ) T

.

Hence it suffices to compute (Aeλ Ko |Ko ) = (Aφλ Ko |Ko ). After this simplification, the Weyl transform eigenvalues (at least for domains of rank 1) have been determined in [AU6]: Theorem 6.3. Let D = {z ∈ Cd : (z|z) < 1} be the unit ball of rank 1. Then the Weyl calculus satisfies    Γ(ν − d2 + λ)Γ(ν − d2 − λ) d−ρ+λ d−ρ−λ α , (Ko |Weλ Ko ) = 2 F1 Γ(ν)Γ(ν − d) α−1 ν where 2 F1 is the Gauss hypergeometric function. Note that the product of Γ-factors, analogous to the Toeplitz-Berezin case, is still present, but has to be supplemented by the hypergeometric function. It is expected that a similar phenomenon holds for higher rank and also for more general functional calculi. A common generalization of both the Toeplitz-Berezin calculus and the Weyl calculus is the so-called interpolating calculus [AU7] which depends on an additional parameter α. For the Fock space we assume α ∈ C, whereas for bounded symmetric domains we assume that α ∈ B, the closed unit disk. Consider the map z → sα (z) := αz on the domain D. By conjugation, we define for any w ∈ D, α −1 sα w := g s g where g ∈ G satisfies g(0) = w. The interpolating calculus for parameter α is defined by the local operator field α α (Aα w ψ)(z) := ψ(sw (z)) j(sw (z))

where j denotes the Jacobian cocycle. In the flat case D = Cd we have sα w (z) = αz + (1 − α) w and therefore ν(1−α)(z−w|w) . (Aα w ψ)(z) = ψ(αz + (1 − α) w) e

If α = 0, (A0w ψ)(z) = ψ(w) eν (z−w|w)

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H. Upmeier

and one obtains the Toeplitz calculus  ν d  (A0f ψ)(z) = dw f (w) ψ(w) e−ν(w|w) eν(z|w) . π Cd

If α = −1,

(z) = 2w − z s(−1) w is the geodesic symmetry and one obtains the Weyl calculus  ν (−1) (Af ψ)(z) = ( )d dw f (w) ψ(2w − z) e−2ν(w|w) e2ν(z|w) . π Cd

It is shown in [AU4] that for α → ∞, the Wick calculus, formally given by hk → Th Tk∗ , is obtained via a suitable limit procedure. Theorem 6.4. For the interpolating α-calculus, in the flat case, the link transform is given by 1 − |α|2 ∆ , A∗α Aα f = exp |1 − α|2 ν where ∆ is the Laplacian. The general set-up for Moyal products of quantized operators (also called ∗-products) is as follows: Let A, B, C be covariant calculi and f, g ∈ C ∞ (D) be symbol functions. Then one has the quantized operators Af , Bg and defines the “weak” Moyal product by f • g := C ∗ (Af Bg ). This is in general not associative. In contrast, the “strong” Moyal product f g ∈ C ∞ (D), uniquely defined by Cf g = Af Bg , is associative on the formal level, but in general there is no such function f g which yields the required operator product. In the Toeplitz operator calculus, one has instead an asymptotic expansion [E]
1 Tf Tg = TCk (f,g) , νk k≥0

where Ck (f, g) are uniquely determined G-covariant bidifferential operators. Therefore, in this case,
1 Ck (f, g). f g = νk k≥0

The Correspondence Principle [BLU] gives the following lower-order terms (i) Tfν Tgν − Tfνg  → 0 implies that C0 (f, g) = f g, ν  → 0 implies that C1 (f, g) is related to the Poisson (ii) ν [Tfν , Tgν ] − i T{f,g} bracket.

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97

In the flat case on the Segal-Bargmann-Fock space one has a more or less complete theory of Moyal products, even in the general setting of the interpolating calculus. Let f, g, f g ∈ C ∞ (Cd , C) be symbol functions such that γ β Aα f Ag = Af g .

Then the Moyal product f g can be expressed as an integral   (f g)(ζ) = dξ dη f (ξ) g(η) M(ξ, η; ζ). Cd

Cd

Equivalently,

 Aα ξ

Aβη

dζ Aγζ M(ξ, η; ζ)

= Cd

for the corresponding operator fields. The following result appears in [AU7]. Theorem 6.5. For α, β, γ in the unit disk, subject to the restrictions αβ = γ and Re

(1 − γ)2 (1 + αβ) > 0, (αβ − γ)(1 + γ)

the 3 × 3 matrix ⎡

(1 − α)(β − γ) (1 − α)(1 − β) γ Φ = ⎣ (1 − α)(1 − β) (1 − β)(α − γ) (γ − 1)(1 − α) (γ − 1) α (1 − β)

⎤ (γ − 1)(1 − α) β (γ − 1)(1 − β) ⎦ (1 − γ)(1 − αβ)

has all rows and columns summing up to 0, and the integral kernel for the Moyal product is given as follows ⎛ ⎞ ξ ν (ξ, η, ζ) Φ ⎝ η ⎠ . M(ξ, η; ζ) = exp αβ − γ ζ Note that M is invariant under U (d) × Cd , acting jointly on ξ, η, ζ.

7. Deformation of real symmetric domains It turns out that similar results hold for the more general class of real symmetric domains, which are defined as follows. Let D = G/K be a hermitian symmetric domain, with G a semi-simple Lie group of hermitian type, and let z → z be an antiholomorphic involution of D. Define the real form DR = {z ∈ D : z = z} = GR /KR , where GR = {g ∈ G : g(z) = g(z)} is a reductive Lie group, with maximal compact subgroup KR = K ∩ GR .

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Example. If D = T (Ω) = Ω + iX is a tube domain, the involution (x + iy)∗ = x − iy is the hermitian adjoint, and DR = Ω considered as a real symmetric domain. In this case, GR is only reductive with a non-trivial center. Example. If D = {z ∈ Cp×q |z ∗ z < I} = U (p, q)/U (p) × U (q) is the complex matrix ball, and z → z = (z ij ) is the conjugation, then DR = {x ∈ Rp×q |x∗ x < I} = O(p, q)/O(p) × O(q) coincides with the real matrix ball. Figure 5. Complex and real symmetric domains DC DC = T (Ω)

DR

DR = Ω

 − d −c , one Example. For D = {z ∈ C2×2 : z ∗ z < I} and the involution ac db := −b a can show that DR ⊂ H is realized as the quaternion unit ball. Example. (product case) In case DR = GR /KR is itself hermitian symmetric, the complexification D = DR × D R is endowed with the flip involution (z1 , z 2 )− := (z2 , z 1 ) and G = GR × GR contains GR as the “diagonal” subgroup. We now introduce the “real” version of a covariant calculus. Let D ⊃ DR be a real symmetric domain, irreducible of rank r, with complexification D not necessarily irreducible. Let Hν2 (D) denote the νth Bergman space over D, in case D is irreducible, whereas in the product case we consider Hν2 (DR × DR ) = Op Hν2 (DR ). Here one could also take pairs (ν1 , ν2 ) not necessarily equal. In both cases we obtain the restricted representation Uν |GR which is not irreducible but has a Plancherel decomposition which is multiplicity free. By definition [AU3], a covariant calculus in the “real” setting is a densely defined linear map A : C ∞ (DR ) → Hν2 (D)

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associating to f a holomorphic function Af , subject to the covariance condition Af ◦g−1 = Uν (g) Af ∀ g ∈ GR . The infinitesimal version corresponds to an integral representation  Af = dµR 0 (ζ) f (ζ) Aζ , DR

where Aζ ∈ O(D) is a field of “states” satisfying the covariance condition Ag(ζ) = Uν (g) Aζ , g ∈ GR . µR 0

Here is a (suitably normalized) GR -invariant measure on DR . Similar to the complex case, the adjoint transformation A∗ : Hν2 (D) → L2 (DR , dµR 0) is defined by duality  ∗ ∗ dµR 0 (ζ) f (ζ) AT (ζ) = (f |AT ) = (Af |T )Hν2 (D) . DR

By composition we obtain the link transform B = A∗ A acting on C ∞ (DR ) which commutes with GR -translations. As in the complex setting, the primary example of a real deformation is the Toeplitz calculus [AU3], [Z] T : C ∞ (DR ) → Hν2 (D), which associates to f the holomorphic function  Tf = dµR 0 (ζ) f (ζ) Tζ DR

explicitly given by Tζ (z) =

Kν (z, ζ) ∈ Hν2 (D), ζ ∈ D. Kν (ζ, ζ)1/2

The normalization of µR 0 is chosen so that T1 = I is the identity operator, and hence depends on ν and also on the type of covariant calculus. The adjoint transformation T ∗ : Hν2 (D) → L2 (DR , µR 0) coincides with the so-called weighted restriction (T ∗ h)(ζ) = Kν (ζ, ζ)1/2 h(ζ). In the product case D = DR × DR , one computes easily that Tζ,ζ (z, w) =

Kν (z, w; ζ, ζ) Kν (z, ζ) Kν (ζ, w) = 1/2 Kν (ζ, ζ) Kν (ζ, ζ; ζ, ζ)

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is just the integral kernel of kζ ⊗ kζ∗ . Hence we recover the “complex” Toeplitz calculus. In order to determine the eigenvalues of the Toeplitz-Berezin transform in the real setting consider, as before, the exponentials λ+ρ

eλ (x + y, v) = NX

(x − {ev ∗ v})

and the corresponding Euler type integral  Kν (e, ξ) ∗ T (λ) = T (e) = eλ (ξ) dµR T eλ 0 (ξ) Kν (ξ, ξ)1/2  =

 dx

 dy

BR

λ+ρ (x − {ev ∗ v}) dv NX

−ν NX⊕Y (e + x − y) −ν/2

NX⊕Y (x − {ev ∗ v})

.

The following result [DP], [N], [Z], [AU3] is the real version of the UnterbergerUpmeier formula: Theorem 7.1. Let DR = {x + y + v ∈ X ⊕ Y ⊕ V : x − {ev ∗ v} ∈ Ω} be a real symmetric domain, realized as a real Siegel domain, where U =X ⊕Y is a semi-simple real Jordan ∗-algebra, the self-adjoint part X = {x ∈ U : x∗ = x} is a Euclidean irreducible Jordan algebra and Ω ⊂ X denotes the symmetric cone. Moreover, Y = {y ∈ U : y ∗ = −y}. In this context, the Toeplitz deformation eigenvalues have the form ∗ T (λ) = T

ΓΩ (κ + λ) ΓΩ (κ − λ) ΓΩ (κ + ρ) ΓΩ (κ − ρ)

where ρ is the half-sum of positive roots and κ =

ν 2r

rank (U ) + ρ +

d r

− p.

In order to treat more general deformations, we need the “product formula” for real symbolic calculi [AU3], [AU6]: Theorem 7.2. Let A, B : C ∞ (DR ) → Hν2 (D) be two covariant symbolic calculi, acting on Hν2 (D). The link transform 2 R A∗ B : L2 (DR , µR 0 ) → L (DR , µ0 )

has the Plancherel eigenvalues Aeλ (0) Beλ (0) ∗ B (λ) = 4 A ∗ T (λ) T ∗ T (λ) > 0 is independent of A, B. In particular for all λ ∈ a = Rr , where T ∗ A (λ) =  A

1 |Aeλ (0)|2 . ∗  T T (λ)

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Note that Aλ (0) =

101

∼ dµR 0 (ζ) φλ (ζ) σK0 (ζ) = σK0 (λ)

DR

coincides with the spherical Fourier transform of σK0 , K0 ∈ Hν2 (D) being the symbol of the kernel vector at the origin, which is a KR -invariant function. On a somewhat deeper level one can also define a “real” Weyl calculus W : C ∞ (DR ) → Hν2 (D), which associates to f the holomorphic function  Wf = dµR 0 (ζ) f (ζ) Wζ DR

given by Kν (sζ (z), z)1/2 Kν (z, ζ)1/2 , Kν (sζ (z), ζ)1/2 where sζ ∈ GR is the symmetry about ζ ∈ DR ⊂ D. In the flat case D = Cn , DR = Rn , the analogous mapping Wζ (z) =

Wν : L2 (Rn ) → Hν2 (Cn ) is the (unitary) Bargmann transform, given explicitly by  ν n/2  ν (Wν f )(z) = dζ f (ζ) exp (2ν(z|ζ) − (z|z) − ν(ζ|ζ)). π 2 Rn

In order to compute the Weyl transform eigenvalues, let m = (m1 ≥ m2 ≥ · · · ≥ mr ≥ 0) be an integer partition, and define the Pochhammer symbol ΓΩ (α + m) ΓΩ (α) and the multivariable hypergeometric function  
(α)m (β)m αβ (x1 , . . . , xr ) = Jm (x1 , . . . , xr ). 2 F1 (γ)m (1)m γ m (α)m =

Here Jm are the well-known Jack polynomials [ST], which are proportional to the spherical functions on the symmetric cone Ω regarded as a real symmetric domain of root type (A), restricted to the “diagonal” C e1 + · · · + C er for a Jordan algebra frame e1 , · · · , er of X. The reason why only “discrete” parameters m occur lies in the fact that one really considers the “compact dual” S of Ω. For r = 1, the symmetric domains in K = R, C, H, O (m = 2, Cayley plane) can be expressed as a hyperbolic space m
D = {x ∈ Km | xi x∗i < 1}. i=1

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H. Upmeier Figure 6. Rank 1 real symmetric domains ZR ZC rC ρ

Rm Cm 1

Cm m Cm × C 2

m−1 4

m 2

Hm C2×2m 2 m + 12

O2 C16 V 2 11 2

In this case 2ρ + 1 = a+d 2 , where a = dim K. It follows that ZR is uniquely determined by d = dim ZR and ρ. Generalizing both the Toeplitz and the Weyl calculus, we may also consider two covariant calculi of interpolating type Aα , Aβ : C ∞ (DR ) → Hν2 (DC ) and the associated link transform A∗ α Aβ : C ∞ (DR ) → C ∞ (DR ) ∗ α Aβ (λ) for λ ∈ a . The which is uniquely determined by the eigenvalues A R product formula gives ∗ α Aβ (λ) = A

where

 Aeλ (0) = DR

β Aα eλ (0) Aeλ (0) A0eλ (0)

dµR 0 (ζ) eλ(ζ) Aζ (0) =



dµR 0 (ζ) φλ (ζ) Aζ (0)

DR

and φλ is the spherical function. In view of the product formula, the following result [AU7], [AU9] solves the eigenvalue problem for rank 1 domains. Theorem 7.3. For real rank 1 domains DR we have d/2−ρ+λ d/2−ρ−λ α  Γ (ν − ρ + λ) Γ (ν − ρ − λ) 2 F1 α−1 d/2+ν−2ρ R α Aeλ (0) = . d/2 d/2−2ρ α  Γ (ν) Γ(ν − 2ρ) 2 F1 α−1 d/2+ν−2ρ For the special case of the Weyl calculus (α = −1) considered in [AU6] we have Theorem 7.4. For real symmetric domains of rank 1, the Weyl transform eigenvalues are determined by the formula d/2−ρ+λ d/2−ρ−λ 1 (2) Γ (ν − ρ + λ) Γ (ν − ρ − λ) 2 F1 d/2+ν−2ρ Weλ (0) = . d/2 d/2−2ρ 1 Γ (ν) Γ(ν − 2ρ) (2) 2 F1 d/2+ν−2ρ

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Remark 7.5. In the complex setting ZR = Cm we have d/2 = 2ρ = m and obtain the simpler formula    m Γ (ν − m m−ρ+λ m−ρ−λ α R α 2 + λ) Γ (ν − 2 − λ) Aeλ (0) = 2 F1 Γ (ν) Γ(ν − m) α−1 ν which generalizes Theorem 6.3 (for α = −1). Analogous to the complex setting, the real symmetric domains give rise to a so-called Moyal restriction [AU7]: For the Fock space and F ∈ C ∞ (Cd , C), the Moyal restriction F ∈ C ∞ (Rd , C) is defined by C

R γ 2 d Aα F I = AF ∈ Hν (C ).

Writing

 dw M(w, ζ) F (w),

(F )(ζ) = Cd

we have C



dζ R Aγζ M(w, ζ)

Aα wI = Rd

where I(ζ) = Kν (ζ, ζ)1/2 . Theorem 7.6. For α, γ in the unit disk, subject to the restrictions α2 = γ and Re the symmetric 3 × 3 matrix ⎡ ⎢ Φ=⎣

(1 − γ)2 (1 + α2 ) > 0, (α2 − γ)(1 + γ)

(1−α) γ 2 α−γ 2 α(γ−1) 2

α−γ 2 1−α 2 γ−1 2

⎤

α(γ−1) 2 γ−1 2

⎥ ⎦

(1 − γ)(1 +

α 2)

has all rows and columns summing up to 0, and ⎞ w ν(1 − α) M(w, ζ) = exp (w w ζ) Φ ⎝ w ⎠ α2 − γ ζ ⎛

for all w ∈ Cd , ζ ∈ Rd .

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8. Vector-valued Bergman spaces and intertwining operators A current area of research in harmonic and complex analysis on symmetric domains concerns vector-valued holomorphic functions. Let D = G/K be a hermitian symmetric domain, and let G = Aut (D) be its holomorphic automorphism group. G is a semi-simple Lie group of non-compact type, and K = {g ∈ G : g(0) = 0} is its maximal compact subgroup. As always, we realize D ⊂ Z as the unit ball of a Jordan triple with triple product {uv ∗ w}. Then K = Aut (Z) = {k ∈ GL (Z) : k {uv ∗ w} = {ku(kv)∗ kw}}. Example. If Z = Cp×q is the matrix triple, uv ∗ w + wv ∗ u {uv ∗ w} = 2 and D = {z ∈ Z : zz ∗ < I} is the matrix ball. Here G = SU (p, q) consists of all Moebius transformations   ab (z) = (az + b)(cz + d)−1 cd 6 5  and K = a0 d0 : a ∈ U (p), d ∈ U (q) consists of all linear isometries z → a z d−1 . Example. In case Z = {z ∈ Cp×p : z t = z} is the Jordan subtriple of all symmetric matrices, G = Sp (2p, R) is the symplectic group and K = U (p). One frequently 0} called Siegel’s uses the unbounded realization D = {z = z t ∈ Cp×p : z + z ∗ half-space. Not let E be a unitary representation space for K, with dimC E < ∞. There π is a homomorphism K −→ U (E) into the corresponding unitary group. Example. For K = SU (2), putting E = z1m z2n−m : 0 ≤ m ≤ n = SU (2)n yields all irreducible representations. Example. If K = U (k) and E = Ck is the defining representation, the exterior powers E = Λj Ck , 0 ≤ j ≤ k, are called the fundamental representations. For the group SU (k), the well-known Schur functor construction Em for integer partitions m = m1 ≥ m2 ≥ · · · ≥ m k ≥ 0 of length ≤ k yields all irreducible representations. Example. If K = U (p) × U (q) is the direct product, we may consider the tensor product E = Ep ⊗ Eq∗ = L(Eq , Ep ) of all Hilbert-Schmidt operators, with its canonical U (p) × U (q)-representation, given by   a0 (T ) = a T d∗ = πp (a) T πq (d)∗ 0d for T ∈ L(Eq , Ep ).

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In order to make contact with symmetric domains, let N : Z × Z → C be the Jordan triple determinant, which is given by N (z, w) = Det (I − zw∗ ) for matrices Z = Cp×q , and let B : Z × Z → End (Z) be the Bergman operator, denoted by z, w → B(z, w), and defined as B(z, w) x = x − 2{zw∗ x} + {z{wx∗ w}∗ z}. Example. In the matrix case Z = Cp×q , it has been shown above that B(z, w) x = (1 − zw∗ ) x (1 − w∗ z) and hence B(z, w) = L1−zw∗ R1−w∗ z . Proposition 8.1. If z, w ∈ D, then B(z, w) ∈ K C ⊂ GL (Z). In particular, we obtain for matrices:   0 1 − zw∗ ∈ K C, B(z, w) = 0 (1 − w∗ z)−1 whereas in case of the unit disk, B(z, w) x = (1 − zw)2 x, and for the unit ball, B(z, w) x = (1 − (z|w))(x − (x|w) z). Recall that N (z, w) = det B(z, w)1/p . Now let π : K → U (E) be a unitary finite-dimensional representation, which is not necessarily irreducible. Then, for z ∈ D, π(B(z, z)) =: B(z, z)π is positive definite and we define as an induced representation the vector-valued Bergman space  Hν2 (D, E) = {ψ : D → E holomorphic: dµ0 (z)N (z, z)ν (ψ(z)|B(z, z)π ψ(z)) < ∞}, D

where dµ0 (z) = N (z, z)−p dz = det B(z, z)−1 dz. If E = C, we recover the scalar-valued Bergman spaces. In case of the unit ball D ⊂ Cd , we have K = U (d) and the inner product has the form  dz (1 − (z|z))ν−d−1 ((1 − (z|z))ψ(z)| (I − z ∗ z)π ψ(z)).  ! " D

∈GL (d)=K C

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If ψ ∈ Hν2 (D, E) and φ ∈ E (corresponding to the constants), one has the reproducing property (φ|ψ(z))E

= (Bz−π φ|ψ)H 2 (D,E)  = dµ0 (w) N (z, z)ν (B(w, z)−π φ|B(w, w)π ψ(w))E . D

This shows that the mapping D × D → GL (E), given by z, w → B(z, w)−π , plays the role of a matrix-valued reproducing kernel. The scalar-valued reproducing kernel N (z, w)−ν = det B(z, w)−ν/p arises as a special case, either by keeping ν and letting π be the trivial representation or, alternatively, by setting ν = 0 and considering the 1-dimensional representation K → U (1) given by k → (det k)ν/p . The full holomorphic discrete series, not necessarily scalar-valued, is related to the following construction. Lemma 8.2. For g ∈ G = Aut (D), z ∈ D the holomorphic derivative satisfies (i) g  (z) ∈ K C ⊂ GL (Z) and the covariance property (ii) g  (z) B(z, w) g  (w)∗ = B(g(z), g(w)). Thus we may define the holomorphically induced representation πν,E : G → U (Hν2 (D, E)) via the formula (πν,E (g −1 ) ψ)(z) = det g  (z)ν/p g  (z)π [ψ(g(z))] ∈ E or, equivalently, as (πν,E (g −1 ) ψ)(z) = j(g, z)π ψ (g, z), where j : G × D → K C is the group-valued holomorphic cocycle j(g, z) = g  (z). The cocycle property means j(g1 g2 , z) = j(g1 , g2 (z)) j(g2 , z). It is well known that for symmetric domains, the group K has a rich and interesting representation theory related to the Hua-Schmid decomposition introduced in Section 3. Let Z be an irreducible hermitian Jordan 7 triple, and consider the algebra P(Z) of all polynomials on Z, identified with sym Z  , where Z  is the dual space. Then K = Aut (Z) acts on P(Z) via (π(k) p)(z) = p(k −1 z). As shown in [S] there is a multiplicity-free K-decomposition
P(Z) = Pm (Z) m

labelled by integer partitions m = (m1 ≥ m2 ≥ · · · ≥ mr ≥ 0)

Toeplitz Operator Algebras and Complex Analysis

107

of length ≤ r = rank (Z). Moreover, in [U3] it has been shown that Pm (Z) has the highest weight vector Nm (z) = N1 (P1 z)m1 −m2 N2 (P2 z)m2 −m3 · · · Nr (Pr z)mr , where Nk is the kth Jordan algebra minor, and Pk is the kth Peirce projection. Example. We have Nm,0,...0 (z) = (z|e1 )m and Nm,m,...m (z) = N (z)m (for tube type domains). Combining the previous concepts, for the compact group U = K, and the holomorphic cocycle j : G × D → K C given by the holomorphic derivative j(g, z) = g  (z) ∈ K C , we obtain a unitary irreducible representation πm : K → U (Pm (Z)) via translation (πm (k) p)(ζ) = p(k −1 ζ). The induced K-homogeneous multiplier representations are realized on the vectorvalued Bergman space Hλ2 (D, Pm (Z)) = {ψ : D → Pm (Z) holomorphic:  dz N (z, z)λ−p (ψ(z, ζ)|ψ(z, B(z, z) ζ))Pm (Z) < ∞} D

where, as above, B(z, w) ζ = (1 − zw∗ ) ζ(1 − w∗ z) is the Bergman operator acting on Z. This space carries a projective unitary G-action, which is irreducible and belongs to the holomorphic discrete series λ (g −1 ) ψ)(z, ζ) = det g  (z)λ/p ψ(g(z), g  (z) ζ). (Um

More generally, let U be a compact Lie group such that K ⊂ U and let J : G × D → UC be a holomorphic multiplier (cocycle), denoted by g, z → J (g, z), which is holomorphic in z ∈ D and satisfies J (g1 g2 , z) = J (g1 , g2 (z)) · J (g2 , z). Let π : U → U (E) be a unitary irreducible representation with dimC E < ∞, and denote by π C : U C → GL (E) its holomorphic extension. Let o be the base point of D and choose a real analytic cross-section γz ∈ G, γz (o) = z. Then K(z, z) = J (γz∗ , o)π J (γz , o)π belongs to GL (E) via π C . We let K(z, w) ∈ GL (E) be its sesqui-holomorphic extension. The induced U -homogeneous multiplier representation is realized on

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H. Upmeier

the vector-valued Bergman spaces



H (D, E) = {Ψ : D → E holomorphic: 2

dµ0 (z)(Ψ(z)|K(z, z) Ψ(z))E D



dµ0 (z)(J (γz , o)π Ψ(z)|J (γz , o)π Ψ(z))E < +∞}

= D

endowed with the G-action (Uπ (g −1 ) Ψ)(z) = J (g, z)π Ψ(g(z)). An important special case is given by the symmetric Grassmann manifolds. It is well known [LO2] that GC = Aut (M ), where D ⊂ Z ⊂ M is the compact symmetric dual space. Consider the finite-dimensional space
P m (Z), P n (Z) = m1 ≤n

with its reproducing kernel K n (ω, ζ) = Kωn (ζ) = N (ζ, −ω)n =




(−n)m (−1)|m | Km (ζ, ω).

m1 ≤n

Figure 7. The partitions occurring in P n (Z) n

r

Define a GC -action on P n (Z) via the kernel vectors γ −1  Kωn = det γ  (ω)−n/p Kγn∗ (ω) . Here GC × M → M is the Moebius type action extending the action of G on D. Let tz ∈ GC be the translation, defined by tz (ζ) = z + ζ and consider the “adjoint” (relative to the compact form U ⊂ GC ) t∗−z ζ = ζ z = B(ζ, z)−1 (ζ − {ζz ∗ ζ}), which is called the quasi-inverse. Example. For matrices Z = Cp×q , we have ζ z = (1 − ζ z ∗ )−1 ζ.

Toeplitz Operator Algebras and Complex Analysis

109

A holomorphic cocycle G × D → GC , denoted by g, z → J (g, z), is defined via C J (g, z) = t−1 g(z) g tz ∈ G .

The cocycle property J (g1 g2 , z) = J (g1 , g2 (z)) J (g2 , z) is easily checked. In this situation, we obtain the “Grassmann homogeneous” vector-valued Bergman space Ψ

Hν2 (D, P n (Z)) = {D −→ P n (Z) : D × Z  z, ζ → Ψ(z, ζ), hol  ν−p dz N (z, z) Ψ(z, B(z, z)1/2 ζ z )2 < +∞} D

since, according to [LO2], γz (ζ) = z + B(z, z)1/2 ζ z ∈ G defines a real-analytic cross-section, i.e., γz (0) = z, with the property J (γz , 0) ζ = B(z, z)1/2 ζ z . The unitary G-action has the form (Unν (g −1 ) Ψ)(z, ζ) = det g  (z)ν/p det g  (z + ζ)−n/p Ψ(g(z), g(z + ζ) − g(z)). Theorem 8.3. There is a multiplicity-free G-decomposition
⊕ Hν2 (D, P n (Z)) = Hν2 (D, Pm (Z)), n≥m1 ≥···≥mr ≥0

showing that the representation is not irreducible. Nevertheless, the Toeplitz C ∗ -algebra T generated by coordinate multiplications acts irreducibly on Hν2 (D, P n (Z)). Conversely, one may ask whether every irreducible T -module is of this kind? This is closely related to the problem of classification of homogeneous holomorphic vector bundles [KM]. The explicit intertwining operators use the Faraut-Koranyi binomial formula
N (z, w)−λ = det (1 − zw∗ )−λ = (λ)m K m (z, w) m

where e(z|w) =




K m (z, w)

m

is expanded into the series of reproducing kernels for Pm (Z). Here r .

a (i − 1))mi 2 i=1   partitions having the is the multi-Pochhammer symbol, and we consider the n+r r form n ≥ m1 ≥ m2 ≥ · · · ≥ mr ≥ 0. (λ)m =

(λ −

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In the simplest (scalar-valued) case the intertwiner is given by
(−n)m (I f )(z, ζ) = Km (ζ, ∂z ) f (z). (λ)m m

9. Analytic continuation of Wallach parameters Another recent generalization of Bergman spaces concerns the analytic continuation of the Wallach set. Let Pm (Z) be the space of all polynomials of type m = m1 ≥ · · · mr ≥ 0, and let K C  Pm (Z) be the irreducible action defined by (h  p)(z) = p(h−1 z). Consider the Bergman operator B : Z × Z → End (Z) defined by B(z, w) = id − 2 z
w∗ + Qz Qw where Qz ζ := {zζ ∗ z}. Then B(z, w) ∈ K C if z, w ∈ D. Example. For matrices Z = Cp×q the Bergman operator is given by B(z, w) x = (1 − zw∗ ) x (1 − w∗ z). As discussed already, the vector-valued Bergman spaces of type m Hν2 (D, Pm (Z))

F

= {D −→ Pm (Z) : D × Z  z, ζ → F (z, ζ)  hol dz N (z, z)ν−p (F (z, ζ)|F (z, B(z, z) ζ)ζ < ∞} D

carry the irreducible projective G-representation ν (Um (g −1 ) F )(z, ζ) = det g  (z)ν/p F (g(z), g  (z) ζ)

induced by the holomorphic cocycle G × D → K C coming from the complex derivative g  (z) and satisfying the chain rule (g1 g2 ) (z) = g1 (g2 (z)) g2 (z). In Jordan theoretic terms, a real analytic cross-section gz ∈ G, satisfying gz (0) = z, can be realized in exponential form gz = exp ((v − zv ∗ z)

∂ ). ∂z

Then gz (0) = B(z, z)1/2 is a positive operator. The classical concept of Grassmann manifolds has the following generalization. Let Z be a hermitian Jordan triple, which is irreducible of rank r, and consider the compact manifold S of all tripotents of rank  ≤ r. As a special case, Sr is the

Toeplitz Operator Algebras and Complex Analysis

111

Shilov boundary. Two tripotents u ∼ v are called Peirce-equivalent iff they have the same Peirce spaces 1 Zk (u) = Zk (v), k = 0, , 1. 2 The quotient space M  = S / ∼ is called the th Jordan Grassmann manifold [LO2]. In general, it is a compact hermitian symmetric space under the semi-simple Lie group K  := K/center. Example. If Z = Cr×s is the matrix triple for r ≤ s, S consists of all partial isometries, satisfying uu∗ u = u, of rank u =  ≤ r. One can show that M ≈ Gr (Cr ) × Grs− (Cs ) is the product of “classical” Grassmannians. In general, let U = Z1 (u) ∈ M be the Peirce 1-space. Then the tangent space TU (M ) = Z1/2 (u) coincides with the Peirce

1 2 -space.

Therefore M carries the tautological bundle Z1 ↓ M

 

U ↓ U

and the tangent bundle Z1/2 ↓ M

 

Z1/2 (u) ↓ U = Z1 (u).

On the analytic side, let 1 (p|q) = (∂p q)(0) = d π



dz e−(z|z) p(z) q(z)

Z

be the Fischer-Fock inner product on P(Z), with reproducing kernel
e(z|w) = K m (z, w) m

expanded into the kernels for Pm (Z). Example. We have P10··0 (Z) = Z  (dual space) and K10··0 (z, w) = (z|w) coincides with the normalized K-invariant scalar product. On the other hand, if Z = X C is of tube type, then Pm···m (Z) = C · N m and hence m 1 Km···m (z, w) = N (z)m N (w) . (d/r)m···m

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The Faraut-Kor´ anyi binomial series expansion
N (z, w)−ν = (ν)m Km (z, w), m

expressed via the Pochhammer symbol (ν)m =

r . ΓΩ (ν + m ) a = (ν − (k − 1))mk , ΓΩ (ν) 2 k=1

can be used to define the discrete part of the Wallach set, the so-called discrete Wallach spaces
H2a/2 (D) = Pm1 ,...,m , 0,...,0 (Z) m+1 =0

as the space of “-harmonic” functions, with inner product
(φm |ψm ) (φ|ψ) a/2 = . ( a/2)m m =0 +1

2 Figure 8. The partitions occurring in Ha/2 (D) n

 r It is an important problem to obtain an integral expression for the inner product [AU1], [AU2], [AU5]. Define GL (Ω)-invariant differential operators on the symmetric cone Ω ⊂ X by ΓΩ (α) n N (x)d/r−α ∂N Tαn = N (x)α+n−d/r ΓΩ (α + n) and consider the unique extension to a holomorphic differential operator T˜αn on X C . By construction [AU5, Section 3.1], we have the eigenvalues (α + n)m T˜αn pm = pm . (α)m Theorem 9.1. For φ, ψ ∈ H2a/2 (D) the inner product can be expressed as an integral  (φ|ψ) a/2 = dU (φ|S(U ) |T˜U ψ|S(U ) )Hardy M

over the th Jordan Grassmann manifold M . Here S(U ) denotes the Shilov boundary of U ∩ D, so that ·  S = U × S(U ) U

Toeplitz Operator Algebras and Complex Analysis

113

as a disjoint union. Moreover, (T˜U )U ∈M is a K-covariant family of holomorphic (pseudo) differential operators on U , which can be constructed (in the sense of spectral theory) from the building blocks T˜αn (restricted to U ) by requiring that the eigenvalues are given by (ra/2)m (d/r)m , ( a/2)m where m runs over all partitions such that m+1 = 0. Note that in contrast to the situation in Proposition 4.3, the union of the respective Shilov boundaries is in fact disjoint since everything is “centered at the origin”, whereas in Proposition 4.3 one considers the translated Shilov boundaries as subsets of S ⊂ ∂D. Another part of the analytic continuation are the generalized Dirichlet spaces [A], defined for 1 ≤ n ∈ N. Let d1 /r = 1 + a(r − 1)/2 correspond to the dimension d1 of the Peirce 1-space of Z. Taking all partitions m1 ≥ m2 ≥ · · · ≥ mr ≥ n, one obtains as a completion
˜ d /r−n = Pm (Z), H 1 mr ≥n

which is a unitarizable G-space called the Dirichlet space, endowed with the inner product
∼ (fm |gm )F (f |g)n = . (1 + a2 (r − 1) − n)m mr ≥n

Figure 9. The partitions occurring in the Dirichlet space n

r

It is proved in [FK1] that, in case Z = X C is of tube type, there is a unitary isomorphism ∂n

N 2 ˜ d/r−n ≈ Hd/r+n (D) H

sending the parameter sense that

d r

− n = λ to p − λ =

d r

+ n, which is G-equivariant in the

j ∂N (Ud/r−n (g) ψ) = Ud/r+n (g) ψ. This is false for non-tube type domains, as pointed out by G. Zhang, who has found a vector-valued generalization using the so-called transvectants and Shimura operators [PZ], [HLZ]. Using the Jordan Grassmann manifolds the intertwiner can

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also be realized in more geometric terms [AU8]. For 1 ≤  ≤ r consider the homogeneous line bundle Ln = S × C = {[u, ξ] = [v, η] : u
u∗ = v
v ∗ , η = Nu (v)n ξ} ∼

over M , i.e., the nth power of the determinant bundle, which is the basic holomorphic line bundle over M . Here Nu is the Jordan determinant of Z1 (u). Theorem 9.2. There exists a unitary G-intertwiner into the vector-valued Bergman space ˜ d /r−n ∼ H (D, Pn···n (Z)). = H2 a 1+ 2 (r−1)+n

1

In case Z if of tube type, Pn···n (Z) = C · N n is 1-dimensional, but if Z is not of tube type, e.g., if D is the unit ball in Cd , d > 1, then dimC Pn···n (Z) > 1. The geometric framework for the intertwiner, using the Jordan-Grassmann manifolds, can be described as follows [AU8]: Proposition 9.3. As a special case of the Borel-Weil-Bott theorem, one may realize K-equi Pn··n (Z) −→ Γhol (Mr , Lnr ) ≈

via holomorphic sections over Mr , using the restriction p → p|Sr . More generally, we have an identification K-equi Pn·· n 0···0 (Z) −→ Γhol (M , Ln )  !" ≈ 

via holomorphic sections over M . Theorem 9.4. Using the Jordan-Grassmann manifold, we obtain the G-equivariant intertwiner I ˜ d /r−n −→ H Hd21 /r+n (D, Pn···n (Z)) ≈ Hd21 /r+n (D, Γhol (Mr , Lnr )) 1 ≈

in the form n (I f )(z, U ) = (∂N f |U )(PU z), u

where U = Z1 (u) for some tripotent u. More generally, for 1 ≤  ≤ r, there are similar intertwiners ˜ 2a H (−1)+1−n (D) 2

I

−→ H 2a (−1)+1+n (D, Γhol (Mk , Ln )) ≈

≈

2

˜ 2a H (−1)+1+n (D, Pn··n 0··0 (Z)). 2

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H. Upmeier H. Upmeier, Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, CBMS Ser. Math. 67, Amer. Math. Soc., Providence RI, 1987. H. Upmeier, Toeplitz C ∗ -algebras and non-commutative duality, J. Oper. Th. 26 (1991), 407–432. H. Upmeier, Symmetric Banach Manifolds and Jordan C ∗ -Algebras, NorthHolland 1985. H. Upmeier, Multivariable Toeplitz Operators and Index Theory, Birkh¨ auser 1996. H. Upmeier, Toeplitz operators and solvable C ∗ -algebras on hermitian symmetric spaces, Bull. Amer. Math. Soc. 11 (1984), 329–332. H. Upmeier, Toeplitz operators on symmetric Siegel domains, Math. Ann. 271 (1985), 401–414. H. Upmeier, Index theory for Toeplitz operators on bounded symmetric domains, Bull. Amer. Math. Soc. 16 (1987), 109–112. H. Upmeier, Fredholm indices for Toeplitz operators on bounded symmetric domains, Amer. J. Math. 110 (1988), 811–832. H. Upmeier, Index theory for multivariable Wiener-Hopf operators, J. reine angew. Math. 384 (1988), 57–79. M. Vergne, H. Rossi, Analytic continuation of holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1975), 1–59. N. Wallach, The analytic continuation of the discrete series I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17, 19–37. A. Wassermann, Alg`ebres d’op´erateurs de Toeplitz sur les groupes unitaires, C.R. Acad. Sci. 299 (1984), 871–874. G. Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769–3787.

Harald Upmeier Fachbereich Mathematik University of Marburg 35032 Marburg Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 121–142 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Rotation Algebras and Continued Fractions Florin P. Boca Abstract. This paper discusses two problems related with the approximation of rotation algebras: (i) estimating the norm of almost Mathieu operators and (ii) studying a certain AF algebra associated with the continued fraction algorithm. The Effros-Shen AF algebras naturally arise as primitive quotients of this algebra. Mathematics Subject Classification (2000). Primary 46L05; Secondary 11A55, 47A30, 47B36. Keywords. Rotation algebras, almost Mathieu operators, continued fractions, AF algebras.

1. Introduction Let Γ be a finitely generated discrete group. Let λ denote the left regular repre2 sentation of Γ on the Hilbert  space  (Γ). For 2any finite set S ⊂ Γ, consider the 1 Markov operator µS = |S| s∈S λs acting on  (Γ) as (µS f )(t) =

1
f (s−1 t). |S| s∈S

Suppose S is a symmetric generating set for Γ. A classical result of Kesten [23] shows that the group Γ is amenable if and only if the spectral radius σ(µS ) of µS is equal to 1. The natural example where Γ is the (non-amenable) free group Fn with generators g1 , . . . , gn and S = {g1±1 , . . . , gn±1√}, n  2, has also been considered by Kesten [22]. In this case σ(µS ) = µS  = n1 2n − 1 and the spectrum of µS is the whole interval [−µS , µS ]. Things are different when the representation λ is twisted by a cocycle. Here we are interested in the case where Γ is the abelian group Z2 . Given θ ∈ [0, 1), the mapping β(m, n) = eπiθm∧n defines a skew-symmetric bicharacter of Z2 × Z2 . The (left) regular representation of Z2 is twisted by β to (πs f )(t) = β(s, t)f (t − s),

s, t ∈ Z2 , f ∈ 2 (Z2 ).

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This formula defines a projective unitary representation π of Z2 on 2 (Z2 ) such that πs∗ = π−s and πs1 πs2 = β(s1 , s2 )πs1 +s2 = β(s1 , s2 )β(s2 , s1 )πs2 πs1 ,

s1 , s2 ∈ Z2 .

The unitaries Uθ = π(1,0) and Vθ = π(0,1) acting on the orthonormal basis {δ(m,n) } of 2 (Z2 ) consisting of Dirac functions as Uθ δ(m,n) = eπinθ δ(m+1,n) , Vθ δ(m,n) = e−πimθ δ(m,n+1) , commute as Uθ Vθ = e2πiθ Vθ Uθ . The rotation algebra Aθ is defined as the universal C ∗ -algebra generated by two unitary operators uθ and vθ such that uθ vθ = e2πiθ vθ uθ . Using the simplicity of Aθ when θ is irrational and the canonical isomorphism between C ∗ (Uθq , Vθq ) and C(T2 ) when θ = pq in lowest terms, the mapping uθ → Uθ , vθ → Vθ , is seen to extend to an isomorphism ρθ : Aθ → C ∗ (Uθ , Vθ ) ⊂ B(2 (Z2 )). The C ∗ -algebra A θ is endowed with  the tracial state τθ m n defined on polynomials in uθ and vθ by τθ α u v m,n (m,n) θ θ = α(0,0) . It is easily seen that ρθ and the GNS representation of Aθ defined by the state τθ are unitarily equivalent. Another representation, σθ : Aθ → B(2 (Z)), is obtained by mapping $θ , V$θ ), where the unitaries U $θ and V$θ act on an orthonormal basis (uθ , vθ ) → (U 2 2πinθ $ $ δn . Every irreducible representation {δn }n of  (Z) by Uθ δn = δn−1 , Vθ δn = e of Ap/q has dimension q and is unitarily equivalent with one of the representations πz1 ,z2 defined as πz1 ,z2 (uθ ) = z1 U0 , πz1 ,z2 (vθ ) = z2 V0 , with ⎤ ⎡ ⎡ ⎤ 0 1 0 ... 0 0 1 ⎥ ⎢0 0 1 . . . 0 0⎥ ⎢ ω ⎥ ⎢ ⎢ ⎥ 2 ⎥ ⎢ .. .. .. . . ⎢ ⎥ . . ω . . = U0 = ⎢ . . . , V ⎥ , ω = e2πip/q . ⎢ ⎥ 0 . . .⎥ ⎥ ⎢ ⎢ . .. ⎦ ⎣0 0 0 . . . 0 1⎦ ⎣ q−1 ω 1 0 0 ... 0 0 Rotation algebras are closely connected with the three-dimensional discrete Heisenberg group H3 (Z) generated by the set S = {x±1 , y ±1 , z ±1 }, with ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 0 1 0 0 1 0 1 x = ⎣0 1 0⎦ , y = ⎣0 1 1⎦ , z = ⎣0 1 0⎦ , 0 0 1 0 0 1 0 0 1 satisfying the commutation relations z = xyx−1 y −1 , zx = xz, and zy = yz. Since H3 (Z) is amenable, its reduced and full C ∗ -algebras coincide, being described as the universal C ∗ -algebra generated by three unitary elements u, v, w (corresponding to λx , λy , λz ) satisfying the commutation relations uv = vuw, wu = uw, and wv = vw. For every θ the mapping u → uθ , v → vθ , w → e2πiθ 1, extends to a representation βθ of C ∗ (H3 (Z)) onto Aθ . Conversely, for any irreducible unitary representation π of H3 (Z) there is some number θ ∈ [0, 1) such that πz = e2πiθ 1. Consequently πx and πy are unitaries such that πx πy = e2πiθ πy πx . When θ is irrational any two such representations are approximately unitarily equivalent as a consequence of Voiculescu’s noncommutative Weyl-von Neumann theorem. When θ = pq in lowest terms, πx and πy generate a copy of the C ∗ -algebra Mq of q × q matrices with complex entries, and there are z1 , z2 ∈ T such that πx = z1 U0

Rotation Algebras and Continued Fractions

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and πy = z2 V0 (see, e.g., [14, Thm.VII.5.1]). These considerations show that the spectrum of any self-adjoint element h ∈ C ∗ (H3 (Z)) coincide with the closure of the union of the spectra of the elements βθ (h) ∈ Aθ , θ ∈ [0, 1). The C ∗ algebra of any finitely generated torsion-free nilpotent group contains no nontrivial projections (cf., e.g., [2, Prop.1]), hence spec(h) is always a compact interval. Using the equality spec(β√ θ (6µS )) = 2 cos(2πθ) + spec(Hθ ), it was proved in [2] that spec(µS ) = [− 13 (1 + 2), 1] for the set of generators S = {x±1 , y ±1 , z ±1 } of Γ = H3 (Z). The “twisted” Markov operator hθ = uθ + u∗θ + vθ + vθ∗ ∈ Aθ corresponding to the set S = {±(1, 0), ±(0, 1)} of generators of Z2 is called the Harper operator. More generally, one considers almost Mathieu operators hθ,λ = uθ +u∗θ + λ2 (vθ +vθ∗ ), λ ∈ R∗ . The norm of hp/q,λ is seen to be given by 8 8 8 8 λ ∗ ∗ 8 (V U + U + + V ) (1.1) hp/q,λ  = π1,1 (hp/q,λ ) = 8 0 0 0 8. 8 0 2

1 0.8

Θ

0.6 0.4 0.2 0 -4

-2

0 spec
hΘ 

2

4

Figure 1. The Hofstadter butterfly (1  q  30) The spectrum of hp/q,λ is the union of q − 1, respectively q, disjoint intervals, according to whether q is even or odd. The union of the sets spec(hθ ), θ ∈ Q, forms the Hofstadter butterfly (see Figure 1). During the last three decades much effort has been put in the spectral analysis of almost Mathieu operators. One of the crowning achievements is the recent result of Avila and Jitomirskaya [1] which shows that spec(hθ,λ ) is a Cantor set for every irrational θ whenever λ = 0, thus

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answering the ‘Ten Martini problem’ of Kac and Simon. In the next section we will discuss some results concerning the norm of almost Mathieu operators. Rotation algebras provide a noticeable role in noncommutative geometry [13]. A classical result of Pimsner and Voiculescu [26] shows that any irrational rotation algebra Aθ can be embedded into the Effros-Shen AF algebra Fθ [15], which encodes the continued fraction expansion of θ. Combined with Rieffel’s construction [29] of projections of trace {nθ} in Aθ this led, via the (ordered) dimension group of Fθ , to the classification of irrational rotation algebras: two irrational rotation algebras Aθ1 and Aθ2 are isomorphic if and only if θ2 = ±θ1 (mod Z). The order on the dimension group of Fθ also plays a role in the Elliott-Evans decomposition [16] of irrational rotation algebras as inductive limits of circle algebras. The results of Rieffel, Pimsner and Voiculescu yield {τθ (p) : p projection in Aθ , p = 1} = {{nθ} : n ∈ Z} for any irrational θ, thus gaps in the spectrum of almost Mathieu operators are canonically labelled by integers. The ‘dry’ Ten Martini problem asks / Q, is it whether all these canonical gap labels occur, i.e., given λ ∈ R∗ and θ ∈ true that for any n ∈ Z∗ there is p spectral projection of hθ,λ in Aθ such that τθ (p) = {nθ}? The answer is known to be positive for θ of Liouville type [12], but the problem is open in general (see also [1] and [28] for some recent thoughts). In Section 3 we will review some results from [6] concerning a certain AF algebra A associated with the continued fraction algorithm. This algebra is related with both the Effros-Shen algebras and the GICAR AF algebra having the Pascal triangle as Bratteli diagram.

2. Norm estimates for almost Mathieu operators The problem of estimating the norm of Harper operators (i.e., to approximate the external boundary of the Hofstadter butterfly) has been first considered in [2] in connection with the study of the spectrum of the Markov operator µS on the group C ∗ -algebra of the discrete Heisenberg group H3 (Z), generated by the set S = {x±1 , y ±1 , z ±1 }. Explicit estimates for the norm of almost Mathieu operators are provided by Theorem 2.1 ([7]). (i) For every λ ∈ R and θ ∈ [0, 12 ], hθ,λ   Mλ (θ),

(2.1)

with

⎧9 2 ⎪ ⎨ 4 + λ + 4|λ|(cos πθ − sin πθ) cos πθ : ; Mλ (θ) =   ⎪ 4 + λ2 − 1 − 1  1 − 1+cos2 4πθ min{4, λ2 } ⎩ tan πθ 2

In particular for every θ ∈ [ 14 , 34 ], hθ,λ  

9 4 + λ2

if θ ∈ [0, 14 ], if θ ∈ [ 14 , 12 ].

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125

(ii) For every θ ∈ [0, 12 ], hθ   m(θ) = max{f1 (θ), f2 (θ), f3 (θ)},

(2.2)

where f1 , f2 , f3  0 are the elementary functions given by f1 (θ)2 = 6 − 1+

1 sin πθ

4 ; + 1+

+ 1 sin πθ

8 cos2 πθ 2 + , 2 1 + 4 sin πθ cos πθ (1 + sin πθ)3/2

2 2 cos2 2πθ + f2 (θ)2 = 4 + 9 1 + | sin 4πθ| 1 + | sin 4πθ| <  2 = 2 = cos 2πθ 16 cos4 πθ sin2 2πθ > 1+ 9 + , +2 1 + | sin 4πθ| (2 + | tan 2πθ|)2 1 + | sin 4πθ| f3 (θ)2 = 4 + ? +

4(cos 2πθ + 2 cos2 2πθ + 2 cos4 2πθ) 5

4(cos 2πθ + 2 cos2 2πθ + 2 cos4 2πθ) 2− 5

2



√ 8 cos 2πθ + 10 + √ 10

2 .

Figure 2. The graphs of the functions M2 , h·  and m on [0, 1/2] The map [0, 1]  θ → hθ,λ  was proved to be Lipschitz by Bellissard [3]. Figure 2 and the inequalities     sup M2 (θ) − m(θ)  0.18962, M2 (θ) − m(θ)  0.16183, sup 0
θ
1/4

1/4
θ
1/2

126

F.P. Boca

show that Theorem 2.1 provides a reasonably accurate numerical approximation for the boundary of the Hofstadter butterfly. It also shows that Kesten’s result does not extend to the case of twisted Markov operators, even in the case of the abelian group Γ = Z2 . Actually the norm of 14 (uθ + u∗θ + vθ + vθ∗ ) is much smaller √ than the norm 23 of 14 (λg1 + λ∗g1 + λg2 + λ∗g2 ) ∈ Cr∗ (F2 ). To prove hθ   fj (θ), one simply uses the trivial inequalities hθ   σθ (hθ ) 

σθ (hθ )ξ2 , ξ2

where ξ = {ξn }n ∈ 2 (Z) is given, according to j = 1, 2, 3, by ⎧ sin √ β ⎪ ⎪ ⎪ % √ 10 ⎪ ⎨ 2√sin β s|k| cos α if n = 2k, 5 ξ ξn = r|n| , ξn = = n ⎪ s|k| sin α if n = 2k + 1, cos β ⎪ ⎪ ⎪ ⎩ 0

if n = ±2, if n = ±1, if n = 0, else,

for appropriate choices of r = r(θ), s = s(θ) ∈ (0, 1), α = α(θ), β = β(θ) ∈ [0, 2π). The continuity of the map θ → hθ,λ  and (1.1) reduce (2.1) to the situation where θ = pq is a rational number in lowest terms, when hθ,λ  = Hθ,λ  with Hθ,λ = π1,1 (hθ,λ ) = U0 + U0∗ + λ2 (V0 + V0∗ ). Let E be an eigenvalue of the selfadjoint q × q matrix Hθ,λ and ξ = {ξm }m∈Zq ∈ Cq = 2 (Zq ) be a unit eigenvector for E. The equality Hθ,λ ξ = Eξ can be written as a three-term recurrence relation ξm+1 = (E − λ cos 2πmθ)ξm − ξm−1 , or in matrix form as 

  E − λ cos 2πmθ ξm+1 = 1 ξm

m ∈ Zq ,

(2.3)

  −1 ξm . 0 ξm−1

 Set S = m∈Zq ξm ξm−1 cos π(2m − 1)θ. After some heavy trigonometry and intensive use of the recurrence relation (2.3), one can express E 2 in two ways, as
(ξm+1 − ξm−1 + λξm sin 2πmθ)2 E 2 =4 + λ2 + 4λS(cos πθ − sin πθ) − m∈Zq (2.4) 2  4 + λ + 4λS(cos πθ − sin πθ), and respectively as
λ2 2 (4 − sin2 2πθ) ξm sin2 2πmθ 4 m∈Zq 2
λ ξm sin 2πθ sin 2πmθ + ξm+1 − ξm−1 − 2

E 2 =4 + 4λ2 + 4λS cos3 πθ −

m∈Zq

 4 + 4λ2 + 4λS cos3 πθ.

(2.5)

Rotation Algebras and Continued Fractions

127

Some more trigonometry and use of Cauchy-Schwarz inequality lead to |S|  cos πθ for every θ ∈ [0, 14 ]. In conjunction with (2.4) this completes the proof of (2.1) in this range of θ. In the range θ ∈ [ 14 , 12 ] we have cos3 πθ  0  cos πθ − sin πθ. So regardless of the actual√sign of S, inequalities (2.4) and (2.5) yield E 2  4 + λ2 , and therefore hθ,λ   4 + λ2 . This upper bound can be improved to that given by Mλ (θ) by further enhancing this kind of arguments. Remark 2.2. (i) The lower bound estimate (2.2) gives in particular √ min hθ   min f1 (θ) = 6.59303 . . . = 2.56769 . . . 1/4
θ
1/2

(2.6)

1/4
θ
1/2

Numerical computations suggest that the left-hand side in (2.6) is only fractionally larger than 2.59. It would be interesting to improve these estimates. (ii) Estimate (2.2) also gives min0
θ
1/4 hθ 2  7.82387 (which is pretty √ close to 8) and hθ   2 2 for all θ ∈ [0, √ 0.23441]. Numerical computations seem to suggest that the inequality hθ   2 2 may hold in the whole range θ ∈ [0, 14 ]. It would be interesting to prove/disprove this. (iii) The spectrum of the non-self-adjoint operator uθ + u∗θ + λvθ , λ ∈ C, was computed for irrational θ in [4, 30]. It would be interesting to find accurate estimates for its norm.

3. An AF algebra associated with the continued fraction algorithm 3.1. Continued fractions and the Farey tessellation The regular continued fraction representation of real numbers establishes a oneto-one correspondence NN  a = (a1 , a2 , . . .) ←→ θ ∈ (0, 1] \ Q.

(3.1)

In one direction define, for given a, the rational numbers [a1 , . . . , an ] =

1 a1 +

1 a2 +...+

= 1 an

pn . qn

(3.2)

The representation (3.2) is not unique because [a1 , . . . , an ] = [a1 , . . . , an − 1, 1]. However, the Euclidean algorithm shows that any rational number in (0, 1] can be uniquely represented as in (3.2), for some n and a1 , . . . , an ∈ N with an  2. Moreover, pn and qn are obtained by plain matrix multiplication as        an 1 qn qn−1 a1 1 a2 1 ··· = . (3.3) 1 0 1 0 1 0 pn pn−1 Applying the determinant on both sides of (3.3) one gets the familiar relation pn−1 qn − pn qn−1 = (−1)n .

(3.4)

128

F.P. Boca

It is easy to infer from (3.3) that qn is greater or equal than the nth Fibonacci number Fn . In conjunction with (3.4) this yields # # # pn+1 1 pn ## 1 # = −  , # qn+1 qn # qn qn+1 Fn Fn+1 showing that { pqnn }n is a Cauchy sequence of rational numbers. Its limit is irrational and defines the number θ from the right-hand side of (3.1). In the opposite direction the digits an of θ are obtained by plain iterations of the Gauss map     1 1 1 = − , x = 0, G : [0, 1] → [0, 1], G(0) = 0, G(x) = x x x as 1 an = n−1 − Gn (θ), n  1. G (θ) The Gauss map acts as G[a1 , a2 , a3 , . . .] = [a2 , a3 , . . .], hence its periodic points are exactly the reduced (purely periodic) quadratic irrational numbers [a1 , . . . , an0 ].

F

B −1 F

BF

AF

A2 F 3

A F

A2 BF

1 4

0 1

ABF

1 3

AB 2 F

ABAF

2 5

1 2

3 5

3 4

2 3

1 1

Figure 3. The Farey tessellation Consider the matrices     1 0 1 1 A= , B= , 1 1 0 1 The equalities (3.3) and   a 1 b M (a)M (b) = 1 0 1

 J=

 0 1 , 1 0

   1 a 1 1 = 0 1 0 0

yield



a1

a2

B A

···B

a2m−1

a2m

A

B a1 Aa2 · · · Aa2m B a2m+1

 M (a) =

a 1

 b = B a Ab , 1

q = 2m p2m  q = 2m p2m

 q2m−1 , p2m−1  q2m+1 . p2m+1

 1 ∈ GL2 (Z). 0

a, b ∈ Z,

(3.5)

(3.6)

Rotation Algebras and Continued Fractions

129

It is now plain to prove that the semigroup G generated by A and B in SL2 (Z) is canonically isomorphic to the free semigroup on two generators F+ 2 . This can be seen geometrically considering the Farey tessellation {gF : g ∈ G} of the upper half-plane H, where F = {0  Re z  1, |z − 12 |  12 } (see Figure 3). However, the group generated by A and B is not free, as one can see from the equality (BA−1 B)4 = I2 . Note also that conjugating by J in (3.6) one gets    a b ∈ SL2 (Z) : 0  a  c, 0  b  d = AG. c d A first estimate [21] on the number Ψ(N ) of elements C ∈ G such that 2 < Tr C  N as N → ∞ was furthered in [5] to   3 ζ  (2) N2 ln N + γ − − + Oε (N 7/4+ε ). (3.7) Ψ(N ) = ζ(2) 2 ζ(2) Interestingly, the contribution to the main term of even length words is which turns out to be much smaller than the contribution ζ
(2) ζ(2) )

N2 ζ(2) (ln N

N 2 ln 2 ζ(2) ,

+ γ − ln 2 −

− of odd length words. The estimate on the number of even length words leads to good estimates of the number of reduced quadratic irrationals according to their natural length [18, 5]. 3 2

3.2. The mediant construction and AF algebras The mediant construction, pictured in Figure 4, associates to every two consecutive

rational numbers pq < pq
with p q − pq  = 1 the new rational number p+p q+q
. Clearly p q

<

p+p
q+q


<

p
q


and p (q + q  ) − (p + p )q  = (p + p )q − p(q + q  ) = 1. p q •

p q •

•

p+p
q+q


•

p
q


•

p
q


Figure 4. The mediant construction Reproducing the diagram from Figure 4 one gets the Pascal triangle with memory G from Figure 5. When removing the edges one gets a familiar table which generates all rational numbers in [0, 1] (cf., e.g., [25, Sect.6.1]). Our point however is that edges are important and it is worth to regard G as a Bratteli diagram, together with a labelling of its vertices. For every n  0 denote by (n, k) n the vertices at floor n and by r(n, k) = p(n,k) q(n,k) the label of (n, k), 0  k  2 . n n We have q(n, 0) = q(n, 2 ) = p(n, 2 ) = 1, p(n, 0) = 0, q(n + 1, 2k) = q(n, k), q(n + 1, 2k + 1) = q(n, k) + q(n, k + 1), and similar recurrence relations for p(n, k).

130

F.P. Boca

 • 1 1

0 • 1 1 2 •

0 • 1 1 3 •

0 • 1 1 4 •

0 • 1 1 5 •

0 • 1 0 1

•

• 1 6

• 1 5

1 • 4 • 2 9

1 • 3

2 3 •

1 2• 2 5 •

1 • 3 2 7 •

• 1 1

3 8 •

3 5 •

1 • 2 3 7 •

2 • 5

• • • • • • • • • • • 1 3 2 3 1 4 3 5 2 5 3 4 11 7 10 3 11 8 13 5 12 7

4 7 •

1 • 2 • 4 9

• 1 2

• 5 9

• 3 5

• 1 1 3 4 •

• 2 3 5 8 •

• 2 3

5 7 •

• 1 1

• 3 4

• • • • • • • • • • • 4 7 3 8 5 7 2 7 5 8 3 7 12 5 13 8 11 3 10 7 11 4

• 7 9

4 5 • • 4 5

• 1 1 • 5 6

•

1 1

Figure 5. The Pascal triangle with memory G x The Farey map F : [0, 1] → [0, 1], defined by F (x) = 1−x if x ∈ [0, 12 ] and 1−x 1 F (x) = x if x ∈ [ 2 , 1], acts on continued fraction expansions as % if a1 = 1, [a2 , a3 , . . .] F [a1 , a2 , a3 , . . .] = [a1 − 1, a2 , a3 , . . .] if a1  2. y For every y ∈ [0, 1] the equation F (x) = y has solutions x = F1 (y) = 1+y and 1 x = F2 (y) = 1+y = 1 − F1 (y). The maps F1 and F2 act on continued fractions by

F1 [a1 , a2 , . . .] = [a1 + 1, a2 , . . .],

F2 [a1 , a2 , . . .] = [1, a1 , a2 , . . .].

Note the equality [a1 + 1, a2 , a3 , . . .] = 1 − [1, a1 , a2 , a3 , . . .]. Remark 3.1. (i) The labels

p q

<

p
q


of two consecutive vertices satisfy

p q − pq  = 1. (ii) The labels of the “new stuff” in the nth floor of G are exactly the rational numbers [a1 , . . . , at ] with a1 + · · · + at = n + 1. For example, the labels 16 < 29 < 3 3 4 5 5 4 5 7 8 7 7 8 7 5 11 < 10 < 11 < 13 < 12 < 9 < 9 < 12 < 13 < 11 < 10 < 11 < 9 < 6 of the new stuff at the fifth floor are [6] < [4, 2] < [3, 1, 2] < [3, 3] < [2, 1, 3] < [2, 1, 1, 2] < [2, 2, 2] < [2, 4] < [1, 1, 4] < [1, 1, 2, 2] < [1, 1, 1, 1, 2] < [1, 1, 1, 3] < [1, 2, 3] < [1, 2, 1, 2] < [1, 3, 2] < [1, 5]. (iii) The labels of the new stuff in G give an enumeration of Q ∩ [0, 1]. (iv) The labels of all vertices (both new and old stuff) in the (n−1)th floor are the rational numbers [a1 , . . . , at ] with a1 + · · · + at  n. They represent exactly the elements of the set F −n ({0}) = {Fiα1 1 · · · Fiαkk (0) : i1 = · · · = ik , α1 + · · · + αk = n}.

Rotation Algebras and Continued Fractions

131

(v) The mapping r(n, k) → 2kn , n  0, 0  k  2n , transforms Q ∩ [0, 1] onto dyadic rationals in [0, 1]. It extends to Minkowski’s question mark function
? : [0, 1] → [0, 1], which is continuous and satisfies ?(0) = 0, ?(1) = 1, ?( p+p q+q
) = p p
1 1 2 ?( q ) + 2 ?( q
)

whenever p q − pq  = 1. Moreover ? transforms quadratic numbers (i.e., solutions of an integer polynomial of degree at most two) from [0, 1] onto Q ∩ [0, 1], and is strictly increasing and singular. On continued fraction expansions ? acts as ∞
(−1)k−1 ?[a1 , a2 , . . .] = . 2a1 +···+ak −1 k=1

•

•

a1

•

•

a2

•

a3

•

...

•

...

Figure 6. The Bratteli diagram D(Fθ ) The Effros-Shen AF algebra (rotation AF algebra) Fθ is defined for every irrational number θ = [a1 , a2 , . . .] ∈ [0, 1] by the Bratteli diagram in Figure 6. It can also be represented by other Bratteli diagrams. The third type of diagram in Figure 7 occurs as a subdiagram in G. Using Bratteli’s diagrammatical description of ideals and primitive ideals in AF algebras (Lemma 3.2. and Theorem 3.8 in [8]) one sees that any irrational rotation AF algebra occurs as a primitive ideal of the AF algebra A defined by the Bratteli diagram G. A more careful analysis shows that the primitive ideals of A are parameterized by the numbers in [0, 1] repeated with multiplicity one for irrational numbers, multiplicity two for 0 and 1, and multiplicity three for rational numbers in (0, 1). More precisely, one has ∼ Fθ . Proposition 3.2. (i) For every θ ∈ [0, 1] \ Q, there is Iθ ∈ Prim A with A/Iθ = (ii) Given θ = pq ∈ Q ∩ (0, 1) in lowest terms, there are Iθ , Iθ+ , Iθ− ∈ Prim A such that A/Iθ ∼ = Mq , A/Iθ− ∼ = A(q,p) , and A/Iθ+ ∼ = A(q,q−p) , where p is the multiplicative inverse of p mod q in {1, . . . , q − 1} and A(q,q
) is some extension of the compact operators by Mq . (iii) There are I0 , I0+ , I1 , I1− ∈ Prim A such that A/I0 ∼ = A/I1 ∼ = C and + ∼ − ∼ $ ∗ A/I0 = A/I1 = K is the C -algebra of compact operators with adjoined unit. (iv) Every primitive ideal of A is of the form Iθ with θ ∈ [0, 1], Iθ+ with θ ∈ [0, 1), or Iθ− with θ ∈ (0, 1]. The Bratteli diagrams of the ideals Iθ and of the quotients A/Iθ are shown in Figures 8, 9 and 10. Remark 3.3. By a result of Bratteli [9], any separable abelian C ∗ -algebra Z can be realized as the center Z(A) of some AF algebra A. Our AF algebra A can be

132

F.P. Boca   

•

a1 + 1

A•

•B

A•

a2

C•

•D

D•

a3

E•

•F

•B

A•

•D

D•

a2

E•

•F

E•

•H

•

a3

G•

•

• •

a2

•B

•

a2

A• D•

•C

a3

•B •C • • ••

•F

a1 + 1

a3

E •• F



a1

C•

a1

E•

 A•

•

H

•

a1 •

a2

B C

a1 + 1

 A•

•

•

D•

•

E •

a3 •

F G

a3

B

•

• •

•

C F

a2

•

• •

H ••G

Figure 7. The AF algebras F[a1 +1,a2 ,a3 ,... ] ∼ = F[1,a1 ,a2 ,a3 ,...] represented by different Bratteli diagrams recovered from that abstract construction by embedding Z = C[0, 1] into the norm closure in L∞ [0, 1] of the linear span of characteristic functions of open intervals  n n ( 2kn , k+1 2n ), 0  k < 2 , and of singleton sets { 2n }, 0    2 . In particular this gives Z(A) = [0, 1]. Remark 3.4. Proposition 3.2 shows some resemblance between the irreducible representations of the the AF algebra A and those of the group algebra C ∗ (H3 (Z)). It would be interesting to investigate the existence of a closer connection between these C ∗ -algebras. The usual topology on [0, 1] can be recovered from the Jacobson topology on Prim A by discarding the “singular” ideals Iθ± . Proposition 3.5. For any sequence {θn }n in [0, 1], θn → θ in [0, 1] if and only if Iθn → Iθ in the Jacobson topology on Prim A. 3.3. The dimension group The Stern-Brocot sequence {θn }∞ n=0 [10, 31] is obtained by enumerating the elements q(n, k) with (n, k) in lexicographic order and n  0, 0  k < 2n . Its first elements are 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, . . .. Note

Rotation Algebras and Continued Fractions

133

 0 1

1 1 1 2 •

0 • 1 0 1 1 4 • 1 5 •

1• 4

2 5 •

• 1 3 2 7 3 11 2 • • 7

2 3 •

1 •2

1 3

0 • 1

• 1 1

3 5 •

3 8 •

1 3 3 4 10 11 • •1 • 3

5 2 5 17 • 14 •7• •

•• • •

Figure 8. The diagrams D(Iθ ) = G \Gθ and D(A/Iθ ) = Gθ when θ = [3, 2, 2, . . .] that θ2N −1 = 1, θ2N = N + 1, θ2N −2 = N , θ2k+1 = θk and θ2k+2 = θk + θk+1 for all k (see Figure 2). The generating function of {θn }∞ n=0 can be expressed [11] as ∞ ∞
. k k+1 θn X n = (1 + X 2 + X 2 ). (3.8) n=0

k=0

The product Θ(X) in the right-hand side of (3.8) satisfies the identity Θ(X) = (1 + X + X 2 )Θ(X 2 ).

(3.9)

Equality (3.9) can be used to give a description of the dimension group of the codimension one ideal C = I1 of A obtained by removing in G all vertices (n, 2n ). The algebra C is the inductive limit of Cn = ⊕0
k<2n Mq
(n,k) with q  defined recursively by q  (n, 0) = q  (n, 2n − 1) = 1, q  (n, 2k) = q  (n − 1, k), q  (n, 2k + 1) = q  (n − 1, k) + q  (n − 1, k + 1), 0  k < 2n . The group K0 (Cn ) identifies with n Z2 , generated by the Murray-von Neumann equivalence classes [e(n,k) ] of minimal projections e(n,k) in the central summand A(n,k) of A, 0  k < 2n . Its positive cone consists of linear combinations with nonnegative integer coefficients of such [e(n,k) ]. We have K0 (C) = lim(K0 (Cn ), αn ) with injective morphisms αn : K0 (Cn ) → −→ K0 (Cn+1 ) given by % if k = 0, [e(n+1,0) ] + [e(n+1,1) ] (3.10) αn [e(n,k) ] = [e(n+1,2k−1) ] + [e(n+1,2k) ] + [e(n+1,2k+1) ] if 1  k < 2n .

134

F.P. Boca

 0 1

1 1

0 1

1 2

0 • 1

1 3 2 7 •

2 3 •

1 2

1  3 1 4 •

• 1 1

• 1 3 3 10 •

3 8

• 2 5

5 4 11 13 • 3• 8

3 5 •

• 1 2

2 5 3 7 •

4 7 •

5 12 •

7 8 19 21 •• 3 8

Figure 9. The diagrams D(I3/8 ) and D(A/I3/8 )

 0 1

1 1

0 1

1 2

0 • 1

1 3 2 7 •

3 8

2 5

3 5 •

• 1 2

2 5

• 1 3 3 10 •

2 3 •

1 2

1  3 1 4 •

• 1 1

3 7 •

4 7 •

5 2 5 4 13 5 12 113 •   • • 8 7 7 7 19 18 17 •• • • 3 8 8 21

+ − Figure 10. The diagrams D(I3/8 ) and D(A/I3/8 )

Rotation Algebras and Continued Fractions

135

 For each n  0 the polynomials c0 + 1
(3.11)

Equalities (3.10) and (3.11) show the commutativity of the diagram φn

K0 (Cn )

/ Pn

αn

 K0 (Cn+1 )



φn+1

βn

/ Pn+1

and as a result K0 (C) is isomorphic to the abelian group P = lim(Pn , βn ). The −→ positive cone P + consists of equivalence classes of polynomials p with nonnegative coefficients. 3.4. Traces on the AF algebra A Let θ = [a1 , a2 , . . .] ∈ [0, 1] \ Q and pn , qn given by (3.3). The AF algebra Fθ is isomorphic to the inductive limit lim(Mqn ⊕ Mqn−1 ) with connecting morphisms −→ Mqn ⊕ Mqn−1  x ⊕ y → (x ⊕ · · · ⊕ x ⊕y) ⊕ x ∈ Mqn+1 ⊕ Mqn . ! "  an times

Traces on Fθ are in one-to-one correspondence (cf., e.g., [17, Section 3.6]) with families {ω(n,k) }n,k of nonnegative numbers, n =  or n  1, k = 1, 2, such that ω = 1,

ω(n−1,1) = ω(n,0) ,

ω(n−1,0) = an ω(n,0) + ω(n,1) ,

n  1.

(3.12)

Letting ωn = ω(n,0) , ω1 = ω, and applying recursively (3.12) we get ωn+1 = (−1)n (qn ω − pn )  0, and thus

p2k+1 p2k ω , q2k q2k+1

n  0,

k  0.

(3.13)

The convergence of the continued fraction algorithm and (3.13) yield ω = θ, hence Fθ has a unique trace τθ , given by τθ (en ) = ω(n,0) = (−1)n−1 (qn−1 θ−pn−1 ), n  1, where en denotes a minimal projection in Mqn . Note the equality qn τθ (en ) + qn−1 τθ (en−1 ) = qn ω(n,0) + qn−1 ω(n,1) = 1.

136

F.P. Boca

 [1] •

[2] •

[3]

[1, 2]

•

•

[4]

[22 ]

•

[5] •

• [6]

[12 , 2]

•

[3, 2] •

[2, 1, 2] •

[1, 3]

•

[12 , 3]

[2, 3] •

•

•

[13 , 2] •

[1, 22 ] •

[1, 4] •

• • • • • • • • • • • • • • • [4, 2] [3, 1, 2] [32 ] [2, 1, 3][2, 12 , 2] [23 ] [2, 4] [12 , 4][12 , 22 ][14 , 2][13 , 3][1, 2, 3][1, 2, 1, 2][1, 3, 2][1, 5]

Figure 11. The diagram T

Traces on the AF algebra A are parameterized by families α = {α(n,k) }n,k of nonnegative numbers, n  −1, 0  k  2n , such that ⎧ α = 1, α(n,0) = α(n+1,0) + α(n+1,1) , n  −1, ⎪ ⎪ ⎨ α(n,2n ) = α(n+1,2n+1 ) + α(n+1,2n+1 −1) , n  0, ⎪ ⎪ ⎩ α(n,k) = α(n+1,2k−1) + α(n+1,2k) + α(n+1,2k+1) , n  1, 0 < k < 2n .

(3.14)

Glancing at the recurrence relations (3.14) and at G we notice that such α is uniquely determined only by α(n,k) with k odd, and that ⎧ ∞  ⎪ α(i,1) , ⎪ ⎨α = 1  i=0

α(0,1) 

∞  i=1

α(i,2i −1) ,

∞   ⎪  ⎪ ⎩α(n,k)  α(n+i,2i k−1) + α(n+i,2i k+1) ,

(3.15) ∀k odd, k < 2n .

i=1

The vertices , (0, 1) and (n, k) with n  1 and k odd, represent precisely the new stuff T in G. As noticed in Remark 3.1 the mapping (n, k) → r(n, k) provides a one-to-one correspondence between the set V (T ) of vertices of the binary tree T and Q ∩ [0, 1] (see Figure 11 where ak denotes a string of k consecutive a’s). L

The maps on V (T ) \ {, (0, 1)} given by (n, k) → (n + 1, 2k − 1) and respecR $ R $ : Q ∩ (0, 1) → [0, 1] defined by tively (n, k) → (n + 1, 2k + 1) induce maps L, $ $ L(r(v)) = r(Lv), R(r(v)) = r(Rv), r(v) ∈ Q ∩ (0, 1). For at  2 one can prove

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inductively the formulae

% [a1 , . . . , at−1 , at − 1, 2] [a1 , . . . , at−1 , at + 1] % $ 1 , . . . , at ]) = [a1 , . . . , at−1 , at + 1] R([a [a1 , . . . , at−1 , at − 1, 2] $ 1 , . . . , at ]) = L([a

if t even, if t odd, (3.16) if t even, if t odd.

From (3.16) we infer that for every vertex (n, k) with label r(n, k) = [a1 , . . . , at ], at  2, the set {r(n + i, 2i k ± 1) : i  1} is given by the union of the sets {[a1 , . . . , at−1 , at − 1, 1, i] : i  1} and {[a1 , . . . , at−1 , at , i] : i  1}. Combining this with (3.15) we obtain Proposition 3.6. There is a one-to-one correspondence between traces τ on the AF algebra A and functions φ : Q ∩ [0, 1] → [0, 1] such that ⎧ ∞ ∞    k    ⎪ ⎪ , φ k1 , φ(1)  φ k+1 ⎨φ(0) = 1  k=1

k=1

∞    ⎪ ⎪ φ[a1 , . . . , at−1 , at − 1, 1, k] + φ[a1 , . . . , at−1 , at , k] , at  2. ⎩φ[a1 , . . . , at ]  k=1

Moreover, we have   k    τ (e(n,k) ) = φ r(n, k) = φ ?−1 n , 2

∀n  0, ∀k odd, 0 < k  2n .

In the case of the trace τ˜θ = τθ πθ on A, where πθ : A → Fθ ⊂ Aθ denotes the quotient map from Proposition 3.2 (i) and θ = [a1 , a2 , . . .], the corresponding function φ is given by φ(0) = 1, φ[a1 , . . . , at−1 , k] = (−1)t−1 (qt−1 θ − pt−1 ) = |qt−1 θ − pt−1 |,

1  k  at ,

and φ(r) = 0 if r ∈ Q is not of this form. 3.5. A description of A by generators and relations The algebraic-combinatorial side of the theory of subfactors requires elaborated calculations in AF algebras. A convenient method of keeping track of matrix units and embeddings of building blocks is provided by the path algebra model for AF algebras [17, 19]. One fundamental example where this was successfully used is that of the GICAR algebra, defined by the classical Pascal triangle as a Bratteli diagram, and the construction of a sequence of projections in this AF algebra satisfying the Temperley-Lieb-Jones commutation relations. Given a Bratteli diagram with only one vertex  on the first floor, consider the set Ω of infinite monotone increasing paths starting at . Denote by Ω[r,s] the set of monotone increasing paths starting on the rth floor and ending on the sth floor, 1  r  s  ∞, and set Ωr] = Ω[1,r] , Ω[r = Ω[r,∞] . For every path ξ ∈ Ω and 1  r  s  ∞ denote its natural truncation by ξ[r,s] ∈ Ω[r,s] . Set ξr] = ξ[1,r] and ξ[r = ξ[r,∞] . Denote by ξ ◦ η the concatenation of two paths ξ ∈ Ωr] and η ∈ Ω[r with ξr = ηr . For each (ξ, η) ∈ Rr := {(ξ, η) ∈ Ωr] × Ωr] : ξr = ηr }, the

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mapping on Ω defined by Tξ,η ω = δ(η, ωr] )ξ ◦ ω[r extends to a norm one bounded linear operator on 2 (Ω). The operators Tξ,η , (ξ, η) ∈ Rr , satisfy the matrix unit relations
∗ Tη,ξ = Tξ,η , Tξ,η Tξ
,η
= δ(η, ξ  )Tξ,η
, Tξ,ξ = 1, ξ∈Ωr]

showing that their linear span Ar forms a finite-dimensional C ∗ -algebra. Moreover,
Tξ◦λ,η◦λ ιr (Tξ,η ) = λ∈Ω[r,r+1] λr =ξr (=ηr ) ιr defines a unital inclusion Ar → Ar+1 and A ∼ (A , ι ). = lim −→ r r We consider the following elements in A:

1. the projection en in An−1,n ⊆ An onto the linear space of edges from N (north) to SW (south-west), n  1. 2. the projection fn in An−1,n ⊆ An onto the linear span of edges from N to SE, n  0. 3. the projection gn = 1 − en − fn in An−1,n ⊆ An onto the linear span of edges from N to S, n  0. 4. the partial isometry vn ∈ An−1,n+1 ⊆ An+1 with initial support vn∗ vn = e˜n = gn fn+1 and final support vn vn∗ = f˜n = fn en+1 , which flips paths in the diamonds of shape N-S-SE-NE, n  0. 5. the partial isometry wn ∈ An−1,n+2 ⊆ An+1 with initial support wn∗ wn = eˇn = gn en+1 and final support wn wn∗ = fˇn = en fn+1 , which flips paths in the diamonds of shape N-S-SW-NW, n  1. The AF-algebra A is generated by G = {en }n1 ∪ {fn }n0 ∪ {vn }n0 ∪ {wn }n1 . Using the path algebra model for A these generators are shown to satisfy the following relations: (R1) e2n = e∗n = en , fn2 = fn∗ = fn , en + fn + gn = 1, em , fn , gk mutually commute. (R2) er , fr , gr commute with vs and ws whenever r < s or r > s + 1. [vs , vr ] = [vs , vr∗ ] = [vs , wr ] = [vs , wr∗ ] = 0 whenever |r − s| > 1. (R3) vn∗ vn = gn fn+1 , vn vn∗ = fn en+1 , wn∗ wn = gn en+1 , wn wn∗ = en fn+1 . (R4) (1 − fn )vn = (1 − en+1 )vn = 0, vn (1 − gn ) = vn (1 − fn+1 ) = 0. (1 − en )wn = (1 − fn+1 )wn = 0, wn (1 − gn ) = wn (1 − en+1 ) = 0. (R5) vn gn = fn vn , vn fn+1 = en+1 vn , wn gn = en wn , wn en+1 = fn+1 wn . ∗ (R6) vn+1 vn = vn2 = vn±1 vn∗ = vn±1 vn = 0, 2 ∗ ∗ wn = 0, wn+1 wn = wn = wn±1 wn = wn±1 vn wn = vn±1 wn = wn vn = wn±1 vn = 0, vn wn∗ = vn±1 wn∗ = vn∗ wn = vn∗ wn−1 = 0. The partial isometries vn vn+1 , wn wn+1 , wn∗ vn+1 , vn∗ wn+1 are the only nonzero ∗ ∗ products ab with a ∈ {vn , vn∗ , wn , wn∗ } and b ∈ {vn+1 , vn+1 , wn+1 , wn+1 } and are pictured in Figure 13.

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139

 f0

g0

v0

f1

g1

v1

f2

g2

e2

v2 f3

g3

v3

e1

w2 e3

w3

f3

g3

v3

w1

f2

g2

g1

e2

v2 e3

f3

g3

w3

g2

w2 e3

v3

f3

g3

w3

e3

v3

g3

w3

Figure 12. The generators of A

vn vn+1 =

wn∗ vn+1 =

wn wn+1 =

vn∗ wn+1 =

Figure 13. The elements vn vn+1 : gn gn+1 fn+2 → fn en+1 en+2 , wn wn+1 : gn gn+1 en+2 → en fn+1 fn+2 , wn∗ vn+1 : en gn+1 fn+2 → gn en+1 en+2 , vn∗ wn+1 : fn gn+1 en+2 → gn fn+1 fn+2 Relations (R6) show that the maps σi → vi−1 and σi → wi define representations of Artin’s braid group # A @ Bn = σ1 , . . . , σn−1 # σi σj = σj σi if |i − j| > 1, σi σi+1 σi = σi+1 σi σi+1 on the algebra A. The equalities vn2 = 0 and vn vn±1 vn = 0 also show that Rn (λ) = 1 + λvn satisfies the Yang-Baxter type relation Rn (λ)Rn+1 (λ + µ)Rn (µ) = Rn+1 (µ)Rn (λ + µ)Rn+1 (λ). Similar relations are satisfied by 1 + λwn .

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λ 1 For every λ > 0 set τ = (1+λ) 2 ∈ (0, 4 ]. Mimicking the construction of Temeperley-Lieb-Jones projections in the GICAR algebra consider the elements √ √ 1 (vn∗ vn + λ vn + λ vn∗ + λvn vn∗ ), n  0, En = 1+λ √ √ 1 (wn∗ wn + λ wn + λ wn∗ + λwn wn∗ ), n  1. Fn = 1+λ Using the commutation relations (R1)–(R6) one shows that En and Fn are selfadjoint projections in A such that

En Fn = 0,

[Em , En ] = [Fm , Fn ] = [Em , Fn ] if |m − n| > 1,

and satisfying the braiding relations En En+1 En = τ En en+2 , Fn Fn+1 Fn = τ Fn fn+2 ,

En+1 En En+1 = τ En gn , Fn+1 Fn Fn+1 = τ Fn gn ,

(3.17)

and also En Fn+1 En = λτ En fn+2 ,

Fn En+1 Fn = λτ Fn en+2 ,

En+1 Fn En+1 = λτ En+1 en , Fn+1 En Fn+1 = λτ Fn+1 fn , En En+1 Fn = En Fn+1 Fn = En+1 En Fn+1 = En+1 Fn Fn+1 = 0. Relations (3.17) are reminiscent of the commutation relations en en±1 en = τ en satisfied by the Jones projections [20], which can be constructed using the path algebra model for the GICAR algebra. It would be interesting to see if the algebra generated by the two sequences of projections {En }n and {Fn }n can be obtained by some “gluing” process from two copies of the Temperley-Lieb algebra generated by the sequence of projections {en }n . The features of the Pascal triangle with memory have been used in the study of certain statistical mechanics models called Farey spin chains (see, e.g., [24, 27] and references therein). It would be interesting to see whether relations (3.17) have any significance in that context.

References [1] A. Avila, S. Jitomirskaya, The ten martini problem, preprint math.DS/0503363, Ann. Math., to appear. [2] C. B´eguin, A. Valette, A. Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper’s equation, J. Geom. Phys. 21 (1997), 337–356. [3] J. Bellissard, Lipshitz continuity of gap boundaries for Hofstadter-like spectra, Comm. Math. Phys. 160 (1994), 599–613. [4] F.P. Boca, On the spectrum of certain discrete Schr¨ odinger operators with quasiperiodic potential, Duke Math. J. 101 (2000), 515–528. [5] F.P. Boca, Products of matrices [ 10 11 ] and [ 11 01 ] and the distribution of reduced quadratic irrationals, J. Reine Angew. Mathematik 606 (2007), 149–165.

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[6] F.P. Boca, An AF algebra associated with the Farey tessellation, Canad. J. Math., to appear. [7] F.P. Boca, A. Zaharescu, Norm estimates of almost Mathieu operators, J. Funct. Anal. 220 (2005), 76–96. [8] O. Bratteli, Inductive limits of finite-dimensional C ∗ -algebras, Trans. Amer. Math. Soc. 171 (1972), 195–235. [9] O. Bratteli, The center of approximately finite-dimensional C ∗ -algebras, J. Funct. Anal. 21 (1976), 195–202. [10] A. Brocot, Calcul des rouages par approximation, Revue Chronom´etrique 3 (1861), 186–194. [11] L. Carlitz, A problem in partitions related to Stirling numbers, Riv. Mat. Univ. Parma (2) 5 (1964), 61–75. [12] M.D. Choi, G.A. Elliott, N. Yui, Gauss polynomials and the rotation algebra, Invent. Math. 99 (1990), 225–246. [13] A. Connes, Noncommutative Geometry, Academic Press, 1994. [14] K. Davidson, C ∗ -Algebras by Example, Fields Institute Monographs Vol. 6, Amer. Math. Soc., Providence, RI, 1996. [15] E.G. Effros, C.-L. Shen, Approximately finite C ∗ -algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), 191–204. [16] G.A. Elliott, D.E. Evans, The structure of the irrational rotation C ∗ -algebra, Ann. Math. 138 (1993), 477–501. [17] D.E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford University Press, 1998. [18] C. Faivre, Distribution of L´evy constants for quadratic numbers, Acta Arith. 61 (1992), 13–34. [19] F. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, Springer-Verlag, 1989. [20] V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1–25. ¨ uk, M. Peter, C. Snyder, On asymptotic properties of a number [21] J. Kallies, A. Ozl¨ theoretic function arising out of a spin chain model in statistical mechanics, Comm. Math. Phys. 222 (2001), 9–43. [22] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. [23] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. [24] A. Knauf, Number theory, dynamical systems and statistical mechanics, Rev. Math. Phys. 11 (1999), 1027–1060. [25] I. Niven, H.S. Zuckerman, An Introduction to the Theory of Numbers, 2nd edition, John Wiley & Sons, Inc., 1966. [26] M. Pimsner, D. Voiculescu, Imbedding the irrational rotation C ∗ -algebras into an AF algebra, J. Operator Theory 8 (1980), 201–210. [27] T. Prellberg, J. Fiala, P. Kleban, Cluster approximation for the Farey fraction spin chain, J. Statist. Phys. 123 (2006), 455–471.

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[28] J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 244 (2004), 297–309. [29] M.A. Rieffel, C ∗ -algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415–429. [30] P. Sarnak, Spectral behaviour of quasiperiodic potentials, Comm. Math. Phys. 84 (1982), 377–401. ¨ [31] M. Stern, Uber eine zahlentheoretische Funktion, J. Reine Angew. Mathematik 55 (1858), 193–220. Florin P. Boca Department of Mathematics University of Illinois Urbana-Champaign Urbana, IL 61801, USA and Institute of Mathematics “Simion Stoilow” of the Romanian Academy P.O. Box 1-764 RO-014700 Bucharest, Romania e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 143–158 c 2008 Birkh¨
auser Verlag Basel/Switzerland

On the Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators with Semi-almost Periodic Symbols Giorgi Bogveradze and Lu´ıs P. Castro Abstract. Conditions for the Fredholm property of Wiener-Hopf plus/minus Hankel operators with semi-almost periodic Fourier matrix symbols are exhibited. Under such conditions, a formula for the sum of the Fredholm indices of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators is derived. Concrete examples are worked out in view of the computation of the Fredholm indices. Mathematics Subject Classification (2000). Primary 47A53; Secondary 47B35, 47A68, 42A75. Keywords. Fredholm index formula, Wiener-Hopf plus/minus Hankel operator, semi-almost periodic function, Fredholm property.

1. Introduction In recent years a great attention has been devoted to Wiener-Hopf plus/minus Hankel operators with different classes of Fourier symbols (cf., e.g., [2]–[5], [7]–[10], [13]–[18], [20], [23]). Most of this interest comes directly from the MathematicalPhysics applications where those operators arise (cf. [5], [6], [16], [24]). In this direction the classical work of Power [19] is also relevant since it includes the study of the spectra and essential spectra of Hankel operators by investigating the C ∗ -algebra generated by the Toeplitz and Hankel operators (in the two cases of piecewise continuous symbols and almost periodic symbols). In the present work we will consider matrix Wiener-Hopf plus/minus Hankel operators WΦ ± HΦ : [L2+ (R)]N → [L2 (R+ )]N , This research was supported by Funda¸c˜ ao para a Ciˆencia e a Tecnologia (Portugal) through Unidade de Investiga¸c˜ ao Matem´ atica e Aplica¸co ˜es of University of Aveiro.

144

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where WΦ and HΦ denote Wiener-Hopf and Hankel operators defined by WΦ = r+ F −1 Φ · F : [L2+ (R)]N → [L2 (R+ )]N , HΦ = r+ F

−1

Φ · FJ :

[L2+ (R)]N

→ [L (R+ )] 2

N

(1.1) ,

(1.2)

respectively. In general, for a Banach algebra/space B (and N ∈ N), we will denote by B N ×N the Banach algebra of all N × N matrices with entries in B, and B N will denote the Banach space of all N -dimensional vectors with entries in B. The notations L2 (R) and L2 (R+ ) are used for the Banach spaces of complex2 valued Lebesgue measurable functions ψ for which |ψ| is integrable on R and R+ , 2 respectively. Additionally, in (1.1)–(1.2), L+ (R) denotes the subspace of L2 (R) formed by all functions supported in the closure of R+ = (0, +∞), the operator r+ performs the restriction from [L2 (R)]N onto [L2 (R+ )]N , F denotes the Fourier $ transformation, J is the reflection operator given by the rule Jφ(x) = φ(x) = φ(−x), x ∈ R, and Φ ∈ [L∞ (R)]N ×N is the so-called Fourier matrix symbol. Within this context, the main goal of the present work is to present a formula for the Fredholm index of the matrix operator which has the following diagonal form   0 WΦ + HΦ : [L2+ (R)]2N → [L2 (R+ )]2N (1.3) DΦ := 0 WΦ − HΦ in the case where the entries of the matrix Φ are semi-almost periodic functions (and under certain conditions in which we will obtain a Fredholm property characterization of DΦ ).

2. Auxiliary material In view of the definition of semi-almost periodic functions, and also to present some of the properties of such elements, we will start by considering the almost periodic functions. Let us take the smallest closed subalgebra of L∞ (R) that contains all the functions eλ with λ ∈ R (eλ (x) := eiλx , x ∈ R). This is denoted by AP and called the algebra of almost periodic functions: AP := algL∞ (R) {eλ : λ ∈ R} . The following subclasses of AP are also of interest: AP+ := algL∞ (R) {eλ : λ ≥ 0}, AP− := algL∞ (R) {eλ : λ ≤ 0} . The almost periodic functions have a great amount of important and well-known properties. Among them, for our purposes the following ones are the most relevant. Proposition 2.1. (cf. [1, Proposition 2.22]) Let A ⊂ (0, ∞) be an unbounded set and let {Iα }α∈A = {(xα , yα )}α∈A

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 145 be a family of intervals Iα ⊂ R such that |Iα | = yα − xα → ∞ as α → ∞. If ϕ ∈ AP, then the limit  1 M (ϕ) := lim ϕ(x)dx α→∞ |Iα | I α exists, is finite, and is independent of the particular choice of the family {Iα }. Definition 2.2. Let ϕ ∈ AP. The number M (ϕ) given by Proposition 2.1 is called the Bohr mean value or simply the mean value of ϕ. In the matrix case the mean value is defined entry-wise. ˙ (with R ˙ = R ∪ {∞}) represent the (bounded and) continuous Let C(R) functions ϕ on the real line for which the two limits ϕ(−∞) := lim ϕ(x), x→−∞

ϕ(+∞) := lim ϕ(x) x→+∞

exist and coincide. The common value of these two limits will be denoted by ϕ(∞). ˙ will stand for the functions ϕ ∈ C(R) ˙ for which ϕ(∞) = 0. Furthermore, C0 (R) ¯ := C(R) ∩ P C(R), ˙ where C(R) is the usual set of continuous funcLet C(R) ˙ is the set of all bounded piecewise continuous tions on the real line and P C(R) ˙ functions on R. As mentioned in the last section, in our operators we will deal with Fourier symbols from the C ∗ -algebra of semi-almost periodic elements which is defined as follows. Definition 2.3. The C ∗ -algebra SAP of all semi-almost periodic functions on R is ¯ the smallest closed subalgebra of L∞ (R) that contains AP and C(R): ¯ . SAP = algL∞ (R) {AP, C(R)} In [21] D. Sarason proved the following theorem which reveals in a different way the structure of the SAP algebra. ¯ be any function for which u(−∞) = 0 and u(+∞) = 1. Theorem 2.4. Let u ∈ C(R) ˙ such that If ϕ ∈ SAP, then there exist ϕ , ϕr ∈ AP and ϕ0 ∈ C0 (R) ϕ = (1 − u)ϕ + uϕr + ϕ0 . The functions ϕ , ϕr are uniquely determined by ϕ, and independent of the particular choice of u. The maps ϕ → ϕ , ϕ → ϕr are C ∗ -algebra homomorphisms of SAP onto AP. Remark 2.5. The last theorem is also valid in the matrix case. For a Banach algebra B, we are going to denote by GB the group of all invertible elements in B. Since our results will be obtained through certain factorizations of the involved matrix functions, we will therefore recall the definitions of the so-called right and left AP factorization.

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Definition 2.6. A matrix function Φ ∈ GAP N ×N is said to admit a right AP factorization if it can be represented in the form Φ(x) = Φ− (x) D(x) Φ+ (x)

(2.1)

for all x ∈ R, with Φ− ∈ GAP−N ×N , Φ+ ∈ GAP+N ×N , and D is a diagonal matrix of the form D(x) = diag(eiλ1 x , . . . , eiλN x ),

λj ∈ R .

The numbers λj are called the right AP indices of the factorization. A right AP factorization with D = IN ×N is referred to be a canonical right AP factorization. In another way, it is said that a matrix function Φ ∈ GAP N ×N admits a left AP factorization if instead of (2.1) we have Φ(x) = Φ+ (x) D(x) Φ− (x) for all x ∈ R, and Φ± and D having the same properties as above. Remark 2.7. It is readily seen from the above definitions that if an invertible $ admits almost periodic matrix function Φ admits a right AP factorization, then Φ −1 a left AP factorization, and also Φ admits a left AP factorization. The vector containing the right AP indices will be denoted by k(Φ), i.e., in the above case k(Φ) := (λ1 , . . . , λN ). If we consider the case with equal right AP indices (k(Φ) = (λ1 , λ1 , . . . , λ1 )), then the matrix d(Φ) := M (Φ− )M (Φ+ ) is independent of the particular choice of the right AP factorization (cf., e.g., [1, Proposition 8.4]). In this case the matrix d(Φ) is called the geometric mean of Φ. In order to relate operators and to transfer certain operator properties between the related operators, we will also need the known notion of equivalence after extension for bounded linear operators. Definition 2.8. We will say that the linear bounded operator S : X1 → X2 (acting between Banach spaces) is equivalent after extension with T : Y1 → Y2 (also acting between Banach spaces) if there exist Banach spaces Z1 , Z2 and boundedly invertible linear operators E and F, such that the following identity holds     T 0 S 0 =E F ; 0 IZ1 0 IZ2 here IZi represents the identity operator in the Banach space Zi , i = 1, 2. Remark 2.9. It is clear that if T is equivalent after extension with S, then T and S have the same Fredholm regularity properties (i.e., the properties that directly depend on the kernel and on the image of the operator). Lemma 2.10. Let Φ ∈ G[L∞ (R)] WΦΦ4 −1 .

N ×N

. Then DΦ is equivalent after extension with

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 147 This lemma has its roots in the Gohberg-Krupnik-Litvinchuk identity [10, 14], from which with additional equivalence after extension operator relations it is possible to find invertible and bounded linear operators E and F such that C B WΦΦ4 0 −1 (2.2) DΦ = E 0 I[L2 (R)]N F . +

A technique about how to construct such equivalence after extension relation is described in [4].

3. The Fredholm property We start by recalling a known Fredholm characterization for Wiener-Hopf operators with SAP matrix Fourier symbols having lateral almost periodic representatives admitting right AP factorizations. Theorem 3.1. (cf. [1, Theorem 10.11] and [11]) Let Φ ∈ SAP N ×N and assume that the almost periodic representatives Φ and Φr admit a right AP factorization. Then the Wiener-Hopf operator WΦ is Fredholm if and only if: (i) Φ ∈ GSAP N ×N ; (ii) The almost periodic representatives Φ and Φr admit canonical right AP factorizations (therefore with k(Φ ) = k(Φr ) = (0, . . . , 0)) ; (iii) sp(d−1 (Φr )d(Φ )) ∩ (−∞, 0] = ∅ , where sp(d−1 (Φr )d(Φ )) stands for the set of the eigenvalues of the matrix d−1 (Φr )d(Φ ) := [d(Φr )]−1 d(Φ ) . The matrix version of Sarason’s Theorem (cf. Theorem 2.4) says that if Φ ∈ GSAP N ×N then this matrix admits the following representation Φ = (1 − u)Φ + uΦr + Φ0 ,

(3.1)

¯ , u(−∞) = 0, u(+∞) = 1, Φ0 ∈ C0 (R). ˙ From where Φ,r ∈ GAP N ×N , u ∈ C(R) (3.1) it follows that 4 −1 = [(1 − u ) + u˜Φ )r + Φ )0 ]−1 , Φ ˜)Φ and 4 −1 = [(1 − u)Φ + uΦ + Φ ][(1 − u ) + u )r + Φ )0 ]−1 . ˜)Φ ˜Φ ΦΦ  r 0

(3.2)

Therefore, from (3.2), we obtain that 4 −1 4 −1 ) = Φ Φ (ΦΦ   r ,

4 −1 4 −1 ) = Φ Φ (ΦΦ r r  .

(3.3)

These representations, and the above relation between the operator DΦ and the pure Wiener-Hopf operator, lead to the following characterization in the case when 4 −1 Φ Φ r admits a right AP factorization.

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4 −1 Theorem 3.2. Let Φ ∈ SAP N ×N , and assume that Φ Φ admits a right AP r factorization. In this case, the operator DΦ is Fredholm if and only if the following three conditions are satisfied: (j) Φ ∈ GSAP N ×N ; 4 −1 admits a canonical right AP factorization; (jj) Φ Φ 

r

4 −1 (jjj) sp[d(Φ Φ r )] ∩ i R = ∅ (where, as before, Φ and Φr are the local representatives at ∓∞ of Φ, cf. (3.1)).

Proof. If DΦ is a Fredholm operator, then by standard arguments (e.g., due to the classical Simonenko technique [22]) it follows that Φ ∈ G[L∞ (R)]N ×N , and therefore condition (j) arises; cf., e.g., [9, §2.4] for a detailed exposition in the case of Toeplitz plus/minus Hankel operators with L∞ symbols. Moreover, the Fredholm property of DΦ also implies that the operator WΦΦ4 −1 is Fredholm (due to the equivalence after extension relation, cf. (2.2)). Employing 4 4 4 −1 ∈ GSAP N ×N , (ΦΦ −1 ) and (ΦΦ −1 ) now Theorem 3.1 we will obtain that ΦΦ  r admit a canonical right AP factorizations and 4 4 −1 ) )d((ΦΦ −1 ) )] ∩ (−∞, 0] = ∅ . sp[d−1 ((ΦΦ r 

(3.4)

4 −1 By virtue of (3.3) we conclude that Φ Φ r admits a canonical right AP factorization. Once again, due to (3.3), from (3.4) we derive that 4 4 −1 −1 sp[d−1 (Φr Φ  )d(Φ Φr )] ∩ (−∞, 0] = ∅ .

(3.5)

4 −1 can be normalized In addition, a canonical right AP factorization of Φ Φ r into 4 −1 Φ Φ r = Ψ− ΛΨ+ ,

(3.6)

where Ψ± have the same factorization properties as the original lateral factors 4 −1 ). of the canonical factorization but with M (Ψ ) = I, and where Λ := d(Φ Φ ±



r

Thus, (3.6) allows  −1 4 4 4 −1 −1 −1 −1 4 = Φ Φ =Ψ Ψ−1 Φr Φ r + Λ − ,  which in particular shows that 4 −1 −1 d(Φr Φ .  )=Λ Thus, (3.5) turns out to be equivalent to sp[Λ2 ] ∩ (−∞, 0] = ∅ which directly from the eigenvalue definition leads to sp[Λ] ∩ iR = ∅ . Therefore the proposition (jjj) is satisfied, and the proof of the “if part” is completed.

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 149 Now we will be concerned with the “only if part”. From the hypothesis that 4 −1 , and therefore this is also invertible in Φ ∈ GSAP N ×N we can consider ΦΦ 4 −1 are given by the formula SAP N ×N . The left and right representatives of ΦΦ 4 −1 4 −1 (3.3). Since Φ Φr = (ΦΦ ) admits a canonical right AP factorization, then  4 −1 ) = Φ ) Φ−1 (ΦΦ  r admits a canonical left AP factorization and  4 −1 4 −1 ) ]−1 = Φ Φ [(ΦΦ  r  admits a canonical right AP factorization. Moreover, using the same reasoning as in the “if part”, these last two canonical right AP factorizations and condition (jjj) imply that 4 4 −1 −1 4 4 −1 ) )d((ΦΦ −1 ) )]∩(−∞, 0] = sp[d−1 (Φ Φ sp[d−1 ((ΦΦ r  r  )d(Φ Φr )]∩(−∞, 0] = ∅. All these facts together with Theorem 3.1 give us that WΦΦ4 −1 is a Fredholm operator. Using now the equivalence after extension relation presented in (2.2), we
obtain that DΦ is a Fredholm operator. Remark 3.3. In regard to the last theorem notice the following: (i) if the symbol Φ belongs to the Wiener sub-algebra of SAP N ×N , i.e., SAP W N ×N (cf., e.g., [1] for a complete definition of this algebra), then we can drop the initial assumption 4 −1 which states that Φ Φ admits a right AP factorization (since this holds always 

r

in such a case); (ii) if the lateral representatives of Φ are N × N constant matrices (having therefore an even more particular situation than in (i)), then the symbol Φ ¯ N ×N , and the theorem provides an alternative Fredholm property belongs to [C(R)] characterization of such operators to the already known characterizations for this particular case (and which were obtained by different methods; cf., e.g., [8] and [15]).

4. Index formula for the sum of Wiener-Hopf plus/minus Hankel operators In the present section we will be concentrated in obtaining a Fredholm index formula for DΦ , i.e., for the sum of Wiener-Hopf plus/minus Hankel operators 4 −1 admits a right W ± H with Fourier symbols Φ ∈ GSAP N ×N such that Φ Φ Φ

Φ



r

AP factorization. Due to this purpose, let us assume from now on that WΦ + HΦ and WΦ − HΦ are Fredholm operators. Let GSAP0,0 denotes the set of all functions ϕ ∈ GSAP for which k(ϕ ) = k(ϕr ) = 0. To define the Cauchy index of ϕ ∈ GSAP0,0 we need the next lemma. Lemma 4.1. [1, Lemma 3.12] Let A ⊂ (0, ∞) be an unbounded set and let {Iα }α∈A = {(xα , yα )}α∈A

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be a family of intervals such that xα ≥ 0 and |Iα | = yα − xα → ∞, as α → ∞. If ϕ ∈ GSAP0,0 and arg ϕ is any continuous argument of ϕ, then the limit  1 1 lim ((arg ϕ)(x) − (arg ϕ)(−x))dx (4.1) 2π α→∞ |Iα | Iα exists, is finite, and is independent of the particular choices of {(xα , yα )}α∈A and arg ϕ. The limit (4.1) is denoted by ind ϕ and is usually called the Cauchy index of ϕ. Moreover, following [18, section 4.3] we can generalize the notion of Cauchy index for SAP functions with k(φ ) + k(φr ) = 0 in the way which was introduced in Lemma 4.1, i.e.,  1 1 lim indϕ = ((arg ϕ)(x) − (arg ϕ)(−x))dx , (4.2) 2π α→∞ |Iα | Iα where ϕ ∈ {φ ∈ SAP : k(φ ) + k(φr ) = 0}. The following theorem is well known, and provides a formula for the Fredholm index of matrix Wiener-Hopf operators with SAP Fourier symbols. Theorem 4.2. [1, Theorem 10.12] Let Φ ∈ SAP N ×N . If the almost periodic representatives Φ , Φr admit right AP factorizations, and if WΦ is a Fredholm operator, then   N 
1 1 1 − − arg ξk IndWΦ = −ind det Φ − (4.3) 2 2 2π k=1

where ξ1 , . . . , ξN ∈ C \ (−∞, 0] are the eigenvalues of the matrix d−1 (Φr )d(Φ ) and {·} stands for the fractional part of a real number. Additionally, when choosing arg ξk in (−π, π), we have IndWΦ = −ind det Φ −

N 1
arg ξk . 2π k=1

We will now be concentrated on a index formula for DΦ (i.e., on a formula for the sum of the Fredholm indices of WΦ + HΦ and WΦ − HΦ ). In fact, it directly follows from the definition of the operator DΦ (cf. formula (1.3)) that IndDΦ = Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] . Now, employing the above equivalence after extension relation (cf. (2.2)), we deduce that IndDΦ = IndWΦΦ4 −1 . Consequently, we have: Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = IndWΦΦ4 −1 .

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 151 Using (4.3) for WΦΦ4 −1 (which is a Fredholm operator because of the assumption made in the beginning of the present section), we obtain that   N 
1 1 1 4 −1 − − arg ηk , (4.4) )] − IndWΦΦ4 −1 = −ind[det(ΦΦ 2 2 2π k=1

4 4 −1 ) )d((ΦΦ −1 ) ) where ηk ∈ C\(−∞, 0] are the eigenvalues of the matrix d−1 ((ΦΦ r  4 4 −1 −1 −1 = d (Φr Φ )d(Φ Φr ), cf. (3.3). However, as pointed out in the proof of TheoD E2 4 4 4 −1 −1 −1 )d(Φ ) = d(Φ ) , and therefore (4.4) can be rewritrem 3.2, d−1 (Φr Φ Φ Φ  r  r  ten as   N 
1 1 1 4 −1 − − arg ζk (4.5) IndWΦΦ4 )] − −1 = −ind[det(ΦΦ 2 2 π k=1

4 −1 where ζk ∈ C \ iR are the eigenvalues of the matrix d(Φ Φ r ). In addition, formula (4.5) is reduced to N 1
4 −1 )] − IndWΦΦ4 = −ind[det(Φ Φ β(ζk ) , −1 π k=1

where

% β(ζk ) :=

arg(ζk ) arg(−ζk )

if %e ζk > 0 if %e ζk < 0

with the argument in both cases being chosen in (−π/2, π/2). 4 −1 )]. Recalling that the matrix Let us now simplify the form of ind[det(ΦΦ 4 −1 Φ Φ has a canonical right AP factorization (due to the assumption made in the 

r

4 −1 beginning of the section), it holds k(Φ Φ r ) = (0, . . . , 0). From here we obtain that 4 −1 (4.6) k(det(Φ Φ r )) = 0. Consequently, this yields: k((det Φ) ) + k((det Φ)r ) = = =

k(det(Φ )) + k(det(Φr )) −1

r ) ) k(det(Φ )) + k(det(Φ 4 −1 )) k(det(Φ )) + k(det(Φ r



=

4 −1 k(det(Φ ) det(Φ r )) 4 −1 )) k(det(Φ Φ

=

0,

=



r

(4.7)

−1 ). Additionally, we have used because for any f ∈ GAP we have k(f ) = k(f4 here the fact that [det Φ] = det Φ (which follows from a direct computation). A

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4 −1 , since from (4.6) we have: similar argument also applies to Φ 4 4 −1 ) ) + k((det Φ −1 ) ) = k((det Φ  r

4 4 −1 −1 k(det(Φ  )) + k(det(Φr )) −1

=

4 −1 ) + k(det(Φ r )) 4 k(det(Φ )) + k(det(Φ−1 r )) 4 −1 k(det(Φ Φ ))

=

0.

= =

 ) k(det(Φ



r

(4.8)

Therefore, due to (4.7), (4.8) and (4.2), we can perform the following computations: 4 4 −1 )] = ind[det Φ det Φ −1 ] ind[det(ΦΦ 4 −1 = ind det Φ + ind det Φ  = ind det Φ + ind[det Φ−1 ] = ind det Φ − ind[det Φ−1 ] = ind det Φ − ind[det Φ]−1 = ind det Φ + ind det Φ = 2 ind det Φ . Consequently, we have just deduced the following result. 4 −1 admits a right AP Corollary 4.3. Let Φ ∈ GSAP N ×N and assume that Φ Φ r factorization. If WΦ ± HΦ are Fredholm operators, then   N 
1 1 1 − − arg ζk , Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = −2 ind det Φ − 2 2 π k=1 (4.9) 4 −1 where ζk ∈ C \ iR are the eigenvalues of the matrix d(Φ Φr ). Moreover, denoting % if %e ζk > 0 arg(ζk ) β(ζk ) = arg(−ζk ) if %e ζk < 0 with the argument in both cases being chosen in (−π/2, π/2), the formula (4.9) simplifies to the following one: Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = −2 ind det Φ −

N 1
β(ζk ) . π k=1

(4.10)

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 153

5. Examples In the present section we provide two concrete examples for the above theory. 5.1. Example 1

1.0

0.5

−20

0

20

Figure 1. The graph of det Φ. Let us assume that Φ(x)



e−i(1+α)x −iαx − 1 + eix e

= (1 − u(x))  ei(1+α)x +u(x) iαx − 1 + e−ix e

0 ei(1+α)x  0

e−i(1+α)x



,

√ where α = ( 5 − 1)/2, and u is the following real-valued function 1 1 (5.1) u(x) = + arctan(x) . 2 π Being clear that Φ ∈ SAP 2×2 , we will start to show that Φ ∈ GSAP 2×2 . To this end we need first of all to compute the determinant of Φ :  (1 − u(x))e−i(1+α)x + u(x)ei(1+α)x det Φ(x) = det (1 − u(x))(e−iαx − 1 + eix ) + u(x)(eiαx − 1 + e−ix )  0 (1 − u(x))ei(1+α)x + u(x)e−i(1+α)x = 1 − 2u(x)(1 − u(x))(1 − cos 2(1 + α)x) . Recalling that u is a real-valued function given by (5.1), we have that   1 2u(x)(1 − u(x)) ∈ 0, , 2

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and where the maximum value 1/2 is attained for x = 0, in view of 1 2 − arctan2 (x) . 2 π2 Due to the reason that 1 − cos 2(1 + α)x ∈ [0, 2] but 1 − cos 2(1 + α)mπ = 2 (for m ∈ Z), we have 2u(x)(1−u(x))(1−cos 2(1+α)x) ∈ [0, 1). Therefore det Φ ∈ (0, 1] , and thus Φ is invertible. In this case, despite of the invertibility of Φ, the Wiener-Hopf operator with symbol Φ is not a Fredholm operator. This happens because the matrix function   e−i(1+α)x 0 ∈ GAP 2×2 e−iαx − 1 + eix ei(1+α)x 2u(x)(1 − u(x)) =

does not have a right AP factorization (cf. [12, pages 284–285] for all the details about this matrix function). However, the Wiener-Hopf plus/minus Hankel operators with the same symbol Φ will have the Fredholm property. Indeed, besides having Φ ∈ GSAP 2×2 , let us observe that:   1 0 4 −1 = I2×2 . Φ Φr = 0 1 4 −1 Consequently, Φ Φ r obviously admits a canonical right AP factorization and 4 −1 d(Φ Φ r ) = I2×2 . Thus the eigenvalues of this matrix are equal to 1 ∈ iR. This allows us to conclude that the operator DΦ is a Fredholm operator (cf. Theorem 3.2). This means that the operators WΦ ± HΦ have the Fredholm property. Let us now calculate the sum of their Fredholm indices. From the above form of det Φ, we have already concluded that det Φ ∈ (0, 1] . Thus, we have that det Φ is a real-valued positive function, and so its argument is zero (the graph of det Φ is given in Figure 1). Altogether we have: Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = 0 , 4 −1 since the eigenvalues of d(Φ Φ r ) are also real (recall that they are equal to 1), and therefore their arguments are also zero. 5.2. Example 2 Let us now take the following matrix-valued function:  ix   −ix   e 0 0 e Ψ(x) = (1 − u(x)) + u(x) + 0 e−ix 0 eix

 −1 , 0 (5.2) where u is the same as in the previous example. A direct computation provides that Ψ is invertible. In fact, 2  x−i 2 2 det Ψ(x) = (1 − u(x)) + u (x) + 2u(x)(1 − u(x)) cos 2x − − 1 , (5.3) x+i 0 x−i x+i − 1

x−i x+i

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 155

2

@@ I

1

−3

−2

−1

0

1 0

−1

@ R @

−2

Figure 2. The (oriented) graph of det Ψ. and hence det Ψ(x) = f (x) + ig(x) , where f (x) = 1 − 2u(x)(1 − u(x))(1 − cos 2x) +

4(x2 − 1) , (x2 + 1)2

and 8x . (x2 + 1)2 From these formulas it follows that f and g do not vanish simultaneously, and ˙ Hence Ψ ∈ GSAP 2×2 . Although having consequently det Ψ(x) = 0 for all x ∈ R. this necessary condition to the Fredholm property of the Wiener-Hopf operator with the symbol Ψ, it is easily seen that WΨ is not Fredholm. The reason for this is based on the fact that although the matrix-valued functions   ix 0 e Ψ (x) = 0 e−ix g(x) = −

and

 Ψr (x) =

e−ix 0

0 eix



have obvious right AP factorizations (with the identity matrix in the role of the lateral minus and plus factors; cf. (2.1)) they do not have a canonical right AP factorization (since k(Ψ ) = (1, −1) and k(Ψr ) = (−1, 1)).

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10−6 8

4

0 0.992

0.994

0.996

0.998

1.0

−4

−8

Figure 3. The oscillation at infinity of det Ψ. Despite this situation, the Wiener-Hopf plus/minus Hankel operators with the same symbol Ψ are Fredholm. Indeed, first of all note that we have already deduced that Ψ is invertible in SAP 2×2 . Moreover, 4 −1 Ψ Ψ r = I2×2

(5.4)

4 −1 and therefore Ψ Ψ r has the trivial canonical right AP factorization. From (5.4) we also obtain that 4 −1 )] = {1} ∩ iR = ∅ . sp[d(Ψ Ψ 

r

These are sufficient conditions for these Wiener-Hopf plus/minus Hankel operators to have the Fredholm property (cf. Theorem 3.2). To calculate the sum of the Fredholm indices of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators we need once again to use the above computed determinant of Ψ. From (5.3) it follows that ind det Ψ = 1 (the graph of the det Ψ is given in the Figure 2, and it is a closed curve; note also that limx→±∞ det Ψ(x) = 1, and in the Figure 3 is shown the oscillation of the function det Ψ at infinity). Therefore, from formula (4.10), we obtain that Ind[WΨ + HΨ ] + Ind[WΨ − HΨ ] = −2 for Ψ in (5.2). Acknowledgment The authors are thankful to the referee for very constructive comments and suggestions.

Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators 157

References [1] A. B¨ ottcher, Yu.I. Karlovich and I.M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Oper. Theory Adv. Appl. 131, Birkh¨ auser Verlag, Basel, 2002. [2] G. Bogveradze and L.P. Castro, Wiener-Hopf plus Hankel operators on the real line with unitary and sectorial symbols, Contemp. Math. 414 (2006), 77–85. [3] G. Bogveradze and L.P. Castro, Invertibility of matrix Wiener-Hopf plus Hankel operators with APW Fourier symbols, Int. J. Math. Math. Sci. 38152 (2006), 1–12. [4] L.P. Castro and F.-O. Speck, Regularity properties and generalized inverses of deltarelated operators, Z. Anal. Anwend. 17 (1998), 577–598. [5] L.P. Castro, F.-O. Speck and F.S. Teixeira, Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip, J. Integral Equations Appl. 15 (2003), 359– 383. [6] L.P. Castro, F.-O. Speck and F.S. Teixeira, On a class of wedge diffraction problems posted by Erhard Meister, Oper. Theory Adv. Appl. 147 (2004), 211–238. [7] L.P. Castro, F.-O. Speck and F.S. Teixeira, A direct approach to convolution type operators with symmetry, Math. Nachr. 269-270 (2004), 73–85. [8] T. Ehrhardt, Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip, J. Funct. Anal. 208 (2004), 64–106. [9] T. Ehrhardt, Factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip, Habilitation Thesis, Technische Universtit¨ at Chemnitz, Chemnitz, 2004. [10] N. Karapetiants and S. Samko, Equations with Involutive Operators, Birkh¨ auser, Boston, 2001. [11] Yu. Karlovich, On the Haseman problem, Demonstratio Math. 26 (1993), 581–595. [12] Yu.I. Karlovich and I.M. Spitkovsky, Factorization of almost periodic matrix functions and the Noether theory of certain classes of equations of convolution type (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 276–308; translation in Math. USSR-Izv. 34 (1990), no. 2, 281–316. [13] V.G. Kravchenko, A.B. Lebre and G.S. Litvinchuk, Spectrum problems for singular integral operators with Carleman shift, Math. Nachr. 226 (2001), 129–151. [14] V.G. Kravchenko and G.S. Litvinchuk, Introduction to the Theory of Singular Integral Operators with Shift, Kluwer Academic Publishers Group, Dordrecht, 1994. [15] A.B. Lebre, E. Meister and F.S. Teixeira, Some results on the invertibility of WienerHopf-Hankel Operators, Z. Anal. Anwend. 11 (1992), 57–76. [16] E. Meister, F.-O. Speck and F.S. Teixeira, Wiener-Hopf-Hankel operators for some wedge diffraction problems with mixed boundary conditions, J. Integral Equations Appl. 4 (1992), 229–255. [17] A.P. Nolasco and L.P. Castro, A Duduchava-Saginashvili’s type theory for WienerHopf plus Hankel operators, J. Math. Anal. Appl. 331 (2007), 329–341. [18] A.P. Nolasco, Regularity Properties of Wiener-Hopf-Hankel Operators, PhD Thesis, University of Aveiro, 2007. [19] S.C. Power, C ∗ -algebras generated by Hankel operators and Toeplitz operators, J. Funct. Anal. 31 (1979), 52–68.

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[20] S. Roch and B. Silbermann, Algebras of Convolution Operators and their Image in the Calkin Algebra, Report MATH (90-05), Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut f¨ ur Mathematik, Berlin, 1990. [21] D. Sarason, Toeplitz operators with semi-almost periodic symbols, Duke Math. J. 44 (1977), 357–364. [22] I.B. Simonenko, Some general questions in the theory of the Riemann boundary problem, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), no. 5, 1138–1146 (in Russian); English translation in Math. USSR-Izv. 2 (1968), no. 5, 1091–1099. [23] F.S. Teixeira, On a class of Hankel operators: Fredholm properties and invertibility, Integral Equations Operator Theory 12 (1989), 592–613. [24] F.S. Teixeira, Diffraction by a rectangular wedge: Wiener-Hopf-Hankel formulation, Integral Equations Operator Theory 14 (1991), 436–454. Giorgi Bogveradze and Lu´ıs P. Castro Research Unit “Matem´ atica e Aplica¸co ˜es” Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 159–172 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Diffraction by a Strip and by a Half-plane with Variable Face Impedances Lu´ıs P. Castro and David Kapanadze Dedicated to W.L. Wendland on the occasion of his 70th birthday Abstract. A study is presented for boundary value problems arising from the wave diffraction theory and involving variable impedance conditions. Two different geometrical situations are considered: the diffraction by a strip and by a half-plane. In the first case, both situations of real and complex wave numbers are analyzed, and in the second case only the complex wave number case is considered. At the end, conditions are founded for the well-posedness of the problems in Bessel potential space settings. These conditions depend on the wave numbers and the impedance properties. Mathematics Subject Classification (2000). Primary 35J05; Secondary 35J25, 35C15, 47H50. Keywords. Wave diffraction, boundary value problem, Helmholtz equation, impedance problem, well-posedness, pseudo-differential equations.

1. Introduction A large variety of diffraction problems for time-harmonic or electromagnetic waves can be formulated as the Helmholtz equation  2  ∂2 ∂2 ∂ 2 u = 0 + + + k ∂x2 ∂y 2 ∂z 2 in R3 \Θ, supplemented by an appropriate boundary condition on the boundary of the obstacle Θ, and Sommerfeld’s radiation condition (see, e.g., [1]–[9], [12]–[16], [19]–[21]). Here k is the wave number (being proportional to the frequency of the corresponding incident wave), and being either a real or a complex number (due This work was supported in part by Unidade de Investiga¸ ca ˜o Matem´ atica e Aplica¸co ˜es of University of Aveiro, and the Portuguese Science Foundation (FCT–Funda¸ca ˜o para a Ciˆ encia e a Tecnologia) through the grant number SFRH/BPD/20524/2004.

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to a non-lossy or lossy medium, respectively). The electromagnetic theory gives to a large spectrum of materials a quasi-homogeneous refracted wave, which propagates perpendicularly to the boundary of Θ regardless of the incident angle. The z-dependence can be therefore cancelled due to the perpendicular wave propagation. This allows us to pass from a R3 to a R2 formulation, with a corresponding transformation on the obstacle Θ. When considering Robin boundary conditions in acoustics, these correspond to the impedance surface properties of the obstacle (which naturally have a direct consequence on the sound propagation). In the real world the impedance typically varies along the structure of the obstacles. As such, it turns to be more significant to study boundary conditions having an impedance contribution in the form of a given function (which will therefore vary along the structure). This will be the case in the present work, which in this way turns out to be more general than previous corresponding studies where only constant parameters were considered in the role of impedance numbers (cf., e.g., [3, 4, 13, 21]). Within the Bessel potential spaces framework, we will in the current paper be concerned with the wave diffraction problem determined by the two-dimensional Helmholtz equation and variable impedance boundary conditions. Such a problem will be analyzed for two different geometrical situations: the strip and the halfline obstacles. In the first case the analysis will be realized for both situations of a complex and a pure real wave number, and in the second case only for the non-real wave number situation. The paper is organized as follows: in the next section the problems are formulated in a rigorous way by means of the appropriated Bessel potential space settings. Therefore, all the notations are implemented in Section 2, and the concrete boundary and radiation conditions are introduced. In Section 3 the corresponding homogeneous problems are considered, and for these the uniqueness of the solution is obtained under certain conditions. In Section 4 some potential operators are introduced. These operators will be of fundamental importance in the next section to translate our problems into systems of integral equations. Finally, also in Section 5, the Fredholm theory of such systems of integral equations is studied, and corresponding consequences for the invertibility of the evolved operators are also derived. This last fact guarantees the existence, uniqueness and continuous dependence of solutions (on the data) for the original problems (in a scale of space smoothness orders).

2. Formulation of the problems In this section we establish the general notation which will allow the mathematical formulation of the problems. We start by introducing the spaces where the problems will be considered. As usual, S(Rn ) denotes the Schwartz space of all rapidly vanishing functions and S  (Rn ) the dual space of tempered distributions on Rn . The Bessel potential

Diffraction by a Strip and by a Half-plane

161

space H s (Rn ), with s ∈ R, is formed by the elements ϕ ∈ S  (Rn ) such that 8 8 s/2 8 −1  8 2 8 1 + |ξ| · Fϕ8 < ∞. ϕH s (Rn ) = 8F 8 L2 (Rn )

As the notation indicates,  · H s (Rn ) is a norm for the space H s (Rn ) which makes it a Banach space. Here, F = Fx→ξ denotes the Fourier transformation in Rn . $ s (D) the closed subspace For a given domain, D, on Rn we denote by H s n of H (R ) whose elements have supports in D, and H s (D) denotes the space of generalized functions on D which have extensions into Rn that belong to H s (Rn ). $ s (D) is endowed with the subspace topology, and on H s (D) we put The space H $ s (Rn \D). Note that the spaces H s (R+ ) the norm of the quotient space H s (Rn )/H $ s (R+ ) can be identified for − 1 < s < 1 ; for the case s = 0 we will denote and H 2 2 them by L2 (R+ ). Let us denote by Σ either a finite interval ]0, a[ (with 0 < a < ∞) or the positive half-line R+ , and define Ω := R2 \{(x, y) ∈ R2 : x ∈ Σ, y = 0} . 1 Recall that Hloc (Ω) denotes the Sobolev space of vector-functions u in Ω such that   12 2 2 (u, u)K = (|u(x)| + |∇u(x)| )dx < ∞, K

with ∇ = (∂1 , ∂2 ), and for any compact set K ⊂ Ω. We will often use the usual restriction operator rΣ : H s (R) → H s (Σ) . We have now introduced sufficient notations to present a differential formulation of the impedance wave diffraction problem in a Bessel potential spaces framework. We are interested in studying the problem of existence and uniqueness of an u ∈ L2 (R2 ), with 1+ u|Ω ∈ Hloc (Ω)

such that



0≤<

for

 ∂2 ∂2 2 u = 0 + + k ∂x2 ∂y 2 ⎧ + = h+ ⎨ u1 − ip+ u+ 0 ⎩

− − u− 1 + ip u0

=

1 , 2

(2.1)

in

Ω,

(2.2)

on

Σ,

(2.3)

h−

where the wave number k ∈ C\{0} is given, as well as p± ∈ L∞ (Σ) and the Dirich± let and Neumann traces are denoted by u± 0 = u|y=±0 and u1 = (∂u/∂y)|y=±0 , ± −1/2 respectively. Finally, the elements h ∈ H (Σ) are arbitrarily given since the

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dependence on the data is to be studied for well-posedness. Note that the multipli−1/2 cations p± u± (Σ) 0 are well defined as the elements of the spaces L2 (Σ) → H and conditions in (2.3) are understood in the distributional sense (cf. [11]). In particular, conditions (2.2)–(2.3) lead to ⎧ + = 0 ⎨ u0 − u− 0 (2.4) on R\Σ . ⎩ + u1 − u− = 0 1 Thus, when computing − + + − − h+ − h− = (u+ 1 − u1 ) − (ip u0 + ip u0 ) ,

we realize that we necessarily need to have $ −1/2 (Σ) . h+ − h− ∈ rΣ H

(2.5)

This occurs because from (2.4) and (2.3) it follows − $ −1/2 (Σ) , u+ 1 − u1 ∈ H

and also due to − − ip+ u+ 0 + ip u0 ∈ L2 (Σ)

$ −1/2 (Σ) is con(which does not change the space in (2.5) because L2 (R+ ) → H tinuously embedded). Therefore, (2.5) is a necessary condition to the above problem to be solvable, and we may recognize it as a compatibility condition between the data, cf. [14]. Note that when the wave number is real (i.e., k ∈ R\{0}) and Σ = ]0, a[ is a finite interval (0 < a < ∞) it is natural to require that the eventual solution of (2.2)–(2.3) should also satisfy the Sommerfeld radiation condition at infinity, u ∈ Som(Ω):   3 ∂ u(x) − i|k|u(x) = O |x|− 2 for |x| → ∞ (2.6) ∂|x| see, e.g., [9]. Let us also note that the case when Σ = R+ and the wave number k is real will not be considered in here. In this way, below we will refer to: (i) Problem PR+ as the one characterized by (2.2), (2.3), and (2.5) in the case of k ∈ C\R and Σ = R+ ; (ii) Problem P]0,a[ as the one characterized either by (2.2), (2.3), (2.6), and (2.5) in the case of k ∈ R\{0} and Σ = ]0, a[ being a finite interval, or by (2.2), (2.3), and (2.5) when k ∈ C\R and Σ = ]0, a[.

3. Uniqueness of solution In this section we will consider the uniqueness of solution to the homogeneous Problem PR+ and homogeneous Problem P]0,a[ . We will start with the half-line case.

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Theorem 3.1. Let Σ = R+ . If &m k = 0,  = 0, and one of the following situations holds almost everywhere in z ∈ R+ : (a)

(%e k)(&m k) > 0 ,

%e p± (z) ≥ 0

(b)

(%e k)(&m k) < 0 ,

%e p± (z) ≤ 0

(c)

|&m k| ≥ |%e k| ,

(d)

%e k = 0 ,

%e p± (z) ≤ &m p± (z)

(e)

%e k = 0 ,

%e p± (z) ≥ −&m p± (z) ,

&m p± (z) ≥ 0

then the homogeneous Problem PR+ has only the trivial solution u = 0 in the space H 1 (Ω). Proof. Let R be a sufficiently large positive number and B(R) be the disk centered at the origin with radius R. Set ΩR := Ω ∩ B(R). Note that the domain ΩR has piecewise smooth boundary SR := (∂B(R) ∩ Ω) ∪ (R+ ∩ B(R)) ∪ (R+ ∩ B(R)) , and denote by n(x) = (n1 (x), n2 (x)) the outward unit normal vector at the point x ∈ SR . Let u be a solution of the homogeneous Problem PR+ . Then the Green formula (see, e.g., [18]) for u and its complex conjugate u¯ in the domain ΩR yields  D  E 2 2 2 |∇u| − k |u| dx = ∂n u u ¯ dSR . (3.1) ΩR



From (3.1) we obtain D E |∇u|2 − k 2 |u|2 dx

SR





= R+ ∩B(R)

ΩR

u+ ¯+ 1u 0 dx



−

R+ ∩B(R)

u− ¯− 1u 0 dx

∂n u u dS + ∂B(R)   + + 2 2 p |u0 | dx + i p− |u− i 0 | dx R+ ∩B(R) R+ ∩B(R)  ∂n u u dS . (3.2) +

=

∂B(R)

F Note that, since &m k = 0, the integral ∂B(R) ∂n u u dS tends to 0 as R → ∞. Indeed, in (R, φ) polar coordinates, we have   2π−δ2  2π ∂n u u dS = R ∂n u u dφ = R lim ∂n u u dφ ∂B(R)

0

δ1 ,δ2 →0+

δ1

and we take in account that the solution u ∈ H (Ω) of the Helmholtz equation exponentially vanishes at infinity in the sector φ ∈ (δ1 , 2π − δ2 ). Therefore, passing 1

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to the limit as R → ∞ in (3.2), it follows  D   E 2 |∇u|2 − k 2 |u|2 dx = i p+ |u+ | dx + i 0 R+

Ω

R+

2 p− |u− 0 | dx.

From the real and imaginary parts of the last identity, we obtain  D  E  |∇u|2 + (&m k)2 − (%e k)2 |u|2 dx = Ω   + + 2 2 − &m p |u0 | dx − &m p− |u− 0 | dx ; R+

 −2(%e k)(&m k)



|u| dx = 2

Ω

R+

(3.3)

R+



%e p

+

2 |u+ 0 | dx

+ R+

2 %e p− |u− 0 | dx .

(3.4)

As a consequence, under the conditions of (a) or (b), formula (3.4) implies that u = 0 on Ω; if (c) holds, then from (3.3) it also follows that u = 0 on Ω; for the case of %e k = 0, formulas (3.3) and (3.4) lead to the identity  D E 2 |∇u|2 + (&m k) |u|2 dx = Ω   + 2 + + 2 − (&m p ± %e p )|u0 | dx − (&m p− ± %e p− )|u− 0 | dx R+

R+

from which we conclude that under the conditions (d) or (e) also holds u = 0 on Ω.
A similar result is also valid for Σ = ]0, a[, and &m k = 0. The corresponding proof is analogous to the previous one (with obvious changes; cf. also [2]), and for this reason it will not be presented in here. The following result contains such a case in the statement (ii). Theorem 3.2. Let Σ = ]0, a[ (with 0 < a < ∞). (i) If &m k = 0, %e p± (z) ≥ 0 (almost everywhere in z ∈ Σ) and  = 0, then the homogeneous Problem P]0,a[ has only the trivial solution u = 0 in the space 1 (Ω) ∩ Som(Ω). Hloc (ii) If &m k = 0,  = 0, and one of the following situations holds almost everywhere in z ∈ Σ: (a)

(%e k)(&m k) > 0 ,

%e p± (z) ≥ 0

(b)

(%e k)(&m k) < 0 ,

%e p± (z) ≤ 0

(c)

|&m k| ≥ |%e k| ,

(d)

%e k = 0 ,

%e p± (z) ≤ &m p± (z)

(e)

%e k = 0 ,

%e p± (z) ≥ −&m p± (z) ,

&m p± (z) ≥ 0

then the homogeneous Problem P]0,a[ has only the trivial solution u = 0 in the space H 1 (Ω).

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Proof of (i). Let &m k = 0, %e p± ≥ 0 (and  = 0). We assume that Σ = ]0, a[ is a part of some smooth and simple curve S which separates the space R2 into two disjoint domains Ω+ and Ω− = R2 \Ω+ , such that Ω+ is a bounded domain and S = ∂Ω± . In this case, let us denote by n(x) = (n1 (x), n2 (x)) the outward unit normal vector at the point x ∈ S = ∂Ω+ . Additionally, let R be a sufficiently large positive number and B(R) be the disk centered at the origin with radius R, such − that Ω+ ⊂ B(R). Set Ω− R := Ω ∩B(R), and let u be a solution of the homogeneous Problem P]0,a[ . Then, Green’s formula for u and its complex conjugate u ¯ (in the ) yields domains Ω+ and Ω− R  D E G H + |∇u|2 − k 2 |u|2 dx = [∂n u]+ , (3.5) S , [u]S S + Ω D  E G H − |∇u|2 − k 2 |u|2 dx = − [∂n u]− + ∂n u u ¯ dS . (3.6) S , [u]S Ω− R

S

∂B(R)

±

Here, the symbols [·] denote the non-tangential limit values on S from Ω± and 1 ·, ·S , ·, ·Σ denote the duality brackets between the dual spaces H − 2 (S) and 1 1 1 1 1 $ − 2 (Σ) and H 2 (Σ), or H − 2 (Σ) and H $ 2 (Σ). For regular functions H 2 (S), or H (e.g., f, g ∈ L2 (M)), we have  @ A f, g M = f g dM , M

for M = S or M = Σ. Note that the interior regularity in Ω of solutions of the Helmholtz equation = [u]− and [∂n u]+ = [∂n u]− . Then, by summing up (2.2) gives us [u]+ S\Σ S\Σ S\Σ S\Σ (3.5) and (3.6), we obtain   D E @ + +A @ − −A 2 2 2 |∇u| − k |u| dx = u1 , u0 Σ − u1 , u0 Σ + ∂n u u dS Ω+ ∪Ω− R

∂B(R)

A @ − − −A @ + = i p+ u + 0 , u0 Σ + i p u0 , u0 Σ

 +

∂n u u dS . ∂B(R)

(3.7) Recall that we are assuming R to be sufficiently large, and so we can apply the Sommerfeld radiation condition on the circle ∂B(R). Let us now separate the imaginary part of the equation (3.7), and use the fact that u ∈ Som(Ω) implies 1 u(x) = O(|x|− 2 ) as |x| → ∞. Then we obtain    2 − − 2 %e p+ |u+ | dΣ + %e p |u | dΣ + |k| |u|2 dS = O(R−1 ) , 0 0 Σ

Σ

∂B(R)

which yields

 |u|2 dS = 0 ,

lim

R→∞

∂B(R)

due to the conditions %e p± ≥ 0. Therefore, from the Rellich–Vekua Theorem [22], it follows that u = 0 in Ω.


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4. Potential operators Let us denote the standard fundamental solution of the Helmholtz equation (in two dimensions) by K(x)

i (1) − H0 (k|x|) , 4

:=

(1)

where H0 (k|x|) is the Hankel function of the first kind of order zero (cf. [9, §3.4]) and introduce the corresponding single and double layer potentials  K(x − y)ψ(y)dy , x ∈ /R, V (ψ)(x) = R

 W (ϕ)(x)

= R

[∂n(y) K(x − y)]ϕ(y)dy ,

x∈ /R,

where ψ and ϕ are density functions. Theorem 4.1 (Cf. [4], Theorem 4.1). The single and double layer potentials V and W defined in the following framework of Bessel potential spaces $ s (R+ ) → H s+1+ 12 (Ω) V :H

and

$ s (R+ ) → H s+ 12 (Ω) W :H

(4.1)

are continuous operators for all s ∈ R. Note that by the standard arguments of Green identities we obtain the following integral representation of a solution u ∈ H 1 (Ω) of the homogeneous Helmholtz equation (cf. [10] and [22]) − + − u(x) = W (u+ 0 − u0 )(x) − V (u1 − u1 )(x) ,

x∈Ω.

(4.2)

In fact, e.g., for the case of Σ = R+ , this may be obtained by passing to the limit (as R → ∞) in the following representation formula:  5 6 u(x) = [∂n(y) K(x − y)][u(y)]+ − K(x − y)[∂n(y) u(y)]+ dS, x ∈ ΩR , SR

where SR and ΩR are the same as in the proof of Theorem 3.1. Let us now recall some properties of the above introduced potentials. The following jump relations are well known (cf. [4, Section 4]): − [V (ψ)]+ R = [V (ψ)]R =: H(ψ), ±

− [∂n W (ϕ)]+ R = [∂n W (ϕ)]R =: L(ϕ) ,

[W (ϕ)]R =: [± 12 I](ϕ), where

1 [∂n V (ψ)]± R =: [∓ 2 I](ψ) ,

(4.3)

 H(ψ)(z)

:=

L(ϕ)(z)

:=

K(z − y)ψ(y)dy , z ∈ R ,  lim ∂n(x) [∂n(y) K(y − x)]ϕ(y)dy ,

(4.4)

R

x→z∈R

and I denotes the identity operator.

R

z ∈ R,

(4.5)

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167

Theorem 4.2 ([10]). The operators (4.4)–(4.5) can be extended or restricted to the following bounded mappings: rΣ H :

$ s (Σ) → H s+1 (Σ) , H

(4.6)

rΣ L :

$ s+1 (Σ) → H s (Σ) , H

(4.7)

for arbitrary s ∈ R. These rΣ H and rΣ L are pseudo-differential operators of order −1 and 1, respectively. Moreover, both operators are invertible provided −1 < s < 0.

5. Existence and regularity of solutions We look for solutions of problems PR+ and P]0,a[ in the form u(x) = W (ϕ)(x) − V (ψ)(x) ,

x∈Ω,

(5.1)

where the unknown densities ϕ and ψ are related to the source u and its normal derivative by the following equations (cf. (4.2)): − ϕ = u+ 0 − u0 ,

− ψ = u+ 1 − u1 .

(5.2)

We start with the simplest case:  = 0. The boundary conditions (2.3) (together with (2.5)) can be equivalently rewritten in the form % + + + − − = f0 u1 − u− 1 − ip u0 − ip u0 , (5.3) + + u+ = f1 1 − ip u0  $ − 12 (Σ)) and f1 := h+ ∈ H − 12 (Σ). where f0 := h+ − h− ∈ rΣ H The representation formula (5.1) together with the jump relations (4.3) and the boundary conditions (5.3) lead to the following system of pseudo-differential equations on Σ with unknown ϕ and ψ: % 5 6 = f0 rΣ ψI − ip+ [( 12 I)ϕ − Hψ] − ip− [(− 12 I)ϕ − Hψ] . (5.4) 5 6 1 + 1 rΣ Lϕ − (− 2 I)ψ − ip [( 2 I)ϕ − Hψ] = f1 Then, from (5.4), we have AΦ = F

on Σ ,

(5.5)

$ (Σ) × H $ (Σ), F := (f0 , f1 ) ∈ (rΣ H $ (Σ)) × H with Φ := (ψ, ϕ) ∈ H and   rΣ I + i(p+ + p− )rΣ H −i 12 p+ rΣ I + 12 ip− rΣ I . A := 1 + +1 r I + ip r H r L − ip r I Σ Σ Σ Σ 2 2 − 12

1 2

− 12

− 12

(Σ),

Lemma 5.1. Let Σ = R+ . If p± ∈ L∞ (R+ ) are such that |p± (z)| ≤ const z−δ (for some δ > 0), then the operator   $ − 12 (R+ ) × H $ 12 (R+ ) → rR+ H $ − 12 (R+ ) × H − 12 (R+ ) T :H

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L.P. Castro and D. Kapanadze

where

 T = (Tjl )j,l=1,2 :=

i(p+ + p− )rR+ H ip+ rR+ H

i − 2 (p

− p+ )rR+ I



− 2i p+ rR+ I

is compact. Proof. Let us show that the operator T11 is compact in the corresponding spaces; then, similar arguments are valid for the remaining entries of T and the proof will be complete. Looking to the structure of T11 , first we have the continuous mapping   $ − 12 (R+ ) → rR+ H $ 12 (R+ ) → L2 (R+ ). rR+ H : H Then, the multiplication by p± gives us a compact operator as a composition of a continuous operator in weighted (at infinity) L2 spaces and a compact embedding (cf. [17, Theorem 1.4.19]):  comp  $ − 12 (R+ ) . p± I : L2 (R+ ) → x−δ L2 (R+ ) → rR+ H Therefore, T11 is a compact operator.




Theorem 5.2. Let  = 0, Σ = R+ , &m k = 0, p± ∈ L∞ (R+ ) such that |p± (z)| ≤ constz−δ , for some δ > 0, and let one of the conditions (a)–(e) of Theorem 3.1 be satisfied, then Problem PR+ has a unique solution in the space H 1 (Ω), which is representable in the form (5.1) with the densities ϕ and ψ defined by the uniquely solvable pseudo-differential equation (5.5). Proof. Let us analyze the invertibility of the matrix operator   $ 12 (R+ ) → rR+ H $ − 12 (R+ ) × H − 12 (R+ ) . $ − 12 (R+ ) × H A:H

(5.6)

Due to Lemma 5.1, the operator A is a compact perturbation of the triangular matrix operator   rR+ I 0 B := 1 rR+ L 2 rR+ I with the invertible operators rR+ I and rR+ L in the main diagonal (provided −1 < s < 0). Thus, (5.6) is a Fredholm operator with zero Fredholm index. Then, Theorem 3.1 implies that the corresponding kernel is trivial, and therefore the invertibility of the operator A in (5.6) is obtained.
The following result is obtained when applying a corresponding analogous procedure to the case where Σ is the finite interval ]0, a[ (having however a significant difference in the solution spaces when k ∈ R, due to the necessity of the Sommerfeld radiation condition at infinity in this case). Theorem 5.3. Let  = 0, and Σ = ]0, a[ (with 0 < a < ∞). (i) If &m k = 0 and %e p± (z) ≥ 0 (a.e. z ∈ Σ), then Problem P]0,a[ has a 1 (Ω) ∩ Som(Ω), which is representable in the form unique solution in the space Hloc

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169

(5.1) with the densities ϕ and ψ defined by the uniquely solvable pseudo-differential equation (5.5). (ii) If &m k = 0 and one of the conditions (a)–(e) of Theorem 3.2 is satisfied, then Problem P]0,a[ has a unique solution in the space H 1 (Ω), which is representable in the form (5.1) with the densities ϕ and ψ defined by the uniquely solvable pseudodifferential equation (5.5). We are now in conditions to analyze the regularity of the solutions u of our wave diffraction problems. Theorem 5.4. Let Σ = R+ , &m k = 0, p± ∈ L∞ (R+ ) with |p± (z)| ≤ constz−δ , for some δ > 0, and let in addition one of the conditions (a)–(e) of Theorem 3.1 be satisfied. If the boundary data satisfy the conditions 1 1 $ − 12 + (R+ ) , h+ − h− ∈ rR+ H (h+ , h− ) ∈ H − 2 + (R+ ) × H − 2 + (R+ ) , for 0 ≤  < 12 , then the unique solution u of Problem PR+ possesses the following regularity u ∈ H 1+ (Ω) . Proof. The solvability result (and also the case  = 0) follows from Theorem 5.2. Let 0 <  < 12 . Then, due to Theorem 4.1 and the representation formula $ − 12 + (R+ ) × H $ 12 + (R+ ). (5.1), it is sufficient to show that (ψ, ϕ) ∈ H $ 12 + (R+ ) → (rR+ H $ − 12 + (R+ )× H $ − 12 + (R+ ))× H − 12 + (R+ ) The operator A : H can be written as A = B + T, where the operators B and T are given above. Due to Theorem 4.2, we have that   $ 12 + (R+ ) → rR+ H $ − 12 + (R+ ) × H − 12 + (R+ ) $ − 12 + (R+ ) × H B:H is an invertible operator for every  ∈ ]0, 12 [. The inverse of B is provided by the formula   0 0 −1 , B = − 21 (rR+ L)−1 (rR+ L)−1 $ − 2 + (R+ ) denotes the zero extension opera$ − 2 + (R+ )) → H where 0 : (rR+ H tor. In addition, the operator T is continuous when acting between the spaces $ − 12 + (R+ ) × H $ 12 + (R+ ) and L2 (R+ ) × L2 (R+ ). This follows from the continuH ity property of the evolved operators (cf. Theorem 4.2), and from the embedding properties of the Bessel potential spaces. Then, from AΦ = F , we have Φ = B −1 F − B −1 T Φ, i.e., 1

1

ψ = f0 − 0 (T11 ψ + T12 ϕ) ,

(5.7)

and −1

ϕ = (rR+ L)

1 1 1 −1 −1 (− f0 + f1 ) − (rR+ L) (T21 − T11 )ψ − (rR+ L) (T22 − T12 )ϕ . 2 2 2 (5.8)

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The equality (5.7) together with $ − 12 (R+ ) × H $ 12 (R+ ) , (ψ, ϕ) ∈ H and the continuity properties of T11 and T12 , imply that T11 ψ, T12 ϕ ∈ L2 (R+ ), $ − 12 + (R+ ). Using now this knowledge in the equality (5.8), and therefore ψ ∈ H −1 and noting that (rR+ L) is a pseudo-differential operator of order −1, we finally $ 12 + (R+ ). obtain that ϕ ∈ H
Using the same technique as above, the same improvement of space smoothness orders is obtained for the case of Σ = ]0, a[: Theorem 5.5. Let Σ = ]0, a[ and either: (i) &m k = 0 and %e p± (z) ≥ 0 (a.e. z ∈ Σ), or (ii) &m k = 0 and one of the conditions (a)–(e) of Theorem 3.2 be satisfied. If the boundary data satisfy the conditions $ − 2 + (Σ) h+ − h − ∈ H

(h+ , h− ) ∈ H − 2 + (Σ) × H − 2 + (Σ), 1

1

1

for 0 ≤  < 12 , then the solution u of Problem P]0,a[ possesses the following regularity 1+ u ∈ Hloc (Ω) ∩ Som(Ω) ,

u ∈ H 1+ (Ω) ,

in the case (i) ,

in the case (ii) .

To end up, let us return to the case Σ = R+ and prove a more global result. Theorem 5.6. Let Σ = R+ , &m k = 0, and p± ∈ L∞ (R+ ) be such that −δ , |p± (z) − p± ∞ | ≤ const z

for some δ > 0 and constants p± ∞ , and let in addition one of the conditions (a)–(e) of Theorem 3.1 be satisfied. If the boundary data satisfy the conditions $ − 12 + (R+ ) h+ − h− ∈ rR+ H

(h+ , h− ) ∈ H − 2 + (R+ ) × H − 2 + (R+ ), 1

1

for 0 ≤  < 12 , then the solution u of Problem PR+ possesses the following regularity u ∈ H 1+ (Ω) Proof. It is sufficient to prove an analogous result as Theorem 5.2. In the present case, the operator A can be rewritten as A = A$ + T$ where

 A$ :=

− rR+ I + i(p+ ∞ + p∞ )rR+ H 1 2 rR+ I

+ ip+ ∞ rR+ H

1 − −i 21 p+ ∞ rR+ I + 2 ip∞ rR+ I 1 rR+ L − ip+ ∞ 2 rR+ I



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171

and T$ := (T$ij )i,j=1,2 , with entries T$11

− := i(p+ + p− − p+ ∞ − p∞ )rR+ H ,

T$12

1 1 − − := −i (p+ − p+ ∞ )rR+ I + i(p − p∞ )rR+ I , 2 2

T$21

:= i(p+ − p+ ∞ )rR+ H ,

1 := i(p+ − p+ ∞ ) rR+ I . 2 Analogously as in Lemma 5.1, we have that the matrix operator   $ − 12 (R+ ) × H $ 12 (R+ ) → rR+ H $ − 12 (R+ ) × H − 12 (R+ ) T$ := (T$ij )i,j=1,2 : H T$22

is compact. Then, arguing as in Theorem 5.2 and applying [4, Theorem 5.3], we obtain that the operator A in (5.6) is invertible.


References [1] L.P. Castro, Solution of a Sommerfeld diffraction problem with a real wave number. In: C. Constanda, M. Ahues, A. Largillier (Eds.), Integral Methods in Science and Engineering, Birkh¨ auser, Boston, MA, 2004, pp. 25–30. [2] L.P. Castro, D. Kapanadze, Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case, Math. Nachr., 12 pp., to appear. [3] L.P. Castro, D. Kapanadze, The impedance boundary-value problem in a strip for Helmholtz equation, Preprint CM I–09 (2006), Department of Mathematics, University of Aveiro, 15pp. [4] L.P. Castro, D. Kapanadze, On wave diffraction by a half-plane with different face impedances, Math. Methods Appl. Sci., 18pp., to appear. [5] L.P. Castro, D. Natroshvili, The potential method for the reactance wave diffraction problem in a scale of spaces, Georgian Math. J. 13 (2006), 251–260. [6] L.P. Castro, F.-O. Speck, Relations between convolution type operators on intervals and on the half-line, Integral Equations Operator Theory 37 (2000), 169–207. [7] L.P. Castro, F.-O. Speck, F.S. Teixeira, On a class of wedge diffraction problems posted by Erhard Meister, Oper. Theory Adv. Appl. 147 (2004), 213–240. [8] S.N. Chandler-Wilde, A.T. Peplow, A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane, ZAMM Z. Angew. Math. Mech. 85 (2005), 79–88. [9] D. Colton, R. Kress, Inverse Acoustic and Electronic Scattering Theory, SpringerVerlag, Berlin, 1998. [10] R. Duduchava, D. Natroshvili, E. Shargorodsky, Boundary value problems of the mathematical theory of cracks, Trudy Instituta Prikladnoj Matematiki Imeni I. N. Vekua 39 1990, 68–84. [11] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

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[12] E. Meister, Some multiple-part Wiener-Hopf problems in mathematical physics. In: Mathematical Models and Methods in Mechanics, Banach Center Publications 15, PWN-Polish Scientific Publishers, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1985, pp. 359–407. [13] E. Meister, F.-O. Speck, Modern Wiener-Hopf methods in diffraction theory, Pitman Res. Notes Math. Ser. 216 (1989), 130–171. [14] A. Moura Santos, F.-O. Speck, F.S. Teixeira, Compatibility conditions in some diffraction problems, Pitman Res. Notes Math. Ser. 361 (1996), 25–38. [15] B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (2nd ed.), Chelsea Publishing Company, New York, 1988. [16] A.F. dos Santos, F.S. Teixeira, The Sommerfeld problem revisited: Solution spaces and the edge conditions, J. Math. Anal. Appl. 143 (1989), 341–357. [17] B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester, 1998. [18] S.L. Sobolev, Partial Differential Equations of Mathematical Physics (English translation), Dover Publications, New-York, 1989. [19] F.-O. Speck, Mixed boundary value problems of the type of Sommerfeld’s half-plane problem, Proc. R. Soc. Edinb., Sect. A, 104 (1986), 261–277. [20] F.-O. Speck, Sommerfeld diffraction problems with first and second kind boundary conditions, SIAM J. Math. Anal. 20 (1989), 396–407. [21] F.-O. Speck, R.A. Hurd, E. Meister, Sommerfeld diffraction problems with third kind boundary conditions, SIAM J. Math. Anal. 20 (1989), 589–607 [22] I. Vekua, On metaharmonic functions, Trudy Tbilisskogo Matematicheskogo Instituta 12 (1943), 105–174 (in Russian; an English translation is in I. Vekua, New Methods for Solving Elliptic Equations, North-Holland Publishing Company, Amsterdam, 1967). Lu´ıs P. Castro and David Kapanadze Research Unit “Matem´ atica e Aplica¸co ˜es” Department of Matematics University of Aveiro 3810-193 Aveiro Portugal e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 173–185 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Factorization Algorithm for Some Special Matrix Functions Ana C. Concei¸c˜ao and Viktor G. Kravchenko Abstract. We will see that it is possible to construct an algorithm that allows us to determine an effective factorization of some matrix functions. For those matrix functions it is shown that its explicit factorization can be obtained through the solutions of two non-homogeneous equations. Mathematics Subject Classification (2000). Primary 47A68; Secondary 47A10. Keywords. Explicit factorization, algorithm, singular integral operator, nonhomogeneous equations.

1. Introduction The explicit factorization of matrix-valued functions has applications in different areas, such as the theory of singular integral operators, boundary value problems and the theory of non-linear differential equations (see, for instance, [2, 7, 10, 11]). It is well known that there exist algorithms to determine explicit factorizations for rational matrix functions (see, for instance, [1] and [12]). However, results about explicit factorization of non-rational matrix functions exist only for some restricted classes of matrix functions (see, for instance, [3, 4, 5, 6, 8]). Let T denote the unit circle and consider the space L2 (T). As usual, −,0 −,0 − L+ 2 (T) = imP+ , L2 (T) = imP− , L2 (T) = L2 (T) ⊕ C,

where P± = (I ± S)/2 denote the Cauchy projection operators associated with the singular integral operator S,  1 ϕ(τ ) Sϕ(t) = dτ, t ∈ T, πi T τ − t and I represents the identity operator. This research was partially supported by the Centre for Mathematics and its Applications at Instituto Superior T´ecnico (Portugal).

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We say that a matrix-valued function A, such that A±1 ∈ [L∞ (T)]n×n , admits a left (right) generalized factorization in L2 (T) if it can be represented as A = A+ ΛA− (A− ΛA+ ), where + ±1 − κ1 κn A±1 + ∈ [L2 (T)]n×n , A− ∈ [L2 (T)]n×n , Λ = diag{t , . . . , t }, −1 κ1 ≥ · · · ≥ κn integers, and A−1 + P+ A− I represents a bounded linear operator in [L2 (T)]n . If κ1 = . . . = κn = 0, then A is said to admit a left (right) canonical generalized factorization.

In the following sections we shall be dealing with the class of matrix functions   1 b (1.1) Aγ (b) = b |b|2 + γ where b is an essentially bounded function and γ is a non-zero complex constant. Some results related with Aγ (b) can be seen in [3, 4, 5, 9, 10, 11]. The books [10] and [11] contain relations between a Hermitian second-order matrix function G, with negative determinant and definite diagonal elements, and A−1 (b): ' It is proved in [10] that the matrix functions G and A−1 (b) admit a generalized factorization in L2 (T) only simultaneously and their partial indices coincide. ' It is also proved that the matrix function A−1 (b) admits a right generalized factorization in L2 (T) if and only if the unity does not belong to the limit spectrum (i.e., the set of limit points of the spectrum and eigenvalues of infinite multiplicity) of the self-adjoint operator N− = Hb Hb∗ (Hb = P− bP+ is a Hankel operator with the symbol b) and its partial indices are ±l, where l is the multiplicity of 1 as an eigenvalue of N− . In [4] we consider the class of matrix-valued functions (1.1). For these matrixvalued functions, when −γ belongs to the resolvent set of the self-adjoint positive operator N+ : L2 (T) → L2 (T), N+ = Hb∗ Hb , we obtain that it is possible to compute a canonical factorization (see Theorem 4.4 in [4]) when the function b is in a certain decomposing algebra of continuous functions and satisfies some additional conditions. The method used therein was based on the construction of the resolvent of the operator N+ . In [5] we generalize our previous result, simplifying some of the conditions imposed before and obtaining a canonical factorization of Aγ (b) (when −γ ∈ ρ(N+ )) through the solutions of the two related non-homogeneous equations, (N+ + γI)u+ = 1

(1.2)

(N+ + γI)v+ = b.

(1.3)

and

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175

In [3] we generalize our previous results for the case when b can be represented through an inner-outer factorization (when the outer function is rational). We construct an algorithm for solving the equations (1.2) and (1.3) (when the solutions exist). Consequently, an explicit canonical generalized factorization of Aγ (b) (when it exists) can be obtained through the solutions u+ and v+ . Now we are able to generalize our previous results for the case when we have a non-canonical factorization, that is, when −γ does not belong to the limit spectrum of N+ . With the purpose of obtaining a factorization of the matrix function Aγ (b), let us assume, with no loss of generality, that b has analytic continuation into the interior of the unit circle. Let us assume that Aγ (b) admits a generalized factorization − Aγ (b) = A+ (1.4) γ ΛAγ . Then its partial indices are κ and −κ, where κ is the multiplicity of −γ as an eigenvalue of N+ .

2. Canonical case For κ = 0 (that is, for −γ ∈ ρ(N+ )), it can be easily seen that: Proposition 2.1. If κ = 0, then the equation Φ+ = Aγ (b) Φ− + Aγ , Φ− (∞) = 02×2 has the only solution − −1 Φ+ = A+ − E. γ , Φ− = (Aγ )

Besides that, when κ = 0, we have the explicit generalized factorization of Aγ (b) through the solutions of two non-homogeneous equations (see Theorem 2.1 in [3]): Theorem 2.2. The matrix function Aγ (b) admits a canonical generalized factorization if and only if −γ ∈ ρ(N+ ). And, in that case, − Aγ (b) = A+ γ Aγ ,

A+ γ = Φ+  Φ+ = γ  Φ− =

−1 A− , γ = (E + Φ− )  u+ v+ , P+ (bu+ ) 1 + P+ (bv+ )

and

P− (bP− (bu+ )) P− (bP− (bv+ )) −P− (bu+ )

−P− (bv+ )

 ,

where (N+ + γI)u+ = 1

and

(N+ + γI)v+ = b.

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3. Non-canonical case Let us consider now the case when −γ ∈ σ(N+ ). Let us represent A− γ (∞) as   a− (∞) b− (∞) − Aγ (∞) = . c− (∞) d− (∞) We obtain that Proposition 3.1. If κ > 0 (that is, if −γ ∈ σ(N+ )), then the equation Φ+ = Aγ (b) Φ− + Aγ , Φ− (∞) = 02×2

(3.1)

is not solvable. Proof. In the proof we used the fact that c− (∞) = 0 ∨ d− (∞) = 0 A− γ (∞)

(for to be invertible) and the Corollary 3.1 of [11], that gives necessary conditions to the solvability of equations of the form (3.1).
Let the matrix function A admit a left generalized factorization. If A admits a canonical generalized factorization, then there exists a generalized factorization A = A+ A− such that A− (∞) = E. If A admits a non-canonical generalized factorization, then a generalized factorization A = A+ ΛA− , such that A− (∞) = E, does not always exist. We can have three different cases: (Case 1)



A− (∞) = (Case 2)



A− (∞) = (Case 3) A− (∞) =



a− (∞) 0

a− (∞) b− (∞) 0 c− (∞)

a− (∞) c− (∞)



b− (∞) d− (∞)

b− (∞) d− (∞)

, a− (∞) = 0, d− (∞) = 0, b− (∞)arbitrary,  , b− (∞) = 0, c− (∞) = 0, a− (∞) arbitrary,

 , c− (∞) = 0,

d− (∞) = 0, a− (∞) and b− (∞)

such that are not simultaneously equal to zero. For each one of these cases, we have different associated problems. Using the Theorem 3.2 of [11], that gives necessary and sufficient conditions for the solvability of a problem of the form ϕ+ (t) = G(t)ϕ− (t) + g(t),

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through the factors of a factorization of the matrix function G, we obtain the following results. Proposition 3.2. (Case 1) Let Aγ (b) admit a left generalized factorization (1.4). Then ∃ rκ (t) = tκ + sκ−1 (t) such that the problem    1 0 Φ+ = Aγ (b) Φ− + , (3.2) 0 rκ Φ− (∞) = 02×2 , is solvable. Proof. In the proof we also used the fact that since d− (∞) = 0, then ∃1 rκ (t) = tκ + sκ−1 (t) : P+ (d− (t)rκ (t)) = d− (∞)tκ .




In a similar way we have the next proposition. Proposition 3.3. (Case 2) Let Aγ (b) admit a left generalized factorization (1.4). Then ∃ rκ (t) = tκ + sκ−1 (t) such that the problem    rκ 0 , (3.3) Φ+ = Aγ (b) Φ− + 0 1 Φ− (∞) = 02×2 , is solvable. Proof. In the proof we also used the fact that since c− (∞) = 0, then ∃1 rκ (t) = tκ + sκ−1 (t) : P+ (c− (t)rκ (t)) = c− (∞)tκ .




For the last case we have an analogously result. Proposition 3.4. (Case 3) Let Aγ (b) admit a left generalized factorization (1.4). Then ∃ ri,κ (t) = tκ + si,κ−1 (t), i = 1, 2 such that the problem    0 r1,κ Φ+ = Aγ (b) Φ− + , (3.4) 0 r2,κ Φ− (∞) = 02×2 , is solvable. Proof. In the proof we also used the fact that since c− (∞) = 0, then ∃1 r1,κ (t) = tκ + s1,κ−1 (t) : P+ (c− (t)r1,κ (t)) = c− (∞)tκ . Since d− (∞) = 0, then ∃1 r2,κ (t) = tκ + s2,κ−1 (t) : P+ (d− (t)r2,κ (t)) = d− (∞)tκ . We can summarize the above results in the following proposition. Proposition 3.5. Let Aγ (b) admit a left generalized factorization (1.4). Then ∃ such that the problem

ri,κ (t) = ai,0 tκ + si,κ−1 (t), i = 1, 2

  r1,κ Φ+ = Aγ (b) Φ− + 0

Φ− (∞) = 02×2 , is solvable.

0 r2,κ

 ,




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To obtain an explicit factorization of Aγ (b) we need to solve two non-homogeneous equations. First, for each case, we have to solve the associated problem. Using a similar method to that described in [4], we obtain the next results that allows us to obtain a non-canonical generalized factorization of the matrix function Aγ (b), through the solutions of two non-homogeneous equations. Theorem 3.6. (Case 1) If the problem (3.2) is solvable, then the equations (N+ + γI)x+ = γ and (N+ + γI)y+ = γ b rκ , are solvable. And, in that case,  Φ+ = 1 Φ− = γ



x+



y+

,

P+ (bx+ ) γrκ + P+ (by+ ) P− (bP− (bx+ ))

P− (bP− (by+ ))

−P− (bx+ )

−P− (by+ )

 ,

and Aγ (b) admits the generalized factorization Aγ (b) = F+ ΛF− , 

where F+ =

⎛ F− = ⎝

x+ ∆ P+ (bx+ ) ∆



y+

γrκ + P+ (by+ )  κ  0 t Λ(t) = , 0 t−κ

t−κ (γrκ −P− (by+ )) γ tκ (P− (bx+ )) γ∆

,

t−κ (γ(brκ −y+ )−bP− (by+ )) γ tκ (γx+ +bP− (bx+ )) γ∆

⎞ ⎠,

κ = dim ker (N+ + γI), and ∆=

1 det Φ+ . γ

Proof. Let

  ± φ11 φ± 12 Φ± = φ± φ± 21 22 be a solution of the problem (3.2). So,  −     + 1 b φ11 + 1 φ− φ11 φ+ 12 12 = , φ+ φ+ b |b|2 + γ φ− φ− 21 22 21 22 + rκ that is,

⎧ + − φ11 = φ− ⎪ 11 + 1 + b φ21 ⎪ ⎪ ⎪ ⎨ φ+ = φ− + b φ− + b rκ 12 12 22 + − ⎪ φ+ ⎪ 21 = b φ11 + γ φ21 ⎪ ⎪ ⎩ + − φ22 = b φ+ 12 + γ φ22 + γ rκ

.

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179

Applying the projection operator P+ to the two first equations and the projection operator P− to the last two, we obtain that ⎧ + φ11 = 1 + P+ (b φ− ⎪ 21 ) ⎪ ⎪ ⎪ + − ⎨ φ12 = P+ (b φ22 ) + b rκ . − ⎪ 0 = P− (b φ+ ⎪ 11 ) + γ φ21 ⎪ ⎪ ⎩ − 0 = P− (b φ+ 12 ) + γ φ22 We get that + + − − (N+ + γI)φ+ 11 = P+ (bP− (b φ11 )) + γ φ11 = −γP+ (b φ21 ) + γ(1 + P+ (b φ21 ) = γ

and + + (N+ + γI)φ+ 12 = P+ (bP− (b φ12 )) + γ φ12 − = −γP+ (b φ− 22 ) + γ(P+ (b φ22 ) + b rκ ) = γ b rκ .

Let us consider x+ = φ+ 11

and

y+ = φ+ 12 .

Besides that, 1 φ− 21 = − P− (b x+ ), γ 1 φ− 22 = − P− (b y+ ), γ 1 P− (b P− (b x+ )), γ 1 − − P− (b P− (b y+ )), = φ+ 12 − b φ22 − b rκ = −P− (b φ22 ) = γ + − φ+ 21 = b φ11 + γ φ21 = P+ (b x+ ),

+ − − φ− 11 = φ11 − 1 − b φ21 = −P− (b φ21 ) =

φ− 12

and

+ − φ+ 22 = b φ12 + γ φ22 + γ rκ = P+ (b y+ ) + γ rκ . Otherwise, we can get the solutions of the problem (3.2) represented through the factors of the generalized factorization (1.4). In fact, using Theorem 3.2 of [11], that describes how to obtain the general solution of a problem of the form

ϕ+ (t) = G(t)ϕ− (t) + g(t), through the factors of a factorization of the matrix function G, we obtain that   P+ (tκ b− (t)rκ (t)) + qκ−1 (t) P+ (tκ a− (t)) + pκ−1 (t) + Φ+ (t) = Aγ (t) 0 d− (∞) and Φ− (t) = −

−1 (A− γ (t))



t−κ [P− (tκ a− (t)) + pκ−1 (t)] t−κ [P− (tκ b− (t)rκ (t)) + qκ−1 (t)] c− (t) P− (d− (t)rκ (t))

 ,

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where pκ−1 (t) and qκ−1 (t) are polynomials of degree less than or equal to κ − 1. Since we can always assume that detA+ γ (t) = γ and   β2κ ακ + Φ+ = Aγ , 0 d− (∞) where ακ is a polynomial of degree κ and β2κ is a polynomial of degree less than or equal to 2κ, we can consider that the detΦ+ = γd− (∞)ακ . We have that    1   ακ  1  0 β2κ 0 β2κ ακ + + ∆ ∆ ∆ = Aγ = Aγ Φ+ 0 1 0 1 0 d− (∞) 0 d− (∞)   1 β2κ d− (∞) . = A+ γ 0 d− (∞) $+ of a factorization of Aγ (b), Aγ (b) = A $+ ΛA $− , It is known that any factor A γ γ γ can be represented as   c1 l2κ + $+ A , = A γ γ 0 c2 where l2κ is a polynomial of degree less than or equal to 2κ, c1 , and c2 are non zero constants. So, we obtain a factorization of Aγ (b), Aγ (b) = F+ ΛF− ,

(3.5)

where the factors are defined in that way  1    κ 0 t 0 ∆ F+ = Φ+ , and , Λ(t) = 0 t−κ 0 1

F− = Λ−1 F+−1 Aγ (b).

According to Theorem 3.8 of [11], that is, since two different factorizations of a matrix function G in Lp are either both generalized factorizations or neither of them is a such one, we get that (3.5) represents a generalized factorization of the matrix function (1.1).
We can obtain a similar result for Case 2. Theorem 3.7. (Case 2) If the problem (3.3) is solvable, then the equations (N+ + γI)x+ = γ rκ and (N+ + γI)y+ = γ b, are solvable. And, in that case,  Φ+ =

y+ x+ P+ (bx+ ) γ + P+ (by+ )

 ,

  1 P− (bP− (bx+ )) P− (bP− (by+ )) , Φ− = −P− (by+ ) −P− (bx+ ) γ and Aγ (b) admits the generalized factorization Aγ (b) = F+ ΛF− ,

Factorization Algorithm for Some Special Matrix Functions where

 F+ =

− y∆+



x+





0

tκ 0

, Λ(t) = t−κ − P+ (by∆+ )+γ P+ (bx+ ) ⎛ ⎞ −κ −κ − (bx+ )) − t (P−γ(bx+ )) − t (γx+ +bP γ ⎠, F− = ⎝ tκ (γ−P (by )) tκ (γb−γy −bP (by )) −

+

+

γ∆

−

181

,

+

γ∆

κ = dim ker (N+ + γI), and ∆=

1 det Φ+ . γ

For the last case, the only difference is that, in the construction of a factorization of Aγ (b), the matrix function F+ appears depending on the constant (∞) − dc− that we have to determine imposing that F+ is a matrix function of the − (∞) type (+), for all ∆ (that is, admits an analytic continuation into the interior of (∞) the unit circle). Considering ρ = − dc− , we obtain the following result. − (∞) Theorem 3.8. (Case 3) If the problem (3.4) is solvable, then the equations (N+ + γI)x+ = γr1,κ and (N+ + γI)y+ = γbr2,κ , are solvable. And, in that case,   y+ x+ , Φ+ = P+ (bx+ ) γr2,κ + P+ (by+ )   1 P− (bP− (bx+ )) P− (bP− (by+ )) , Φ− = −P− (by+ ) −P− (bx+ ) γ and Aγ (b) admits the generalized factorization Aγ (b) = F+ ΛF− , where

 F+ =

x+ +ρy+ ∆ P+ (bx+ )+ρP+ (by+ )+γρr2,κ ∆  κ 0

Λ(t) =  F− =

1 γ

t−κ [γr2,κ − P− (by+ )] 1 κ γ∆ t g

t 0 1 γ

t

y+

 ,

P+ (by+ ) + γr2,κ  , −κ

t−κ [−bP− (by+ ) + γbr2,κ − γy+ ] 1 κ γ∆ t {b g − γ(x+ + ρy+ )}

κ = dim ker (N+ + γI), 1 ∆ = det Φ+ , γ and g = P− (bx+ ) + ρP− (by+ ) − γρ r2,κ .

 ,

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4. About factorization algorithm If the matrix function Aγ (b) admits a generalized factorization, then, for the case when b can be represented through an inner-outer factorization (when the outer function is rational), we can use the algorithm (or a similar one) described in [3] to solve the non-homogeneous equations that appear in the last Theorems. Step 1: Find κ. Step 1.1: Find ker(N+ + γI) using an algorithm similar to that described in [3] to solve the homogeneous equation (N+ + γI)ϕ+ = 0. Go to Step 1.2. Step 1.2: If γ is such that κ = 0, that is, ϕ+ (t) ≡ 0, then −γ ∈ ρ(N+ ) and the matrix function Aγ (b) admits a canonical generalized factorization. Go to Step 2. Otherwise, the matrix function Aγ (b) admits a non-canonical generalized factorization. Go to Step 3. Step 2: Obtain a canonical generalized factorization. Step 2.1: Use the algorithm described in [3] to solve the equations that appear in Theorem 2.2. A canonical generalized factorization of Aγ (b) is obtained. Step 3: Find the two non-homogeneous equations that we need to solve to obtain a generalized factorization. Since N+ is a self-adjoint operator and γ is a real constant, because −γ ∈ σ(N+ ), we have that L2 = im(N+ + γI) ⊕ ker(N+ + γI). + Step 3.1: If 1, ϕ+ j  = 0, ∀ ϕj ∈ ker(N+ + γI), then

(N+ + γI)u+ = 1 is solvable. Otherwise, (N+ + γI)u+ = 1 is not solvable. And, in that case, + ∃ r1,κ : r1,κ , ϕ+ j  = 0, ∀ ϕj ∈ ker(N+ + γI),

that is, ∃ r1,κ : (N+ + γI)x+ = γr1,κ is solvable. Go to Step 3.2. + Step 3.2: If b, ϕ+ j  = 0, ∀ ϕj ∈ ker(N+ + γI), then

(N+ + γI)v+ = b is solvable. Otherwise, (N+ + γI)v+ = b is not solvable. And, in that case, + ∃ r2,κ : r2,κ , ϕ+ j  = 0, ∀ ϕj ∈ ker(N+ + γI),

Factorization Algorithm for Some Special Matrix Functions

183

that is, ∃ r2,κ : (N+ + γI)y+ = γbr2,κ is solvable. Go to Step 3.3. Step 3.3: Solve the solvable non-homogeneous equations found in the Step 3.1 and in the Step 3.2, using an algorithm similar to that described in [3]. Go to Step 4. Step 4: Obtain a non-canonical generalized factorization. Step 4.1: Obtain a non-canonical generalized factorization of the matrix function Aγ (b) using (depending on the non-homogeneous equations that we solved in Step 3.3) Theorem 3.12, Theorem 3.13, or Theorem 3.14.

5. Example We will now present an example to clarify the obtained results. Let us consider the case where b(t) = (t − µ)θ(t), with |µ| > 1 and θ(t) an inner function. Let p1,2 be the zeros of γ + |t − µ|2 . Let θ(t) be a function defined in a neighborhood of p1 and in a neighborhood of p2 . Using an algorithm similar to that described in [3] we can solve the homogeneous equation (N+ + γI)ϕ+ = 0. We obtain that, i) if γ = −1 or if θ(p1 ) = θ(p2 ), then ϕ+ (t) ≡ 0, ii) if γ = −1 and θ(p1 ) = θ(p2 ) = 0, then ϕ+ (t) = iii) if γ = −1 and θ(p1 ) = θ(p2 ) = 0, then ϕ+ (t) =

µ t(t − µ)θ(t) A, µ + µt(t − µ) [µθ(p1 ) + µt(t − µ)θ(t)] A, θ(p1 )[µ + µt(t − µ)]

where A is an arbitrary constant. For the case i) we get a canonical generalized factorization of the matrix function Aγ (b). For the others cases we get a non-canonical generalized factorization of the matrix function A−1 (b). − √π

1+t

Let us consider the case iii) (for instance, θ(t) = e 7 1−t ). The function |t − µ|2 − 1 have a zero with multiplicity two or two different zeros. Then, we have to analyze which zeros belong to the interior region of the unit circle, which zeros belong to the exterior region of T, and which belong to the unit circle. So, for this simple example, we can get different systems to solve, depending on the value of

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A.C. Concei¸c˜ao and V.G. Kravchenko

µ (see [3]). To obtain results in a simple form, we will consider a concrete value of µ. Let us consider, for instance, the case when µ = 32 . In that case, we have √ √ that p1 = 14 (3 − i 7) and p2 = 14 (3 + i 7). Following the factorization algorithm described in the last section, we get that Step 1:

' [2θ(p ) + t(−3 + 2t)θ(t)] ( 1 Step 1.1: ker(N+ − I) = span θ(p1 )[2 − 3t + 2t2 ) κ=1 Step 1.2: A−1 (b) admits a non-canonical generalized factorization

Step 3: Step 3.1: (N+ − I)u+ (t) = 1 is not solvable (N+ − I)x+ (t) = −[t − 32 (1 − θ(p1 )θ(0))] is solvable Step 3.2: (N+ − I)v+ (t) = b(t) is not solvable (N+ − I)y+ (t) = −b(t)t is solvable Step 3.3: x+ (t) = y+ (t) =

−2t(−3 + 2t)(θ(p1 ) − θ(t)) − 3θ(p1 )tθ(0)h(t) + 3Bh(t) , 3θ(p1 )(2 − 3t + 2t2 )

θ(p1 )[4C + 6tθ(t) − 13t2 θ(t) + 12t3 θ(t) − 4t4 θ(t)] − 6tCθ(t) + 4t2 Cθ(t) 2θ(p1 )(2 − 3t + 2t2 ) and h(t) = 2θ(p1 ) + t(−3 + 2t)θ(t), where B and C are arbitrary constants.

Step 4: A non-canonical generalized factorization of the matrix function A−1 (b) is obtained using Theorem 3.14 of Section 3.

References [1] H. Bart, I. Gohberg, and M.A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Operator Theory: Advances and Applications 1, Birkh¨ auser Verlag, Basel-Boston, 1979. [2] K. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications 3, Birkh¨ auser Verlag, BaselBoston, 1981. [3] A.C. Concei¸ca ˜o and V.G. Kravchenko, About explicit factorization of some classes of non-rational matrix functions, to appear in Mathematische Nachrichten 280, No. 9-10, (2007), 1022–1034. [4] A.C. Concei¸ca ˜o, V.G. Kravchenko, and F.S. Teixeira, Factorization of matrix functions and the resolvents of certain operators, in Operator Theory: Advances and Applications, Birkh¨ auser Verlag 142 (2003), 91–100.

Factorization Algorithm for Some Special Matrix Functions

185

[5] A.C. Concei¸ca ˜o, V.G. Kravchenko, and F.S. Teixeira, Factorization of some classes of matrix functions and the resolvent of a Hankel operator, in Factorization, Singular Operators and Related Problems, Kluwer Academic Publishers (2003), 101–110. [6] T. Ehrhardt, and F.-O. Speck, Transformation techniques towards the factorization of non-rational 2×2 matrix functions, Linear Algebra and its applications 353 (2002), 53–90. [7] L.D. Faddeev and L.A. Takhtayan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin, 1987. [8] I. Feldman, I. Gohberg, and N. Krupnik, An Explicit Factorization Algorithm, Integral Equations and Operator Theory, Birkh¨ auser Verlag, Basel 49 ( 2004), 149–164. [9] V.G. Kravchenko and A.I. Migdal’skii, A regularization algorithm for some boundaryvalue problems of linear conjugation (English), Dokl. Math. 52 (1995), 319–321. [10] G.S. Litvinchuk, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift, Mathematics and its Applications 523, Kluwer Academic Publishers, Dordrecht, 2000. [11] G.S. Litvinchuk and I.M. Spitkovskii, Factorization of Measurable Matrix Functions, Operator Theory: Advances and Applications 25, Birkh¨ auser Verlag, Basel, 1987. [12] S. Pr¨ ossdorf, Some Classes of Singular Equations, North-Holland, Amsterdam, 1978. Ana C. Concei¸ca ˜o and Viktor G. Kravchenko ´ Area Dep. de Matem´ atica Universidade do Algarve Campus de Gambelas 8000-810 Faro, Portugal e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 187–199 c 2008 Birkh¨
auser Verlag Basel/Switzerland

On a Radon Transform Ekaterina Gots and Lev Lyakhov Abstract. In this article a special type of Radon transform (Kipriyanov-Radon transform Kγ ) is considered and some properties of this transform are proved. The main results of this work are the inversion formulas of Kγ , which were obtained with a help of general B-hypersingular integrals. Mathematics Subject Classification (2000). Primary 44A12, 42B20; Secondary 26A33. Keywords. Radon transform, inversion formulas, B-hypersingular integrals.

1. Introduction Let x=(x , x ) ∈ RN , N ≥ 2, x =(x1 , . . . , xn ), x =(xn+1 , . . . , xN ), where n ≤ + N is fixed, and EN is the part of the space RN determined by the inequalities x1 >0, . . . , xn >0. Suppose that a function f is defined in RN , even with  respect γ to each variable xi , and absolutely integrable with the weight (x )γ = nj=1 xj j : F  γ + |f (x)| (x ) dx < ∞, where γ1 , . . . , γn are the fixed positive numbers. EN Following [1] we define a special case of Radon transform. ; By a function 9 2 2 + z 2 , x ), f (x) we construct the rotation function f˜(z)=f ( z + z , . . . , z 2 1

2

2n−1

2n

+ . Then consider the (N + n − 1)−dimension hyperz=(z1 , z2 , . . . , z2n , x ) ∈ EN N +n+n plane Γ = {z : z, ξ= k=1 zi ξi =p, |ξ|=1}, where ξ = (ξ1 , 0, ξ2 , 0, , . . . , ξn , 0, ξ  ) is a normal vector. Thus, a Radon transform of the rotation function constructed by the plane Γ will be called Kipriyanov-Radon transform (KR-transform) of a function f :  n . γ −1 Kγ [f ](ξ; p) = C(γ) f$(z, x ) z2jj dΓ, (1.1)

C(γ) =

n

 Γ

γk +1 2

k=1 Γ( γk ) Γ( 1 ) , 2 2

1, 2, . . . , n.

Γ



dΓ =

(−1)i−1 ξi

j=1

dz1 . . . dzi−1 dzi+1 . . . dzN +n

i = 2j, j =

188

E. Gots and L. Lyakhov From definition (1.1) parity and homogeneity properties follow. Kγ [f ](−ξ; −p) = Kγ [f ](ξ; p),

Kγ [f ](αξ; αp) =

1 Kγ [f ](ξ; p). |α|

(1.2)

The main results of this work are the inversion formulas of KR-transform. To simplify we prove these formulas for Schwartzian functions that are even with respect to each variable xi , where i = 1, 2, . . . , n. Such class of functions we denote + by Sev (EN ) (see [2]). It is worth noting that there is a class of the “worst” functions, for which we can apply the inversion formulas we obtained. It is the space of Riesz B-potentials. One can find the description of such spaces in the work [3]. For the classical Radon transform similar results were obtained by Ilham A. Aliev and Boris Rubin (see [4, 5, 6, 7]). Also it should be pointed out that we don’t consider the problem of the KR-transform range in this work, although it could be derived from the formula which shows close connection to KR-transform and Fourier-Bessel and Fourier transforms (see below) and their ranges.

2. Fourier, Fourier-Bessel and Kipriyanov-Radon transforms + For functions from Sev (EN ) the mixed Fourier-Bessel transform of the following form is defined and invertible (see its properties in [2]): 



FB [ϕ](ξ) = ϕ(ξ) I = jγ (x ; ξ  ) e−i x ,ξ  ϕ(x) (x )γ dx , (2.1) + EN

I = (2π)n−N FB−1 [ϕ](x)

n . j=1

1 2γj −1

In the formulas above (x )γ =

n  i=1

Γ2



γj +1 2





jγ (x ; ξ  ) ei x





,ξ



ϕ(ξ) I (ξ  )γ dξ .

+ EN

xγi i , jγ (x ; ξ  )=

n j=1

j γj −1 (xj ξj ) , and the func2

tion jν (t) are related to the corresponding Bessel functions of the first kind Jν (t) by the formula jν (t) = 2ν Γ(ν + 1) Jνtν(t) . The representation of the function jν in terms the Poisson integral ([8, p. 117])  π Γ(ν + 1) e−it cos Θ sin2ν Θ dΘ jν (t) = Γ (ν + 1/2) Γ(1/2) 0 results (2.1) in  n . γ −1 z2jj dz, f˜(z)e−iz,ξ FB [f ](ξ) = C(γ) + EN +n

j=1

; 9 2 2 , x ), z = (z , z , . . . , z , x ) ∈ E + + z2n where f˜(z) = f ( z12 + z22 , . . . , z2n−1 1 2 2n N +n , + ξ = (ξ1 , 0, ξ2 , 0, . . . , ξn , 0, ξ  ) ∈ EN +n . Now let |ξ| = 1. Extracting the integral taken

On a Radon Transform

189

over the hyperspace  z, ξ  = p, we obtain +∞  FB [f ](ξ) = C(γ) e−ip −∞

 f˜(z)

z,ξ=p

n .

γ −1

z2jj

dΓ dp.

j=1

The inner integral here coincides (under the condition |ξ| = 1) with the right part of (1.1), hence +∞  e−ip Kγ [f ](ξ; p) dp . (2.2) FB [f ](ξ) = −∞

We can give another meaning to (2.2) if we make the substitution αξ, α = 0 for any unit vector ξ and p for α p in the integral. Using property (1.2) we obtain +∞  FB [f ](αξ) = fI(αξ) = e−iαp Kγ [f ](ξ; p) dp;

(2.3)

−∞

Kγ [f ](ξ, p) =

1 2π

+∞ 

eiαp FB [f ](αξ) dα.

(2.4)

−∞

Formulas (2.3) and (2.4) show a close connection between the KR-transform and the Fourier-Bessel transform in the real space, viz one transform is obtained from the other by applying the Fourier transform (directly in (2.3) or inversely in (2.4)).

3. KR-transform of a generalized shift, a generalized convolution, and B-differentiation + We assume hereafter that f (x) ∈ Sev (EN ).


1. KR-transform of a generalized shift. Consider the generalized shift Txy
, which acts on each of the variables xi (i = 1, 2, . . . , n) by the rule (see [2],[8])   π   ; Γ γi2+1 yi Txi f (x) =  γi   1  f . . . , x2i + yi2 − 2xi yi cos αi , . . . , x sinγi −1 αi dαi Γ 2 Γ 2 0

(3.1) and the mixed generalized shift


f → (T y f )(x) = (Txy
f )(x , x + y  ). And consider the Poisson operator  π  π. n Pxγ
f (x)=C(γ) ... sinγk −1 βk f (x1 cos β, . . . , xn cos βn , x ) dβ1 . . . dβn , 0

0

k=1

(3.2)

190

E. Gots and L. Lyakhov

where

  Γ γi2+1    . C(γ) = Γ γ2i Γ 12 i=1 n .

+ Theorem 3.1. If f (x) ∈ Sev (EN ) then the following formulas are valid

Kγ [(T y f )(x)](ξ; p) = Pyγ
Kγ [f ](ξ, p + y, ξ) , 1 Kγ [(T f )(x)] (ξ; p) = 2π

+∞ 

y

eiαy





,ξ



jγ (αy  ; ξ  )fI(αξ) eiαp dα.

(3.3) (3.4)

−∞

Proof. Since (T y jν )(x) = jν (x)jν (y) (see [2, p. 19]), then FB [(T y f )(x)](αξ) = eiαy





,ξ



jγ (αy  ; ξ  ) fI(αξ).

From this and formula (2.4) it follows (3.4). To prove (3.3) we substitute jγ in (3.4) for the Poisson integral, then π π . n C(γ) y ... sinγj −1 βj dβ1 . . . dβn Kγ [(T f )(x)](ξ; p) = 2π j=1 0

0

+∞  n



× eiα(p+ j=1 yj ξj cos βj +y ,ξ ) fI(αξ) dα. −∞

Using (2.4) again, we have π y

Kγ [(T f )(x)] (ξ; p) = C(γ)

... 0

×Kγ [f ](p +

n


π . n

sinγj −1 βj

0 j=1

yj ξj cos βj + y  , ξ  ) dβ1 . . . dβn .

j=1

Now formula (3.3) follows from the definition of the Poisson operator.




2. KR-transform of a generalized convolution. A generalized convolution, generated by the shift (3.1), has the following form (see [2])  (u ∗ v)γ (x) = u(y)(T −y v)(x)(y  )γ dy. + EN

+ ) then the Theorem 3.2. If functions u(x) and v(x) belong to the space Sev (EN following formula is valid +∞  Kγ [(u ∗ v)γ ](ξ; p) = Kγ [u](ξ, t) Kγ [v](ξ; p − t) dt. −∞

On a Radon Transform

191

This formula is proved by applying a well-known fact about the Fourier-Bessel transform of a generalized convolution FB [(u ∗ v)γ ](ξ)=FB [u](ξ) FB [v](ξ) (see [2]) and formula (2.4). 3. With respect to the variables xi , i = 1, 2, . . . , n, we will apply differentiation n  by means of the Bessel differential operators (B-differentiation) Bx
= Bxi = i=1   n  ∂2 ∂ , γi > 0. In particular, we will consider the operator + xγii ∂x ∂x2 i

i=1

i

∆B = Bx
+ ∆ , where ∆ =

N  i=n+1

∂2 . ∂x2i

(The operator ∆B and its fractional degrees were investi-

gated by I.A. Kipriyanov in his book [2].) 


Let Pm ((x )2 , x ) = ak (x )2k (x )k be a homogeneous polyno2|k
|+|k

|=m

mial of order m with constant coefficients, even with respect to the first n variables and Pm (Bx
, Dx

) be the singular differentiation operator of order m = 2|k  | + |k  | corresponding to this polynomial, where k  = (k1 , k2 , . . . , kn ), k  = (kn+1 , kn+2 , . . . , kN ) integer-valued multi-indices of length |k  | = k1 + k2 + · · ·+ kn , |k  | = kn+1 + kn+2 + · · · + kN . The following formulas are valid Kγ [Pm (Bx
, Dx

)f ](ξ; p) = Pm ((ξ  )2 , ξ  ) Pm (Bξ
, Dξ

)Kγ [f ](ξ; p) = (−1)|k





|∂

m

∂ m Kγ [f ](ξ; p) , ∂pm

Kγ [Pm ((x )2 , x )f (x)](ξ; p) . ∂pm

(3.5) (3.6)

Proof. From (2.4) and taking into account that FB [Bxj ](ξ)=(iξj )2 FB [f ](ξ), j=1, n,

FB [

∂f ](ξ)=(iξj )FB [f ](ξ), j=n + 1, N, ∂xj

we obtain

∂ 2 Kγ [f ] (ξ; p), j = 1, 2, . . . , n, Kγ [Bxj ](ξ; p) = ξj2 ∂p2   ∂Kγ [f ] ∂f Kγ (ξ; p), j = n + 1, n + 2, . . . , N. (ξ; p) = ξj ∂xj ∂p From this we have (3.5). Formula (3.6) is proved in a similar way just taking into account the following equalities ∂ 2 Kγ [x2j f (x)] (ξ; p), j = 1, 2, . . . , n, ∂p2 ∂Kγ [xj f (x)] ∂Kγ [f ] (ξ; p), j = n + 1, n + 2, . . . , N. (ξ; p) = − ∂ξj ∂p

Bξj Kγ [f ](ξ; p) =




192

E. Gots and L. Lyakhov

In particular, for B-differentiation, carrying out by the operator ∆B , the following formulas are valid: Kγ [(∆B f )(x)](ξ; p) = |ξ|2

∂ 2 Kγ [f ] (ξ; p), ∂p2

∆B Kγ [f ](ξ; p) =

∂ 2 Kγ [|x|2 f (x)] (ξ; p). ∂p2

4. Presentation of some functions in the form of weighted spherical integrals Following [2], let us introduce the function of “weighted plane wave” type. Let f (t) be a one-variable function, Pxγn is the Poisson operator (3.2). A function of the scalar product f (ξ, x) is usually called a “plane wave” (see [9]). We will call the function Pxγ
f (ξ, x) weighted plane wave. The meaning of such a notation is that the corresponding transformation of a weighted integral expression (with a weight (x )γ ), containing Pxγ
f (ξ, x), transforms this function into a function of ordinary scalar product, i.e., a function of an (ordinary) plane wave. Let us consider the expression  + J = Pxγ
f (ξ, x)(x )γ dx, BN (r) = {x : |x| < r; x1 > 0, . . . , xn > 0}. + BN (R)

The “antipolar” coordinate transform z2j−1 = xj cos αj , z2j = xj sin αj

j = 1, 2, . . . , n

dz = x1 . . . xn dx dα

(4.1)

reduces to the following  J = C(γ)

f (ξ, z)

n .

γ −1

z2jj

dz,

(4.2)

j=1

+ BN +n (r)

+  where BN +n (r) = {z : |z| < r; z2j > 0, j = 1, 2, . . . , n}, z = (z1 , . . . , z2n , x ) ∈ +  EN +n , ξ = (ξ1 , 0, ξ2 , 0, . . . , ξn , 0, ξ ), and C(γ) is the constant which normalizes the Poisson operator. Let ξN +n = 0 and let us use a coordinate transform, such that the direction of the vector ξ coincides with the directing vector of OzN +n . In addition, the equation of the plane ξ, z = p|ξ| changes to the equation zN +n = p. If we single out integration over zN +n , we will obtain

r J = C(γ) −r

 f (p|ξ|) dp + BN +n−1 (

n .

√

i=1 r 2 −p2 )

γi −1 z2i dz1 . . . dzN +n−1 .

On a Radon Transform

193

9 9 + + 2 2 BN r2 − p2 in the space EN +n−1 ( r − p ) is a sphere of radius +n−1 . After some easy calculation we achieve formula:  Pξγ
f (σ, ξ)

n .

1 σiγi

dS(σ) = C(γ)|SN +n−1 ||γ|−1

i=1

+ SN (1)

f (|ξ|p)(1 − p2 )

N +|γ|−3 2

−1

π C(γ)|SN +n−1 ||γ|−1 =

N −n−1 2

2n−1 Γ

n 

Γ

dp;

(4.3)

 γi +1  2

i=1

N +|γ|−1 2

 .

From formula (4.3) we can obtain simple formulas for calculating some spherical integrals. 1. Let f (t) = |t|k , then  Pξγ
|σ, ξ|k

n .

π

N −n−1 2

σiγi dS(σ) =

i=1

+ SN (1)

Γ

n  k+1   2

2n−1 Γ



Γ

 γi +1 

i=1  N +|γ|+k 2

2

|ξ|k .

(4.4)

2. Let f (t) = |t|k ln |t|. Then 

π

N −n−1 2

Pξγ
|σ, ξ|k ln |σ, ξ|(σ  )γ dS(σ) =

+ SN (1)

Γ

n  k+1  

2n−1 Γ

2



Γ

 γi +1 

i=1  N +|γ|+k 2

     N + |γ| + k k+1 −F , × ln |ξ| + F 2 2 F (z) =

2

|ξ|k

(4.5)

Γ (z) . Γ(z)

5. General B-hypersingular integrals Let us consider mixed generalized centered finite differences (g.f. differences), noncentered and centered types. We use the same notation for centered and mixed differences and specify which of them are meant when necessary. Centered mixed g.f. differences have the form (

l t ϕ)(x)

=

l


(−1)k Clk



 l T ( 2 −k)t ϕ (x)

k=0

=

l
k=0

(−1)k Clk

   l   l ( −k)t
− k t . Tx
2 ϕ x , x + 2

(5.1)

194

E. Gots and L. Lyakhov

Noncentered mixed g.f. differences have the form (

l t ϕ)(x)

=

l


(−1)k Clk (T kt ϕ)(x) =

k=0

l





(−1)k Clk Txkt
ϕ(x , x + kt ).

(5.2)

k=0

Let us also consider general B-hypersingular integrals  1 ( lt ϕ)(x)  γ (t ) dt , (Dγα ϕ) (x) = dN,γ,l(α) |t|N +|γ|+α

(5.3)

+ EN

where dN,γ,l (α) is a normalizing constant whose value is chosen so that construction (5.3) does not depend on l for l > α . The divergent integral on the right-hand side of (5.3) is regularized by applying mixed generalized finite differences of the form (5.1) or (5.2). The Fourier-Bessel gives us the following representation of the   transform −1 α operator (5.3): Dα ϕ (x) = F [|x| FB [ϕ]]. The integral degrees of the singular γ B differential operator (−∆B )m have the same representation for α = 2m, hence Dα γ is of the form of a fractional degree of the operator (−∆B ). Truncated general B-hypersingular integrals are obtained by integrating over + the domain {|t| > ε}+ ={|t| > ε, t∈EN } . We use the notation   γ  ( lt ϕ)(x)  γ γ (Eα ϕ) (x) = lim Eα,ε ϕ (x) = lim (t ) dt . (5.4) ε→0 ε→0 |t|N +|γ|+α {|t|>ε}+

+ ). Consider mixed g.f. difference (5.2) in (5.3) and (5.4). Take ϕ(x)∈Sev (EN Applying the mixed integral Fourier-Bessel transform to construction (5.4), we obtain:    G l

H 
J K (t )γ −i kt

, ξ|ξ|  ξ e kt I (−1)k Ckl j , dt. FB Eγα,ε ϕ (ξ)=|ξ|α ϕ(ξ) γ |ξ| |t|N +|γ|+α k=0

{|t|>ε}+

This shows that the role of the normalizing constant in the definition of general B-hypersingular integral (5.3) (with noncentered generalized finite difference) must be played by the coefficient    l 




(t )γ k l  ξ dN,γ,l (α) = lim e−ikt ,ξ /|ξ| dt. (−1) Ck jγ kt , N +|γ|+α ε→0 |ξ| |t| k=0

{|t|>ε}+

(5.5) As in [10, 11] the normalizing factor (5.5) does not depend on ξ for |ξ| = 1. Separating integration with respect to the weight variable in (5.5), we obtain n γj −1 l  γ +1     n 1 − e−it,ξ/|ξ| 1 . Γ i2 j=1 tj   dt dt . dN,γ,l (α) = n/2 N +|γ|+α γi |t|N +α π Γ 2 (1 + |t|2 ) 2 i=1 + EN

En

(5.6)

On a Radon Transform

195

The first integral in this expression is well known (see [11, 12]); this is the normalizing coefficient for the ordinary hypersingular integral with noncentered finite-difference regularization. As a function of the parameter α, it was studied by Samko in [11]. The special feature of this coefficient is that it vanishes at a positive integer α smaller than l. We refer to the first integral in (5.6) as the Samko coefficient and denote it by SN,l (α). The second integral in (5.6) can be evaluated. We denote it by   n  γi +1  Γ N +α i=1 Γ 2 2  . (5.7) C(N, γ) = n/2 π 2n Γ N +|γ|+α 2 The normalizing constant of the general B-hypersingular integral (5.3) with mixed g.f. difference (5.2) has the form dN,γ,l (α) = SN,l (α) C(N, γ). This implies that the normalizing coefficients dN,γ,l (α) and SN,l (α) have the same properties. The application of mixed g.f. difference (5.1) in (5.3) and (5.4) yields the following representation of the normalizing coefficient [cf. (5.6)]:  t l t n γj −1  γi +1   i 21 −i 21  n e − e . Γ 2 1 j=1 tj  γi  dt dt . dN,γ,l (α) = n/2 N +|γ|+α N +α |t| π Γ 2 (1 + |t|2 ) 2 i=1 EN

+ En

(5.8) Here, the first integral is again the Samko coefficient normalizing ordinary hypersingular integrals with centered finite-difference regularization (see [11, 12]). Therefore, as well as in the case of ordinary hypersingular integrals, the general Bhypersingular integral identically vanishes if a mixed centered g.f. difference of odd order is applied: Eγα ≡ 0. Thus, in the construction of operator (5.3), only mixed centered g.f. differences of even order should be taken. For such differences, the normalizing coefficient (5.8) is surely nonzero and can be evaluated by the formula dN,γ,l (α) = SN,l (α) C(N, γ), where SN,l (α) is the Samko coefficient normalizing the ordinary hypersingular integral with centered finite-difference regularization (see [11] or [12, pp. 372-376]) and the coefficient C(N, γ) is defined by (5.7), as in (5.6). The following theorem takes place. Theorem 5.1. General hypersingular integrals (5.3) converge absolutely on the functions having all bounded derivatives (with respect to the variables xn+1 , . . . , xN ) and B-derivatives (with respect to x1 , . . . , xn ) up to order m = 2k  + k  = [α] + 1.

6. Inversion formulas of KR-transform In [13] the inversion formulas of Kipriyanov-Radon were proved in the case of one single variable. In this article the case of n weighted variables is considered and general inversion formulas of Kipriyanov-Radon are proved.

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E. Gots and L. Lyakhov

We recall the representation of the general B-hypersingular integral (5.3) in the Fourier-Bessel images J K Dβγ ϕ = (−∆B )β/2 ϕ(x) = FB−1 |x|β FB [ϕ] . (6.1) + ) according to (6.1) the following semigroup property takes For ϕ ∈ Sev (EN place: if β = β1 + β2 , then Dβγ = Dβγ 1 Dβγ 2 . In particular, if k is a positive integer k/2

and β = k + α, then Dβγ = Dα γ ∆B . Let us introduce weighted Riesz-potentials Uβγ of the following form  (Uβγ f )(x) = f (y)(T −y kγ )(x)(y  )γ dy, β > 0 + EN

with the kernel



|x|β−N −|γ| , β − N − |γ| = 0, 2, 4, . . . , AN,γ (β) |x|β−N −|γ| ln |x|, β − N − |γ| = 0, 2, 4, . . . , where constants AN,γ (β) are selected in such a way that J K FB Uβγ ϕ (ξ) = |ξ|−β ϕ(ξ) I . 1

kγ (x) =

(6.2)

Comparison of (6.1) and (6.2) allows to name weighted Riesz-potentials Uβγ by fractional B-derivatives of negative order. We use general B-hypersingular integrals, because they reverse weighted + ), then from (6.1) and (6.2) Dβγ (Uβγ ϕ) = ϕ Riesz-potentials, viz if ϕ ∈ Sev (EN β/2

is following. The fundamental solution J K of equation ∆B ϕ = 0 function Eγ,β (x)=FB−1 (−1)β/2 |x|−β . Calculating, we obtain Eγ,β (x)=AN,γ (β) |x|β−N −|γ| , (1)

(1)

β − N − |γ| = 0, 2, 4, . . . ;

Eγ,β (x)=AN,γ (β) |x|β−N −|γ| ln |x|, (2)

(β > 0) is the

(2)

β − N − |γ| = 0, 2, 4, . . . ,

(6.3)

  Γ γi2+1 for β−N −γ =0, 2, 4, 6, . . . and N −n  γi +1   β−N −|γ|  β−n−1 (2) π 2 ! for β−N −γ=0, 2, 4, . . . . Γ 2 AN,γ (β) = 2iN +|γ|+2 i=1 Γ 2 2 −β Now we can formulate the main result of this work. (1)

where AN,γ (β) =

π

N −n 2 Γ

β β β−n 2 i 2 N +|γ|−β 2

( )

Γ(

)   n β

n

i=1

Theorem 6.1. Let γ = (γ1 , . . . , γn ) be a multi-index, consisted of fixed positive + numbers, f ∈ Sev (EN ) and Kγ [f ] is KR-transform of the function f (x), see (1.1). The general inversion formula of such a transform is  22n−|γ|−N π n+1−N +|γ|−1  γi +1  DN f (η) = N +|γ|−1 n Pηγ
Kγ [f ](ξ; η, ξ)(ξ  )γ dS(ξ), (6.4) γ 2 i Γ i=1 2 + SN (1)

+ (1) – the surface of the halfwhere Dβγ is the B-hypersingular integral (6.1), SN sphere |ξ| = 1, ξ1 > 0, . . . , ξn > 0.

On a Radon Transform

197

If the number |γ| is natural then: a) if N + |γ| is negative, the following formula is valid  N +|γ|−1 22n−|γ|−N π n+1−N  γi +1  ∆B 2 f (η) = N +|γ|−1 n Pηγ
Kγ [f ](ξ; η, ξ)(ξ  )γ dS(ξ). (6.5) 2 i Γ i=1 2 + SN (1)

b) if N + |γ| is even, then the following formulas are valid p=+∞     N +|γ|−2 22n−N −|γ| 1 γ  γ 2 dKγ [f ](ξ; p), ∆ (ξ ) dS(ξ) P f (η) = −
η n   B  η, ξ − p π N −n Γ2 γi2+1 + p=−∞ S (1) N

i=1

f (η) = −

2

(6.6) +∞  N +|γ|  (N + |γ| − 1)! 1 (ξ  )γ dS(ξ) Pηγ
Kγ [f ](ξ; p)dp. n  γi +1   η, ξ − p Γ2 2 + −∞ S (1) 

2n−N −|γ|

π N −n

N

i=1

In formulas (6.5–7) ∆B =

n  j=1

(6.7) ∂2 ∂x2j

+

γ ∂ xn xn .

Proof. 1. Let N + |γ| = 2m, m = 1, 2, 3, . . .. In equation (4.4) replace the vector ξ  by the vector ξ  − η  , multiply by
+ (T ηξ
f )(ξ), and integrate over ξ ∈ EN . Then on the left side of the equality we    substitute ξ − η for ξ , and use self-adjointness of the generalized shift. Then    (T η f )(ξ) Pξγ
|ξ, ξ|k (ξ  )γ dS(ξ)(ξ  )γ dξ = CN,n,k (γ) T −η |(ξ)|k f (ξ)(ξ  )γ dξ. + EN

+ SN (1)

+ EN

In this equality the generalized shift T η is specified by (3.1). Let us choose β in (1) such a way that k=β − N − |γ|=1. The expression on the right, divided by AN,γ (β) is a B-potential. That is why   (T η f )(ξ) Pξγ
|ξ, ξ|k (ξ  )γ dS(ξ)(ξ  )γ dξ = C1 f (η), (5.8) Dβγ + EN

+ SN (1)

  N +|γ|+1  (1) n 2 γi +1 . where C1 = CN,n,β,1 (γ) = CN,n,1 (γ)AN,γ (N +|γ|+1)=− π(2i) n+1−N 22n i=1 Γ 2 If we invert the order of integration in the left part of the equality (5.8) and use a transformation of the coordinates (4.1), then we obtain  M = C(γ)Dβγ (ξ  )γ dS(ξ)  ×

+ SN (1)

; ; n . γ −1 $ 2 2 , ξ  )|ξ $ , ξ| $ (T −η f )( ξ$12 + ξ$22 , . . . , ξ$2n−1 + ξ$2n ξ$2jj dξ,

+ EN +n

j=1

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where ξ$ = (ξ1 , 0, ξ2 , 0, . . . , ξn , 0, ξ  ), ξ$ = (ξ$1 , ξ$2 , . . . ξ$2n , ξ  ) belong to the half+ space EN +n . In the inner integral we single out integration over the hyperspace  $ = p|ξ| and integration over p, then obtain ξ$ , ξ  M = Dβγ (ξ  )γ dS(ξ) + SN (1)

+∞   ; ; n . γ −1 2 2 , ξ  ) ξ$2jj dΓdp. |p|C(γ) (T η f )( ξ$12 + ξ$22 , . . . , ξ$2n−1 + ξ$2n × −∞

j=1

$ ξ$
,ξ=p

By definition (1.1) the inner integral is the KR-transform of the function (T η f )(ξ):  M=

+∞  (ξ ) dS(ξ) |p| Kγ [ (T η f ) (ξ) ] (ξ; p) dp .  γ

Dβγ + SN (1)

−∞

We apply (3.4) to the KR-transform of the generalized shift, then use the presentation of KR-transform (2.4), and after it isolate the operator ∆B from the operator Dβγ , and apply B-differentiation by means of the operator ∆B . In the result we have +∞   ∂2 β−2  γ M = Dγ (ξ ) dS(ξ) |p|Pηγ
2 Kγ [f ](ξ; p + η, ξ)dp ∂p + SN (1)

 = Dβ−2 γ

−∞

+∞  ∂2 (ξ  )γ dS(ξ)Pηγ
|p| 2 Kγ [f ](ξ; p + η, ξ)dp. ∂p

+ SN (1)

−∞

Integrating by parts the inner integral we obtain: +∞  ∂2 |p| Pηγ
2 Kγ [f ](ξ, p + ξ, η) dp = 2Pηγ
Kγ [ f ] (ξ, ξ, η). ∂p

−∞

Thus, the general inversion formula of KR-transform (6.4) follows. From (6.4) we obtain formula (6.5) for the whole numbers |γ|, when N + |γ| is an odd natural number. + In (6.5) it is enough, that f ∈ Sev (EN ). + ), N + |γ| = 2m is an even natural number. The same 2. Assume that f ∈ Sev (EN actions reduce to the inversion formula of the KR-transform for the whole |γ| (6.6). Note that formula (6.6) has a singularity in the integral and should be interpreted as a principal value. After formal differentiating this equality we have (6.7). And now it should be interpreted only in the sense of the regularized value of a divergent integral (see [14, p. 31]) or in the sense of a f.p.-integral (see [15, p. 424]).


On a Radon Transform

199

References [1] Kipriyanov I.A., Lyakhov L.N., On Fourier, Fourier-Bessel, and Radon transforms, Doklady Akademii Nauk, 360:2 (1998), 157–160 [Dokl. Math. 57, 361–364 (1998)]. [2] Kipriyanov I.A., Singular Elliptic Boundary Value Problems, Nauka, Moscow, 1996. [3] Lyakhov L.N., On the one class of hypersingular integrals, Russian Acad. Sci. Dokl. Math. 49 (1), 83–87 (1994). [4] Aliev I.A., Rubin B., Wavelet-like trans- forms for admissible semi-groups; inversion formulas for potentials and Radon transforms, J. Fourier Anal. Appl. 11 (2005), no. 3, 333–352. [5] Rubin B., Reconstruction of functions from their integrals over k-planes, Israel J. Math. 141 (2004), 93–117. [6] Rubin B., Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math. 29 (2002), no. 3, 471–497. [7] Rubin B., Helgason-Marchaud inversion formulas for Radon transforms, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3017–3023. [8] Levitan B.M., Expansion by Bessel functions in the Fourier series and integrals, Usp. Mat. Nauk, 4:2(1951), 102–143. [9] John F., Plane Waves and Spherical Means, Wiley (Interscience), New York, 1955. [10] Lyakhov L.N., On the one class of hypersingular integrals, Dokl. Akad. Nauk SSSR 315 (2), 291–296 (1990). [11] Samko S.G., On the Riesz-potential Spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 40 (5), 1443–1472 (1976). [12] Samko S.G., Kilbas A.A., and Marichev O.I., Integrals and Derivatives of Fractional Orders and Some of Their Applications Nauka i Tekhnika, Minsk, 1987. [13] Lyakhov L.N., Kipriyanov-Radon Transform, Trudy Mat. Inst. Steklov, 248 (2005), 153–163. [14] Gelfand I.M., Graev M.I. and Vilenkin I.Ya., Integral Geometry and Representation Theory, GIFML, Moscow, 1962. [15] Edwards R.E., Functional Analysis. Theory and Application, Mir, Moscow, 1969. Ekaterina Gots Institut f¨ ur Mathematik Universit¨ at Z¨ urich, Winterthurerstr. 190 CH-8057 Z¨ urich Switzerland e-mail: [email protected] Lev Lyakhov Voronezh State Technological Academy, Revolution avenue, 19 Voronezh, Russia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 201–206 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Extensions of σ-C ∗-algebras Rachid El Harti Abstract. Let A be a σ-C ∗ -algebra. The bounded part b(A) of A introduced by Konrad Schm¨ udgen in [4] is a C ∗ -algebra for some C ∗ -norm. We shall show that if A is a split extension of a σ-C ∗ -algebra B by a closed two-sided ideal I then b(A) will be a split extension of the C ∗ -algebra b(B) by the closed two-sided b(I). A number of results concerning the bounded part of a σ-C ∗ -algebra are established. Mathematics Subject Classification (2000). Primary 46L05; Secondary 46K05, 46J40, 46M15. Keywords. C ∗ -algebra, split extension, multipliers algebra, spectrum.

1. Introduction and Notations Throughout this paper, a σ-C ∗ -algebra A shall be a complete topological ∗-algebra whose topology arises from an increasing countable family { . n , n ≥ 1} of C ∗ seminorms satisfying (the Hausdorff condition) 0 ker  . n = {0} n≥1

where ker  . n = {a ∈ A : an = 0}. Since every C ∗ -seminorm is submultiplicative (see [8]) and so, ∗-preserving, each set ker  . n is a closed two-sided ∗-ideal of A and the algebra An = A/ ker  . n equipped with the C ∗ -norm  . n , is automatically complete [7, Folg. 5.4], so that it is a C ∗ -algebra. For m ≥ n, we have the canonical ∗-homomorphisms φn : A → An , φm : A → Am and φm,n : Am → An such that φn = φm,n ◦ φm . Thus A is an inverse limit of C ∗ -algebras (An , n ≥ 1) (see [1], [4], [5]). We write: A = lim(An , n ≥ 1). ←

Note that a closed ∗-subalgebra of a σ-C ∗ -algebra is a σ-C ∗ -algebra. The bounded part of a σ-C ∗ -algebra A is the subalgebra b(A) = {a ∈ A : a∞ = supn≥1 an < +∞}. It is a C ∗ -algebra with norm  . ∞ . Moreover, (b(A),  . ∞ )

202

R. El Harti

is continuously embedded in A with dense range [7]. An element a of b(A) will be called a bounded element of A. Let I be a closed two-sided of a σ-C ∗ -algebra A. It is clear that I is a ∗-subalgebra of A and thus, it is a σ-C ∗ -algebra. Note also that A/I is a σC ∗ -algebra. Let B and I be σ-C ∗ -algebras. An extension of B by I is a σ-C ∗ algebra A together with a topological injection ∗-homomorphism i : I → A the image of which is a two-sided ideal in A, and a topologically surjective (=open) ∗-homomorphism σ : A → B, of which this two-sided ideal is the kernel. We can say that there is a triple (A, σ, i) for which the sequence i

σ

0 −→ I −→A−→B −→ 0 is exact. This extension is said to be split if σ has a right inverse ∗-homomorphism ρ : B → A whose range as a closed ∗-subalgebra of A is a topological direct sum of I. The main of this paper is to show that if (A, σ, i) is an extension of B by I, then (b(A), σ ˜ , ˜i) is an extension of b(B) by b(I) where σ ˜ and ˜i are respectively the restriction of σ and i on b(B) and b(I). We show also that (b(A), σ ˜ , ˜i) is split whenever (A, σ, i) is split.

2. Preliminaries Let (A,  . n , ≥ 1) be a σ-C ∗ -algebra. For each a ∈ A, the spectrum spA (a) of a is the set of λ ∈ C such that λ.1A − a is non-invertible in A. The real number rA (a) = sup {|λ| : λ ∈ spA (a)} is called the spectral radius of a. With the above notation, an element a ∈ A is invertible in A if, and only if, πn (a) is invertible in An for each n ≥ 1, and so  spAn (πn (a)) ∀a ∈ A. (1) spA (a) = n∈1

We say that an element a of A is spectrally bounded if its spectrum is bounded. Now, we recall a collection of elementary properties of a σ-C ∗ -algebra indicating certain references in the literature for each of them. Proposition 2.1. [1] Let A be a σ-C ∗ -algebra. Then every closed two-sided ideal of A is a ∗-subalgebra of A and it is a σ-C ∗ -algebra. We can use Lemma 3.1 of this paper to show the last proposition. In general, a ∗-homomorphism between topological ∗-algebras is not necessarily continuous. In the case of σ-algebras, we have: Theorem 2.1. [4] Let A be a σ-C ∗ -algebra. Then, for each σ-C ∗ -algebra B, every ∗-homomorphism φ : A → B is continuous. Theorem 2.2. [4] Let φ : A → B be a ∗-homomorphism of σ-C ∗ -algebra. Then φ(b(A)) ⊆ B and hence, φ : b(A) → b(B) is a continuous ∗-homomorphism.

Extensions of σ-C ∗ -algebras

203

Like in the category of C ∗ -algebras, the continuous functional calculus has been well established. Theorem 2.3. [4] Let A be a unital σ-C ∗ -algebra. For each normal element a ∈ A, there exists a unique continuous ∗-homomorphism θ : f → f (a) from C(spA (a)) to A, sending the identity function z → z to a such that θ(Cb (spA (a))) ⊆ b(A). Moreover, if σ : A → B is ∗-homomorphism from A into another σ-C ∗ -algebra B, then σ(f (a)) = f (σ(a)).

3. Extensions of σ-C ∗ -algebras We start with some statements that we need to prove the main results. Lemma 3.1. Let X = lim← (Xn , n ≥ 1) be a inverse limit of Banach spaces with the canonical linear maps πn : X → Xn , and assume that Y is a closed subset of X . Then, Y = lim← ((πn (Y), n ≥ 1) is a inverse limit of closed sets (πn (Y), πn ), where each πn : Y → πn (Y) is the associate canonical linear map. Proof. The family {πn−1 (Vn ), n ≥ 1, Vn is an open set of Xn } forms a basis for ˜m,n the restriction the topology of X . Set Yn = πn (Y) and for m > n, denote by π of πm,n on Ym into Ym . It is easy to check that (Ym , π ˜m,n , n ≥ 1) is a projective system and the inverse limit is ˜m,n (ym ) = yn , ∀ m > n ≥ 1}. lim(Yn , πn , n ≥ 1) = {y = (yn )n≥1 : πn (y) = π ←

Notice that this space is equal to X ∩ (∩{πn−1 (Yn ), n ≥ 1}). It is easy to see that / Y. Since X \Y is Y ⊆ ∩{πn−1 (Yn ), n ≥ 1}∩X . To prove the other inclusion, let x ∈ open, there exists a neighborhood Vn ∈ Xn of πn (x) such that πn−1 (Vn )∩Y = ∅, and / πn (Y). It follows that x ∈ / ∩{πn−1 (Yn ), n ≥ 1}. thus, Vn ∩ πn (Y) = ∅. Then πn (x) ∈ This complete the proof.
As a first consequence, we obtain the following result that is known in the C ∗ -algebras case. Proposition 3.1. Let (A,  . n ) be a σ-C ∗ -algebra and B be a closed ∗-subalgebra. Then each element a ∈ B satisfies spB (a) = spA (a). Proof. Let a ∈ B. By Lemma 3.1, we have B = lim← (πn (B), πn ). Set πn (B) = Bn . It is clear that Bn is a closed star subalgebra of the C ∗ -algebra An and thus,
spBn (πn (a)) = spAn (πn (a)). By (1), the result follows. Remark 3.1. Let A be a σ-C ∗ -algebra with the identity. It is easy to check that if a ∈ b(A), then the spectrum spA (a) is bounded. The following remark show that the converse is not true. Let Mn (C) be the unital algebra of all n × n complex n ∗ matrices and let An = k=1 ⊕Mn (C) be the C -algebra of all block matrices

204

R. El Harti

with increasing blocks of size 1 to n. Then for each m ≥ n, the canonical ∗homomorphism πmn : Am −→ An is the projection on the first n block of the top left-hand corner. We can easily check that (An , πmn , n) is an projective system of C ∗ -algebras such that the inverse limit is the σ-algebra Πn≥1 Mn (C) where the canonical homomorphism πn from Πn≥1 Mn (C) to An is the projection onto the first n blocks. Consider the infinite matrix ⎛

⎜ ⎜ N =⎜ ⎜ ⎝

0 0 . . .

1 0 . . .

0 2 . . .

. 0 . 0 .

. . . n .

. . . 0 .

⎞ . . ⎟ ⎟ ⎟. ⎟ . ⎠ .

Then N is a quasinilpotent element of the σ-C ∗ -algebra Πn≥1 Mn (C) and ; N ∞ ≥ ρ(πn (N )πn (N )∗ ) ≥ (n − 1) for all n ≥ 1. The algebra Πn≥1 Mn (C) is a non trivial σ-algebra having many spectrally unbounded elements. In using the equation (1), we can check that every spectrally bounded normal element of a σ-C ∗ -algebra A is bounded and hence, these two properties are equivalent in the commutative case. Moreover, b(A) is generated as a linear space by the Hermitian spectrally bounded element of A. Proposition 3.2. Let A and B be two σ-C ∗ -algebras. If φ : A → B is an injective ∗-homomorphism from A into B with a closed image, then b(A) ∼ = b(B) ∩ φ(A). In particular, if A is a closed star subalgebra of B, then b(A) = b(B) ∩ A. Proof. This proposition follows from the fact that the algebra of bounded part b(A) of a σ-C ∗ -algebra A is the space generated by hermitian elements of A for which the spectrum is finite [7].
The action b → b(A) is a covariant functor from the category of σ-C ∗ -algebras to the category of C ∗ -algebras. Now, we prove the main. Theorem 3.1. Let I be a closed two-sided ideal of a σ-C ∗ -algebra A, then b(I) is a closed two-sided ideal of b(A). Moreover, if the following sequence i

σ

0 → I −→A−→B → 0 is an extension of B by I, where i is the identity map and σ a surjective ∗homomorphism, then ˜i

σ

0 −→ b(I)−→b(A)−→b(B) −→ 0 is an extension of b(A) by b(I). Moreover, if the first extension is split, then the second will be too.

Extensions of σ-C ∗ -algebras

205

Proof. By Proposition 3.2, b(I) = b(A) ∩ I. In particular, it is a two-sided ideal of b(A). The C ∗ -algebra b(I) embeds into the C ∗ -algebra b(A), and thus it is closed there. It is easily seen that im(˜i) ⊆ ker(˜ σ ). To show the other inclusion, let a an element of b(A) such that σ ˜ (a) = 0. Then a ∈ I, and therefore a ∈ b(I) = b(A) ∩ I. It remains to prove that σ ˜ : b(A → b(B)) is surjective. Indeed, let b = b∗ ∈ b(B), then there exist a = a∗ in A such that σ(a) = b. Consider a bounded continuous function f = f ∗ in Cb (R) such that f = z on the compact set spb(B) (b). Then f (b) = b. Using Theorem 2.3, we deduce that f (a) ∈ b(A) and b = f (b) = f (σ(a)) = σ(f (a)). Finally, if the extension (A, σ, i) of B by I is split, then the splitting homomorphism ρ : B → A for this extension sends b(B) into b(A) and thus, the restriction ρ˜ of ρ on b(B) is the splitting homomorphism for the extension (b(A), σ ˜ , ˜i) of b(B) by b(I). This completes the proof.
Corollary 3.1. Let I be a closed two-sided ideal of a σ-algebra A, then the sequence i

σ

0 −→ b(I)−→b(A)−→b(A/I) −→ 0 is exact where i is the identity map and σ : A → Ap is the canonical projection. Corollary 3.2. If p is a continuous C ∗ -seminorm on a σ-C ∗ -algebra A, then ∼ Ap . b(A)/b(ker(p)) = Proof. Applying Theorem 3.1 for I = ker(p) and B = Ap , we obtain the result.
Corollary 3.3. Let A be a σ-C ∗ -algebra such that b(A) is simple, then A is C ∗ algebra. Corollary 3.4. Let (A,  . n , n ≥ 1) be a σ-C ∗ -algebra. Then 1. For each n ≥ 1, A and b(A) are respectively the extensions of An by ker  . n and b(ker  . n ) and thus An = b(A)/b(ker  . n ). 2. If there exist closed two-sided ideals I and J such that A = I ⊕ J then b(A) = b(I) ⊕ b(J ). Indeed, consider the following extension 0 → I → A → J → 0. This extension is split and the splitting homomorphism is the ∗-homomorphism j : J → A which sends J into J . The same splitting homomorphism sends b(J ) into b(J ). It follows that the sequence 0 → b(I) → b(A) → b(J ) → 0 is a split extension.

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A σ-C ∗ -algebra A is to be prime if it contains no-zero orthogonal closed twosided ideals, i.e., if I and J are two closed two-sided ideals of A with IJ = 0 then I = 0 or J = 0. Proposition 3.3. Let A be a σ-C ∗ -algebra. Then, the bounded part b(A) is prime if and only if A is prime. Let A0 be a σ-unital C ∗ -algebra without identity and let A00 the Pedersen ideal of A0 (it is a dense two-sided ideal which is minimal among all dense hereditary ideals of A0 , see [3]). Consider Γ(A00 ) the σ-C ∗ -algebra of the multipliers on A00 , which is well studied by N.C. Phillips in [6], where it is presented as a inverse limit of unital C ∗ -algebras. Indeed, let (en ) be a countable bounded approximate identity of the a σ-unital C ∗ -algebra A0 contained in A+ 00 . Put Ien = (A0 .en )(en .A0 ). Then Ien is closed two-sided ideal of A0 singly generated by en and Γ(A00 ) = lim M (Ien ) ←−

where M (Ien ) is the multipliers algebra of Ien . Proposition 3.4. A σ-unital C ∗ -algebra A0 is prime if and only if Γ(A00 ) is prime. Proof. First note that the bounded part of Γ(A00 ) is the multipliers algebra M (A0 ) of A0 . By using the Proposition and Lemma 2.2 of [2], we deduce the result.


References [1] A. Inoue, Locally C ∗ -algebras, Mem. Fac. Sci. Kyushu Univ. Ser. A, 25: 197–235 (1971). [2] M. Mathieu, Elementary operators on prime C ∗ -algebras I, Math. Ann. 284(2): 223– 244 (1989). [3] G.K. Pedersen, Measure theory for C ∗ -algebras, Math. Scand. 19:131–143 (1966). [4] N.C. Phillips, Inverse limits of C ∗ -algebras, J. Operator Theory 19(1): 159–195 (1988). [5] N.C. Phillips, Inverse limits of C ∗ -algebras, in Operator algebras and applications, Vol. I, London Math. Soc., Lecture Note Ser. 135, pages 127–185, Cambridge University Press, Cambridge (1988). [6] N.C. Phillips, A new approach to the multipliers of Pedersen’s ideal, Proc. Amer. Math. Soc. 104(3): 861–867 (1988). ¨ [7] Konrad Schm¨ udgen, Uber LMC-Algebren, Math. Nachr. 68: 167–182 (1975). [8] Z. Sebesty´en, Every C ∗ -seminorm is automatically submultiplicative, Period. Math. Hangar. 10(1): 1–8 (1979). Rachid El Harti University Hassan I. FST de Settat. Bp 577. Settat. Morocco e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 207–228 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized H¨ older Spaces Alexei Yu. Karlovich Abstract. We prove higher-order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized H¨ older spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of B¨ ottcher and Silbermann and generalize their results for symbols in standard H¨ older spaces. Mathematics Subject Classification (2000). Primary 47B35; Secondary 15A15, 47B10, 47L20, 47A68. Keywords. Block Toeplitz matrix, determinant, trace, Szeg˝ o-Widom limit theorems, decomposing algebra, canonical Wiener-Hopf factorization, generalized H¨ older space, Bari-Stechkin class.

1. Introduction 1.1. Finite block Toeplitz matrices Let Z, N, Z+ , and C be the sets of integers, positive integers, nonnegative integers, and all complex numbers, respectively. Suppose N ∈ N. For a Banach space X, let XN and XN ×N be the spaces of vectors and matrices with entries in X. Let T be the unit circle. For 1 ≤ p ≤ ∞, let Lp := Lp (T) and H p := H p (T) be the standard Lebesgue and Hardy spaces of the unit circle. For a ∈ L1N ×N one can define  2π 1 ak = a(eiθ )e−ikθ dθ (k ∈ Z), 2π 0 the sequence of the Fourier coefficients of a. Let I be the identity operator, P be the Riesz projection of L2 onto H 2 , Q := I − P , and define I, P , and Q on L2N The author is supported by F.C.T. (Portugal) grants SFRH/BPD/11619/2002 and FCT/ FEDER/POCTI/MAT/59972/2004.

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elementwise. For a ∈ L∞ a(t) := a(1/t) and (Ja)(t) := t−1 $ a(t). N ×N and t ∈ T, put $ Define Toeplitz operators T (a) := P aP |Im P,

T ($ a) := JQaQJ|Im P

and Hankel operators H(a) := P aQJ|Im P,

H($ a) := JQaP |Im P.

The function a is called the symbol of T (a), T ($ a), H(a), H($ a). We are interested in the asymptotic behavior of finite block Toeplitz matrices Tn (a) := (aj−k )nj,k=0 generated by (the Fourier coefficients of) the symbol a as n → ∞. Many results about asymptotic properties of Tn (a) as n → ∞ are contained in the books by Grenander and Szeg˝ o [13], B¨ ottcher and Silbermann [5, 6, 7], Hagen, Roch, and Silbermann [15], Simon [24], and B¨ ottcher and Grudsky [2]. 1.2. Szeg˝ o-Widom limit theorems 1/2,1/2

)N ×N be the Let us formulate precisely the most relevant results. Let (K2,2 ∞ 2 satisfying a Krein algebra [19] of matrix functions a in L∞ k  |k| < ∞, N ×N k=−∞ where  ·  is any matrix norm on CN ×N . The following beautiful theorem about the asymptotics of finite block Toeplitz matrices was proved by Widom [27]. 1/2,1/2

Theorem 1.1. (see [27, Theorem 6.1]). If a ∈ (K2,2 )N ×N and the Toeplitz 2 operators T (a) and T ($ a) are invertible on HN , then T (a)T (a−1 ) − I is of trace class and, with appropriate branches of the logarithm, log det Tn (a) = (n + 1) log G(a) + log det1 T (a)T (a−1 ) + o(1) where det1 is defined in Section 2.1 and    2π 1 G(a) := lim exp log det I ar (eiθ )dθ , r→1−0 2π 0

I ar (eiθ ) :=

as ∞


n → ∞, (1)

an r|n| einθ .

n=−∞

The proof of the above result in a more general form is contained in [5, Theorem 6.11] and [7, Theorem 10.30]. (n) (n) Let λ1 , . . . , λ(n+1)N denote the eigenvalues of Tn (a) repeated according to their algebraic multiplicity. Let sp A denote the spectrum of a bounded linear operator A and trM denote the trace of a matrix M . Theorem 1.1 is equivalent to the assertion
(n) log λi = tr log Tn (a) = (n + 1) log G(a) + log det1 T (a)T (a−1 ) + o(1). i

Widom [27] noticed that Theorem 1.1 yields even a description of the asymptotic behavior of trf (Tn (a)) if one replaces f (λ) = log λ by an arbitrary function f analytic in an open neighborhood of the union sp T (a) ∪ sp T ($ a) (we henceforth call such f simply analytic on sp T (a) ∪ sp T ($ a)).

Asymptotic Formulas for Toeplitz Matrices 1/2,1/2

Theorem 1.2. (see [27, Theorem 6.2]). If a ∈ (K2,2 on sp T (a) ∪ sp T ($ a), then

trf (Tn (a)) = (n + 1)Gf (a) + Ef (a) + o(1)

209

)N ×N and if f is analytic as

n → ∞,

(2)

where

 2π 1 (trf (a))(eiθ )dθ, 2π 0  1 d log det1 T [a − λ]T [(a − λ)−1 ]dλ, Ef (a) := f (λ) 2πi ∂Ω dλ det1 is defined in Section 2.1, and Ω is any bounded open set containing the set sp T (a) ∪ sp T ($ a) on the closure of which f is analytic. Gf (a)

:=

The proof of Theorem 1.2 for continuous symbols a is also given in [7, Section 10.90] and in [6, Theorem 5.6]. In the scalar case (N = 1) Theorems 1.1 and 1.2 go back to Gabor Szeg˝o (see [13] and historical remarks in [5, 6, 7, 15, 24]). 1.3. Smoothness effects Fisher and Hartwig [11] were probably the first to draw due attention to higherorder correction terms in asymptotic formulas for Toeplitz determinants. B¨ ottcher and Silbermann [4] obtained analogs of Theorem 1.1 for symbols belonging to γ γ H¨older-Zygmund spaces CN ×N , 0 < γ < ∞. If γ > 1/2, then CN ×N is properly 1/2,1/2

γ )N ×N , and for a ∈ CN contained in (K2,2 ×N , formula (1) is then valid with o(1) 1−2γ ). Nowadays this result can be proved almost immediately replaced by O(n by using the so-called Geronimo-Case-Borodin-Okounkov formula (see [8]). The γ author [18] proved that if γ > 1/2 and a ∈ CN ×N , then (2) holds with o(1) replaced by O(n1−2γ ). That is, for very smooth symbols the remainders in (1) and (2) go to zero with high speed (depending on the smoothness). On the other hand, B¨ ottcher and Silbermann [4] (see also [5, Sections 6.15– 6.20] and [7, Sections 10.34–10.38]) observed that if 0 < γ ≤ 1/2, then (1) requires a correction involving additional terms and regularized operator determinants. This is the effect of “insufficient smoothness”. They also studied the same problems for Wiener algebras with power weights [4], [5, Sections 6.15–6.20], [7, Sections 10.34–10.38]. Recently the author [16] extended their higher-order versions of Theorem 1.1 to Wiener algebras with general weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. Corresponding higher-order asymptotic trace formulas are proved in [17] (see also [7, Section 10.91]). Very recently, it was observed in [3] that the approach of [4] with some improvements of [16] is powerful enough to deliver higher-order asymptotic formulas α,β for Toeplitz determinants with symbols in generalized Krein algebras (Kp,q )N ×N with 1 < p, q < ∞, 0 < α, β < 1, and 1/p + 1/q = α + β ∈ (0, 1). This is another example of “insufficient smoothness” because one cannot guarantee that α,β )N ×N . Notice that generalized T (a)T (a−1 ) − I is of trace class whenever a ∈ (Kp,q Krein algebras contain discontinuous functions in contrast to H¨ older-Zygmund spaces and weighted Wiener algebras, which consist of continuous functions only.

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1.4. About this paper In this paper, we will study asymptotics of Toeplitz matrices with symbols in generalized H¨older spaces following the approach of [4]. Our results improve earlier γ results by B¨ottcher and Silbermann for CN ×N , 0 < γ < 1, because the scale of generalized H¨older spaces is finer than the scale of H¨older spaces C γ , 0 < γ < 1 (although we will not consider generalizations of the case γ ≥ 1). The paper is organized as follows. Section 2 contains definitions of Schattenvon Neumann classes and regularized operator determinants, as well as definitions of the Bari-Stechkin class and generalized H¨older spaces Hω and their subspaces H0ω . Our main results refining Theorems 1.1 and 1.2 are stated in the end of Section 2. In Section 3, we present an abstract approach from [4] (see also [16]) to higher-order asymptotic formulas for block Toeplitz matrices. To apply these results it is necessary to check that the symbol admits canonical left and right bounded Wiener-Hopf factorizations, at least one of the factors is continuous, and some products of Hankel operators belong to the Schatten-von Neumann class 2 ) for m ∈ N. In Section 4, we collect necessary information about WienerCm (HN Hopf factorization in decomposing algebras of continuous functions and verify that the algebras (Hω )N ×N and (H0ω )N ×N have the factorization property. In Section 5, we prove simple sufficient conditions for the membership in the Schattenvon Neumann classes of products of Hankel operators with symbols in (Hω )N ×N . These results are based on the classical Jackson theorem on the best uniform approximation. In Section 6, we prove our asymptotic formulas on the basis of the results of Sections 3–5.

2. Preliminaries and the main results 2.1. Schatten-von Neumann classes and operator determinants Let H be a separable Hilbert space, B(H) be the Banach algebra of all bounded linear operators on H, C0 (H) be the set of all finite-rank operators, and C∞ (H) be the closed two-sided ideal of all compact operators on H. Given A ∈ B(H) define sn (A) := inf{A − F B(H) : F ∈ C0 (H), dim F (H) ≤ n} for n ∈ Z+ . For 1 ≤ p < ∞, the collection of all operators K ∈ B(H) satisfying  1/p
p KCp (H) := sn (K) <∞ n∈Z+

is denoted by Cp (H) and referred to as a Schatten-von Neumann class. Note that C∞ (H) = {K ∈ B(H) : sn (K) → 0 as n → ∞} and KC∞ (H) = sup sn (K) = KB(H) . n∈Z+

The operators belonging to C1 (H) are called trace class operators. Let A ∈ B(H) be an operator of the form I + K with K ∈ C1 (H). If {λj (K)}j≥0 denotes the sequence of the nonzero eigenvalues of K counted up to

Asymptotic Formulas for Toeplitz Matrices algebraic multiplicity, then the product The determinant of A is defined by



j≥0 (1 + λj (K))

det A = det(I + K) =

.

211

is absolutely convergent.

(1 + λj (K)).

j≥0

If K ∈ Cm (H), where m ∈ N \ {1}, one can still define a determinant of I + K, but for classes larger than C1 (H), the above definition requires a regularization. A simple computation (see [23, Lemma 6.1]) shows that then   m−1
(−K)j Rm (K) := (I + K) exp − I ∈ C1 (H). j j=1 Thus, it is natural to define det1 (I + K) := det(I + K),

detm (I + K) := det(I + Rm (K)) for m ∈ N \ {1}.

One calls detm (I + K) the m-regularized determinant of A = I + K. For more information about Schatten-von Neumann classes and regularized operator determinants, see [12, Chap. III–IV] and also [23]. 2.2. The Bari-Stechkin class A real-valued function ϕ is said to be almost increasing on an interval I of R if there is a positive constant A such that ϕ(x) ≤ Aϕ(y) for all x, y ∈ I such that x ≤ y. One says that ω : (0, π] → [0, ∞) belongs to the Bari-Stechkin class (see [1, p. 493] and [14, Chap. 2, Section 2]) if ω is almost increasing on (0, π], ω(x) > 0 for all x ∈ (0, π], and  x  π 1 ω(y) x ω(y) dy < ∞, sup dy < ∞. lim ω(x) = 0, sup x→0+0 ω(x) y ω(x) y2 x>0 x>0 0 x To give an example of functions in the Bari-Stechkin class, let us define inductively the sequence of functions k on (xk , ∞) by 1 (x) := log x, x1 := 1 and for k ∈ N \ {1}, k (x) := log(k−1 (x)) and xk such that k−1 (xk ) = 1. Elementary computations show that for every γ ∈ (0, 1) and any finite sequence β1 , . . . , βm ∈ R there exists a set of positive constants b1 , . . . , bm such that the function   m . bk , 0 < x ≤ π, (3) βk k ω(x) = xγ x k=1

belongs to the Bari-Stechkin class. In particular, ω(x) = xγ , 0 < γ < 1, is a trivial example of a function in the Bari-Stechkin class. 2.3. Generalized H¨ older spaces The modulus of continuity of a bounded function f : T → C is defined by ω(f, x) := sup sup |f (ei(y+h) ) − f (eiy )|, |h|≤x y∈R

0 ≤ x ≤ π.

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Let ω belong to the Bari-Stechkin class. The generalized H¨older space Hω is defined as the set of all continuous functions f : T → C satisfying |f |ω := sup 0<x≤π

ω(f, x) < ∞. ω(x)

We will consider also the subspace H0ω of functions f ∈ Hω such that lim

x→0+0

ω(f, x) = 0. ω(x)

It is well known that Hω and H0ω are Banach algebras under the norm f Hω := f C + |f |ω . 2.4. Higher-order asymptotic formulas for determinants 2 For a ∈ L∞ N ×N and n ∈ Z+ , define the operators Pn and Qn on HN by Pn :

∞


ak tk →

k=0

n


a k tk ,

Qn := I − Pn .

k=0

2 2 → Pn HN may be identified with the finite block The operator Pn T (a)Pn : Pn HN n Toeplitz matrix Tn (a) := (aj−k )j,k=0 . If A is a unital algebra, then its group of all invertible elements is denoted by GA. For 1 ≤ p ≤ ∞, put H p := {f ∈ Lp : f ∈ H p }. Suppose

v− ∈ (H ∞ )N ×N , u− ∈

∞ v+ ∈ HN ×N , ∞ GHN ×N ,

G(H ∞ )N ×N ,

u+ ∈

b := v− u−1 + ,

c := u−1 − v+ .

(4) (5)

and define Theorem 2.1 (Main result 1). Let ω, ψ belong to the Bari-Stechkin class. Suppose a ∈ L∞ N ×N can be factored as a = u− u+ with u− ∈ G(Hω ∩ H ∞ )N ×N , and suppose T ($ a) is invertible on

2 . HN

u+ ∈ G(Hψ ∩ H ∞ )N ×N ,

(6)

Then the following statements hold.

(a) The function a admits a factorization a = v+ v− , where v− ∈ G(H ∞ )N ×N ∞ and v+ ∈ GHN ×N . (b) If     ∞
1 1 ψ < ∞, (7) ω k k k=1

then T (a)T (a

−1

lim

n→∞

2 ) − I and T ($ c)T ($b) − I belong to C1 (HN ) and

1 det Tn (a) = det1 T (a)T (a−1 ) = . G(a)n+1 det1 T ($ c)T ($b)

Asymptotic Formulas for Toeplitz Matrices (c) If m ∈ N \ {1} and

∞     m
1 1 ψ ω < ∞, k k

213

(8)

k=1

2 ) and then T ($ c)T ($b) − I ∈ Cm (HN ⎧ ⎡ j ⎤⎫ m−1 m−1 ⎨ ⎬

1 det Tn (a) 1 ⎣ ⎦ = tr exp − F (b, c) lim , n,k n→∞ G(a)n+1 ⎩ ⎭ detm T ($ j c)T ($b) j=1 k=0

(9)

where

 k c)Qn Qn T (b)Pn Fn,k (b, c) := Pn T (c)Qn Qn H(b)H($

(n, k ∈ Z+ ).

(d) Suppose m ∈ N \ {1}. If (8) is fulfilled and %         & n m−1
1 1 1 1 ψ ψ = 0, ω lim ω n→∞ n n j j j=1 then one can remove Fn,m−1 (b, c) in (9), that is, ⎧ ⎡ j ⎤⎫ m−2 ⎨ m−1 ⎬
1
det Tn (a) 1 ⎣ ⎦ = tr lim exp − F (b, c) . n,k n+1 n→∞ G(a) ⎩ ⎭ detm T ($ j c)T ($b) j=1 k=0

(10)

(11)

(e) If m ∈ N and (8) is fulfilled, then there exists a nonzero constant E(a) such that log det Tn (a) = (n + 1) log G(a) + log E(a) ⎡  j ⎤ n m−1

1 m−j−1
G,k (b, c) ⎦ + tr ⎣ j (12) =1 j=1 k=0  ∞       m
1 1 ψ ω +O k k k=n+1

as n → ∞, where

 k G,k (b, c) := P0 T (c)Q Q H(b)H($ c)Q Q T (b)P0

(, k ∈ Z+ ).

(f) If, under the assumptions of part (e), u− ∈ G(H0ω ∩ H ∞ )N ×N

or

u+ ∈ G(H0ψ ∩ H ∞ )N ×N ,

then (12) holds with O(. . . ) replaced by o(. . . ). Let α, β ∈ (0, 1) and ω(x) = xα , ψ(x) = xβ . If α + β > 1, then (7) holds. If α + β > 1/m for some m ∈ N \ {1}, then (8) and (10) are fulfilled and we arrive at the theorem of B¨ottcher and Silbermann [7, Theorems 10.35(ii) and 10.37(ii)] for standard H¨ older spaces. It seems that part (f) is new even for standard H¨older spaces.

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2.5. Refinements of the Szeg˝ o-Widom limit theorems The case of ω = ψ in Theorem 2.1 is of particular importance. In this case we will prove the following refinement of the Szeg˝ o-Widom limit theorems. Theorem 2.2 (Main result 2). Let ω belong to the Bari-Stechkin class and let H be either Hω or H0ω . Suppose ∞   2
1 ω <∞ (13) k k=1

and put

⎧  ∞   2 
⎪ 1 ⎪ ⎪ if O ω ⎪ ⎨ k k=n+1   δ(n, H) := ∞   2
⎪ 1 ⎪ ⎪ if ω o ⎪ ⎩ k

H = Hω , H = H0ω .

k=n+1

1/2,1/2 (K2,2 )N ×N .

(a) We have HN ×N ⊂ 2 a) are invertible on HN , (b) If a ∈ HN ×N and the Toeplitz operators T (a) and T ($ then (1) holds with o(1) replaced by δ(n, H). (c) If a ∈ HN ×N and f is analytic on sp T (a) ∪ sp T ($ a), then (2) holds with o(1) replaced by δ(n, H). For Hω = C γ with γ ∈ (1/2, 1) and O(n1−2γ ) in place of δ(n, H), parts (a) and (b) are already in [4] (see also [8]) and part (c) is in [18]. Notice that the scale of generalized H¨ older spaces is finer than the scale of standard H¨older spaces. For instance, for every γ ∈ (0, 1) there exist functions ω1 and ω2 of the form (3) such that  0 C γ+ε ⊂ Hω1 ⊂ C γ ⊂ Hω2 ⊂ C γ−ε , 0<ε<1−γ

0<ε<γ

where each of the embeddings is proper (see [14, Section II.3]). Hence, Theorems 2.1 and 2.2 refine corresponding results for standard H¨ older spaces.

3. Higher-order asymptotic formulas: the approach of B¨ ottcher and Silbermann 3.1. Asymptotic formulas involving regularized operator determinants The following result goes back to B¨ottcher and Silbermann [4] (see also [5, Sections 6.15 and 6.20] and [7, Sections 10.34 and 10.37]). Theorem 3.1. Suppose a ∈ L∞ N ×N satisfies the following assumptions: (i) there are two factorizations a = u− u+ = v+ v− , where u− , v− ∈ G(H ∞ )N ×N ∞ and u+ , v+ ∈ GHN ×N ; (ii) u− ∈ CN ×N or u+ ∈ CN ×N . Then the following statements are true.

Asymptotic Formulas for Toeplitz Matrices

215

2 (a) If H(a)H($ a−1 ) ∈ C1 (HN ), then

lim

n→∞

det Tn (a) = det1 T (a)T (a−1). G(a)n+1

2 ) for some m ∈ N, then (9) is (b) If H(b)H($ c) and H($ c)H(b) belong to Cm (HN fulfilled. 2 (c) If H(b)H($ c) and H($ c)H(b) belong to Cm (HN ) for some m ∈ N \ {1} and

lim trFn,m−1 (b, c) = 0,

n→∞

then (11) holds. Proof. Part (a) follows from [7, Corollary 10.27]. Part (b) is proved in the present form in [16, Theorem 15]. Part (c) follows from part (b) and [16, Propositions 6,13, and 14].
Notice that hypothesis (ii) can be replaced by a weaker hypothesis (see [7, Section 10.34]), which allows us to work with two discontinuous factors u− and α,β )N ×N (see [3]). u+ . This is useful in the case of generalized Krein algebras (Kp,q 3.2. Decomposition of the logarithm of Toeplitz determinants The following lemma is an important step in the proof of Theorems 2.1(e), (f) and Theorem 2.2. It was obtained in [4] (see also [5, Section 6.16] and [7, Section 10.34]). Lemma 3.2. Suppose a ∈ L∞ N ×N satisfies hypotheses (i) and (ii) of Theorem 3.1. Suppose for all sufficiently large n (say, n ≥ n0 ) there exists a decomposition & % ∞
Gn,k (b, c) = −tr Mn + sn , tr log I − k=0

of N × N matrices and {sn }∞ where {Mn }∞ n=n0 is a sequence n=n0 is a sequence ∞ of complex numbers. If n=n0 |sn | < ∞, then there exists a constant E(a) = 0, depending on {Mn }∞ n=n0 and arbitrarily chosen N × N matrices M1 , . . . , Mn0 −1 , such that for all n ≥ n0 , ∞
sk . log det Tn (a) = (n + 1) log G(a) + tr(M1 + · · · + Mn ) + log E(a) + k=n+1

4. Wiener-Hopf factorization in decomposing algebras of continuous functions 4.1. Definitions and general theorems Let A be a Banach algebra continuously embedded in C. Suppose A contains the set of all rational functions without poles on T and A is inverse closed in C, that is, if a ∈ A and a(t) = 0 for all t ∈ T, then a−1 ∈ A. The sets A− := A ∩ H ∞ and A+ := A ∩ H ∞ are subalgebras of A. The algebra A is said to be decomposing if every function a ∈ A can be represented in the form a = a− + a+ where a± ∈ A± .

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A.Y. Karlovich

Let A be a decomposing algebra. A matrix function a ∈ AN ×N is said to admit a right (resp. left ) Wiener-Hopf factorization in AN ×N if it can be represented in the form a = a− Da+ (resp. a = a+ Da− ), where a± ∈ G(A± )N ×N ,

D(t) = diag{tκ1 , . . . , tκN },

κi ∈ Z,

κ1 ≤ · · · ≤ κ N .

The integers κi are usually called the right (resp. left ) partial indices of a; they can be shown to be uniquely determined by a. If κ1 = · · · = κN = 0, then the respective Wiener-Hopf factorization is said to be canonical. The following result was obtained by Budjanu and Gohberg [9, Theorem 4.3] and it is contained in [10, Chap. II, Corollary 5.1] and in [20, Theorem 5.7’]. Theorem 4.1. Suppose the following two conditions hold for the algebra A: (a) the Cauchy singular integral operator  1 ϕ(τ ) (Sϕ)(t) := v.p. dτ πi τ −t T

(t ∈ T)

is bounded on A; (b) for any function a ∈ A, the operator aS − SaI is compact on A. Then every matrix function a ∈ AN ×N such that det a(t) = 0 for all t ∈ T admits a right Wiener-Hopf factorization in AN ×N . Notice that (a) holds if and only if A is a decomposing algebra. The following theorem follows from a more general result due to Shubin (see [20, Theorem 6.15]). Theorem 4.2. Let A be a decomposing algebra and let  ·  be a norm in the algebra AN ×N . Suppose a, d ∈ AN ×N admit canonical right and left Wiener-Hopf factorizations in the algebra AN ×N . Then for every ε > 0 there exists a δ > 0 such that if a − d < δ, then for every canonical right Wiener-Hopf factorization (r) (r) (l) (l) a = a− a+ and for every canonical left Wiener-Hopf factorization a = a+ a− (r) (r) one can choose a canonical right Wiener-Hopf factorization d = d− d+ and a (l) (l) canonical left Wiener-Hopf factorization d = d+ d− such that (r)

(r)

[a± ]−1 − [d± ]−1  < ε,

(l)

(l)

[a± ]−1 − [d± ]−1  < ε.

a± − d±  < ε, a± − d±  < ε,

(r) (l)

(r)

(l)

4.2. Verification of the hypotheses of Theorem 4.1 for generalized H¨ older spaces Theorem 4.3. Let ω belong to the Bari-Stechkin class and let H be either Hω or H0ω . Then (a) a ∈ H is invertible in H if and only if a(t) = 0 for all t ∈ T; (b) S is bounded on H; (c) for a ∈ H, the operator aS − SaI is compact on H.

Asymptotic Formulas for Toeplitz Matrices

217

Proof. (a) Obviously, GH ⊂ GC. Conversely, if f ∈ GC, then for all x ∈ (0, π], ω(1/f, x) ≤ 1/f 2C ω(f, x). From this inequality we see that if f ∈ GC ∩ H, then 1/f ∈ H. Part (a) is proved. (b) For Hω this result follows from the well-known Zygmund estimate (see [28] and also [1, p. 492], [14, p. 10]):  π  x ω(f, y) ω(f, y) dy + cx dy, 0 < x ≤ π, (14) ω(Sf, x) ≤ c y y2 0 x with a positive constant c independent of f ∈ Hω . For a self-contained proof of the boundedness of S on Hω (in a more general situation of moduli of smoothness ωα (f, x) of order α > 0), see S. Samko and A. Yakubov [22, Theorem 2]. If f ∈ H0ω and  x  π ω(f, y) ω(f, y) dy, F2 (f, x) := x F1 (f, x) := dy, y y2 0 x then, by [14, Section IV.4, Lemma 1], lim

x→0+0

F1 (f, x) F2 (f, x) = lim = 0. x→0+0 ω(x) ω(x)

(15)

From (14) and (15) it follows that Sf ∈ H0ω whenever f ∈ H0ω . That is, S is bounded on H0ω , too. Part (b) is proved. (c) For a ∈ Hω , the compactness of aS − SaI on Hω was proved by Tursunkulov [26] (see also a survey by N. Samko [21, Corollary 4.8]). If a ∈ H0ω , then aS − SaI is bounded on H0ω by part (b) and is compact on ω H by what has just been said above. Since H0ω ⊂ Hω , it is easy to see that the
operator aS − SaI is also compact on H0ω . 4.3. Wiener-Hopf factorization in generalized H¨ older spaces Theorem 4.4. Let ω belong to the Bari-Stechkin class and let H be either Hω or H0ω . Suppose a ∈ HN ×N . 2 (a) If T (a) is invertible on HN , then a admits a canonical right Wiener-Hopf factorization in HN ×N . 2 (b) If T ($ a) is invertible on HN , then a admits a canonical left Wiener-Hopf factorization in HN ×N . Proof. We follow the proof of [18, Theorem 2.4]. 2 (a) If T (a) is invertible on HN , then det a(t) = 0 for all t ∈ T (see, e.g., [10, Chap. VII, Proposition 2.1]). Then, by [10, Chap. VII, Theorem 3.2], the matrix function a admits a canonical right generalized factorization in L2N , that ±1 2 2 is, a = a− a+ , where a±1 − ∈ (H )N ×N , a+ ∈ HN ×N (and, moreover, the operator −1 2 a− P a− I is bounded on LN ). On the other hand, from Theorems 4.1 and 4.3 it follows that a ∈ HN ×N admits a right Wiener-Hopf factorization a = u− Du+ in HN ×N . Then 2 u±1 − ∈ (H− )N ×N ⊂ (H )N ×N ,

2 u±1 + ∈ (H+ )N ×N ⊂ HN ×N ,

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A.Y. Karlovich

that is, a = u− Du+ is a right generalized factorization in L2N . By the uniqueness of the partial indices in a right generalized factorization in L2N (see, e.g., [20, Corollary 2.1]), D = 1. Part (a) is proved. (b) In view of Theorem 4.3(a), a−1 ∈ HN ×N . By [7, Proposition 7.19(b)], 2 2 is equivalent to the invertibility of T (a−1 ) on HN . the invertibility of T ($ a) on HN −1 In view of part (a), there exist f± ∈ G(H± )N ×N such that a = f− f+ . Put −1 . Then v± ∈ G(H± )N ×N and a = v+ v− is a canonical left Wiener-Hopf v± := f± factorization in HN ×N .


5. Some applications of approximation theory 5.1. The best uniform approximation For n ∈ Z+ , let P n be the set of all Laurent polynomials of the form n
p(t) = αj tj , αj ∈ C, t ∈ T. j=−n

By the Chebyshev theorem (see, e.g., [25, Section 2.2.1]), for f ∈ C and n ∈ Z+ , there is a Laurent polynomial pn (f ) ∈ P n such that f − pn (f )C = infn f − pC . p∈P

(16)

Each such polynomial pn (f ) is called a polynomial of best uniform approximation. By the Jackson theorem (see, e.g., [25, Section 5.1.2]), there exists a constant A > 0 such that for all f ∈ C and all n ∈ Z+ ,   1 . (17) inf f − pC ≤ Aω f, p∈P n n+1 5.2. Norms of truncations of Toeplitz and Hankel operators Let X be a Banach space. For definiteness, let the norm of a = (aα,β )N α,β=1 in XN ×N be given by aXN ×N := N max aα,β X . 1≤α,β≤N

We will simply write a∞ , aC , and aω instead of aL∞ , aCN ×N , and N ×N a(Hω )N ×N , respectively. Denote by A the norm of a bounded linear operator 2 . A on HN Put ∆0 := P0 and ∆j := Pj − Pj−1 for j ∈ {0, . . . , n}. Lemma 5.1. Let n ∈ Z+ . Suppose v± and u± satisfy (4), (5), and u−1 ± ∈ CN ×N . Then there exists a positive constant AN depending only on N such that for all n ∈ Z+ and all j ∈ {0, . . . , n},   1 −1 , (18) Qn T (b)∆j  ≤ AN v− ∞ max ω [u+ ]α,β , 1≤α,β≤N n−j+1   1 , (19) ∆j T (c)Qn  ≤ AN v+ ∞ max ω [u−1 − ]α,β , 1≤α,β≤N n−j +1

Asymptotic Formulas for Toeplitz Matrices Qn H(b) ≤ H($ c)Qn  ≤

219

  1 −1 , AN v− ∞ max ω [u+ ]α,β , 1≤α,β≤N n+1   1 . AN v+ ∞ max ω [u−1 − ]α,β , 1≤α,β≤N n+1

(20) (21)

Proof. The idea of the proof is borrowed from [7, Theorem 10.35(ii)] (see also −1 ∞ ∞ [18, Proposition 3.2]). Since u−1 + , v+ ∈ HN ×N and u− , v− ∈ (H )N ×N , by [7, Proposition 2.14], T (b) = T (v− )T (u−1 + ),

T (c) = T (u−1 − )T (v+ ), 4 H($ c) = H(u−1 )T (v ).

H(b) = T (v− )H(u−1 + ),

−

+

It is easy to see that Qn T (v− )Pn = 0 and Pn T (v+ )Qn = 0. Hence =

Qn T (v− )Qn T (u−1 + )∆j ,

(22)

=

(23)

Qn H(b) =

∆j T (u−1 − )Qn T (v+ )Qn , Qn T (v− )Qn H(u−1 + ),

(24)

H($ c)Qn

−1 H(u4 − )Qn T (v+ )Qn .

(25)

Qn T (b)∆j ∆j T (c)Qn

=

n−j −1 Let pn−j (u−1 + ) and pn−j (u− ) be the polynomials in PN ×N of best uniform approx−1 imation of u−1 + and u− , respectively, where j ∈ {0, . . . , n}. Simple computations show that

Qn T [pn−j (u−1 + )]∆j = 0,

∆j T [pn−j (u−1 − )]Qn = 0

(26)

for all j ∈ {0, . . . , n} and Qn H[pn (u−1 + )] = 0,

H[(pn (u−1 − ))$]Qn = 0.

(27)

From (22) and (26) we get Qn T (b)∆j 

−1 ≤ Qn T (v− )Qn  QnT [u−1 + − pn−j (u+ )]∆j  −1 ≤ const v− ∞ u−1 + − pn−j (u+ )C .

Combining this inequality with (16)–(17), we arrive at (18). Inequalities (19)– (21) can be obtained in the same way by combining (26)–(27) and representations (23)–(25), respectively.
5.3. The asymptotic of the trace of Fn,m−1 (b, c) Lemma 5.2. Let ω, ψ belong to the Bari-Stechkin class. Suppose v± and u± satisfy (4) and (6). If m ∈ N\{1} and (10) is fulfilled, then trFn,m−1 (b, c) → 0 as n → ∞. Proof. Since ∆j Fn,m−1 (b, c)∆j is an N × N matrix for each n ∈ Z+ and each j ∈ {0, . . . , n}, we have |tr∆j Fn,m−1 (b, c)∆j | ≤ CN ∆j Fn,m−1 (b, c)∆j ,

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A.Y. Karlovich

where CN is a positive constant depending only on N . Hence # # #
# n
# n # # |trFn,m−1 (b, c)| = #tr ∆j Fn,m−1 (b, c)∆j ## ≤ CN ∆j Fn,m−1 (b, c)∆j . (28) # j=0 # j=0 Taking into account that ∆j Pn = Pn ∆j = ∆j for j ∈ {0, . . . , n}, we obtain ∆j Fn,m−1 (b, c)∆j  ≤ ∆j T (c)Qn  (Qn H(b) H($ c)Qn )m−1 Qn T (b)∆j . (29) From Lemma 5.1 and the definition of the semi-norms | · |ω and | · |ψ it follows that for all n ∈ Z+ and all j ∈ {0, . . . , n},     1 −1 Qn T (b)∆j  ≤ AN v− ∞ , (30) max |[u+ ]α,β |ψ ψ 1≤α,β≤N n−j+1     1 ∆j T (c)Qn  ≤ AN v+ ∞ , (31) max |[u−1 − ]α,β |ω ω 1≤α,β≤N n−j+1     1 , (32) max |[u−1 Qn H(b) ≤ AN v− ∞ + ]α,β |ψ ψ 1≤α,β≤N n+1     1 H($ c)Qn  ≤ AN v+ ∞ . (33) max |[u−1 − ]α,β |ω ω 1≤α,β≤N n+1 Combining (28)–(33), we get   |trFn,m−1 (b, c)| = O ω

1 n+1



 ψ

1 n+1

 m−1 n+1
1 1 ψ ω j j j=1

as n → ∞. This implies that if (10) holds, then trFn,m−1 (b, c) → 0 as n → ∞.




5.4. Products of Hankel operators in Schatten-von Neumann classes Lemma 5.3. Let 1 ≤ p < ∞, let ω, ψ belong to the Bari-Stechkin class, and let ∞     p
1 1 ψ ω < ∞. k k k=1

(a) Suppose a ∈ L∞ N ×N admits a factorization a = u− u+ with u± satisfying (6). 2 ). Then H(a)H($ a−1 ) ∈ Cp (HN c)H(b) and H(b)H($ c) belong (b) Suppose v± and u± satisfy (4) and (6). Then H($ 2 to Cp (HN ). Proof. This statement is proved by analogy with [7, Lemma 10.36]. Let us prove only part (a). By [7, Proposition 2.14], H(a) = T (u− )H(u+ ) and H($ a−1 ) = −1 )H(u−1 ). For n ∈ Z , let p (u ) and p (u−1 ) be the polynomials in P n T (u4 +

−

+

n

+

n

−

of best uniform approximation of u+ and u−1 − , respectively. Observe that dim Im (T (u− )H[pn (u+ )]) ≤ n + 1,

N ×N

−1 −1 dim Im (T (u4 + )H[pn (u− )]) ≤ n + 1,

whence sn+1 (H(a)) ≤ T (u− )H(u+ ) − T (u−)H[pn (u+ )] ≤ O(u+ − pn (u+ )C )

(34)

Asymptotic Formulas for Toeplitz Matrices

221

−1 sn+1 (H($ a−1 )) ≤ O(u−1 − − pn (u− )C ).

(35)

and similarly From (16), (17), and the definition of the seminorms | · |ω and | · |ψ it follows that     1 , (36) u+ − pn (u+ )C ≤ AN max |[u+ ]α,β |ψ ψ 1≤α,β≤N n+1     1 −1 u−1 , (37) max |[u−1 ≤ AN − − pn (u− )C − ]α,β |ω ω 1≤α,β≤N n+1 where AN is a positive constant depending only on ω, ψ, N . Combining (34)–(37), we get     a−1 )) = O ω(1/n) (n ∈ N). (38) sn (H(a)) = O ψ(1/n) , sn (H($ From (38) and Horn’s theorem (see, e.g., [12, Chap. II, Theorem 4.2]) it follows that  ∞       ∞ ∞ p

 p 
 1 1 p p −1 −1 ψ , ω a ) ≤ a ) =O sk H(a)H($ sk H(a) sk H($ k k k=1

k=1

k=1

which finishes the proof of part (a). Part (b) is proved similarly.




6. Proofs of the main results 6.1. Decomposition of the trace of the logarithm of one matrix series Lemma 6.1. Let ω, ψ belong to the Bari-Stechkin class and let Σ be a compact set in the complex plane. Suppose v− : Σ → (H ∞ )N ×N ,

∞ v+ : Σ → HN ×N ,

±1 ω ψ ∞ ∞ u±1 − : Σ → (H ∩ H )N ×N , u+ : Σ → (H ∩ H )N ×N

are continuous functions. If m ∈ N, then there exist a constant Cm ∈ (0, ∞) and a number n0 ∈ N such that for all λ ∈ Σ and all n ≥ n0 , ⎡  % & j ⎤ ∞ m−1

1 m−j−1
    ⎦ = sn (λ) trlog I − Gn,k b(λ), c(λ) +tr⎣ Gn,k b(λ), c(λ) j j=1 k=0

k=0

and |sn (λ)|

≤

 m Cm v− (λ)∞ v+ (λ)∞ m   1 −1 × max ω [u− (λ)]α,β , 1≤α,β≤N n+1 m   1 −1 × max ω [u+ (λ)]α,β , . 1≤α,β≤N n+1

222

A.Y. Karlovich

Proof. From Lemma 5.1 it follows that 8  8   8Gn,k b(λ), c(λ) 8 ≤ A2N v− (λ)∞ v+ (λ)∞ k+1 k+1   1 −1 × max ω [u− (λ)]α,β , 1≤α,β≤N n+1 k+1   1 × max ω [u−1 (λ)] , . α,β + 1≤α,β≤N n+1 for all n, k ∈ Z+ and all λ ∈ Σ. Moreover, A2N v− (λ)∞ v+ (λ)∞   × max ω [u−1 − (λ)]α,β ,

 1 max ω 1≤α,β≤N 1≤α,β≤N n+1       1 1 −1 −1 2 ψ . ≤ AN max v− (λ)∞ v+ (λ)∞ u− (λ)ω u+ (λ)ψ ω λ∈Σ n+1 n+1 1 n+1

 



[u−1 + (λ)]α,β ,

Since ω(1/n) → 0 and ψ(1/n) → 0 as n → ∞, there exists a number n0 ∈ N such that the left-hand side of the latter inequality is less than one for all λ ∈ Σ and all n ≥ n0 . Now the proof can be developed by analogy with [17, Proposition 3.3].
6.2. Proof of Theorem 2.1 Proof of part (a). Since ω and ψ belong to the Bari-Stechkin class, there exist α, β ∈ (0, 1) such that ω(x)/xα and ψ(x)/xβ are almost increasing (see [1, Lemma 2] or [14, p. 54]). Hence there exists a constant A > 0 such that ω(x) ≤ Axγ and ψ(x) ≤ Axγ for all x ∈ (0, π], where γ := min{α, β} ∈ (0, 1). Therefore γ γ ψ γ u− ∈ (Hω )N ×N ⊂ CN ×N and u+ ∈ (H )N ×N ⊂ CN ×N , where C is the stanγ 2 a) is invertible on HN and dard H¨ older space generated by h(x) = x . Since T ($ γ a = u− u+ ∈ CN ×N , by Theorem 4.4(b), the function a admits a canonical left γ ∞ Wiener-Hopf factorization a = v+ v− in CN ×N . In particular, v− ∈ G(H )N ×N ∞ and v+ ∈ GHN .
×N Proof of parts (b) and (c). From Theorem 2.1(a) it follows that hypotheses (i) and (ii) of Theorem 3.1 are satisfied. Suppose m ∈ N and (8) holds. In view of Lemma 5.3 and [7, Proposition 2.14], 2 I−T (a)T (a−1 ) = H(a)H($ a−1 ) ∈ Cm (HN ),

2 I−T ($ c)T ($b) = H($ c)H(b) ∈ Cm (HN ),

2 ). Hence Theorems 2.1(b) and 2.1(c) follow from Theoand H(b)H($ c) ∈ Cm (HN rems 3.1(a) and 3.1(b).


Proof of part (d). By Lemma 5.2, trFn,m−1 (b, c) as n → ∞. Hence the statement follows from the arguments of the proof of part (c) and Theorem 3.1(c).
Proof of part (e). Suppose Σ consists of one point λ only (and we will not write the dependence on it). From Lema 6.1 it follows that there exist a positive constant

Asymptotic Formulas for Toeplitz Matrices

223

Cm and a number n0 ∈ N such that for all n ≥ n0 , ⎡  & j ⎤ % ∞ m−1

1 m−j−1
Gn,k (b, c) = −tr ⎣ Gn,k (b, c) ⎦ + sn , tr log I − j j=1 k=0

(39)

k=0

where

  −1  u  ω |sn | ≤ Cm v− ∞ v+ ∞ u−1 − ω + ψ

1 n+1



 ψ

1 n+1

m .

(40)

∞ From (8) and (40) we get n=n0 |sn | < ∞. Applying Lemma 3.2 to the decomposition (39), we conclude that there exists a constant E(a) = 0 such that for all n ≥ n0 , log det Tn (a) =(n + 1) log G(a) + log E(a) ⎡  j ⎤ n m−1 ∞

1 m−j−1

G,k (b, c) ⎦ + sk . + tr ⎣ j j=1 =1

From (40) we get



∞


sk = O

k=n+1

k=0

(41)

k=n+1

    m  ∞
1 1 ψ ω k k

(n → ∞).

(42)

k=n+1

Combining (41) and (42), we arrive at (12). Part (e) is proved. Proof of part (f). In view of (41), it is sufficient to show that  ∞       ∞ m

1 1 ψ (n → ∞). ω sk = o k k k=n+1




(43)

k=n+1

By Lemma 6.1, for all k ≥ n0 ,  m |sk | ≤ Cm v− ∞ v+ ∞ m   1 −1 × max ω [u− ]α,β , 1≤α,β≤N k+1 m   1 −1 × max ω [u+ ]α,β , . 1≤α,β≤N k+1

(44)

If u− ∈ G(H0ω ∩ H ∞ )N ×N , then for every ε > 0 there exists a number n1 (ε) ≥ n0 such that for all k ≥ n1 (ε),     1 1 −1 < εω . (45) max ω [u− ]α,β , 1≤α,β≤N k+1 k+1 From (44) and (45) it follows that for all n ≥ n1 (ε), ∞
k=n+1

|sk | ≤ ε

m



m Cm v− ∞ v+ ∞ u−1 + ψ

    m ∞
1 1 ψ ω , k k

k=n+1

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A.Y. Karlovich

that is, (43) holds. If u+ ∈ G(H0ψ ∩ H ∞ )N ×N , then one can show as above that (43) is fulfilled.
6.3. Auxiliary lemma Lemma 6.2. Let ω belong to the Bari-Stechkin class. If Σ is a compact set in the complex plane and a : Σ → (H0ω )N ×N is a continuous function, then %    &  −1 1 1 = 0. ω sup max ω [a(λ)]α,β , lim n→∞ n n λ∈Σ 1≤α,β≤N Proof. Assume the contrary. Then there exist a constant C > 0 and a sequence {nk }∞ k=1 such that %    &  −1 1 1 ω = C. sup max ω [a(λ)]α,β , lim k→∞ nk nk λ∈Σ 1≤α,β≤N Hence there exist a number k0 ∈ N and a sequence {λk }∞ k=k0 such that for all k ≥ k0 ,   −1   C 1 1 > 0. (46) ω ≥ max ω [a(λk )]α,β , 1≤α,β≤N nk nk 2 ∞ Since {λk }∞ k=k0 is bounded, there is its convergent subsequence {λkj }j=1 . Let λ0 be the limit of this subsequence. Clearly, λ0 ∈ Σ because Σ is closed. Since the function a : Σ → (H0ω )N ×N is continuous at λ0 , for every ε ∈ (0, C/2), there exists a ∆ > 0 such that |λ − λ0 | < ∆, λ ∈ Σ implies a(λ) − a(λ0 )ω < ε. Because λkj → λ0 as j → ∞, for that ∆ there exists a number J ∈ N such that |λkj − λ0 | < ∆ for all j ≥ J, and thus a(λkj ) − a(λ0 )ω < ε

for all j ≥ J.

On the other hand, (46) implies that −1     C 1 1 > 0 for all j ≥ J. ≥ ω max ω [a(λkj )]α,β , 1≤α,β≤N nkj nkj 2

(47)

(48)

It is easy to see that if f, g ∈ Hω , then for all x ∈ (0, π], ω(g, x) ω(f, x) ≤ + |f − g|ω . ω(x) ω(x) Hence, for all j ≥ J, −1     1 1 ω max ω [a(λkj )]α,β , 1≤α,β≤N nkj nkj −1     1 1 + a(λkj ) − a(λ0 )ω . max ω [a(λ0 )]α,β , ≤ ω 1≤α,β≤N nkj nkj From (47)–(49) we get for all j ≥ J, −1     C 1 1 − ε > 0. ≥ max ω [a(λ0 )]α,β , ω 1≤α,β≤N nkj nkj 2

(49)

Asymptotic Formulas for Toeplitz Matrices

225

It follows that there exist a pair α0 , β0 ∈ {1, . . . , N } and a subsequence {ms }s∈N of {nkj }∞ j=J such that for all s ∈ N, −1     C 1 1 − ε > 0. ≥ ω [a(λ0 )]α0 ,β0 , ω ms ms 2 This contradicts the fact that [a(λ0 )]α0 ,β0 ∈ H0ω .




6.4. Proof of Theorem 2.2 Proof of part (a). Similarly to the proof of Lemma 5.3 one can show that if a belongs to (Hω )N ×N , then         sn H(a) = O ω(1/n) , sn H($ a) = O ω(1/n) (n ∈ N). (50) 2 ). It is well known that Combining (13) and (50), we get H(a), H($ a) ∈ C2 (HN 5 6 1/2,1/2 2 (K2,2 )N ×N = a ∈ L∞ a) ∈ C2 (HN ) N ×N : H(a), H($

(see [6, Section 5.1], [7, Sections 10.8–10.11]), which finishes the proof.




Proof of part (b). By Theorem 4.4, the function a admits canonical right and left Wiener-Hopf factorizations in HN ×N . From Theorem 2.1(b) we get log det Tn (a) = (n + 1) log G(a) + log det1 T (a)T (a−1 ) + o(1) (n → ∞).

(51)

On the other hand, from Theorem 2.1(e), (f) it follows that there exists a nonzero constant E(a) such that log det Tn (a) = (n + 1) log G(a) + log E(a) + δ(n, H) (n → ∞). From (51) and (52) we deduce that E(a) = det1 T (a)T (a (1) with o(1) replaced by δ(n, H).

−1

(52)

), that is, we arrive at


Proof of part (c). This statement is proved by analogy with [17, Theorem 1.5] and [18, Theorem 1.4], although the idea of this proof goes back to [27, Theorem 6.2]. Let Ω be any bounded open set containing the set sp T (a)∪sp T ($ a) on the closure of which f is analytic and let Σ be a closed neighborhood of its boundary ∂Ω such that Σ ∩ (sp T (a) ∪ sp T ($ a)) = ∅. Let λ ∈ Σ. Then T (a) − λI = T [a − λ] and T ($ a) − λI = 2 T [(a − λ)$] are invertible on HN . By Theorem 4.4, a − λ admits canonical right and left Wiener-Hopf factorizations a − λ = u− (λ)u+ (λ) = v+ (λ)v− (λ) in HN ×N . Since a − λ : Σ → HN ×N is a continuous function with respect to λ, in view of Theorem 4.2, these factorizations can be chosen so that the functions ±1 ∞ u±1 − , v− : Σ → (H ∩ H )N ×N ,

±1 ∞ u±1 + , v+ : Σ → (H ∩ H )N ×N

are continuous. From Lemma 6.1 with m = 1 it follows that there exist a constant C1 ∈ (0, ∞) and a number n0 ∈ N such that for all λ ∈ Σ and all n ≥ n0 ,    1 −1 max ω [u− (λ)]α,β , |sn (λ)| ≤C1 v− (λ)∞ v+ (λ)∞ 1≤α,β≤N n+1    (53) 1 , (λ)] , × max ω [u−1 α,β + 1≤α,β≤N n+1

226

A.Y. Karlovich

where

% sn (λ) = tr log I −

∞


Gn,k



 b(λ), c(λ)

& .

k=0

If a ∈ (Hω )N ×N , then from (53) we get for all λ ∈ Σ and all n ≥ n0 , 2     1 −1 −1 |sn (λ)| ≤ C1 max v− (λ)ω v+ (λ)ω u− (λ)ω u+ (λ)ω ω . (54) λ∈Σ n+1 If a ∈ (H0ω )N ×N , then from Lemma 6.2 it follows that for every ε > 0 there exists a number n1 (ε) ≥ n0 such that for all λ ∈ Σ and all n ≥ n1 (ε),     1 1 < εω . (55) max ω [u−1 (λ)] , α,β ± 1≤α,β≤N n+1 n+1 Combining (53) and (55), we obtain for all λ ∈ Σ and all n ≥ n1 (ε), 2     1 |sn (λ)| ≤ ε2 C1 max v− (λ)ω v+ (λ)ω ω . λ∈Σ n+1 From (54) and (56) we get for all λ ∈ Σ, ∞ ∞   2

1 ω |sk (λ)| ≤ const k k=n+1 k=n+1 2   ∞ ∞

1 2 ω |sk (λ)| ≤ ε const k k=n+1

(56)

if

a ∈ (Hω )N ×N , n ≥ n0 ,

if

a ∈ (H0ω )N ×N , n ≥ n1 (ε). (58)

(57)

k=n+1

From Lemma 3.2 and Theorem 2.1(b) it follows that for all λ ∈ Σ and n ≥ n0 , log det Tn (a− λ) = (n+ 1) log G(a− λ)+ log det1 T [a− λ]T [(a− λ)−1]+

∞


sk (λ).

k=n+1

Multiplying this equality by −f  (λ) and then integrating over ∂Ω by parts, we get   d d log det Tn (a − λ)dλ = (n + 1) log G(a − λ)dλ f (λ) f (λ) dλ dλ ∂Ω ∂Ω  ∞   (59)
 + 2πiEf (a) − f (λ) sk (λ) dλ. ∂Ω

k=n+1

It was obtained in the proof of [27, Theorem 6.2] (see also [6, Theorem 5.6] and [7, Section 10.90]) that  d 1 log det Tn (a − λ)dλ = trf (Tn (a)), f (λ) (60) 2πi ∂Ω dλ  d 1 log G(a − λ)dλ = Gf (a). f (λ) (61) 2πi ∂Ω dλ

Asymptotic Formulas for Toeplitz Matrices From (57) and (58) it follows that  ∞  
 − f (λ) sk (λ) dλ = δ(n, H) (n → ∞). ∂Ω

227

(62)

k=n+1

Combining (59)–(62), we arrive at (2) with o(1) replaced by δ(n, H).




References [1] N.K. Bari and S.B. Stechkin, Best approximation and differential properties of two conjugate functions. Trudy Moskovsk. Matem. Obshch., 5 (1956), 483–522 (in Russian). [2] A. B¨ ottcher and S.M. Grudsky, Spectral Properties of Banded Toeplitz Operators. SIAM, Philadelphia, PA, 2005. [3] A. B¨ ottcher, A.Yu. Karlovich, and B. Silbermann, Generalized Krein algebras and asymptotics of Toeplitz determinants. arXiv:math.FA/0612529. [4] A. B¨ ottcher and B. Silbermann, Notes on the asymptotic behavior of block Toeplitz matrices and determinants. Math. Nachr., 98 (1980), 183–210. [5] A. B¨ ottcher and B. Silbermann, Invertibility and Asymptotics of Toeplitz Matrices. Akademie-Verlag, Berlin, 1983. [6] A. B¨ ottcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. Springer-Verlag, New York, 1999. [7] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators. 2nd edition. SpringerVerlag, Berlin, 2006. [8] A. B¨ ottcher and H. Widom, Szeg˝ o via Jacobi. Linear Algebra Appl., 419 (2006), 656–667. [9] M.S. Budjanu and I.C. Gohberg, General theorems on the factorization of matrixvalued functions. II. Some tests and their consequences. Amer. Math. Soc. Transl. (2), 102 (1973), 15–26. [10] K.F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators. Birkh¨ auser Verlag, Basel, 1981. [11] M.E. Fisher, R.E. Hartwig, Asymptotic behavior of Toeplitz matrices and determinants. Arch. Rational Mech. Anal., 32 (1969), 190–225. [12] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators. AMS, Providence, RI, 1969. [13] U. Grenander and G. Szeg˝ o, Toeplitz Forms and Their Applications. University of California Press, Berkeley, Los Angeles, 1958. [14] A.I. Guseinov and H.Sh. Mukhtarov, Introduction to the Theory of Nonlinear Singular Integral Equations. Nauka, Moscow, 1980 (in Russian). [15] R. Hagen, S. Roch, and B. Silbermann, C ∗ -Algebras and Numerical Analysis. Marcel Dekker, Inc., New York, 2001. [16] A.Yu. Karlovich, Higher order asymptotics of Toeplitz determinants with symbols in weighted Wiener algebras. J. Math. Anal. Appl., 320 (2006), 944–963.

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[17] A.Yu. Karlovich, Asymptotics of determinants and traces of Toeplitz matrices with symbols in weighted Wiener algebras. Z. Anal. Anwendungen, 26 (2007), 43–56. [18] A.Yu. Karlovich, Higher order asymptotic formulas for traces of Toeplitz matrices with symbols in H¨ older-Zygmund spaces. Proceedings of IWOTA 2005, Operator Theory: Advances and Applications, to appear. [19] M.G. Krein, Certain new Banach algebras and theorems of the type of the WienerL´evy theorems for series and Fourier integrals. Amer. Math. Soc. Transl. (2), 93 (1970), 177–199. [20] G.S. Litvinchuk and I.M. Spitkovsky, Factorization of Measurable Matrix Functions. Birkh¨ auser Verlag, Basel, 1987. [21] N. Samko, On compactness of integral operators with a generalized weak singularity in weighted spaces of continuous functions with a given continuity modulus. Proc. Razmadze Math. Institute, 136 (2004), 91–113. [22] S. Samko and A. Yakubov, A Zygmund estimate for the modulus of continuity of fractional order of the conjugate function. Izv. Vyssh. Uchebn. Zaved. (1985), no. 12, 66–72 (in Russian). [23] B. Simon, Notes on infinite determinants of Hilbert space operators. Advances in Math., 24 (1977), 244–273. [24] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. AMS, Providence, RI, 2005. [25] A.F. Timan, Theory of Approximation of Functions of a Real Variable. Pergamon Press, Oxford, 1963. [26] B. Tursunkulov, Completely continuous operators in generalized H¨ older spaces. Dokl. Akad. Nauk Uzbek. SSR (1982), no. 12, 4–6 (in Russian). [27] H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II. Advances in Math., 21 (1976), 1–29. [28] A. Zygmund, O module ciaglo´sci sumy szeregu spz¸ez˙ onego z szeregiem Fouriera. Prace Mat.-Fiz., 33 (1924), 125–132 (in Polish). Alexei Yu. Karlovich Departamento de Matem´ atica, Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049–001, Lisbon, Portugal e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 229–261 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Nonlocal Singular Integral Operators with Slowly Oscillating Data Yuri I. Karlovich Abstract. The paper is devoted to studying the of(nonlocal)    Fredholmness a− singular integral operators with shifts N = a+ g V g P+ + g Vg P− p on weighted Lebesgue spaces L (Γ, w) where 1 < p < ∞, Γ is an unbounded slowly oscillating Carleson curve, w is a slowly oscillating Muckenhoupt weight, the operators P± = 12 (I ± SΓ ) are related to the Cauchy singular integral operator SΓ , a± g are slowly oscillating coefficients, Vg are shift operators given by Vg f = f ◦ g, and g are slowly oscillating shifts in a finite subset of a subexponential group G acting topologically freely on Γ. The Fredholm criterion for N consists of two parts:  ±of an invertibility criterion for polynomial functional operators A± = ag Vg in terms of invertibility of corresponding discrete operators on the space lp (G), and of a condition of local Fredholmness of N at the endpoints of Γ established by applying Mellin pseudodifferential operators with compound slowly oscillating V (R)-valued symbols where V (R) is the Banach algebra of absolutely continuous functions of bounded total variation on R. Mathematics Subject Classification (2000). Primary 47G10, 47A53; Secondary 45E05, 47G30. Keywords. Singular integral operator with shifts, slowly oscillating data, Fredholmness, weighted Lebesgue space, Mellin pseudodifferential operator, compound symbol.

1. Introduction Let B(X) be the Banach algebra of all bounded linear operators acting a Banach space X, and let K(X) be the closed two-sided ideal of all compact operators in B(X). An operator A ∈ B(X) is said to be Fredholm, if its image is closed and the spaces ker A and ker A∗ are finite-dimensional or, equivalently, if the coset A + K(X) is invertible in the Calkin algebra B(X)/K(X) (see, e.g., [13]). Partially supported by the SEP-CONACYT Project No. 25564 (M´exico).

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Let Γ be an oriented rectifiable simple open arc in the complex plane, and let Lp (Γ, w) be the weighted Lebesgue space with the norm  1/p |f (τ )|p |w(τ )|p |dτ | f Lp(Γ,w) := Γ

where 1 < p < ∞ and w : Γ → [0, ∞] is a measurable function with w−1 ({0, ∞}) of measure zero. As is known (see, e.g., [15], [11], [12] and also [6]), the Cauchy singular integral operator SΓ , given for f ∈ L1 (Γ) and almost all t ∈ Γ by  5 6 1 f (τ ) dτ, Γ(t, ε) := τ ∈ Γ : |τ − t| < ε , (1.1) (SΓ f )(t) := lim ε→0 πi Γ\Γ(t,ε) τ − t is bounded on the space Lp (Γ, w) if and only if w ∈ Ap (Γ), that is,  1/p   1/q 1 w(τ )p |dτ | w(τ )−q |dτ | < ∞, sup sup ε>0 t∈Γ ε Γ(t,ε) Γ(t,ε)

(1.2)

where 1/p + 1/q = 1. If (1.2) holds, then H¨ older’s inequality implies that sup sup |Γ(t, ε)|/ε < ∞,

(1.3)

ε>0 t∈Γ

where |Γ(t, ε)| stands for the Lebesgue (length) measure of Γ(t, ε). Condition (1.2) is called the Muckenhoupt condition. The curves Γ satisfying (1.3) are named Carleson or Ahlfors-David curves (see [6]). Let G be a discrete group of orientation-preserving diffeomorphisms of Γ onto itself that satisfy the conditions ln |g  | ∈ Cb (Γ),

(ω ◦ g)/ω, ω/(ω ◦ g) ∈ L∞ (Γ)

for all g ∈ G,

(1.4)

where Cb (Γ) is the C ∗ -algebra of all bounded continuous complex-valued functions on Γ. If (1.4) holds, then for every g ∈ G, the shift operator Vg given by Vg f = f ◦g is bounded on the space Lp (Γ, w). We say that the action of the group G on Γ is topologically free (cf. [1]) if for each finite set F ⊂ G and each arc γ ⊂ Γ there exists a point τ ∈ γ such that the points g(τ ) (g ∈ F ) are pairwise distinct. A group G is called subexponential [1] if for every finite subset F ⊂ G, lim |F n |1/n = 1 where n→∞

|F n | is the number of different words of the length n constituted by the elements g ∈ F . In what follows we assume that the group G is subexponential and G acts on Γ topologically freely. The paper is devoted to studying the Fredholmness of (nonlocal) singular integral operators with shifts on weighted Lebesgue spaces Lp (Γ, w) where 1 < p < ∞, Γ is an unbounded Carleson curve, and w ∈ Ap (Γ). Here we consider nonlocal singular integral operators of the form 
  
− N= P P− V + V (1.5) a+ a g + g g g where P± = 12 (I ± SΓ ), a± g ∈ Cb (Γ), Vg are shift operators, and g runs through a finite subset F ⊂ G. Below we assume that all the data (the contour, the weight, the coefficients, and the shifts) are slowly oscillating.

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Singular integral operators with piecewise continuous coefficients and cyclic infinite groups of shifts, as well as algebras of such operators, were studied by different methods by V.G. Kravchenko and the author [24]–[25], A.G. Myasnikov and L.I. Sazonov [32]–[33], and A.P. Soldatov [42]–[43] (also see [31] and the references therein). The Fredholmness on the spaces Lp (Γ) (1 < p < ∞) for singular integral operators with discrete subexponential groups of shifts acting topologically freely on Γ and Banach algebras of such operators were studied in [19] and [29] in the case of piecewise smooth contours, piecewise continuous coefficients and piecewise smooth shifts. The present paper generalizes the Fredholm results of [19] and [29] to singular integral operators with discrete subexponential groups of shifts on weighted Lebesgue spaces Lp (Γ, w) in the case of slowly oscillating data. Singular integral operators with a backward slowly oscillating involutive shift were earlier investigated in [26], singular integral operators of the form Vα P+ + GP− with slowly oscillating coefficients G and slowly oscillating shifts α : R+ → R+ generating infinite cyclic groups were studied in [23]. To study nonlocal operators of the form (1.5), we apply the general method developed in [19] and [28] and generalizing the method of [25]. The Fredholm criterion for N consists of two parts: of an invertibility criterion for the functional  operators A± = a± g Vg and of a condition of local Fredholmness for the operator N at the endpoints of the contour Γ. The local invertibility theory for functional operators A± is constructed in terms of invertibility of corresponding discrete operators on the space lp (G) by analogy with [29]. The local Fredholmness of N at the endpoints of Γ is studied by applying the results of [21]–[23] on Mellin (and Fourier) pseudodifferential operators with usual and compound (double) slowly oscillating V (R)-valued symbols where V (R) is the Banach algebra of absolutely continuous functions of bounded total variation on R. The crucial role plays the fact that the product of each shift operator Vg and an integral operator RΓ having point singularities at the endpoints of Γ is similar to a Mellin pseudodifferential operator to within compact operators. Studying algebras of Mellin pseudodifferential operators with H¨ ormander’s classes of slowly oscillating symbols and its applications to studying algebras of singular integral operators (without shifts) with slowly oscillating data was earlier carried out by V.S. Rabinovich [35]–[37] and by A. B¨ ottcher, V.S. Rabinovich and the author [7]–[9] (also see [39] and the references therein). Pseudodifferential operators on Rn with shifts slowly varying at infinity were studied in [38]. Another approach suitable for studying C ∗ -algebras of singular integral operators with discrete subexponential and amenable groups of shifts, which are associated with C ∗ -dynamical systems, was developed by A.B. Antonevich and A.V. Lebedev [1], [4] (for applications see [1]–[3]) and by the author [17], [20] (see applications in [5] and [18]). The paper is organized as follows. In Section 2 we introduce slowly oscillating data. In Section 3 we collect necessary results on Mellin pseudodifferential operators. Section 4 deals with studying properties of special slowly oscillating functions and symbols. In Section 5 we apply the results on Mellin pseudodifferential opera-

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tors described in Section 3 to singular integral operators. In Section 6 we study the invertibility of functional operators with slowly oscillating coefficients and slowly oscillating shifts on the weighted Lebesgue space Lp (Γ, w) where 1 < p < ∞, Γ is an unbounded slowly oscillating Carleson curve, and w is a slowly oscillating Muckenhoupt weight. In Section 7 we establish a Fredholm criterion for singular integral operators (1.5) with shifts on the space Lp (Γ, w) in the case of slowly oscillating data. As a corollary, an explicit Fredholm criterion for the operator T = Vα P+ + GP− ∈ B(Lp (Γ, w)) with slowly oscillating data is obtained (cf. [23]).

2. Slowly oscillating data Following [9], we introduce the slowly oscillating data. Slowly oscillating functions. Let R+ = (0, ∞) and let SO(R+ ) stand for the set of all functions a ∈ Cb (R+ ) := C(R+ ) ∩ L∞ (R+ ) which are slowly oscillating at 0 and ∞, that is (see, e.g., [40]), satisfy the condition ' ( lim max |a(x) − a(y)| : x, y ∈ [r, 2r] = 0, for s ∈ {0, ∞}. (2.1) r→s

It is clear that (2.1) is equivalent to the condition ' ( lim max |a(r) − a(νr)| : ν ∈ [λ−1 , λ] = 0, r→s

for s ∈ {0, ∞},

(2.2)

with any λ > 1. Obviously, SO(R+ ) is a unital C ∗ -subalgebra of L∞ (R+ ). Let M (A) be the maximal ideal space of a unital commutative C ∗ -algebra A. Identifying the points t ∈ R+ := [0, +∞] with the evaluation functionals t(f ) = f (t) for f ∈ C(R+ ), we get M (C(R+ )) = R+ . Consider the fibers # 5 6 Ms (SO(R+ )) := ξ ∈ M (SO(R+ )) : ξ #C(R+ ) = s of the maximal ideal space M (SO(R+ )) over the points s ∈ {0, ∞}. Applying the map R+ → R, x → ln x, we immediately infer the following two assertions from [21, Propositions 2.4 and 2.5] (also see [9, Section 2]). Proposition 2.1. We have M := M0 (SO(R+ )) ∪ M∞ (SO(R+ )) = clos SO∗ R+ \ R+

(2.3)

where clos SO∗ R+ is the weak-star closure of R+ in SO(R+ ), the dual space of SO(R+ ). Proposition 2.2. Let {ak }∞ k=1 be a countable subset of SO(R+ ) and s ∈ {0, ∞}. For each ξ ∈ Ms (SO(R+ )) there is a sequence {xn } ⊂ R+ such that xn → s as n → ∞ and (2.4) ξ(ak ) = lim ak (xn ) f or all k = 1, 2, . . . n→∞

Conversely, if {xn } ⊂ R+ , xn → s as n → ∞, and the limits limn→∞ ak (xn ) exist for all k = 1, 2, . . . , then there is a ξ ∈ Ms (SO(R+ )) such that (2.4) holds.

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In what follows we write a(ξ) := ξ(a) for every a ∈ SO(R+ ) and every ξ ∈ M. Slowly oscillating curves. An unbounded oriented simple open arc Γ with the starting point t is called a slowly oscillating curve (at t and ∞) if ' ( Γ = τ = t + reiθ(r) : r ∈ R+ (2.5) where θ is a real-valued function in C 3 (R+ ) and the functions (rDr )j θ belong  j ∞ to SO(R+ ) for all j = 1, J 2, 3 (here K (rDr )θ = rθ (r)). Hence, (rDr ) θ ∈ L (R+ ) j for j = 1, 2, 3, and lim (rDr ) θ (r) = 0 for j = 2, 3 and s ∈ {0, ∞}. Note that r→s

t, ∞ ∈ / Γ and the function θ(r) may be unbounded as r → 0 or r → ∞. Since 9 |dτ | = 1 + |rθ (r)|2 dr and |rθ (r)| ≤ M < ∞ for all r ∈ R+ , we infer from (2.5) that  |τ −t|+ε |Γ(τ, ε)| ≤

max{|τ −t|−ε,0}

9 9 1 + |rθ (r)|2 dr ≤ 2ε 1 + M 2

for τ ∈ Γ,

and thus Γ is a Carleson curve. Slowly oscillating weights. Let Γ be a slowly oscillating curve given by (2.5). We call a function w : Γ → (0, ∞) a slowly oscillating weight (at t and ∞) if   (2.6) w t + reiθ(r) = ev(r) for r ∈ R+ , where v is a real-valued function in C 3 (R+ ) and the functions (rDr )j v are in SO(R+ ) for all j = 1, 2, 3. One can show (see, e.g., [6, Theorem 2.36] and [27, Section 5]) that w ∈ Ap (Γ) if and only if −1/p < lim inf rv  (r) ≤ lim sup rv  (r) < 1/q r→s

r→s

for s ∈ {0, ∞}.

We denote by ASO the set of all pairs (Γ, w) such that Γ is a slowly oscillating p curve and w is a slowly oscillating weight in Ap (Γ). and let α be an orientation-preserving Slowly oscillating shifts. Let (Γ, w) ∈ ASO p diffeomorphism of Γ onto itself such that ln |α | ∈ Cb (Γ),

(w ◦ α)/w, w/(w ◦ α) ∈ L∞ (Γ),

(2.7)

which implies the boundedness of the shift operator Vα given by Vα f = f ◦ α and its inverse operator Vα−1 on the space Lp (Γ, w). We call α a slowly oscillating shift (at t and ∞) if    for r ∈ R+ , (2.8) α(t + reiθ(r) ) = t + reω(r) exp iθ reω(r) where ω is a real-valued function in C 3 (R+ ) and the functions (rDr )j ω belong to SO(R+ ) for all j = 0, 1, 2, 3. Observe that the slow oscillation of ω at t and ∞ is equivalent to the property: lim rω  (r) = 0,

r→0

lim rω  (r) = 0.

r→∞

(2.9)

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  If α is a slowly oscillating shift, then the function r → α t + reiθ(r) , where    ω(r)   ω(r)  1 + rω  (r)  ω(r)  1 + ire re exp ω(r) + iθ re − iθ(r) (2.10) α (τ ) = θ 1 + irθ (r) for τ = t + reiθ(r) ∈ Γ, belongs to SO(R+ ). Since ln |α | ∈ Cb (Γ), we conclude due to (2.9) that   inf 1 + rω  (r) > 0. (2.11) r∈R+

Slowly oscillating coefficients. Let (Γ, w) ∈ ASO p . We denote by SO(Γ) the set of all functions cΓ : Γ → C such that   cΓ t + reiθ(r) = c(r) for r ∈ R+ , (2.12) and c ∈ SO(R+ ). Below we assume that the coefficients of singular integral operators with shifts are in SO(Γ).

3. Mellin pseudodifferential operators In this section we collect necessary results on Mellin pseudodifferential operators. Let V (R) be the set of all absolutely continuous functions a : R → C of bounded total variation V (a) where % n &
# # # # V (a) := sup a(xk ) − a(xk−1 ) : −∞ < x0 < x1 < · · · < xn < +∞, n ∈ N . k=1

Hence (see, e.g., [14, Chapter 9]), there exist finite one-sided limits a(±∞) = limx→±∞ a(x), and therefore F the function a is continuous on R = [−∞, +∞], a ∈ L1 (R), and V (a) = R |a (x)|dx. Clearly, V (R) is a unital Banach algebra with the norm (3.1) aV := aL∞ (R) + V (a). For n ∈ N, let Cb (Rn+ , V (R)) be the set of all functions a : Rn+ × R → C such that r → a(r, ·) is a bounded continuous V (R)-valued function on Rn+ . Then the function r → a(r, ·)V belongs to Cb (Rn+ ). The set Cb (Rn+ , V (R)) with the norm 8 8 8 8 8a8 := sup 8a(r, ·)8V (3.2) C (Rn ,V (R)) b

+

r∈Rn +

becomes a Banach algebra. Let dµ() = d/ be the (normalized) invariant measure on R+ . As usual, let C0∞ (R+ ) be the set of all infinitely differentiable functions of compact support on R+ . By [22, Theorem 6.1] (also see [21, Theorem 3.1]), we have the following. Theorem 3.1. If a ∈ Cb (R+ , V (R)), then the Mellin pseudodifferential operator OP (a), defined for functions u ∈ C0∞ (R+ ) by the iterated integral  iλ   J K d 1 r dλ a(r, λ) u() , f or r ∈ R+ , (3.3) OP (a)u (r) = 2π R   R+

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extends to a bounded linear operator on every Lebesgue space Lp (R+ , dµ) with 1 < p < ∞, and there is a number Cp ∈ (0, ∞) depending only on p and such that 8 8 8 8 8OP (a)8 p ≤ Cp 8a8C (R+ ,V (R)) . B(L (R+ ,dµ)) b

The boundedness of Mellin pseudodifferential operators with compound symbols a(r, , λ) : R2+ ×R → C is given by the following result (see [22, Theorem 6.2]). Theorem 3.2. If a compound symbol a(r, , λ) satisfy the conditions ∂λk (∂ )j a ∈ Cb (R2+ , V (R))

f or all k, j = 0, 1, 2,

then the pseudodifferential operator OP (a) defined for functions u ∈ C0∞ (R+ ) by the iterated integral  iλ   J K d 1 r dλ a(r, , λ) u() , f or r ∈ R+ , (3.4) OP (a)u (r) := 2π R   R+ extends to a bounded linear operator on every Lebesgue space Lp (R+ , dµ) with 1 < p < ∞, and there is a constant Cp ∈ (0, ∞) depending only on p and such that  8 8 8 8 5 68 5 68 $  2 a(r, , ·) 8 + 28∂λ D $  2 a(r, , ·) 8 8OP (a)8 p ≤ Cp sup 38D B(L (R+ ,dµ))

V

r,∈R+

8 5 68  $  2 a(r, , ·) 8 , + 8∂λ2 D V

V

$  2 := I − ( ∂ )2 . with D

Let S(Rn+ , V (R)) be the Banach algebra of all functions a ∈ Cb (Rn+ , V (R)) such that the V (R)-valued functions (x1 , . . . , xn ) → a(ex1 , . . . , exn , ·) are uniformly continuous on Rn and 8 8 lim sup 8a(r, ·) − ah (r, ·)8V = 0 (3.5) |h|→0 r∈Rn

+

where ah (r, λ) = a(r, λ + h) for all (r, λ) ∈ Rn+ × R. Following [22, Remark 6.4], we say that a function a ∈ Cb (Rn+ , V (R)) is slowly oscillating if lim

r1 ,...,rn →0

cmVr (a) = 0,

lim

r1 ,...,rn →∞

cmVr (a) = 0

where, for r = (r1 , . . . , rn ) ∈ Rn+ , '8 ( 8 cmVr (a) = max 8a(r, ·) − a(νr, ·)8V : ν ∈ [1/2, 2] .

(3.6)

(3.7)

According to (2.2) we may substitute ν ∈ [1/2, 2] in (3.7) by ν ∈ [λ−1 , λ] with any λ > 1. Clearly, the set SO(Rn+ , V (R)) of all slowly oscillating functions in Cb (Rn+ , V (R)) is a Banach subalgebra of Cb (Rn+ , V (R)). Finally, by [22, Section 2], the set E(Rn+ , V (R)) := S(Rn+ , V (R)) ∩ SO(Rn+ , V (R))

(3.8)

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$ n , V (R)), consisting of functions a(r, λ) for which and its subset E(R +  lim

M→−∞

M

sup r∈Rn +

−∞

# # #[∂λ a](r, λ)#dλ =

 lim

M→+∞

+∞

sup r∈Rn +

# # #[∂λ a](r, λ)#dλ = 0,

(3.9)

M

also are Banach subalgebras of Cb (Rn+ , V (R)).

$ + , V (R)), we Passing to the function (x, λ) → a(ex , λ) for every a(r, λ) ∈ E(R immediately get the following analogue of Proposition 2.2 from [21, Lemma 2.9].

$ + , V (R)) and s ∈ {0, ∞}, then for every point ξ ∈ Lemma 3.3. If a(r, λ) ∈ E(R Ms (SO(R+ )) there are a sequence {xn } ⊂ R+ and a function a(ξ, ·) ∈ V (R) such that xn → s and 8 8 lim 8a(xn , ·) − a(ξ, ·)8 = 0; (3.10) n→∞

V

and, conversely, every sequence yn → s contains a subsequence xn → s such that (3.10) holds for some ξ ∈ Ms (SO(R+ )). Since the Mellin pseudodifferential operators OP (a) ∈ B(Lp (R+ , dµ)) are connected with the Fourier pseudodifferential operators b(x, D) ∈ B(Lp (R)) by the relation OP (a) = Eb(x, D)E −1 where b(x, λ) = a(ex , λ) for (x, λ) ∈ R × R and E is the isometric isomorphism of Lp (R) onto Lp (R+ , dµ) given by (Ef )(x) = f (ln x) for x ∈ R+ , we infer from [21, Theorem 4.4 and Corollary 8.4] the following two compactness results. Theorem 3.4. If a(r, λ) ∈ S(R+ , V (R)) and

lim

ln2 r+λ2 →∞

a(r, λ) = 0, then the Mellin

pseudodifferential operator OP (a) is compact on every Lebesgue space Lp (R+ , dµ) with 1 < p < ∞. Theorem 3.5. If a(r, λ), b(r, λ) ∈ E(R+ , V (R)), then the commutator OP (a)OP (b) − OP (b)OP (a) is a compact operator on every Lebesgue space Lp (R+ , dµ) with 1 < p < ∞. According to [22, Theorem 6.3], we have the following. Theorem 3.6. If a(r, , λ) is a compound symbol in the non-closed algebra ' ( I 2 , V (R)) := a(r, , λ) : ∂ k ( ∂ )j a ∈ E(R2 , V (R)) f or k, j = 0, 1, 2 , (3.11) E(R + λ + then the Mellin pseudodifferential operator OP (a) is represented in the form OP (a) = OP (I a) + K where I a(r, λ) = a(r, r, λ) and K is a compact operator on the space Lp (R+ , dµ).

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4. Properties of slowly oscillating functions and symbols By analogy with the Banach subalgebras S(Rn+ , V (R)), SO(Rn+ , V (R)), E(Rn+ , V (R)) of Cb (Rn+ , V (R))

(n ∈ N),

we introduce the Banach subalgebras S(Rn+ ), SO(Rn+ ), E(Rn+ ) of Cb (Rn+ ) as follows: S(Rn+ ) consists of all functions a : Rn+ → C for which the functions a(ex1 , . . . , exn ) ∈ Cb (Rn ) are uniformly continuous on Rn ; SO(Rn+ ) consists of all functions a ∈ Cb (Rn+ ) such that '# ( # lim max #a(r1 , . . . , rn )−a(νr1 , . . . , νrn )# : ν ∈ [1/2, 2] = 0 for s ∈ {0, ∞}; r1 ,...,rn →s

and E(Rn+ ) = S(Rn+ )∩SO(Rn+ ). Note that E(R+ ) = SO(R+ ) but E(Rn+ ) = SO(Rn+ ) for n > 1. For a real-valued function a ∈ C 1 (R+ ), let ma (r, ) = (a(r) − a())/(ln r − ln ).

(4.1)

According to the proofs of Lemma 8.1 and Corollary 8.2 in [22], we have the following. Lemma 4.1. If a ∈ C 3 (R+ ) and (rDr )j a ∈ SO(R+ ) for all j = 1, 2, 3, then the functions (∂ )j ma ∈ E(R2+ ) for all j = 0, 1, 2. Lemma 4.2. If ω ∈ SO(R+ ), a, b ∈ E(R2+ ), and ' ( inf Re a(r, ) : (r, ) ∈ R2+ > 0, ' ( 5 6 clos λa(r, ) + b(r, ) : (r, , λ) ∈ R2+ × R ∩ nπi : n ∈ Z = ∅,

(4.2) (4.3)

then the functions ψk (r, , λ) :=

λk exp(iλω(r)) sinh[λa(r, ) + b(r, )]

(k = 0, 1, 2, . . .)

(4.4)

belong to the Banach algebra E(R2+ , V (R)). Proof. It follows from (4.2)–(4.4) that, for k = 0, 1, 2, . . ., '# ( # sup # coth[λa(r, ) + b(r, )]# : (r, , λ) ∈ R2+ × R =: M < ∞, ' ( # #−1 sup |λ|k # sinh[λa(r, ) + b(r, )]# : (r, , λ) ∈ R2+ × R =: Ak < ∞,  |λ|k dλ =: Bk < ∞. sup r,∈R+ R | sinh[λa(r, ) + b(r, )]|

(4.5) (4.6) (4.7)

Since ω ∈ SO(R+ ) and a, b ∈ E(R2+ ), for every ε > 0 there is a δ > 0 such that # # # # # # #ω(r1 ) − ω(r2 )# < ε, #a(r1 , 1 ) − a(r2 , 2 )# < ε, #b(r1 , 1 ) − b(r2 , 2 )# < ε (4.8)

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whenever | ln r1 − ln r2 | < δ and | ln 1 − ln 2 | < δ. Moreover, in view of (4.5)– (4.7), there exists an ε > 0 such that, for every θ ∈ [0, 1], every λ ∈ R and all pairs (r1 , 1 ), (r2 , 2 ) ∈ R2+ satisfying (4.8), we obtain    K J (4.9) coth λa(r1 , 1 ) + b(r1 , 1 ) θ + λa(r2 , 2 ) + b(r2 , 2 ) (1 − θ) ≤ 2M, # J    K# −1 |λ|k # sinh λa(r1 , 1 ) + b(r1 , 1 ) θ + λa(r2 , 2 ) + b(r2 , 2 ) (1 − θ) # ≤ 2Ak , (4.10)  k |λ| # J    K# dλ ≤ 2Bk . # sinh λa(r ,  ) + b(r ,  ) θ + λa(r2 , 2 ) + b(r2 , 2 ) (1 − θ) # 1 1 1 1 R (4.11) Since according to (4.4), J  K exp(iλω(r)) kλk−1 + λk iω(r) − a(r, ) coth[λa(r, ) + b(r, )] [∂λ ψk ](r, , λ) = sinh[λa(r, ) + b(r, )] (4.12) and since, due to (4.5), ( ' # # sup |ω(r)| + |a(r, )|# coth[λa(r, ) + b(r, )]# : (r, , λ) ∈ R2+ × R ≤ ωCb (R+ ) + aCb (R2+ ) M =: C < ∞,

(4.13)

we infer from (4.6) and (4.7) that  8 # # 8 # # 8ψk (r, , ·)8 = sup #ψk (r, , λ)# + #[∂λ ψk ](r, , λ)#dλ V λ∈R R    k|λ|k−1 + |λ|k |ω(r)|+|a(r, )|| coth[λa(r, ) + b(r, )]| dλ ≤ Ak + | sinh[λa(r, ) + b(r, )]| R ≤ Ak + kBk−1 + Bk C < ∞. (4.14) Thus, the V (R)-valued functions (r, ) → ψk (r, , ·) are bounded on R2+ . Further, we deduce from (4.4) that # # #ψk (r1 , 1 , ·) − ψk (r2 , 2 , ·)# # # # # λk exp(iλω(r2 )) λk exp(iλω(r1 )) # # − =# sinh[λa(r1 , 1 ) + b(r1 , 1 )] sinh[λa(r2 , 2 ) + b(r2 , 2 )] # # ## #−1  ≤ |λ|k # exp iλω(r1 )) − exp(iλω(r2 ))## sinh[λa(r1 , 1 ) + b(r1 , 1 )]# # −1  −1 ## # + |λ|k # sinh[λa(r1 , 1 ) + b(r1 , 1 )] − sinh[λa(r2 , 2 ) + b(r2 , 2 )] # # # # # −1 ≤ |λ|k+1 #ω(r1 ) − ω(r2 )## sinh[λa(r1 , 1 ) + b(r1 , 1 )]# # J K J K## # + |λ|k #λ a(r1 , 1 ) − a(r2 , 2 ) + b(r1 , 1 ) − b(r2 , 2 ) # J K#  1 ## coth (λa(r1 , 1 ) + b(r1 , 1 ))θ + (λa(r2 , 2 ) + (b(r2 , 2 ))(1 − θ) # # J K# dθ. × # sinh (λa(r1 , 1 ) + b(r1 , 1 ))θ + (λa(r2 , 2 ) + (b(r2 , 2 ))(1 − θ) # 0 (4.15)

Nonlocal Singular Integral Operators with Slowly Oscillating Data Hence, by (4.6), (4.9) and (4.10), we obtain 8 # 8 # 8ψk (r1 , 1 , ·) − ψk (r2 , 2 , ·)8 ∞ ≤ Ak+1 #ω(r1 ) − ω(r2 )# L (R) # # # #  + 4M Ak+1 #a(r1 , 1 ) − a(r2 , 2 )# + Ak #b(r1 , 1 ) − b(r2 , 2 )# .

239

(4.16)

On the other hand, (4.12) implies that   V ψk (r1 , 1 , ·) − ψk (r2 , 2 , ·)  J K  # # exp(iλω(r1 )) kλk−1 + λk iω(r1 ) − a(r1 , 1 ) coth[λa(r1 , 1 ) + b(r1 , 1 )] # = # sinh[λa(r1 , 1 ) + b(r1 , 1 )] R  J k−1 K # k + λ iω(r2 ) − a(r2 , 2 ) coth[λa(r2 , 2 ) + b(r2 , 2 )] ## exp(iλω(r2 )) kλ − #dλ sinh[λa(r2 , 2 ) + b(r2 , 2 )] #  # # # exp(iλω(r2 )) exp(iλω(r1 )) # # ≤ # sinh[λa(r1 , 1 ) + b(r1 , 1 )] − sinh[λa(r2 , 2 ) + b(r2 , 2 )] # R D # #E  × k|λ|k−1 + |λ|k |ω(r1 )| + |a(r1 , 1 )|# coth[λa(r1 , 1 ) + b(r1 , 1 )]# dλ    |λ|k |ω(r1 ) − ω(r2 )| + |a(r1 , 1 ) − a(r2 , 2 )|| coth[λa(r1 , 1 )+b(r1 , 1 )]| # # dλ + # sinh[λa(r2 , 2 ) + b(r2 , 2 )]# R # # |λ|k |a(r2 , 2 )|# coth[λa(r1 , 1 ) + b(r1 , 1 )] − coth[λa(r2 , 2 ) + b(r2 , 2 )]# # # + dλ # sinh[λa(r2 , 2 ) + b(r2 , 2 )]# R =: I1 + I2 + I3 . 

By (4.15) and (4.13) we have    |λ| k|λ|k−1 + |λ|k C # # dλ I1 ≤ |ω(r1 ) − ω(r2 )| # # R sinh[λa(r1 , 1 ) + b(r1 , 1 )]     |λ||a(r1 , 1 ) − a(r2 , 2 )| + |b(r1 , 1 ) − b(r2 , 2 )| k|λ|k−1 + |λ|k C + R J K#  1 ## coth (λa(r1 , 1 ) + b(r1 , 1 ))θ + (λa(r2 , 2 ) + b(r2 , 2 ))(1 − θ) # # J K# × # sinh (λa(r1 , 1 ) + b(r1 , 1 ))θ + (λa(r2 , 2 ) + b(r2 , 2 ))(1 − θ) # dθdλ. 0 Changing the order of integration in the second summand and applying (4.7), (4.9) and (4.11), we get # # #  #   I1 ≤ #ω(r1 ) − ω(r2 )# kBk + Bk+1 C + #a(r1 , 1 ) − a(r2 , 2 )#4M kBk + Bk+1 C # #   + #b(r1 , 1 ) − b(r2 , 2 )#4M kBk−1 + Bk C . (4.17) In its turn we deduce from (4.5), (4.7) and the estimate # # # # # coth[λa(r1 , 1 ) + b(r1 , 1 )] − coth[λa(r2 , 2 ) + b(r2 , 2 )]#  # # # # ≤ 4A20 |λ|#a(r1 , 1 ) − a(r2 , 2 )# + #b(r1 , 1 ) − b(r2 , 2 )#

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that

# # # # I2 ≤ #ω(r1 ) − ω(r2 )#Bk + #a(r1 , 1 ) − a(r2 , 2 )#M Bk , (4.18) #  # # # I3 ≤ 4A20 aCb (R2+ ) #a(r1 , 1 ) − a(r2 , 2 )#Bk+1 + #b(r1 , 1 ) − b(r2 , 2 )#Bk . (4.19)

Combining (4.17)–(4.19) we obtain #J  # K  V ψk (r1 , 1 , ·) − (ψk (r2 , 2 , ·) ≤ #ω(r1 ) − ω(r2 )# (k + 1)Bk + Bk+1 C # #J  K + #a(r1 , 1 ) − a(r2 , 2 )# Bk M (4k + 1) + 4Bk+1 M C + A20 acb (R2+ ) #J #  K (4.20) + #b(r1 , 1 ) − b(r2 , 2 )# 4M kBk−1 + 4Bk M C + A20 aCb (R2+ ) . It follows from (4.14), (4.16), (4.20) and (3.6)–(3.7) that ψk ∈ SO(R2+ , V (R)) and the V (R)-valued function (x, y) → ψk (ex , ey , ·) is uniformly continuous on R2 for every k = 0, 1, 2, . . .. It remains to prove that 8 8 lim sup 8ψk (r, , ·) − ψkh (r, , ·)8V = 0. (4.21) |h|→0 r,∈R+

Applying (4.6), (4.7) and (4.13), we deduce from (4.12) that for |h| < ε, # # #[∂λ ψk ](r, , λ + θh)# ≤ kAk−1 + Ak C,  # # #[∂λ ψk ](r, , λ + θh)#dλ ≤ kBk−1 + Bk C,

(4.22) (4.23)

R

where C is given by (4.13). Making use of (4.22) we obtain #  1 # 8 # # 8 J K 8ψk (r, , ·) − ψ h (r, , ·)8 ∞ #h # ∂ (r, , λ + θh)dθ = sup ψ λ k k # # L (R) λ∈R

0

≤ |h|(kAk−1 + Ak C).

(4.24)

In view of (4.12), we get   K [∂λ2 ψk (r, , λ) = k[∂λ ψk−1 ](r, , λ) + iω(r) − a(r, ) coth[λa(r, ) + b(r, )] K  −2 J ψk (r, , λ). (4.25) × ∂λ ψk (r, , λ) + a2 (r, ) sinh[λa(r, ) + b(r, )] Applying (4.25), (4.23) and (4.7), we obtain #  #  1 # #   h 2 # V ψk (r, , ·) − (ψk (r, , ·) = [∂λ ψk ](r, , λ + θh)dθ##dλ #h R 0 D  E    ≤ |h| k (k − 1)Bk−2 + Bk−1 C + C kBk−1 + Bk C + A20 a2Cb (R2 ) Bk . (4.26) +

Combining (4.24) and (4.26), we obtain (4.21). Therefore, ψk ∈ S(R2+ , V (R)) and thus, by (3.8), ψk ∈ E(R2+ , V (R)). Analogously, and even simpler, one can prove that the function (r, , λ) →
coth[λa(r, ) + b(r, )] also belongs to E(R2+ , V (R)).

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241

Lemma 4.3. If ω ∈ SO(R+ ), the functions ( ∂ )j a, ( ∂ )j b are in E(R2+ ) for all j = 0, 1, 2, and (4.2), (4.3) hold, then the functions   5 6 exp(iλω(r)) ∂λk ( ∂ )j coth[λa(r, ) + b(r, )] , ∂λk ( ∂ )j sinh[λa(r, ) + b(r, )] belong to the Banach algebra E(R2+ , V (R)) for all k, j = 0, 1, 2. Proof. Since E(R2+ , V (R)) is a Banach subalgebra of Cb (R2+ , V (R)) and the functions exp(iλω(r)) (r, , λ) → coth[λa(r, ) + b(r, )], ψ0 (r, , λ) = sinh[λa(r, ) + b(r, )] are in E(R2+ , V (R)) in view of Lemma 4.2, and since −a 2a2 coth[λa + b] 2 , ∂ , {coth[λa + b]} = λ sinh2 [λa + b] sinh2 [λa + b]    2 a2 ψ0 , ∂λ ψ0 = iω − a coth[λa + b] ψ0 , ∂λ2 ψ0 = iω − a coth[λa + b] ψ0 + sinh2 [λa + b]

∂λ {coth[λa + b]} =

where ω ∈ SO(R+ ) = E(R+ ) and a ∈ E(R2+ ), it remains to prove that the functions 5 6 5 6 (4.27) ( ∂ )j coth[λa(r, ) + b(r, )] , ( ∂ )j ψ0 (r, , λ) (with ω = 0 and ω ∈ SO(R+ )) are in E(R2+ , V (R)) for j = 1, 2. Simple calculations give the formulas 5 6 λ( ∂ )a + ( ∂ )b ( ∂ ) coth[λa + b] = − , sinh2 [λa + b] 5 6 λ( ∂ )2 a + ( ∂ )2 b [λ( ∂ )a + ( ∂ )b]2 +2 ( ∂ )2 coth[λa + b] = − 2 sinh [λa + b] sinh2 [λa + b] × coth[λa + b]. (4.28) Taking into account (4.4) we also obtain J K ( ∂ )ψ0 = − ψ1 ( ∂ )a + ψ0 ( ∂ )b coth[λa + b], K J  ( ∂ )2 ψ0 = ψ2 ( ∂ a)2 + 2ψ1 ( ∂ a)( ∂ b) + ψ0 ( ∂ b)2 2 coth2 [λa + b] − 1 J K − ψ1 ( ∂ )2 a + ψ0 ( ∂ )2 b coth[λa + b]. (4.29) Since ( ∂ )j a, ( ∂ )j b ∈ E(R2+ ) for j = 1, 2, we infer from the formulas (4.28)– (4.29) and Lemma 4.2 that the functions (4.27) are in E(R2+ , V (R)) for j = 1, 2, which completes the proof.
Given real-valued functions θ(r), reω(r) and mθ (r, ) defined by (4.1), we introduce the real-valued functions    ω(r)   ω(r)  θ reω(r) − θ() γ(r) = θ re (4.30) , η(r, ) = mθ re , = ln r + ω(r) − ln  for r,  ∈ R+ . By (4.30), η(r, ) = mθ (r, ) if ω(r) = 0 for all r ∈ R+ .

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Lemma 4.4. If (rDr )j θ ∈ SO(R+ ) for j = 1, 2, 3 and (rDr )j ω ∈ SO(R+ ) for j = 0, 1, 2, 3, then (rDr )j γ ∈ SO(R+ ) (∂ ) η ∈ j

E(R2+ )

f or j = 1, 2, 3,

(4.31)

f or j = 0, 1, 2.

(4.32)

Proof. Let β(r) := reω(r) for r ∈ R+ . Since for any ν > 0 and all r ∈ R+ , νe−2M ≤ β(νr)/β(r) ≤ νe2M where M = ωCb (R+ ) , we conclude that for each r ∈ R+ , ' ( ' K( J β(νr) : ν ∈ [1/2, 2] ⊂ ν$β(r) : ν$ ∈ 2−1 e−2M , 2e2M , which implies due to (2.2) that for any a ∈ SO(R+ ), '# ( # lim max #(a ◦ β)(r) − (a ◦ β)(νr)# : ν ∈ [1/2, 2] = 0 r→s

(4.33)

for s ∈ {0, ∞},

and therefore a ◦ β ∈ SO(R+ ). Consequently, for all j = 1, 2, 3, the functions [(rDr )j θ](β(r)) are in SO(R+ ) along with the functions (rDr )j θ. In view of (4.30), we obtain J K J K J K (rDr )γ (r) = (rDr )θ (β(r)) · 1 + rω  (r) , K J K J K2 J K J K J (rDr )2 γ (r) = (rDr )2 θ (β(r)) · 1 + rω  (r) + (rDr )θ (β(r)) · (rDr )2 ω (r), K J K J K3 J K J K J (rDr )3 γ (r) = (rDr )3 θ (β(r)) · 1 + rω  (r) + 3 (rDr )θ (β(r)) · 1 + rω  (r) J K J K J K × (rDr )2 ω (r) + (rDr )θ (β(r)) · (rDr )3 ω (r). (4.34) Since SO(R+ ) is a C ∗ -algebra and the functions [(rDr )j θ](β(r)) and (rDr )j ω are in SO(R+ ) for all j = 1, 2, 3, we immediately deduce (4.31) from formulas (4.34). Further, from (4.30) it follows that J K K J (∂ )j η (r, ) = (∂ )j mθ (β(r), ) (j = 0, 1, 2), (4.35) and since (∂ )j mθ ∈ SO(R2+ ) for j = 0, 1, 2 in view of Lemma 4.1, we infer making use of (4.33) and (4.35) that (∂ )j η ∈ SO(R2+ ) for these j too. Finally, as (∂ )j mθ ∈ S(R2+ ) for j = 0, 1, 2 according to Lemma 4.1, we conclude that for every ε > 0 there is a δ > 0 such that #J # K K J # ( ∂ )j mθ (r1 , 1 ) − ( ∂ )j mθ (r2 , 2 )# < ε (4.36) whenever | ln(r1 /r2 )| < δ and | ln(1 /2 )| < δ. Since 6 5 |ω(r1 ) − ω(r2 )|/| ln r1 − ln r2 | ≤ sup |rω  (r)| : r ∈ R+ =: M0 < ∞ and therefore # # # # # # # ln[$ α(r1 )/$ α(r2 )]# = # ln r1 + ω(r1 ) − ln r2 − ω(r2 )# ≤ (1 + M0 )# ln(r1 /r2 )#, (4.37) it remains to take | ln(r1 /r2 )| < δ0 and | ln(1 /2 )| < δ0 where δ0 := δ/(1 + M0 ), which implies in view of (4.35)–(4.37) that ( ∂ )j η ∈ S(R2+ ) for all j = 0, 1, 2.
This completes the proof of (4.32) because S(R2+ ) ∩ SO(R2+ ) = E(R2+ ).

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243

5. Applications of Mellin pseudodifferential operators Let 1 < p < ∞ and (Γ, w) ∈ ASO p . In this section we apply the results on Mellin pseudodifferential operators collected in Section 3 to singular integral operators acting on the weighted Lebesgue space Lp (Γ, w). Below we need the following integral operator RΓ given for f ∈ Lp (Γ, w) by  1 f (ξ) dξ (τ ∈ Γ), (5.1) (RΓ f )(τ ) = πi Γ ξ + τ − 2t where Γ is given by (2.5). Obviously, RΓ has two fixed singularities: at t and ∞. Lemma 5.1. The operator RΓ is bounded on the space Lp (Γ, w). $ = Γ ∪ Γ− where Γ− := {2t − τ : τ ∈ Γ} and t is the starting point of Proof. Let Γ $ ) = w(τ ) if τ ∈ Γ, w(2t $ − τ ) = w(τ ) Γ and the terminating point of Γ− , and let w(τ if τ ∈ Γ− . Since w ∈ Ap (Γ), it is easily seen (cf. [6, Theorem 2.7]) and [27, $ whence the Cauchy singular integral operator S$ is Lemma 4.1]) that w $ ∈ Ap (Γ), Γ $ w). bounded on the weighted Lebesgue space Lp (Γ, $ Consider the isomorphism   ϕ(t) $ w) σ : Lp (Γ, $ → Lp2 (Γ, w), (σϕ)(τ ) = , τ ∈ Γ, ϕ(2t − τ ) where Lp2 (Γ, w) is the Banach space of vectors {ϕi }2i=1 with entries in Lp (Γ, w) 8  8 1/p . Since and the norm 8{ϕi }2i=1 8 = ϕ1 pLp (Γ,w) + ϕ2 pLp (Γ,w)   SΓ −RΓ σSΓ$ σ −1 = ∈ B(Lp2 (Γ, w)), RΓ −SΓ we conclude that RΓ ∈ B(Lp (Γ, w)).




where Γ and w are given by (2.5) and (2.6), Following [9], for (Γ, w) ∈ respectively, we introduce the isomorphisms   Φ : Lp (Γ, w) → Lp (R+ , dµ), (Φf )(r) = ev(r) r1/p f t + reiθ(r) , r ∈ R+ , (5.2) ASO p

Ψ : B(Lp (Γ, w)) → B(Lp (R+ , dµ)), Vα±1

A → ΦAΦ−1 .

(5.3)

p

Since the shift operators are bounded on the space L (Γ, w) due to (2.7), and Lemma 5.1 that all the operators we infer from the condition (Γ, w) ∈ ASO p −1 SΓ , Vα SΓ Vα , RΓ , Vα RΓ are also bounded on the space Lp (Γ, w). As we will see, the Ψ-images of these operators are Mellin pseudodifferential operators of the form (3.4) with compound symbols. and a slowly oscillating shift Theorem 5.2. Let 1 < p < ∞, and let (Γ, w) ∈ ASO p α : Γ → Γ be given by (2.5), (2.6) and (2.8), respectively, where ω ∈ SO(R+ ), (rDr )j θ, (rDr )j v, (rDr )j ω ∈ SO(R+ )

f or j = 1, 2, 3,

(2.7) and (2.11) hold, and     0 < inf 1/p + mv (r, ) ≤ sup 1/p + mv (r, ) < 1. r,∈R+

r,∈R+

(5.4) (5.5)

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If cΓ ∈ SO(Γ), then Ψ(cΓ I) = cI with c ∈ SO(R+ ). For SΓ , Vα SΓ Vα−1 , RΓ , Vα RΓ ∈ B(Lp (Γ, w)), we have Ψ(SΓ ) = OP (σ),

Ψ(Vα SΓ Vα−1 ) = OP (σα ),

(5.6)

Ψ(RΓ ) = OP (ν),

Ψ(Vα RΓ ) = OP (να ),

(5.7)

where for (r, , λ) ∈

R2+

× R,

  λ + i(1/p + mv (r, )) 1 + iθ () coth π , (5.8) 1 + imθ (r, ) 1 + imθ (r, )   λ + i(1/p + mv (r, )) 1 + ω  () + iγ  () σα (r, , λ) := coth π , (5.9) 1 + mω (r, ) + imγ (r, ) 1 + mω (r, ) + imγ (r, )   −1 1 + iθ () λ + i(1/p + mv (r, )) ν(r, , λ) := sinh π , (5.10) 1 + imθ (r, ) 1 + imθ (r, )   −1 λ + i(1/p + mv (r, )) 1 + iθ () sinh π να (r, , λ) := 1 + iη(r, ) 1 + iη(r, )  J K × exp iω(r) λ + i(1/p + mv (r, )) , (5.11) σ(r, , λ) :=

the functions ma (r, ) and γ(r), η(r, ) are given by (4.1) and (4.30), and all the I 2 , V (R)) defined by (3.11). functions (5.8)–(5.11) belong to the algebra E(R + Proof. Since ω ∈ SO(R+ ) and (5.4) holds, from Lemmas 4.1 and 4.4 it follows that the functions 1 + iθ () 1 + ω  () + iγ  () ϕ1 (r, , λ) := , ϕ2 (r, , λ) := , 1 + imθ (r, ) 1 + mω (r, ) + imγ (r, ) J K 1 + iθ () ϕ3 (r, , λ) := exp − ω(r)(1/p + mv (r, )) 1 + iη(r, ) I 2+ , V (R)). On the other hand, the second belong to the non-closed algebra E(R multipliers in (5.8)–(5.11) have one of the following forms: coth[λa(r, ) + b(r, )],

1 , sinh[λa(r, ) + b(r, )]

exp(iλω(r)) . sinh[λa(r, ) + b(r, )]

Moreover, in virtue of (5.4)–(5.5) and (2.11), we conclude that (4.2), (4.3) and all other conditions of Lemmas 4.2 and 4.3 are fulfilled. Hence, by Lemma 4.3, the seI 2 , V (R)) cond multipliers in (5.8)–(5.11) also belong to the non-closed algebra E(R + and therefore, by Theorem 3.2, the Mellin pseudodifferential operators OP (σ), OP (σα ), OP (ν), OP (να ) with compound symbols (5.8)–(5.11) are bounded on the space Lp (R+ , dµ). Equalities (5.6) with σ and σα given by (5.8) and (5.9), respectively, were proved in [8, Theorem 3.1] and [22, Theorem 8.3]. Note that condition (8.10) in [22, Theorem 8.3] for σα is superfluous in view of the Mellin transform identity  1 1 δ xµ = coth [π(λ + iµ)/δ] xiλ dλ (x > 0, 0 < µ < 1, Re δ > µ). πi 1 − xδ 2π R

Nonlocal Singular Integral Operators with Slowly Oscillating Data

245

A straightforward computation on the basis of (2.5), (2.6), (2.8) and (5.1)– (5.3) shows that for all r > 0 and all f ∈ Lp (R+ , dµ),  J K 1 ev(r)−v() (r/)1/p (1 + iθ ())eiθ() Ψ(Vα RΓ )f (r) = f ()d πi R+ eiθ() + reω(r)+iγ(r)  1 d (r/)1/p+mv (r,) (1 + iθ ()) = (5.12) f () . πi R+  1 + (eω(r) r/)1+iη(r,) Taking into account the Mellin transform identity   −1 iλ 1 1 δxµ sinh[π(λ + iµ)/δ] = x dλ (x > 0, 0 < µ < 1, Re δ > µ) πi 1 + xδ 2π R (see, e.g., [7, Theorem 2.2.1] and [6, Theorem 10.22]) and setting δ = 1 + iη(r, ),

µ = 1/p + mv (r, ),

x = eω(r) r/,

we infer from (5.12) that for r ∈ R+ ,   −1   J K λ + i(1/p + mv (r, )) 1 1 + iθ () sinh π Ψ(Vα RΓ )f (r) = dλ 2π R 1 + iη(r, ) R+ 1 + iη(r, )  J K r iλ d × exp iω(r) λ + i(1/p + mv (r, )) f () ,   which gives by (3.4) the second equality in (5.7) with να defined by (5.11). Because η(r, ) = mθ (r, ) if ω(r) = 0, we obtain the first equality in (5.11) with ν given by (5.10) from the second equality in (5.11) by setting ω = 0.
I 2 , V (R)) Since the compound symbols (5.8)–(5.11) are in the algebra E(R + defined by (3.11), we infer from (5.6)–(5.7) and Theorem 3.6 the following. Theorem 5.3. Under the conditions of Theorem 5.2, Ψ(Vα SΓ Vα−1 ) = OP (I σα ) + Kα , I Iα, Ψ(RΓ ) = OP (I ν ) + K, Ψ(Vα RΓ ) = OP (I να ) + K

Ψ(SΓ ) = OP (I σ ) + K,

I K I α are compact operators on the space Lp (R+ , dµ), where K, Kα , K,   λ + i(1/p + rv  (r)) , σ I(r, λ) = coth π 1 + irθ (r)   λ + i(1/p + rv  (r)) σ Iα (r, λ) = coth π , 1 + rω  (r) + irγ  (r)   −1 λ + i(1/p + rv  (r)) νI(r, λ) = sinh π , 1 + irθ (r)   −1 λ + i(1/p + rv  (r)) 1 + irθ (r) sinh π νIα (r, λ) = 1 + iη(r, r) 1 + iη(r, r)  J K × exp iω(r) λ + i(1/p + rv  (r)) ,

(5.13)

(5.14) (5.15) (5.16)

(5.17)

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Yu.I. Karlovich

where

J   KL η(r, r) = θ reω(r) − θ(r) ω(r), and the symbols (5.14)–(5.17) belong the Banach algebra E(R+ , V (R)).

(5.18)

According to (5.18) we obtain η(r, r) − rθ (r) =

1 ω(r)



reω(r)

r

J  K dτ . τ θ (τ ) − rθ (r) τ

(5.19)

Since the function r → rθ (r) belongs to SO(R+ ) and ω ∈ Cb (R+ ), we infer from (5.19) and the mean value theorem that for s ∈ {0, ∞}, J J K K (5.20) lim η(r, r) − rθ (r) = lim ζ(r)θ (ζ(r)) − rθ (r) = 0, r→s

r→s

where ζ(r) belongs to the segment with the endpoints r and reω(r) . Setting −1    J K λ + i(1/p + rv  (r)) ν$α (r, λ) = sinh π exp iω(r) λ + i(1/p + rv  (r))  1 + irθ (r) (5.21) and applying (5.20), we deduce from (5.17) and (5.21) that J K νIα (r, λ) − ν$α (r, λ) = 0. (5.22) lim 2 ln r+λ2 →∞

Analogously, we infer from (2.9), (5.15) and (5.14) that J K lim σ I(r, λ) − σ Iα (r, λ) = 0. 2 ln r+λ2 →∞

(5.23)

Since νIα − ν$α , σ I−σ Iα ∈ E(R+ , V (R)), we infer from (5.22), (5.23) and Theoσ−σ Iα ) rem 3.4 that the Mellin pseudodifferential operators OP (I να − ν$α ) and OP (I are compact on the space Lp (R+ , dµ), which gives the following. Corollary 5.4. Under the conditions of Theorem 5.2, $ $α Ψ(Vα SΓ Vα−1 ) = OP (I σ ) + K, Ψ(Vα RΓ ) = OP ($ να ) + K

(5.24)

where the symbols σ I ∈ E(R+ , V (R)) and ν$α ∈ E(R+ , V (R)) are given by (5.14) and $ K $ α are compact operators on the space Lp (R+ , dµ). (5.21), respectively, and K, Remark 5.5. It is easily seen that actually the functions (5.14)–(5.16) and (5.21) $ + , V (R)) consisting of symbols a(r, λ) ∈ E(R+ V (R)) are in the Banach algebra E(R satisfying (3.9).

6. Functional operators Let (Γ, w) ∈ ASO and let G be a discrete group of slowly oscillating shifts g that p are orientation-preserving diffeomorphisms of Γ onto itself given by      for r ∈ R+ , (6.1) g t + reiθ(r) = t + reωg (r) exp iθ reωg (r) where ωg are real-valued functions in C 3 (R+ ), the functions (rDr )j ωg are in SO(R+ ) for all j = 0, 1, 2, 3, ln |g  | ∈ Cb (Γ) and (w ◦ g)/w, w/(w ◦ g) ∈ L∞ (Γ) for

Nonlocal Singular Integral Operators with Slowly Oscillating Data

247

all g ∈ G. In what follows we assume that the group G is subexponential and G acts on Γ topologically freely. Let A denote the non-closed algebra of functional operators of the form
A= ag,Γ Vg ∈ B(Lp (Γ, w)) (6.2) g∈F

where ag,Γ ∈ SO(Γ), Vg are shift operators given by Vg f = f ◦ g, and F is a finite $g = |g  |1/p Vg are isometric on the space Lp (Γ). subset of G. Then the operators U The operator (6.2) is invertible on the space Lp (Γ, w) if and only if the operator
$g , with $ $= $ ag,Γ U ag,Γ = ag,Γ |g  |−1/p w/(w ◦ g), (6.3) A g∈F

is invertible on the space Lp (Γ). As the weight w and the shifts g are slowly oscillating, we infer from the proof of [23, Lemma 9.3] that w/(w ◦ g) ∈ SO(Γ). Hence $ ag,Γ ∈ SO(Γ) for all g ∈ G. Analogously to [29], with the operator (6.3) we associate the discrete operators Aτ (τ ∈ Γ) acting on the space lp (G) by the rule
$ ag,Γ [h(τ )] f (hg) (h ∈ G) (6.4) (Aτ f )(h) = g∈F

where f ∈ l (G) and [hg](·) = g[h(·)]. Given a bounded linear operator B acting 5 on a Banach space6 X, we consider := inf Bf X : f X = 1 . the lower norm of B defined by B0 By analogy with [29, Lemma 3.1], one can prove the following. p

Lemma 6.1. If 1 < p < ∞, (Γ, w) ∈ ASO p , and the group G acts on Γ topologically freely, that for every τ ∈ Γ the mapping A → Aτ is an (algebraic and topological) homomorphism of A into B(lp (G)), and for every A ∈ A and every τ ∈ Γ, Aτ  ≤ A,

A0 ≤ Aτ 0 .

By analogy with [29, Corollary 3.8], slightly modifying the proof of Theorem 3.7 in [29] and applying the scheme of [30, Theorem 3.2] and [41, Proposition 2.1 and Corollary 2.4] for studying discrete operators, we obtain the following invertibility criterion for functional operators (6.2). and G is a subexponential group of Theorem 6.2. If 1 < p < ∞, (Γ, w) ∈ ASO p slowly oscillating shifts acting on Γ topologically freely, then a functional operator A of the form (6.2) with slowly oscillating coefficients and shifts is invertible on Aτ (τ ∈ Γ) given the space Lp (Γ, w) if and only if all the discrete 8 58 operators 6 by (6.4) are invertible on the space lp (G) and sup 8(Aτ )−1 8B(lp (G)) : τ ∈ Γ < ∞. For any invertible operator A ∈ A of the form (6.2), its inverse operator A−1 belongs to the Banach algebra ' (

AW := A = ag,Γ Vg : ag,Γ ∈ SO(Γ), AW := ag,Γ Vg B(Lp (Γ,w)) < ∞ . (6.5) Lemma 6.3. Functional operators A ∈ AW with slowly oscillating coefficients and slowly oscillating shifts commute with the Cauchy singular integral operator SΓ to within compact operators.

248

Yu.I. Karlovich

Proof. It is sufficient to show that for every function cΓ ∈ SO(Γ) and every slowly oscillating shift g given by (6.1), the operators cΓ SΓ −SΓ cΓ I and Vg SΓ Vg−1 −SΓ are compact on the space Lp (Γ, w) or , equivalently, the operators Ψ(cΓ SΓ − SΓ cΓ I) and Ψ(Vg SΓ Vg−1 − SΓ ), with Ψ given by (5.2)–(5.3), are compact on the space Lp (R+ , dµ). By (5.3), (2.12), and the first equality in (5.13), we obtain σ ) − OP (I σ )OP (c) + K0 (6.6) Ψ(cΓ SΓ − SΓ cΓ I) = OP (c)OP (I   iθ(r) where the functions σ I (r, λ) given by (5.14) and c(r, λ) := cΓ t + re belong to the Banach algebra E(R+ , V (R)), and K0 ∈ K(Lp (R+ , dµ)). Applying Theorem 3.5, we conclude from (6.6) that Ψ(cΓ SΓ − SΓ cΓ I) ∈ K(Lp (R+ , dµ)) too. Finally, the compactness of the operator Ψ(Vg SΓ Vg−1 −SΓ ) follows from (5.24) with α = g and from the first equality in (5.13).
According to Section 2, we set α(ξ) = ξ(a) for every a ∈ SO(R+ ) and every point ξ ∈ M, where M = M0 (SO(R+ )) ∪ M∞ (SO(R+ )). Let ξv  (ξ) := ξ[rv  (r)].  Lemma 6.4. For every functional operator A = g ag,Γ Vg ∈ AW with coefficients ag,Γ ∈ SO(Γ) and slowly oscillating shifts g : Γ → Γ, we have '
J  K(
sup |ag (ξ)| exp −ωg (ξ) 1/p+ξv  (ξ) ≤ $ ag,Γ Cb (Γ) = AW , (6.7) ξ∈M

g

g

  where ag (r) = ag,Γ t + reiθ(r) for r ∈ R+ , and AW is defined in (6.5). $g = |g  |1/p Vg , we deduce from (6.5) and (6.3) that Proof. As U
8
8
8 8 8 8 8ag,Γ Vg 8 p 8$ 8$ $g 8 p ag,Γ U ag,Γ 8C = = AW = B(L (Γ,w)) B(L (Γ))

b (Γ)

. (6.8)

In view of (6.3) and (2.10), we obtain   ; #    #2 −1/p 1 + rωg (r) iθ(r) ω (r) ω (r)  # # g g $ ag (r) := $ = ag (r) 9 ag,Γ t + re 1 + re θ re 1 + |rθ (r)|2 K J  (6.9) × exp(−ωg (r)/p) exp v(r) − v reωg (r) , where, for s ∈ {0, ∞}, limr→s rωg (r) = 0, ;  #  #2 M9 1 + #reωg (r) θ reωg (r) # 1 + |rθ (r)|2 = 1, lim r→s

and, by the proof of Lemma 9.3 in [23],  D  E  lim v reωg (r) − v(r) − ωg (r)rv  (r) = lim

r→s

r→s

r

reωg (r)

D

E d = 0. v  () − rv  (r)  (6.10)

Hence, for every ξ ∈ M, we infer from (6.9) that J  K $ ag (ξ) = ag (ξ) exp − ωg (ξ) 1/p + ξv  (ξ) for all ξ ∈ M.

(6.11)

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249

Finally, from Proposition 2.2 it follows that for every ξ ∈ M, |$ ag (ξ)| ≤ $ ag Cb (R+ ) = $ ag,Γ Cb (Γ) ,


which implies (6.7) due to (6.11) and (6.8).

The invertibility of operators A± ∈ A in the operator N = A+ P+ + A− P− , where P± = 12 (I ± SΓ ), follows from the next property (see [19] and [28]). (A) For any noninvertible operator A ∈ A and any ε > 0, there exists an operator Aε ∈ A such that A − Aε  < ε and one of the following conditions holds: (i) for every P ∈ {P± } there exists a sequence {fn } ⊂ X := Lp (Γ, w) such that fn X = 1, the sequences {Aε fn } and {P+ P− fn } converge in X, but the sequence {P fn } does not contain subsequences convergent in X; (ii) for every P ∈ {P± } there exists a sequence {fn } ⊂ X ∗ := Lq (Γ, w−1 ) such that fn X ∗ = 1, the sequences {A∗ε fn } and {P−∗ P+∗ fn } converge in X ∗ , but the sequence {P ∗ fn } does not contain subsequences convergent in X ∗ . Modifying the proof of [29, Theorem 5.2] we establish the following. Theorem 6.5. Functional operators A ∈ A possess the property (A).  Proof. Let A = g∈F ag,Γ Vg ∈ A where F is a finite subset of G, and assume that A is not invertible on the space Lp (Γ, w). Fix ε > 0 and set Aε = A. Obviously, there is a compact arc γ ⊂ Γ such that the G-orbit of γ coincides with Γ. Since A is not invertible,    inf (Aτ )∗ 0 = 0. inf Aτ 0 τ ∈γ

τ ∈γ

Indeed, otherwise all the operators Aτ (τ ∈ γ) are invertible on the space lp (G) and then, by Theorem 6.2, the functional operator A is invertible, which contradicts the initial assumption. Assume for definiteness that inf Aτ 0 = 0.

(6.12)

τ ∈γ

We will prove that condition (i) of (A) holds, for example, if P = P+ . By (6.12), there are points τn ∈ γ and functions ϕn ∈ lp (G), n = 1, 2, . . . , such that ϕn  = 1,

Aτn ϕn  < n−1

(6.13)

and the functions ϕn have finite supports Qn ⊂ G. Let χl stand for the characteristic function of a set l ⊂ Γ. Obviously, for every closed set l ⊂ Γ, the operator P+ P− χl I = 14 (I − SΓ2 )χl I is compact on the space Lp (Γ, w). −1 )n :=  Let Q . In view of the topologically free action of G on g∈F Qn g Γ and the continuity of the coefficients ag,Γ of the operator A, there exist open $ n the closures neighborhoods un ⊂ Γ of points τn ∈ Γ such that for all h ∈ Qn ∪ Q h(un ) ⊂ Γ are disjoint and, in view of (6.13), Aτ ϕn  ≤ Aτn ϕn  + n−1 < 2n−1

for all τ ∈ un ∩ γ.

(6.14)

250

Yu.I. Karlovich We may assume that the choice of un is subjected to the condition  h(un ). mes γn < 2−n where γn :=

(6.15)

h∈Qn

Since the operators P+ P− χγn I are compact on the space Lp (Γ, w) and s-lim χlm I = 0 on Lp (Γ, w) if lm ⊂ γn and mes lm → 0 as m → ∞, we conm→∞ clude that lim P+ P− χlm IB(Lp (Γ,w)) = lim P+ P− χγn χlm IB(Lp (Γ,w)) = 0.

m→∞

m→∞

(6.16)

Therefore we may without loss of generality assume due to (6.15) and (6.16) that P+ P− χγn I < n−1 .

(6.17)

Let u $n be a smooth Jordan curve containing the arc un ⊂ Γ. According to [34], there are functions ψn+ ∈ Lp ($ un ) such that 8 8 8 8 (n) (6.18) ψn+ ∈ Im P+ , 8ψn+ 8Lp (un ) = 1, 8ψn+ 8Lp ($un \un ) < n−1 (n)

where P+ = (I + Su$ n )/2 and Su$ n is defined by (1.1) with Γ = u $n . Consider the function fn ∈ Lp (Γ, w) given by % J K ϕn (h) Uh−1 (w−1 ψn+ ) (τ ) if τ ∈ h(un ), h ∈ Qn ,  (6.19) fn (τ ) = 0 if τ ∈ Γ \ h∈Qn h(un ), where Ug = |g  |1/p [(w◦g)/w]Vg is an isometric shift operator on the space Lp (Γ, w). $g := wUg w−1 I is an isometric shift operator on the space Lp (Γ). We infer Then U from (6.19), (6.13) and (6.18) that
 # # #J 8 8p K # #ϕn (h)#p # U −1 (w−1 ψn+ ) (τ )#p w(τ )p |dτ | 8fn 8 p = h L (Γ,w) =

h∈Qn




h(un )

# # # −1 +  #p 8 8 $ ψ (τ )# |dτ | = 8ϕn 8pp #ϕn (h)#p # U n h

h∈Qn

l (G)

h(un )

8 + 8p 8ψn 8 p L (u

n)

= 1. (6.20)

Applying (6.19), (6.14) and (6.18), we obtain
 8 8 # # 8Afn 8p p #(Afn )(τ )#p w(τ )p |dτ | = L (Γ,w) =


 $n h∈Q

 =

un

 =

un

$n h∈Q

h(un )


 # #p #[A(wf $ # )](τ ) |dτ | = n

h(un )

$n h∈Q

# #p  #[A(wf $ # n )](h(τ ) |h (τ )||dτ |

un

# #p 
#
# + # # $ a (h(τ ))ϕ (hg)ψ (τ ) |dτ | = g,Γ n n # #

$ n g∈F h∈Q

un


# # # # #(Aτ ϕn )(h)#p #ψn+ (τ )#p |dτ | $n h∈Q

8 8 # 8 8 # 8Aτ ϕn 8pp #ψn+ (τ )#p |dτ | < (2n−1 )p 8ψn+ 8p p = (2n−1 )p , l (G) L (un )

and hence Afn Lp (Γ,w) → 0 as n → ∞.

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Further, in view of (6.15), (6.19), (6.17) and (6.20), we get 8 8 8 8 8 8 8P+ P− fn 8 p = 8P+ P− χγn fn 8Lp (Γ,w) ≤ 8P+ P− χγn I 8 · fn Lp (Γ,w) < n−1 , L (Γ,w) and hence the sequence {P+ P− fn } converges. It remains to prove that the sequence {P+ fn } does not contain convergent subsequences. Let mn := card Qn . Consider the isometric isomorphisms 5 6 σn : Lp (γn , w|γn ) → Lpmn (un ), (σn f )(τ ) = w(h(τ ))(Uh f )(τ ) h∈Q , τ ∈ un . n

Since the weight w is continuous on every set γ n , we infer that χγn P+ χγn I = σn−1 χun σn P+ σn−1 χun σn = σn−1 diag {χun P+ χun }σn + χγn Kn χγn I

(6.21)

where Kn is a compact operator on the space Lp (γn , w|γn ) and analogously to (6.17) we may assume that χγn Kn χγn I < n−1 .

(6.22)

The rest of the proof runs as in [29, Theorem 5.2]. For the sake of completeness we give it here. Suppose for a moment that the sequence {P+ fn } converges (without loss of generality we take all the sequence instead of its subsequence). Then, in view of the5equality fn =6 χγn fn5, it follows from (6.21), (6.15) 6 and (6.22) that the sequences χγn P+ χγn fn and σn−1 diag{χun P+ χun }σn fn also converge. But σn−1 diag{χun P+ χun }σn fn = σn−1 diag {χun P+ χun ψn+ }{ϕn (h)}h∈Qn 6 5 (n) = σn−1 diag χun P+ χun ψn+ {ϕn (h)}h∈Qn 6 5 (n) = fn − σn−1 diag χun P+ χu$ n \un ψn+ {ϕn (h)}h∈Qn . (6.23) 8 (n) 8 Since we can achieve the uniform boundedness of 8P+ 8B(Lp ($u )) by an approprin ate choice of the contours u $n , we infer from (6.18) and (6.13) that 8 8 8 8 −1 6 5 8σn diag χun P+(n) χu$ \u ψn+ {ϕn (h)}h∈Qn 8 ≤ sup 8P+(n) 8 p n n B(L ($ u

n ))

· n−1 . (6.24)

Then from (6.23) and (6.24) it follows that the sequence {fn } converges in Lp (Γ, w). ∞ Let χn be the characteristic function of the set k=n γk , and let f∞ = limn→∞ fn ∈ Lp (Γ, w). Then, due to (6.15) and (6.19), f∞  = lim (1 − χn )f∞  = lim (1 − χn )(f∞ − fn ) ≤ 2 lim fn − f∞  = 0, n→∞

n→∞

n→∞

which contradicts (6.20). Thus, condition (i) in assumption (A) is fulfilled, which completes the proof of the theorem.


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7. Fredholm theory for SIO’s with shifts In this section we study the Fredholmness of singular integral operators with shifts N = A+ P+ + A− P− ∈ B(Lp (Γ, w))

(7.1)

where P± = 12 (I ± SΓ ) and A± are functional operators of the form
A± = a± g,Γ Vg ∈ A,

(7.2)

g∈F

with coefficients a± g,Γ ∈ SO(Γ) and a finite set F ⊂ G. Suppose G is an subexponential group of shifts acting topologically freely on Γ, and all the data are slowly oscillating (see Section 2). According to Section 6, with the operators A± ∈ B(Lp (Γ, w)) given by (7.2) p we associate the families of discrete operators A± τ ∈ B(l (G)), τ ∈ Γ, given by
(A± $ a± (7.3) τ f )(h) = g,Γ [h(τ )]f (hg) (h ∈ G) g∈F ±  −1/p w/(w ◦ g) and f ∈ lp (G). where $ a± g,Γ = ag,Γ |g | By Lemma 6.3, the commutators A± SΓ − SΓ A± are compact the space Lp (Γ, w). Hence, setting H = P+ P− = 14 (I − SΓ2 ) and applying Theorem 6.5, we immediately deduce the following Fredholm result from [28, Theorem 2.1] (also see [25, Theorem 1.1] and [19]).

Theorem 7.1. The operator N given by (7.1) is Fredholm on the weighted Lebesgue space Lp (Γ, w) if and only if (i) the functional operators A± are invertible on the space Lp (Γ, w);  (ii) the operator N is H-Fredholm, that is, there exist operators NH , NH ∈ p B(L (Γ, w)) such that NH N = H + K,

 N NH = H + K ,

with K, K  ∈ K(Lp (Γ, w)).

(7.4)

For the operator N given by (7.1), we introduce the functions N (ξ, λ) := A+ (ξ, λ)P+ (ξ, λ) + A− (ξ, λ)P− (ξ, λ)

for (ξ, λ) ∈ M × R,

where M is given by (2.3), 
J K  A± (ξ, λ) = a± (ξ) exp iω (ξ) λ + i(1/p + ξv (ξ)) , g g g∈F

P± (ξ, λ) =

(7.6)

   λ + i(1/p + ξv  (ξ) 1 1 ± coth π . 2 1 + iξθ (ξ)

Consider the Mellin transform and its inverse given by  M : L2 (R+ , dµ) → L2 (R), (M f )(λ) = f ()−iλ d/, R+  (M −1 ϕ)(r) = ϕ(λ)riλ dλ, M −1 : L2 (R) → L2 (R+ , dµ), R

(7.5)

(7.7)

λ ∈ R, r ∈ R+ .

Nonlocal Singular Integral Operators with Slowly Oscillating Data

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Fix p ∈ (1, ∞). Since the functions P± (ξ, ·), A± (ξ, ·) and hence N (ξ, ·) are Fourier multipliers on Lp (R) for every ξ ∈ M, we conclude that the Mellin convolution operators M −1 N (ξ, ·)M , defined first on functions f ∈ L2 (R+ , dµ) ∩ Lp (R+ , dµ), extend to bounded linear operators on the whole space Lp (R+ , dµ). Along with (7.5)–(7.7) we also consider the functions N (r, λ) := A+ (r, λ)P+ (r, λ) + A− (r, λ)P− (r, λ) defined for (r, λ) ∈ R+ × R, where 
J K  a± A± (r, λ) := g (r) exp iωg (r) λ + i(1/p + rv (r)) ,

(7.8)

(7.9)

g∈F

   λ + i(1/p + rv  (r)) 1 1 ± coth π . P± (r, λ) := 2 1 + irθ (r)

(7.10)

Given k > 0, we introduce the unitary shift operators Wk ∈ B(Lp (R+ , dµ)) defined by (Wk f )(r) = f (kr) for r ∈ R+ . Lemma 7.2. Under the conditions of Theorem 5.2, for every s ∈ {0, ∞}, every ξ ∈ Ms (SO(R+ )) and every operator N ∈ B(Lp (Γ, w)) given by (7.1), there is a sequence {kn } ⊂ R+ such that lim kn = s, and n→∞

lim N (kn , λ) = N (ξ, λ)

n→∞

f or every λ ∈ R,

s-lim Wkn Ψ(N )Wk−1 = M −1 N (ξ, ·)M. n n→∞

(7.11) (7.12)

Proof. Fix s ∈ {0, ∞}. For every λ ∈ R the function r → N (r, λ) belong to SO(R+ ). Therefore, by Proposition 2.2, for every ξ ∈ Ms (SO(R+ )) there is a sequence {kn } ⊂ R+ such that kn → s and the limits (7.11) exist for every rational λ ∈ R. Since P± ∈ E(R+ , V (R)) ⊂ S(R+ , V (R)), we infer from (3.5) that the functions P± (r, ·) are equicontinuous for all r ∈ R+ . Obviously, the functions A± (r, ·) and hence N (r, ·) also are equicontinuous for all r ∈ R+ , which implies that (7.11) holds for every λ ∈ R. If cΓ ∈ SO(Γ), then Ψ(cΓ I) = cI with c ∈ SO(R+ ), whence   = s-lim c[kn (·)]I = c(ξ)I. (7.13) s-lim Wkn cWk−1 n n→∞

n→∞

If K ∈ K(L (R+ , dµ)), then we deduce from [10, Lemma 18.9] that   = 0. s-lim Wkn KWk−1 n p

n→∞

(7.14)

Hence, taking into account (7.14), the first equality in (5.13) and the relations $ + , V (R)) (j = 0, 1, 2), we infer from [21, Lemma 11.4] that I(r, λ) ∈ E(R ∂λj σ   = OP (I σ (ξ, ·)) = M −1 σ I (ξ, ·)M. (7.15) s-lim Wkn Ψ(SΓ )Wk−1 n n→∞   It remains to calculate s-lim Wkn Ψ(Vg )Wk−1 for g ∈ G. Obviously, n n→∞

Ψ(Vg ) = cg V$βg

(g ∈ G)

(7.16)

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K J  where the function cg (r) := exp v(r) − v reωg (r) exp(−ωg (r)/p) is in SO(R+ ), the shift operator V$βg ∈ B(Lp (R+ , dµ)) is given by V$βg f = f ◦ βg , and βg (r) := r exp(ωg (r)) for r ∈ R+ . By (6.10) and (7.13), we obtain   J K = exp − ωg (ξ)(1/p + ξv  (ξ)) I. (7.17) s-lim Wkn cg Wk−1 n n→∞

Hence, we only need to prove that   s-lim Wkn V$βg W −1 = Wexp(ω n→∞

kn

g (ξ))

.

(7.18)

    f (r) = f r exp[ωg (kn r)] , taking f ∈ C0∞ (R+ ) and applying Since Wkn V$βg Wk−1 n [16, Lemma 4.8], we obtain (7.18) by analogy with [16, Theorem 4.13]. Thus, (7.16)–(7.18) imply that   J K = exp − ωg (ξ)(1/p + ξv  (ξ)) Wexp(ωg (ξ)) . (7.19) s-lim Wkn Ψ(Vg )Wk−1 n n→∞

Finally, as Wexp(ωg (ξ)) = M −1 exp(i(·)ωg (ξ))M , we immediately obtain (7.12) from (7.13), (7.15), (7.19) and (7.5)–(7.7).
Lemma 7.3. If for the function N (ξ, λ) given by (7.5)–(7.7), 5 6 m := inf |N (ξ, λ)| : (ξ, λ) ∈ M × R > 0, then

8 58 6 ν (ξ, ·)8V : ξ ∈ M < ∞, CV := sup 8N −1 (ξ, ·)I

(7.20) (7.21)

where νI(r, λ) is given by (5.16). Proof. By (7.5)–(7.7) and (6.7), we obtain   |∂λ N (ξ, λ)| ≤ sup ωg L∞ (R) A+ W P+ L∞ (R+ ×R) + A− W P− L∞ (R+ ×R) g∈F

  ν 2L∞ (R+ ×R) =: CN < ∞. + 2−1 A+ W + A− W I

(7.22)

Hence, in view of (7.20) and (7.22), we conclude that 8 −1 8 8 8 8N (ξ, ·)I ν (ξ, ·)8L∞ (R) ≤ m−1 8νI(ξ, ·)8L∞ (R) ,  # ## #     −1 1 #∂λ N (ξ, λ)##νI(ξ, λ)#dλ + 1 V νI(ξ, ·) ν (ξ, ·) ≤ 2 V N (ξ, ·)I m R m  # #   CN #νI(ξ, λ)#dλ + 1 V νI(ξ, ·) , ≤ 2 m R m which implies by (3.1) and (3.2) that  8 −1 8 8 # # 8 CN 8N (ξ, ·)I #νI(ξ, λ)#dλ + 1 8νI(ξ, ·)8 8 ν (ξ, ·) V ≤ 2 V m R m  8 8 # # CN #νI(r, λ)#dλ + 1 8νI8 ≤ 2 sup < ∞, Cb (R+ ,V (R)) m r∈R+ R m and therefore gives (7.21).




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Lemma 7.4. If the function N (ξ, λ) given by (7.5)–(7.7) satisfies (7.20), then there $ (r, λ) ∈ E(R $ + , V (R)) such that exists a function N $ (ξ, λ) = N −1 (ξ, λ)I ν (ξ, λ) N

f or all (ξ, λ) ∈ M × R.

(7.23)

Proof. Clearly, there is an increasing sequence {rn }n∈N ⊂ R+ such that ln rn+1 ≥ ln rn + 2 > 2 and for every n ∈ N, ' ( inf |N (r, λ)| : | ln r| ≥ ln rn , λ ∈ [−n, n] ≥ m/2 > 0, (7.24) $ (ξ, λ) := N −1 (ξ, λ)I ν (ξ, λ) is well where m is given by (7.20). Then the function N defined for (ξ, λ) ∈ M × R. Fix ξ ∈ M. Setting     $ (r, λ) := N −1 (r, λ)I R+ \ (rn−1 , rn ) × [−n, n] , (7.25) ν (r, λ) for (r, λ) ∈ N n∈N

we may assume without loss of generality that rn > 0 are chosen in such a manner that for every n ∈ N,     −n+1 $ −1 $ (r−1 , ·)− N $ (ξ, ·) +V n N $ (ξ, ·) ≥ 2−n if ξ ∈ M0 (SO(R+ )), N (rn , ·)− N V−n n−1 n     −n+1 $ $ (rn , ·) − N $ (ξ, ·) +V n N $ (ξ, ·) ≥ 2−n if ξ ∈ M∞ (SO(R+ )), N (rn , ·) − N V−n n−1 (7.26) #   F # $ (r, ·) = k #∂λ N $ (r, λ)#dλ for m < k, and for all r ∈ R+ \ (r−1 , rn ), where Vmk N n m # #   $ (r, λ)# + V n N $ (r, ·) max #N −n λ∈[−n,n] ' ( # #   $ (ξ, λ)# + V n N $ (ξ, ·) : ξ ∈ M + 1 ≤ CV + 1, ≤ sup max #N (7.27) −n λ∈[−n,n]

where CV is given by (7.21). Applying the recurrent relations $ (r, λ) = N ⎧ $ $ (rk+1 , λ) − N $ (rk+1 , k) N (r, k) + N if (r, λ) ∈ [r1 , rk+1 ] × [k, k + 1], ⎪ ⎪ ⎪ ⎪ ⎪ $ (r, −k) + N $ (rk+1 , λ) − N $ (rk+1 , −k) if (r, λ) ∈ [r1 , rk+1 ] × [−k − 1, −k], ⎨N −1 ⎪ $ (r, k) + N $ (r−1 , λ) − N $ (r−1 , k) ⎪ N if (r, λ) ∈ [rk+1 , r1−1 ] × [k, k + 1], ⎪ k+1 k+1 ⎪ ⎪ ⎩ $ $ (r−1 , λ) − N $ (r−1 , −k) if (r, λ) ∈ [r−1 , r−1 ] × [−k − 1, −k], N (r, −k) + N k+1 k+1 k+1 1 (7.28)   $ (r, λ) given by (7.25) to the set R+ \ (r−1 , r1 ) × R. we extend the function N 1 Finally, setting −1 $ (r1 , λ) r − r1 $ (r, λ) = N $ (r−1 , λ) r1 − r + N N 1 r1 − r1−1 r1 − r1−1

$ (r, λ) to all R+ × R. for (r, λ) ∈ (r1−1 , r1 ) × R, we extend the function N

(7.29)

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$ ∈ E(R+ , V (R)). For definiteness, assume that r ≥ r1 . If Let us prove that N r ∈ [rk , rk+1 ], then in view of (7.26) and (7.28), we obtain  
∞  −n     k $ (r, ·) + $ (r, ·) $ (r, ·) = V−k N V−n−1 + Vnn+1 N V N n=k  
∞  −n   k $ $ (rn+1 , ·) V−n−1 + Vnn+1 N = V−k N (r, ·) + n=k E  
∞ D −n   k $ (r, ·) + $ (ξ, ·) + 2−n−1 , V−n−1 + Vnn+1 N ≤ V−k N n=k

'# 8 8 #
∞   $ (r, ·)8 ∞ ≤ max #N $ (rn+1 , ·) , $ (r, −k)# + 8N V −n N L (R) n=k −n−1
∞ # #  ( $ (r, λ)#, |N $ (r, k)| + $ (rn+1 , ·) max #N Vnn+1 N n=k λ∈[−k,k] E # #
∞ D −n  $ (r, λ)# + $ (ξ, ·)) + 2−n−1 , V−n−1 + Vnn+1 (N ≤ max #N n=k

λ∈[−k,k]

which implies due to (7.27) and (7.21) that for r ∈ [rk , rk+1 ] and k = 1, 2, . . .,  # # 8 8     k #N $ (r, λ)# + V−k 8N $ (r, ·)8 ≤ $ (r, ·) + 2V N $ (ξ, ·) + 1 N max V λ∈[−k,k]

≤ 3CV + 2 < ∞.

(7.30)

$ (r, ·) is bounded on R+ . By (7.30) and (7.29), the V (R)-valued function r → N $ (r, λ) ∈ SO(R+ , V (R)). Clearly, it is sufficient to consider Let us prove that N −1 , rk−1 ] is analogous). Then we deduce from r,  ∈ [rk , rk+1 ] (the proof for r,  ∈ [rk+1 (7.28) that % $ (r, λ) − N $ (, λ) = N

$ (r, k) − N $ (, k) N for λ ≥ k, $ $ N (r, −k) − N (, −k) for λ ≤ −k,

whence # # 8 8   $ (r, λ)− N $ (, λ)# +V k N $ (r, ·)− N $ (, ·) . (7.31) 8N $ (r, ·)− N $ (, ·)8 = max #N −k V λ∈[−k,k]

Since the functions νI(r, λ) and N (r, λ)I ν (r, λ) belong to E(R+ , V (R)) (see Theorem 5.3) and since for r,  ∈ [rk , rk+1 ] and λ ∈ [−k, k], $ (r, λ) − N $ (, λ) = νI(r, λ) − νI(, λ) N N (r, λ) N (, λ)   J K 1 N (r, λ)I ν (r, λ) − N (, λ)I ν (, λ) 1 + − = νI(r, λ) − νI(, λ) N (r, λ) N (, λ) N (r, λ)N (, λ) (7.32)

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and therefore   J K J K 1 1 $ $ ∂λ N (r, λ) − N (, λ) = ∂λ νI(r, λ) − ∂λ νI(, λ) + N (r, λ) N (, λ)   J K ∂λ N (r, λ) ∂λ N (, λ) + − νI(r, λ) − νI(, λ) N 2 (r, λ) N 2 (, λ) K J K J N (r, λ)I ν (r, λ) − N (, λ)I ν (, λ) ∂λ N (r, λ)I ν (r, λ) − N (, λ)I ν (, λ) + − N (r, λ)N (, λ) N 2 (r, λ)N 2 (, λ) K J (7.33) × N (, λ)∂λ N (r, λ) + N (r, λ)∂λ N (, λ) , we infer from (7.24) and the estimate |∂λ N (r, λ)| ≤ CN < ∞ (cf. (7.22)) that # #   k $ (r, λ) − N $ (, λ)# + V−k $ (r, ·) − N $ (, ·) N max #N λ∈[−k,k]

8 8 8 48 8νI(r, ·) − νI(, ·)8 + 4 8N (r, ·)I ν (r, ·) − N (, ·)I ν (, ·)8V V m  m2  # # # # 16CN 8CN # # #N (r, λ)I νI(r, λ) − νI(, λ) dλ + ν (r, λ) − N (, λ)I ν (, λ)#dλ. + m2 R m3 R $ ∈ SO(R+ , V (R)). The latter estimate, (7.31) and (7.29) imply that N Analogously, applying Theorem 5.3, (7.28), (7.29), the relations (7.32), (7.33) and the estimate |∂λ N (r, λ)| ≤ CN < ∞, one can prove that 8 8 $ (r, ·) − N $ h (r, ·)8 = 0. lim sup 8N V ≤

|h|→0 r∈R+

$ ∈ E(R+ , V (R)). Finally, because due to (7.20) and (5.16), Thus, N ( '   −M +∞ $ (ξ, ·) : ξ ∈ M = 0, lim sup V−∞ N + VM M→+∞

$ ∈ E(R $ + , V (R)). This implies we conclude from (7.28) and (7.29) that actually N by Lemma 3.3 that for every s ∈ {0, ∞} and every ξ ∈ Ms (SO(R+ 8 8 )) there is a $ (kn , ·) − N $ (ξ, ·)8 = 0.
sequence {kn } ⊂ R+ such that kn → s and lim 8N V n→∞

Applying now Theorems 7.1 and 6.2, we establish the following Fredholm criterion for singular integral operators with shifts. Theorem 7.5. Let the conditions of Theorem 5.2 hold for all g ∈ G in place of α, where G is a subexponential group of slowly oscillating shifts acting on Γ topologically freely. Then the operator N given by (7.1) is Fredholm on the weighted Lebesgue space Lp (Γ, w) if and only if (i) all the discrete operators A± τ (τ ∈ Γ) given by (7.3) are invertible on the space lp (G) and 8 8 8 8 −1 8 −1 8 sup 8(A+ < ∞, sup 8(A− < ∞; p p τ ) τ ) τ ∈Γ

B(l (G))

τ ∈Γ

B(l (G))

(ii) the function N (ξ, λ) given by (7.5)–(7.7) satisfies the condition 5 6 inf |N (ξ, λ)| : (ξ, λ) ∈ M × R > 0.

(7.34)

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Proof. Necessity. If the operator N is Fredholm on the space Lp (Γ, w), then condition (i) is fulfilled due to Theorems 7.1 and 6.2. On the other hand, applying Lemma 7.2 to the operator N and its adjoint operator N ∗ ∈ B(Lq (Γ, w−1 )), we infer by analogy with [10, Corollary 18.11] that all the limit operators M −1 N (ξ, ·)M , defined for ξ ∈ M by (7.12), are invertible on the space Lp (R+ , dµ), which, in its turn, is equivalent to the condition: for every ξ ∈ M, 5 6 inf |N (ξ, λ)| : λ ∈ R > 0. (7.35) If (7.34) is not fulfilled then, due to stability of Fredholmness, there is an ε6 > 0 5 such that the operator Nε := N +εI is Fredholm, but inf |Nε (ξ0 , λ)| : λ ∈ R = 0 for some ξ0 ∈ M, which contradicts (7.35) for N = Nε and proves condition (ii). Sufficiently. If (i) holds then, by Theorem 6.2, the functional operators A± are invertible on the space Lp (Γ, w). According to Theorem 7.1, it only remains  to prove the existence of operators NH , NH ∈ B(Lp (Γ, w)) that satisfy (7.4). $ (r, λ) ∈ E(R $ + , V (R)) satisfying By Lemmas 7.3 and 7.4, there is a function N (7.23) and (7.21). Then we infer from Theorem 3.1 that the Mellin pseudodiffe$ ) given by (3.3) is bounded on the space Lp (R+ , dµ). Let rential operator OP (N J K J K $ ) RΓ , N  = −4−1 RΓ Ψ−1 OP (N $) , (7.36) NH = −4−1 Ψ−1 OP (N H  ∈ where Ψ and RΓ are given by (5.3) and (5.1), respectively. Clearly, NH , NH p B(L (Γ, w)). From Theorems 5.2, 5.3 and Corollary 5.4 it follows that

N RΓ = Ψ−1 [OP (N νI)] + K1 ,

RΓ N = Ψ−1 [OP (N νI)] + K2 ,

(7.37)

where N is given by (7.8)–(7.10), νI is given by (5.16), N νI ∈ E(R+ , V (R)), and K1 , K2 ∈ K(Lp (Γ, w)). Passing to Fourier pseudodifferential operators, we infer from [21, Theorem 8.3] that $ ) = OP (I $1, OP (N νI)OP (N ν2) + K

$ )OP (N νI) = OP (I $ 2, OP (N ν2) + K

(7.38)

$ 2 ∈ K(Lp (R+ , dµ)). Since H := P+ P− = 4−1 (I − S 2 ) = −4−1 R2 $ 1, K where K Γ Γ and RΓ = Ψ−1 [OP (I ν )], we immediately deduce from (7.36) to (7.38) that the  operators NH N − H and N NH − H are compact on the space Lp (Γ, w), which gives (7.4).
If the operator N given by (7.1) is Fredholm on the space Lp (Γ, w), then, by Theorem 7.1 and [25, Theorem 1.1], the operator −1 −1 −1  N (−1) := P+ A−1 + + P− A− − NH (A+ A− + A− A+ − 2I),  given by (7.36), is a right regularizer for N : N N (−1) − I ∈ K(Lp (Γ, w)). with NH Consequently, N (r, λ)N (−1) (r, λ) = 1, which implies that

lim inf inf |N (r, λ)| > 0 r→s

λ∈R

for s ∈ {0, ∞}.

On the other hand, (7.39) implies due to (7.11) that |N (ξ, λ)| = lim |N (kn , λ)| ≥ lim inf inf |N (r, λ)| > 0 kn →s

r→s

λ∈R

(7.39)

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for every (ξ, λ) ∈ Ms (SO(R+ )) × R and every s ∈ {0, ∞}, which gives (7.34). Hence, Theorem 7.5 remains true with (7.34) replaced by (7.39). Theorem 7.5 directly implies the following corollary (cf. [23, Theorem 9.5]). Theorem 7.6. If 1 < p < ∞, (Γ, w) ∈ ASO p , α is a slowly oscillating shift on Γ, and all the conditions of Theorem 5.2 are satisfied, then the operator T = Vα P+ + GP− with a coefficient G ∈ SO(Γ) is Fredholm on the space Lp (Γ, w) if and only if inf |G(t)| > 0,

t∈Γ

where

lim inf inf |T (r, λ)| > 0 r→s

λ∈R

(s ∈ {0, ∞}),




T (r, λ) := eiω(r)(λ+i(1/p+rv (r))) P+ (r, λ) + G(r)P− (r, λ) and the functions P± (r, λ) are given by (7.10). Other applications of Theorem 7.5 will be considered in a forthcoming paper. Acknowledgment The author is grateful to the referee for useful comments and suggestions.

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[28] Yu.I. Karlovich and B. Silbermann, Local method for nonlocal operators on Banach spaces. Operator Theory: Advances and Applications 135 (2002), 235–247. [29] Yu.I. Karlovich and B. Silbermann, Fredholmness of singular integral operators with discrete subexponential groups of shifts on Lebesgue spaces. Math. Nachr. 272 (2004), 55–94. [30] V.G. Kurbatov, Algebra of difference operators. Deposited in VINITI, No. 1017-82 Dep., Voronezh, 1982 [Russian]. [31] V.G. Kravchenko and G.S. Litvinchuk, Introduction to the Theory of Singular Integral Operators with Shift. Kluwer Academic Publishers, Dordrecht, 1994. [32] A.G. Myasnikov and L.I. Sazonov, Singular integral operators with non-Carleman shift. Soviet Math. (Iz. VUZ) 24 (1980), No. 3, 22–31. [33] A.G. Myasnikov and L.I. Sazonov, On singular operators with a non-Carleman shift and their symbols. Soviet Math. Dokl. 22 (1980), 531–535. [34] V.S. Pilidi, A priori estimates for one-dimensional singular integral operators with continuous coefficients. Math. Notes 17 (1975), 512–515. [35] V.S. Rabinovich, Singular integral operators on composed contours and pseudodifferential operators. Math. Notes 58 (1995), 722–734. [36] V.S. Rabinovich, Algebras of singular integral operators on compound contours with nodes that are logarithmic whirl points. Russ. Acad. Sci. Izv. Math. 60 (1996), 1261– 1292. [37] V.S. Rabinovich, Mellin pseudodifferential operators techniques in the theory of singular integral operators on some Carleson curves. Operator Theory: Advances and Applications 102 (1998), 201–218. [38] V.S. Rabinovich, Pseudodifferential operators on Rn with variable shifts. Zeitschrift f¨ ur Analysis und ihre Anwendungen 22 (2003), No. 2, 315–338. [39] V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Operator Theory: Advances and Applications 150, Birkh¨ auser, Basel, 2004. [40] D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26 (1977), 817–838. [41] M.A. Shubin, Pseudodifference operators and their Green’s functions. Math. USSR Izvestiya 26 (1986), 605–622. [42] A.P. Soldatov, On the theory of singular operators with a shift. Differential Equations 15 (1979), No. 1, 80–91. [43] A.P. Soldatov, Singular integral operators on the line. Differential Equations 16 (1980), No. 1, 98–105. Yuri I. Karlovich Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos, M´exico e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 263–282 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Poly-Bergman Projections and Orthogonal Decompositions of L2-spaces Over Bounded Domains Yuri I. Karlovich and Lu´ıs V. Pessoa Abstract. The paper is devoted to obtaining explicit representations of polyBergman and anti-poly-Bergman projections in terms of the singular integral operators SD and SD∗ on the unit disk D, studying relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk that are realized by the operators SD and SD∗ , establishing two new orthogonal decompositions of the space L2 (U, dA) (in terms of poly-Bergman and antipoly-Bergman spaces) for an arbitrary bounded open set U ⊂ C with the Lebesgue area measure dA, considering violation of Dzhuraev’s formulas and establishing explicit forms of the Bergman and anti-Bergman projections for several open sectors. Mathematics Subject Classification (2000). Primary 32A25, 46E22; Secondary 30G30, 31A10, 47G10. Keywords. Poly-Bergman and anti-poly-Bergman spaces and projections, singular integral operators, bounded domain, Dzhuraev’s formulas, orthogonal decomposition.

1. Introduction Let U be a domain in C equipped with the Lebesgue area measure dA(z) = dxdy, and let A2n (U ) and A$2n (U ) (n ∈ N) denote the normed linear subspaces of L2 (U ) = L2 (U, dA) that consist of n-differentiable functions such that, respectively, ∂zn f = 0 and ∂zn f = 0 where     1 ∂ ∂ 1 ∂ ∂ ∂ ∂ := +i , ∂z := := −i . ∂z := ∂z 2 ∂x ∂y ∂z 2 ∂x ∂y All the authors were partially supported by FCT project POCTI/MAT/59972/2004 (Portugal). The first author was also supported by the SEP-CONACYT project No. 25564 (M´exico).

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As is known (see, e.g., [5]), A2n (U ) and A$2n (U ) are Hilbert subspaces of L2 (U ) with the inner product ·, · induced by L2 (U ). These spaces are related by the anti-linear norm one operator C : L2 (U ) → L2 (U ), Cf = f .

(1.1)

Obviously, C(A2n (U )) = A$2n (U ) because ∂zn f = ∂zn f . The poly-Bergman and anti-poly-Bergman projections of order n ∈ N are $U,n of L2 (U ) onto A2 (U ) and defined to be the orthogonal projections BU,n and B n 2 A$n (U ), respectively. In the case n = 1, i.e., for the Bergman and anti-Bergman $U . Clearly, B $U,n = CBU,n C. projections, we will use the notation BU and B According to [2, Chapter 2], for a bounded multiply connected domain U with sufficiently smooth boundary, the poly-Bergman and anti-poly-Bergman projections of order n ∈ N are represented in the form $U,n = I − SU,n SU,−n + K $ n, BU,n = I − SU,−n SU,n + Kn , B (1.2) where the singular integral operators SU,n for n ∈ Z\{0} are defined by  (−1)n |n| (w − z)n−1 f (w) dA(w) for f ∈ L2 (U ), (SU,n f )(z) = n+1 π U (w − z)

(1.3)

$ n are compact operators on the space L2 (U ). Observe that S ∗ = SU,−n and Kn , K U,n for n ∈ Z\{0}. By [2, Lemma 2.9.1], SU,−n − (SU )n ∈ K,

SU,n − (SU∗ )n ∈ K,

(1.4)

where K stands for the closed two-sided ideal of all compact operators on the space L2 (U ) and, according to [6], SU and SU∗ are singular integral operators on the space L2 (U ) given for an arbitrary domain U ⊂ C by  1 f (w) (SU f )(z) = − dA(w), π U (w − z)2  1 f (w) (SU∗ f )(z) = − dA(w). π U (w − z)2 Clearly, SU∗ = CSU C is the adjoint operator for SU , and SU,−1 = SU , SU,1 = SU∗ . The paper is devoted to obtaining explicit representations of the poly-Berg$D,n in terms of the man projections BD,n and anti-poly-Bergman projections B singular integral operators SD and SD∗ on the unit disk D, studying relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk that are realized by the operators SD and SD∗ , establishing two new orthogonal decompositions of the space L2 (U ) (in terms of poly-Bergman and anti-polyBergman spaces) for an arbitrary bounded open set U ⊂ C, considering violation of Dzhuraev’s formulas (1.2) and establishing explicit forms of the Bergman and anti-Bergman projections for several open sectors. The paper is organized as follows. In Section 2 we study relations of polyBergman and anti-poly-Bergman projections with singular integral operators on the unit disk. To this end we obtain explicit formulas for the images SD∗ ηk,m of

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the functions ηk,m (z) := z k z m (k, m = 0, 1, 2, . . .), which allows us to prove that if $ n = 0 in (1.2) for all n ∈ N (this was known only for n = 1, U = D, then Kn = K see [2, Chapter 2]). For the upper half-plane such result was earlier obtained in [8, Theorem 1.1] (see also [10]). In Section 3 we prove that the poly-Bergman projections BD,k and anti$D,j commute and their products B $D,j BD,k are finitepoly-Bergman projections B dimensional projections, and establish explicit formulas for the one-dimensional $D BD,(k) and BD B $D,(k) are orthogonal projec$D,(k) where BD,(k) and B projections B 2 tions of the space L (D) onto the true poly-Bergman and true anti-poly-Bergman spaces A2(k) (D) and A$2(k) (D) given by (3.4) and (3.5), respectively. Making use of these formulas we describe relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk that are realized by the operators SD and SD∗ , which is the main result of the section (see Theorem 3.5). In contrast to an analogous result for the upper half-plane Π (see [5, Theorem 2.4] and also [10]), the operator SD is a partial isometry of the spaces A2(k) (D) onto A2(k+1) (D) and (D) into A$2 (D) for all k ∈ N, while SΠ is an isometry of the the spaces A$2 (k+1)

(k)

corresponding spaces over Π. In Section 4 we establish orthogonal decompositions of the space L2 (U ) in terms of poly-Bergman spaces and in terms of anti-poly-Bergman spaces for an arbitrary bounded open set U ⊂ C (see Theorem 4.3 and Corollary 4.4), which are different of those obtained in [7]. The proof is based on the following density results: $U,n = I. s-lim BU,n = I, s-lim B n→+∞

n→+∞

In Section 5 we show that, for bounded domains possessing Dzhuraev’s formulas (1.2), the corresponding slitted domains do not have such formulas. There we also establish explicit forms of the Bergman and anti-Bergman projections for the open sectors 5 6 Km = z = reiθ : r > 0, θ ∈ (0, π/m) (m = 2, 3, . . .). These formulas generalize those for the upper half-plane Π (cf. [9, Lemma 7.5] and [4, Proposition 2.4]).

2. Relations of poly-Bergman and anti-poly-Bergman projections with singular integral operators on the unit disk Let Z+ := N ∪ {0} and let D = {z ∈ C : |z| < 1} be the open unit disk in C. In this subsection we will show that the poly-Bergman and anti-poly-Bergman projections $D,n on the space L2 (D) belong to the non-closed ∗ -algebra generated BD,n and B by the two-dimensional singular integral operator SD and its adjoint operator SD∗ . Let C(U ) be the set of all complex-valued functions continuous on the closure U of an open set U ⊂ C.

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Lemma 2.1. For every k ∈ N, the sets 6 5 Pk := span z l z j : l = 0, 1, . . . , k − 1; j ∈ Z+ ⊂ A2k (D), 6 5 P$k := span z l z j : l = 0, 1, . . . , k − 1; j ∈ Z+ ⊂ A$2k (D) are dense in the Hilbert spaces A2k (D) and A$2k (D), respectively. Proof. Let us first treat the case of poly-Bergman spaces. Given f ∈ L2 (D) and 0 < t < 1, define ft (z) = f (tz) for z ∈ D. Because C(D) is dense in L2 (D), there is a g ∈ C(D) such that f − g < ε. Thus √ f − ft  ≤ f − g + g − gt  + gt − ft  ≤ ε(1 + t−1 ) + π g − gt ∞ , (2.1) where · and ·∞ are norms in the spaces L2 (D) and C(D), respectively. Because g is uniformly continuous on D, inequality (2.1) implies that limt→1− f − ft  = 0. Let f ∈ A2k (D). By [1, Section 1.1], there exist analytic functions ψj (j = k−1 k−1 0, 1, . . . , k − 1) such that f (z) = j=0 z j ψj (z). Hence ft (z) = j=0 z j tj (ψj )t (z) for all t ∈ (0, 1). Since (ψj )t (j = 0, 1, . . . , k − 1) are analytic functions on an open set that contains D, the Taylor series of these functions uniformly converge on D and thus converge in the norm of L2 (D). Given ε > 0 we fix 0 < t < 1 such that f −ft  < ε/2. For every j = 0, 1, . . . , k −1 we choose a polynomial pj (z) such that k−1 (ψj )t − pj  < ε/(2k). Consider now the function φt (z) = j=0 z j tj pj (z) ∈ Pk . Then the density of Pk in A2k (D) follows from the estimate f − φt  ≤ f − ft  + ft − φt  < ε/2 +

k−1


(ψj )t − pj  < ε.

j=0

$k = C(Pk ) is dense in A$2 (D). Finally, since A$2k (D) = C(A2k (D)), the set P k




Let Dε (z) ⊂ C denote the open disk of radius ε centered at a point z. For simplicity we use the abbreviation ηk,m (z) := z k z m (k, m = 0, 1, 2, . . .). Lemma 2.2. For every k, m ∈ Z+ and every z ∈ D, 

 SD∗ ηk,m (z) =

k min{0, m + 1 − k} k−m−2 z k−1 z m+1 + z . m+1 m+1

Proof. Fix z ∈ D. Applying the Green formulas   1 ∂u (w) dA(w) = − u(ξ) dξ, 2i ∂U U ∂w   1 ∂u (w) dA(w) = u(ξ) dξ, 2i ∂U U ∂w

u ∈ C 1 (U ), u ∈ C 1 (U )

(2.2)

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in the case U = D \ Dε (z) and sufficiently small ε > 0, we obtain    ∗ 1 w k wm dA(w) SD ηk,m (z) = − lim π ε→0 D\Dε (z) (w − z)2       k 1 ∂ wk−1 wm+1 ∂ w k wm − dA(w) = lim π ε→0 D\Dε (z) ∂w w − z (m + 1) ∂w w−z    1 w k wm w k wm = dw − lim dw ε→0 ∂D (z) w − z 2πi ∂D w − z ε    k−1 m+1 k w w w k−1 wm+1 dw − lim dw . (2.3) + ε→0 ∂D (z) 2πi(m + 1) w−z w−z ∂D ε Since w = w−1 for w ∈ ∂D, we infer from the Cauchy theorem and from the residue theorem that   k 1 w k wm w k−1 wm+1 dw + dw 2πi ∂D w − z 2πi(m + 1) ∂D w − z    min{0, m + 1 − k} k−m−2 k wm−k+1 1 dw = z . (2.4) = 1− m + 1 2πi ∂D 1 − wz m+1 Further,  lim

ε→0

∂Dε (z)

w k wm dw = lim ε→0 w−z

 0

2π

(z + εe−iϕ )k (z + εeiϕ )m ie2iϕ dϕ



2π

e2iϕ dϕ = 0.

= z k z mi

(2.5)

0

Finally, for k ∈ N, we get   2π w k−1 wm+1 dw = lim lim (z + εe−iϕ )k−1 (z + εeiϕ )m+1 (−i) dϕ ε→0 ∂D (z) ε→0 0 w − z ε = −2πiz k−1 z m+1 . Substituting (2.4) to (2.6) into (2.3), we immediately obtain (2.2).

(2.6)


From [11, p. 61] it follows that if f ∈ C ∞ (U ) and U is a bounded multiply connected domain with C ∞ boundary, then SU f, SU∗ f ∈ C ∞ (U ) and ∂z SU f = ∂z f,

∂z SU∗ f = ∂z f

(2.7)

for all z ∈ U . As we will show in (4.4) and (4.5), these formulas remain true for arbitrary open sets U ⊂ C and all points z ∈ U if f ∈ C ∞ (U ) ∩ L2 (U ). Theorem 2.3. If D ⊂ C is the unit disk, then for every n ∈ N the following Dzhuraev formulas hold: BD,n = I − (SD )n (SD∗ )n ,

$D,n = I − (SD∗ )n (SD )n . B

(2.8)

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Proof. From Lemmas 2.1 and 2.2 it follows that, for every n ∈ N, (SD∗ )n f = 0 for f ∈ A2n (D).

(2.9)

Define the operators Pn := I − (SD )n (SD∗ )n Let us prove that

  Pn L2 (D) ⊂ A2n (D)

(n ∈ N).

(2.10)

for n ∈ N.

(2.11) ∞

Clearly, it is sufficient to prove (2.11) for the dense subset C (D) ⊂ L (D). Let f ∈ C ∞ (D). Then, by (2.7), for z ∈ D we obtain 2

∂zn (SD )n (SD∗ )n f = ∂zn−1 ∂z (SD )n−1 (SD∗ )n f = ∂z ∂zn−1 (SD )n−1 (SD∗ )n f = . . . = ∂zn (SD∗ )n f = ∂zn−1 ∂z (SD∗ )n−1 f = ∂z ∂zn−1 (SD∗ )n−1 f = . . . = ∂zn f. Hence, in view of (2.10), it follows for f ∈ C ∞ (D) and z ∈ D that   ∂zn Pn f = ∂zn f − (SD )n (SD∗ )n f = ∂zn f − ∂zn f = 0, which implies (2.11) for all n ∈ N. Because Pn2 = Pn − (SD )n (SD∗ )n Pn , from (2.11) and (2.9) it follows that Pn2 = Pn . As Pn∗ = Pn due to (2.10), we conclude that Pn is an orthogonal projection. Again (2.9) and (2.10) imply that Pn f = f for f ∈ A2n (D), which together with $D,n = (2.11) gives Pn = BD,n and proves the first equality in (2.8). Applying now B ∗ CBD,n C and SD = CSD C we get the second equality in (2.8).
Lemma 2.2 also allows us to obtain a new proof of the following important assertion being a strengthening of (1.4) for U = D (see [2, Chapter 2]). Corollary 2.4. The following representations hold: SD,−n = (SD )n ,

SD,n = (SD∗ )n .

(2.12)

Proof. It is sufficient to prove only the second formula in (2.12) because then SD,−n = CSD,n C = C(SD∗ )n C = SDn . By definition, SD∗ = SD,1 . By induction suppose that (SD∗ )n = SD,n for n ∈ N and let us prove this formula for n + 1. Let f ∈ C ∞ (D). Applying the relation ∂w SD∗ f = ∂w f and the Green formulas, we infer that (SD,n+1 f )(z) − ((SD∗ )n+1 f )(z) = (SD,n+1 f )(z) − (SD,n SD∗ f )(z)    (w − z)n (−1)n+1 (w − z)n−1 ∗ (n + 1) = f (w) + n (S f )(w) dA(w) π (w − z)n+2 (w − z)n+1 D D       ∂ (−1)n (w − z)n (w − z)n ∂ ∗ f (w) − (S f )(w) dA(w) = π ∂w (w − z)n+1 ∂w (w − z)n+1 D D    (−1)n (w − z)n (w − z)n ∗ = f (w) dw + (SD f )(w) dw n+1 n+1 2πi ∂D (w − z) ∂D (w − z)  K (−1)n (w − z)n wn+1 J f (w) − w−2 (SD∗ f )(w) dw. = n+1 2πi ∂D (1 − wz)

Poly-Bergman Projections and Orthogonal Decompositions Hence, to obtain (2.12) we need to prove that  K (w − z)n wn+1 J ηk,m (w) − w−2 (SD∗ ηk,m )(w) dw = 0 n+1 (1 − wz) ∂D

269

(2.13)

for all ηk,m (z) = z k z m (k, m = 0, 1, 2, . . .). For w ∈ ∂D, from Lemma 2.2 it follows that min{k, m + 1} m−k+2 w , (SD∗ ηk,m )(w) = m+1 whence  %  k wm−k if k = 0, 1, . . . , m, 1 − m+1 −2 ∗ ηk,m (w) − w (SD ηk,m )(w) = 0 if k = m + 1, m + 2, . . . . Thus, according to (2.13) it remains to show that    k (w − z)n wn+1+m−k dw = 0 1− n+1 m+1 ∂D (1 − wz) for k = 0, 1, . . . , m. But the latter follows from the Cauchy theorem because the (w − z)n wn+1+m−k is analytic with respect to w ∈ D, for z ∈ D.
function (1 − wz)n+1

3. Relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk The Stone-Weierstrass theorem implies the density of finite linear combinations of the functions z l z s (l, s ∈ Z+ ) in C(D). Because C(D) is dense in L2 (D) and the uniform convergence on D implies convergence in the quadratic mean, we get the density of the set z l z s (l, s ∈ Z+ ) in L2 (D). Applying the Gram-Schmidt 5 orthonormalization process to the sets of linearly independent functions z l+s z l : 6 5 l l+s 6 : l5 ∈ Z+ where s6 ∈ Z+5, we obtain, respectively, the sets l ∈ Z+ and z z 6 of orthonormal vectors ϕl+s,l : l ∈ Z+ and ϕl,l+s : l ∈ Z+ for s ∈ Z+ that satisfy the relations 6 5 6 5 = span ϕl+s,l : l = 0, 1, . . . , k , span z l+s z l : l = 0, 1, . . . , k 6 5 6 5 (3.1) = span ϕl,l+s : l = 0, 1, . . . , k span z l z l+s : l = 0, 1, . . . , k for all k, s ∈ Z+ . Here, by orthonormalization, : : s+1 s s+1 s z , ϕ0,s (z) = ϕs,0 (z) = z π π Taking into account the fact that

for all s ∈ Z+ .

z l1 z s1 , z l2 z s2  = z l1 +s2 , z l2 +s1  = 0 if and only if l1 − s1 = l2 − s2 , we conclude that ϕl+s,l , ϕm+k,m  = 0 and ϕl+s,l , ϕm,m+n  = 0

(3.2)

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for all l, m ∈ Z+ and all s, k, n ∈ Z+ such that k = s and n + s > 0. On the other hand, by construction,5ϕl+s,l , ϕm+s,m 6 = 0 for all l, m, s ∈ Z+ such that l = m. Consequently, the set ϕl,s : l, s ∈ Z+ is a Hilbert base of the space L2 (D). In addition, Lemma 2.1 and (3.1) imply that 6 5 (3.3) clos span ϕl,s : l ∈ Z+ , s = 0, 1, . . . , k − 1 = A2k (D), 6 5 k − 1 is a Hilbert base for the polyand thus the set ϕl,s : l ∈ Z+ , s = 0, 1, . . . , 5 6 2 Bergman space Ak (D). Analogously, the set ϕl,s : s ∈ Z+ , l = 0, 1, . . . , k − 1 is a Hilbert base for the anti-poly-Bergman space A$2k (D). Let PN denote the orthogonal projection of a Hilbert space H onto its closed subspace N . For every k, j ∈ N, let Nj,k be the finite-dimensional subspace of L2 (D) generated by the functions z l z s (l = 0, 1, . . . , j − 1; s = 0, 1, . . . , k − 1). $D,j BD,k = PN . In particular, the projections B $D,j Lemma 3.1. If j, k ∈ N, then B j,k and BD,k commute. 6 5 Proof. Let ϕl,s : l, s ∈ Z+ be the Hilbert base of L2 (D) introduced above. For l ∈ Z+ and s = 0, 1, . . . , k − 1, we obtain
$D,j ϕl,s , ϕν,µ ϕν,µ $D,j ϕl,s = B B ν=0,1,...,j−1; µ∈Z+

=




$D,j ϕν,µ ϕν,µ = ϕl,s , B

ν=0,1,...,j−1; µ∈Z+

=




ν=0,1,...,j−1; µ∈Z+




ϕl,s , ϕν,µ ϕν,µ

ν=0,1,...,j−1; µ∈Z+

δl,ν δs,µ ϕν,µ =




δl,ν δs,µ ϕν,µ ,

µ=0,1,...,k−1 ν=0,1,...,j−1

$D,j BD,k f = PN f for all f ∈ A2 (D). where δj,k is the Kronecker symbol. Thus B j,k k $D,j BD,k f = 0 for all f ∈ A2 (D)⊥ . Since PN f, g = f, PN g = 0 for Clearly, B j,k j,k k all f ∈ A2k (D)⊥ and all g ∈ L2 (D), we conclude that PNj,k f = 0 for all f ∈ A2k (D)⊥ . $D,j BD,k = PN on the space L2 (D) and dim Im B $D,j BD,k = kj. Consequently, B j,k Finally, because the operator PNj,k is self-adjoint, we infer that the projections $D,j and BD,k commute for all j, k ∈ N. B
Lemma 3.1 in the case j = k = 1 gives the following nice formula known from the description of the harmonic Bergman projection (see, e.g., [3, Proposition 9.9]). Corollary 3.2. For every f ∈ L2 (D) and every z ∈ D,    $D BD f (z) = 1 B f (w) dA(w). π D Proof. Since ϕ1,1 = π −1/2 is a norm one generator of the space N1,1 , we obtain      $D BD f (z) = PN1,1 f (z) = f, ϕ1,1 ϕ1,1 = 1 B f (w) dA(w), π D which completes the proof.


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For a domain U ⊂ C, the true poly-Bergman spaces of order n are given by J K⊥ := A2n (U )∩ A2n−1 (U ) for n > 1, A2(1) (U ) := A21 (U ) := A2 (U ), (3.4)

A2(n) (U )

and the true anti-poly-Bergman spaces of order n are defined by J K⊥ A$2(n) (U ) := A$2n (U ) ∩ A$2n−1 (U ) for n > 1, A$2(1) (U ) := A$21 (U ) := A$2 (U ) (3.5) (see, e.g., [9]). This implies that, for every n ∈ N, A2n (U ) =

n N

A2(k) (U ), A$2n (U ) =

k=1

n N

A$2(k) (U ).

k=1

Such definitions have an equivalent in the context of projections. Indeed, setting BU,(n) := BU,n − BU,n−1 (n > 1), $U,(n) := B $U,n − B $U,n−1 (n > 1), B

BU,(1) := BU,1 := BU , $U,(1) := B $U,1 := B $U , B

we deduce that for every n ∈ N, Im BU,(n) = A2(n) (U ), and BU,n =

n


$U,(n) = A$2 (U ), Im B (n)

$U,n = BU,(k) , B

k=1

n


$U,(k) . B

(3.6)

(3.7)

k=1

Let us study special products of Bergman type projections that we need to establish relations between different true poly-Bergman or different true anti-polyBergman spaces on the unit disk in terms of the operators SD and SD∗ , respectively. As a result, we obtain a generalization of Corollary 3.2. $D BD,(k) and BD B $D,(k) are oneLemma 3.3. For every k ∈ N, the operators B 2 dimensional projections given for g ∈ L (D) by $D BD,(k) g = k g , z k−1 z k−1 B π

$D,(k) g = k g , z k−1 z k−1 . BD B (3.8) π $D BD,k − B $D BD,k−1 is the orthogo$D BD,(k) = B Proof. Fix k ∈ N. By Lemma 3.1, B 2 nal projection of the space L (D) onto the space N1,k ) N1,k−1 (here BD,0 := 0 and orthogonal N1,0 := {0}). Since the linear space N1,k is generated by the pairwise 5 6 functions z m (m = 0, 1, . . . , k − 1), we infer that N1,k ) N1,k−1 = λz k−1 : λ ∈ C . $D BD,(k) is the one-dimensional Thus, taking into account (3.2) we conclude that B 2 projection given for g ∈ L (D) by and

$D BD,(k) g = k BD,(k) g , z k−1  z k−1 = k g , z k−1  z k−1 , B π π which gives the first formula in (3.8). The second formula in (3.8) follows from $D BD,(k) C where the operator C is $D,(k) = C B the first in view of the equality BD B given by (1.1).
The next assertion related to (3.8) allows one to estimate norms of evaluation functionals for the derivatives ∂zn g (n = 0, 1, . . . , k − 1) of functions g ∈ A2k (D).

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Y.I. Karlovich and L.V. Pessoa

Lemma 3.4. If k ∈ N and g ∈ A2k (D), then 1 ∂ k−1 g k g , z k−1  = (0). π (k − 1)! ∂z k−1

(3.9)

Proof. Suppose g ∈ A2k (D). Taking t ∈ (0, 1), by the Lebesgue dominated convergence theorem and by the Green formulas, we obtain  k−1 g , z  = lim− g(w)wk−1 dA(w) t→1 tD    K ∂ J ∂g g(w)wk−1 w dA(w) − (w)wk−1 w dA(w) = lim t→1− tD ∂w tD ∂w    k−1
(−1)j ∂ ∂j g k−1 j+1 lim− dA(w) (w)w w = (j + 1)! t→1 tD ∂w ∂wj j=0 1
(−1)j lim = 2i j=0 (j + 1)! t→1− k−1

=

1 2i

k−1
j=0

(−1)j lim (j + 1)! t→1−

 ∂(tD)

 ∂(tD)

∂j g (w)wk−1 wj+1 dw ∂wj ∂j g (w)wk−j−2 t2j+2 dw. ∂wj

(3.10)

According to [1, Section 1.1], for g ∈ A2k (D) there exist analytic functions gl (l = k−1 0, 1, · · · , k − 1) such that g(z) = l=0 z l gl (z) for z ∈ D. Then
l! ∂j g z l−j gl (z) for z ∈ D and j = 0, 1, . . . , k − 1, j (z) = (l − j)! ∂z l=j k−1

(3.11)

∂j g (0) = j!gj (0). Consequently, substituting (3.11) in (3.10) and applying ∂z j the Cauchy theorem and the residue theorem, we infer that  k−1 k−1 k

k (−1)j l! g , z k−1  = lim wk−l−2 t2l+2 gl (w) dw π 2πi j=0 (j + 1)! (l − j)! t→1− ∂(tD) whence

l=j

k = 2πi =

k−1
j=0

k−1
j=0

(−1)j (k − 1)! lim (j + 1)! (k − 1 − j)! t→1−

 ∂(tD)

gk−1 (w) dw w

(−1)j k! gk−1 (0) = gk−1 (0), (j + 1)! (k − 1 − j)!

which gives (3.9) because, by (3.11), gk−1 (0) =

1 ∂ k−1 g (0). (k − 1)! ∂z k−1

Let us now define the one-dimensional spaces 5 6 5 6 $ k := λz k−1 : λ ∈ C . Lk := λz k−1 : λ ∈ C and L




(3.12)

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273

From (3.2) and (3.3) it follows that $ k ⊂ A$2 (D). L (k)

Lk ⊂ A2(k) (D),

(3.13)

In [5, Theorem 2.4] (see also [10] for another proof of this result) we proved that the singular integral operator SΠ over upper half-plane Π is an isometric isomorphism of the true poly-Bergman space A2(k) (Π) onto A2(k+1) (Π) and of the true anti-poly-Bergman space A$2(k+1) (Π) onto A$2(k) (Π) for every k ∈ N, with   ∗ was also obtained SΠ A$2(1) (Π) = {0} (a dual result for the adjoint operator SΠ there). Now we establish an analogous result in the case of the unit disk D. Theorem 3.5. For every k ∈ N, the operators SD∗ : A2(k+1) (D) → A2(k) (D) ) Lk ,

(3.15)

$ k → A$2 SD∗ : A$2(k) (D) ) L (k+1) (D),

(3.16)

SD :

) Lk →

(3.14)

A2(k+1) (D),

SD :

A2(k) (D)

A$2(k+1) (D)

→

A$2(k) (D)

$k )L

(3.17)

are isometric isomorphisms. In addition, SD∗ : A2(1) (D) → {0},

SD : A$2(1) (D) → {0}.

Proof. Fix k ∈ N. From (2.8) it follows that f ∈ A2k (D) if and only if (SD∗ )k f = 0. Indeed, by (2.8), f ∈ A2k (D) if and only if (SD )k (SD∗ )k f = 0, which is equivalent to 82 A @ A 8 @ 0 = (SD )k (SD∗ )k f, (SD )k (SD∗ )k f = (SD )k (SD∗ )k f, f = 8(SD∗ )k f 8 . Hence, because (SD∗ )k SD∗ g = (SD∗ )k+1 g = 0 for every g ∈ A2k+1 (D), we conclude that   (3.18) SD∗ A2k+1 (D) ⊂ A2k (D). For f ∈ A2k (D), by the derivation formulas (2.7), we obtain ∂zk+1 SD f = ∂zk ∂z f = ∂z ∂zk f = 0 Consequently,

for all z ∈ D.

  SD A2k (D) ⊂ A2k+1 (D),

(3.19)

and thus SD∗ f, g = f, SD g = 0

for all f ∈ A2(k+1) (D) and g ∈ A2k−1 (D),

(3.20)

which together with (3.18) implies that   SD∗ A2(k+1) (D) ⊂ A2(k) (D).

(3.21)

By Lemma 2.2, SD∗ z m = 0 for every m ∈ Z+ , and therefore     SD∗ A2(1) (D) = SD∗ A21 (D) = {0}.

(3.22)

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Y.I. Karlovich and L.V. Pessoa

  By (3.19), SD A2(k) (D) ⊂ A2k+1 (D). On the other hand, analogously to (3.20), we infer from (3.18) that SD f, g = f, SD∗ g = 0 Hence,

for all f ∈ A2(k) (D) and g ∈ A2k (D).

  SD A2(k) (D) ⊂ A2k+1 (D) ) A2k (D) = A2(k+1) (D). A2(k+1) (D) 

(3.23)

As the space is orthogonal to the space A (D) = Im BD , we conclude  that BD A2(k+1) (D) = 0. Therefore, if g ∈ A2(k+1) (D), then from (3.21) it follows that SD∗ g ∈ A2(k) (D), whence SD (SD∗ g) = g − BD g = g. Consequently, instead of (3.23) we obtain the stronger relation   SD A2(k) (D) = A2(k+1) (D). (3.24) 2

Further, by (3.13), Lk ⊂ A2(k) (D). On the other hand, by (3.12), Lk ⊂ A$2(1) (D) and due to (3.22),     SD A$2(1) (D) = CSD∗ A2(1) (D) = {0}. Hence SD (Lk ) = {0} and we deduce from (3.24) that SD : A2(k) (D) ) Lk → A2(k+1) (D) . Let now f, g ∈

A2(k) (D)

(3.25)

) Lk . Then from (2.8) and (3.8) it follows that

$D )g = f , g − f , B $D BD,(k) g SD f , SD g = f , SD∗ SD g = f , (I − B = f , g − (k/π)f , z k−1 g , z k−1  = f , g, which together with (3.24) and (3.25) means that SD is an isometric isomorphism of the space A2(k) (D) ) Lk onto the space A2(k+1) (D). Hence SD∗ is an isometric isomorphism of the space A2(k+1) (D) onto the space A2(k) (D) ) Lk . Finally, (3.16) and (3.17) follows, respectively, from (3.15) and (3.14) in view   of the relations CSD C = SD∗ and C A2(k) (D) = A$2(k) (D).


4. Orthogonal decompositions of L2 -spaces over bounded domains In this section we will obtain two orthogonal decompositions of the space L2 (U ) over a bounded open set U ⊂ C in terms of true poly-Bergman spaces and in terms of true anti-poly-Bergman spaces. For U = D, these decompositions are different of those established in [7]. Given an arbitrary open set U ⊂ C, for every n ∈ N, we consider the operators $ U,n := I − SU,n SU,−n , DU,n := I − SU,−n SU,n , D (4.1) where SU,n for n ∈ Z\{0} are the singular integral operators defined by (1.3). Fix z ∈ U and choose δ > 0 such that Dδ (z) ⊂ U . Applying translation and dilatation operators we infer from (2.12) that n n   ∗ SDδ (z),−n = SDδ (z) , SDδ (z),n = SD . (4.2) δ (z)

Poly-Bergman Projections and Orthogonal Decompositions

275

Let f ∈ C ∞ (U ) ∩ L2 (U ). Fix a function ϕ ∈ C ∞ (U ) such that ϕ = 1 on Dδ/2 (z) and ϕ = 0 on U \ Dδ (z). Taking into account (4.2), for all ξ ∈ Dδ (z) we obtain       SU,−n f (ξ) = SU,−n (ϕf ) (ξ) + SU,−n (f − ϕf ) (ξ)  n    = SD (ϕf ) (ξ) + SU,−n (f − ϕf ) (ξ). (4.3) δ (z) The integral defining SU,−n (f − ϕf ) is not singular in a neighborhood of the point z, and therefore it defines an n-analytic function there. So, applying the results of [11, Chapter 1] to the function SDδ (z) (ϕf ), we infer that the function SU,−n f has all the derivatives ∂zk ∂zj SU,−n f (z) (k, j ∈ Z+ ). Because z is an arbitrary point in U , we conclude that SU,−n f ∈ C ∞ (U ). Thus, taking into account (4.3) and (2.7), for every z ∈ U we get  n   n   n−1    ∂  ∂  n−1 ∂ ∂ n S S SU,−n f (z) = (ϕf ) (z) = (ϕf ) (z) ∂z n ∂z n Dδ (z) ∂z n−1 ∂z Dδ (z)   ∂nf ∂ ∂ n−1  n−1 S (ϕf ) (z) = · · · = (z). (4.4) = ∂z ∂z n−1 Dδ (z) ∂z n Similar arguments depending only on local properties of derivatives and applied to the function SU,n f , with f ∈ C ∞ (U ) ∩ L2 (U ), imply that SU,n f ∈ C ∞ (U ) and  n   ∂nf ∂  S f (z) = (z) for all z ∈ U. (4.5) U,n n ∂z ∂zn In particular, for n = 1 we deduce from (4.4) and (4.5) that formulas (2.7) are valid for arbitrary open sets U ⊂ C, functions f ∈ C ∞ (U ) ∩ L2 (U ) and all z ∈ U . So, if f ∈ C ∞ (U ) ∩ L2 (U ), we infer from (2.7) that ∂n ∂n ∂n S S f = S f = f. (4.6) U,−n U,n U,n ∂z n ∂z n ∂z n Consequently, using density of C ∞ (U ) ∩ L2 (U ) in L2 (U ), we conclude from (4.1) and (4.6) that for arbitrary open sets U ⊂ C, Im DU,n ⊂ A2n (U ) for all n ∈ N.

(4.7)

The same type arguments also imply that $ U,n ⊂ A$2 (U ) for all n ∈ N. Im D n Let χV denote the characteristic function of a set V ⊂ C. Theorem 4.1. Let U be an arbitrary bounded open subset of C. The following formulas hold on the space L2 (U ): s-lim DU,n = I, n→∞

$ U,n = I. s-lim D n→∞

(4.8)

Proof. Choose r > 0 such that U ⊂ Dr (0). Then we infer from Corollary 2.4 that  n  n SU,−n = χU SDr (0),−n χU I = χU SDr (0) χU I = χU χDr (0) SR2 χDr (0) I χU I, whence SU,−n  ≤ 1. Therefore for f ∈ L2 (U ) it follows from (4.1) that f − DU,n f  = SU,−n SU,n f  ≤ SU,n f 

(4.9)

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Y.I. Karlovich and L.V. Pessoa

where

8 8 8 8 SU,n f  = 8χU SDr (0),n χU f 8 ≤ 8SDr (0),n χU f 8 .

Fix f ∈ L (Dr (0)) and let us show that 8 8 lim 8f − BDr (0),n f 8 = 0.

(4.10)

2

n→∞

(4.11)

Since the set C(Dr (0)) is dense in L2 (Dr (0)), for every ε > 0 there is a function g ∈ C(Dr (0)) such that f − g < ε/2. By the Stone-Weierstrass theorem, there exists a polynomial
p(z, z) = ck,j z k z j , k,j=0,1,...,n0

with coefficients ck,j ∈ C, such that g − p(z, z)∞ < π −1/2 ε/2. Then we obtain √ f − p(z, z) ≤ f − g + g − p(z, z) < ε/2 + g − p(z, z)∞ π < ε. Because p(z, z) ∈ A2n0 +1 (Dr (0)) and BDr (0),n f is the element in A2n (Dr (0)) with minimal distance to f , for every n ≥ n0 + 1 we infer that 8 8 8f − BD (0),n f 8 ≤ f − p(z, z) < ε, r which gives (4.11). Since I − BDr (0),n is an orthogonal projection on the space L2 (Dr (0)) and ∗ SDr (0),−n = SD , we deduce from (4.11), Theorem 2.3 and Corollary 2.4 that r (0),n for every f ∈ L2 (Dr (0)), 8 82 @ A 0 = lim 8f − BDr (0),n f 8 = lim f − BDr (0),n f, f n→∞ n→∞ 8 82 @ A = lim SDr (0),−n SDr (0),n f, f = lim 8SDr (0),n f 8 . (4.12) n→∞

n→∞

Combining (4.9), (4.10) and (4.12), we infer that s-limn→∞ DU,n = I on the space L2 (U ). If C is the anti-linear operator (1.1), then we obtain $ U,n = C s-lim DU,n C = I, s-lim D n→∞

n→∞




which completes the proof.

Corollary 4.2. Let U ⊂ C be an arbitrary bounded open set. The following formulas hold on the space L2 (U ): s-lim BU,n = I, n→∞

$U,n = I, s-lim B n→∞

$U,n BU,m = I. s-lim B

n,m→∞

Proof. Fix f ∈ L2 (U ). By (4.7) and (4.8), we obtain BU,n f − f  =

BU,n (f − DU,n f + DU,n f ) − f 

≤

BU,n (f − DU,n f ) + BU,n DU,n f − f 

=

f − DU,n f  + DU,n f − f  → 0 as n → ∞,

(4.13) (4.14)

Poly-Bergman Projections and Orthogonal Decompositions whence, applying (1.1), we get 8 8 $U,n f − f 8 = BU,n Cf − Cf  → 0 8B

277

as n → ∞.

Thus, relations (4.13) are fulfilled, which immediately implies (4.14).




Corollary 4.2 and formulas (4.13), (3.6) and (3.7) immediately imply the following important result. Theorem 4.3. For any bounded open set U ⊂ C, the Hilbert space L2 (U ) equipped with the Lebesgue area measure dA(z) = dxdy admits the following two orthogonal decompositions: ∞ ∞ N N L2 (U ) = A2 (U ) = A$2 (U ). (k)

k=1

(k)

k=1

For the upper half-plane Π, the situation is quite different, and Corollary 4.2 and Theorem 4.3 are not true (see [9] and [10]). Applying Theorem 3.5 we obtain the following corollary from Theorem 4.3. Corollary 4.4. The orthogonal decompositions ∞ ∞ N N L2 (D) = A2(k) (D) = A$2(k) (D) k=1

k=1

of the space L2 (D) = L2 (D, dA) can be rewritten in the following equivalent form: ∞ ∞     N N L2 (D) = SDk A2 (D) = (SD∗ )k A$2 (D) . k=0

k=0

5. Violation of Dzhuraev’s formulas In what follows we will say that a domain U ⊂ C possesses Dzhuraev’s formulas if, for every n ∈ N, $U,n = D $ U,n + K $n, BU,n = DU,n + Kn , B (5.1) $ U,n are given by (4.1) and Kn , K $ n belong to the where the operators DU,n and D ideal K of all compact operators on the space L2 (U ). Theorem 5.1. Let U ⊂ C be a bounded domain possessing Dzhuraev’s formulas and let Lz be a ray outgoing from a point z ∈ U . Then the domain Uz = U \ Lz does not possess Dzhuraev’s formulas, that is, for every n ∈ N, $Uz ,n − D $ Uz ,n ∈ BUz ,n − DUz ,n ∈ / K and B / K. (5.2) Proof. Since the Lebesgue area measure of the set U \ Uz equals zero, we conclude that L2 (U, dA) = L2 (Uz , dA). Thus we may consider SUz ,k and SU,k as operators acting on the same space, which implies that SUz ,k = SU,k for all k ∈ Z \ {0}. Hence, by (4.1), for every n ∈ N, DUz ,n = I − SUz ,−n SUz ,n = I − SU,−n SU,n = DU,n , $ Uz ,n = I − SUz ,n SUz ,−n = I − SU,n SU,−n = D $ U,n . D

(5.3)

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Y.I. Karlovich and L.V. Pessoa

By (5.3) and (5.1), we obtain BUz ,n − DUz ,n = BUz ,n − DU,n = BUz ,n − BU,n + Kn

(n ∈ N),

(5.4)

where Kn ∈ K. Suppose there exists an n ∈ N such that I n ∈ K. BUz ,n − BU,n = K

(5.5) 2

Since BUz ,n and BU,n are orthogonal projections on L (U ) and BUz ,n BU,n = BU,n I n also is an orthogonal because A2n (U ) ⊂ A2n (Uz ) ⊂ L2 (U ), we conclude that K 2 I projection on the space L (U ). Because Kn is a compact operator, we infer that I n is a finite-dimensional projection. K We now consider the sequence of functions gk (ξ) = (ξ − z)1/k ∈ A2 (Uz ) (k ∈ N). These functions are linearly independent. Indeed, for every m ∈ N, m


ck gk (ξ) =

k=1

m


 m!/k ck (ξ − z)1/m!

k=1

is a polynomial in the variable w = (ξ − z)1/m! 5 with coefficients c6k ∈ C, and therefore this polynomial equals zero for all w ∈ (ξ − z)1/m! : ξ ∈ Uz if and only if ck = 0 for all k = 1, 2, . . . , m. Further, the functions ψk = gk − BU,n gk = BUz ,n gk − BU,n gk I n for all k ∈ N. On the other hand, m ck ψk = 0 or, equivalently, are in Im K k=2 m


ck g k =

k=2

m


ck BU,n gk

(5.6)

k=2

if and only if ck = 0 for all k = 2, 3, . . . , m. Indeed, the right-hand side of (5.6) belongs to the space A2n (U ) and therefore is continuous on U for all ck ∈ C, but the left-hand side of (5.6) is not continuous on U if at least one of the coefficients I n are linearly independent for all ck is not zero. Thus the functions ψk ∈ Im K I n and k = 2, 3, . . ., which contradicts the finite-dimensionality of the projection K / K, the first relation in (5.2) follows hence contradicts (5.5). Since BUz ,n − BU,n ∈ from (5.4). Taking into account the equality $Uz ,n − D $ Uz ,n = C(BUz ,n − DUz ,n )C B


with C given by (1.1), we complete the proof of (5.2).

Finally, we establish integral representation for the Bergman and anti-Berg$U if U is the open sector man projections BU and B 5 6 (m = 2, 3, . . .). Km = z = reiθ : r > 0, θ ∈ (0, π/m) Obviously, K1 coincides with the upper half-plane Π = {z ∈ C : Im z > 0}, and ∗ , BΠ = I − SΠ SΠ

(see, e.g., [9, Lemma 7.5]).

∗ $Π = I − SΠ B SΠ

(5.7)

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279

Lemma 5.2. If m ∈ N and εm = e2πi/m , then m−1
k=0

εkm (xm − 1)2 = m2 xm−1 . (x − εkm )2

(5.8)

Proof. Since the left-hand side of (5.8) is a polynomial of degree 2m − 2, to prove (5.8), it is sufficient to show that the polynomial m−1
k=0

εkm (xm − 1)2 − m2 xm−1 (x − εkm )2

(5.9)

has zeros εsm (s = 0, 1, 2, . . . , m − 1) of multiplicity 2. Applying l’Hospital’s rule, we obtain  m−1   s m 
εk (xm − 1)2 εm (x − 1)2 m 2 m−1 = lim − m2 ε−s − m x lims m = 0. x→εm x→εsm (x − εkm )2 (x − εsm )2 k=0

Analogously, for the first derivative of the polynomial (5.9), we infer that  m−1   s m 
εk (xm − 1)2 εm (x − 1)2 m 2 m−1 lims − m x = lim − m2 (m − 1)ε−2s m x→εm x→εsm (x − εkm )2 (x − εsm )2 k=0  m    x −1 (m − 1)xm − mεsm xm−1 + 1 lim − m2 (m − 1)ε−2s = 2εsm lims m s s )2 x→εm x − εs x→ε (x − ε m m m   m(m − 1)ε−2s m − m2 (m − 1)ε−2s = 2εsm mε−s m m = 0. 2 Thus, the numbers εsm (s = 0, 1, 2, . . . , m − 1) are zeros of multiplicity 2 for the polynomial (5.9), and therefore this polynomial identically equals zero.
For every m ∈ N and every k = 0, 1, 2, . . . , m − 1, we consider the integral ∗ given for f ∈ L2 (Km , dA) and all z ∈ Km by operators Rm,k , Rm,k  1 εkm f (w) dA(w), (Rm,k f )(z) = − π Km (w − εkm z)2 (5.10)  1 ε−k m f (w) ∗ (Rm,k f )(z) = − dA(w). 2 π Km (w − ε−k m z) ∗ These operators are bounded on the space L2 (Km , dA) because Rm,k is the adjoint operator for Rm,k and Rm,k = χKm Vεkm SR2 χKm I where (Vε f )(z) = εf (εz) for ∗ z, ε ∈ C. It is easily seen that Rm,0 = SKm , Rm,0 = SK∗ m , and ∗ = CRm,k C Rm,k

(k = 1, 2, . . . , m − 1).

(5.11)

Clearly, in contrast to the singular integral operators SKm and SK∗ m having singu∗ larities at every point of Km , the operators Rm,k and Rm,k for k = 1, 2, . . . , m − 1 have fixed singularities only at the points 0 and ∞.

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Y.I. Karlovich and L.V. Pessoa

Theorem 5.3. For every m ∈ N, the Bergman and anti-Bergman projections on the space L2 (Km , dA) have the form BKm = I −

m−1
m−1


∗ Rm,k Rm,s ,

$Km = I − B

k=0 s=0

m−1
m−1


∗ Rm,k Rm,s ,

(5.12)

k=0 s=0

∗ where the operators Rm,k and Rm,k are given by (5.10).

Proof. Setting x = w/z in (5.8), we immediately obtain the equality m−1
εkm m2 wm−1 z m−1 = . (wm − z m )2 (w − εkm z)2

(5.13)

k=0

Consider the isometric shift operator Wm : L2 (Π, dA) → L2 (Km , dA),

(Wm f )(z) = mz m−1 f (z m ) (z ∈ Km ).

Since ϕm : z → z m is a conformal mapping of Km onto Π for m ∈ N, we infer from (5.7) that −1 −1 ∗ −1 = I − (Wm SΠ Wm )(Wm SΠ Wm ). BKm = Wm BΠ Wm

Analogously to [6] and [4], for z ∈ Km we obtain  1 m2 z m−1 w m−1 −1 (Wm SΠ Wm f )(z) = − lim f (w) dA(w), →0 π K (z) (wm − z m )2 m  1 m2 z m−1 wm−1 ∗ −1 (Wm SΠ Wm f )(z) = − lim f (w) dA(w) →0 π K (z) (w m − z m )2 m

(5.14)

(5.15)

where Km (z) = Km \ D (z). Then from (5.14), (5.15) and (5.13) it follows that (BKm f )(z) − f (z)     m−1 1 m2 z m−1 w m−1 1 m2 wm−1 ζ lim f (ζ)dA(ζ) dA(w) = − lim m →0 π K (z) (wm − z m )2 α→0 π Kα (w) (ζ − w m )2 m m     m−1 1 m2 z m−1 wm−1 1 m2 wm−1 ζ = − lim lim f (ζ)dA(ζ) dA(w) m →0 π K (z) (wm − z m )2 α→0 π Kα (w) (ζ − w m )2 m m   m−1 m−1
1
1 εkm εsm f (ζ) = − lim lim dA(ζ) dA(w) →0 π Km (z) (w − εkm z)2 α→0 s=0 π Kαm (w) (ζ − εsm w)2 k=0  m−1  m−1
1
1 εkm ε−s m f (ζ) =− dA(ζ) dA(w) (z ∈ Km ), 2 π Km (w − εkm z)2 s=0 π Km (ζ − ε−s m w) k=0 which proves the first formula in (5.12), according to (5.10). The second formula $Km = CBKm C. in (5.12) follows from the first in view of (5.11) and B
Theorem 5.3 immediately implies the following.

Poly-Bergman Projections and Orthogonal Decompositions

281

Corollary 5.4. For the sector K2 := {z = x + iy : x, y > 0}, the Bergman and anti-Bergman projections have the form       ∗ $K2 = I − S ∗ − R∗ SK2 − RK2 BK2 = I − SK2 − RK2 SK∗ 2 − RK , B K2 K2 2 ∗ where the operators RK2 and RK are given for f ∈ L2 (K2 , dA) by 2   1 1 f (w) f (w) ∗ dA(w), (R f )(z) = − dA(w). (RK2 f )(z) = − K2 2 π K2 (w + z) π K2 (w + z)2

Finally, finding explicit forms of the Bergman and anti-Bergman kernels for the sectors Km , we obtain other (different of (5.12)) integral representations for $Km . the Bergman and anti-Bergman projections BKm and B Lemma 5.5. For every m ∈ N, the Bergman and anti-Bergman projections on the space L2 (Km , dA) are given for f ∈ L2 (Km , dA) and z ∈ Km by m−1  1
εnm f (w) dA(w), (BKm f )(z) = − π n=0 Km (w − εnm z)2 (5.16) m−1 −n
 1 ε m $Km f )(z) = − (B f (w) dA(w). 2 π n=0 Km (w − ε−n m z) Proof. Since ϕm : Km → Π, z → z m is a conformal mapping, we infer that the Bergman kernel KKm (z, w) for the sector Km (m = 2, 3, . . .) has the form   m2 wm−1 z m−1 . KKm (w, z) = KΠ ϕm (w), ϕm (z) ϕm (w) ϕm (z) = − π (wm − z m )2 Hence KKm (w, z) = KKm (z, w) and, by (5.13), m−1 1
εnm . KKm (w, z) = − π n=0 (w − εnm z)2

(5.17)

Substituting (5.17) into the formulas  KKm (w, z)f (w)dA(w), (BKm f )(z) = Km  $Km f )(z) = (B KKm (w, z)f (w)dA(w), Km

we obtain (5.16).




References [1] M.B. Balk, Polyanalytic Functions. Akademie Verlag, Berlin, 1991. [2] A. Dzhuraev, Methods of Singular Integral Equations. Longman Scientific & Technical, 1992. [3] H. Hedelman, B. Korenblum, and K. Zhu, Theory of Bergman Spaces. SpringerVerlag, New York, 2000.

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[4] Yu.I. Karlovich and L. Pessoa, Algebras generated by Bergman and anti-Bergman projections and by multiplications by piecewise continuous coefficients. Integral Equations and Operator Theory 52 (2005), 219–270. [5] Yu.I. Karlovich and L.V. Pessoa, C ∗ -algebras of Bergman type operators with piecewise continuous coefficients. Integral Equations and Operator Theory 57 (2007), 521–565. [6] S.G. Mikhlin and S. Pr¨ ossdorf, Singular Integral Operators. Springer-Verlag, Berlin, 1986. [7] L. Peng, R. Rochberg, and Z. Wu, Orthogonal polynomials and middle Hankel operators on Bergman spaces. Studia Math. 102(1) (1992), 57–75. [8] J. Ram´ırez and I.M. Spitkovsky, On the algebra generated by a poly-Bergman projection and a composition operator. Factorization, Singular Operators and Related Problems, Proc. of the Conf. in Honour of Professor Georgii Litvinchuk (eds. S. Samko, A. Lebre, and A.F. dos Santos), Kluwer Academic Publishers, Dordrecht, 2003, 273–289. [9] N.L. Vasilevski, Toeplitz operators on the Bergman spaces: Inside-the-domain effects. Contemporary Mathematics 289 (2001), 79–146. [10] N.L. Vasilevski, Poly-Bergman spaces and two-dimensional singular integral operators. In: The Extended Field of Operator Theory (ed. M.A. Dritschel), Operator Theory: Advances and Applications 171 (2006), 349–359. [11] I.N. Vekua, Generalized Analytic Functions. Pergamon Press, Oxford, 1962. Yuri I. Karlovich Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos M´exico e-mail: [email protected] Lu´ıs V. Pessoa Departamento de Matem´ atica Instituto Superior T´ecnico Av. Rovisco Pais, 1049 - 001 Lisboa Portugal e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 283–293 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Vekua’s Generalized Singular Integral on Carleson Curves in Weighted Variable Lebesgue Spaces Vakhtang Kokilashvili and Stefan Samko Abstract. For a Carleson curve Γ we establish the boundedness, in weighted Lebesgue spaces Lp(·) (Γ, ) with variable exponent p(·), of the generalized singular integral operator which arises in the theory of I.N.Vekua generalized analytic functions. The obtained result is an extension of the known results even in the case of constant p. We also show that Vekua’s generalized singular integral exists a.e. for f ∈ L1 (Γ) on an arbitrary rectifiable curve. Mathematics Subject Classification (2000). Primary 42B25; Secondary 47B38. Keywords. singular integrals, generalized analytic functions, weighted Lebesgue spaces, variable exponent, Carleson curve, Zygmund conditions, BaryStechkin class.

1. Introduction We prove the weighted boundedness of the I.N. Vekua generalized singular operator known in the theory of generalized analytic functions, within the frameworks of Lebesgue spaces with variable exponent p(t). We also show that the I.N. Vekua generalized singular integral exists a.e. for every integrable function f even in the case of an arbitrary rectifiable curve, thus proving that the existence properties of the I.N. Vekua generalized singular operator are the same as of the usual singular operator. The obtained results are new even in the case of constant p. The paper is organized as follows. In Section 2 we give a certain background related to the problem and introduce necessary definitions and auxiliary statements. In Section 3 we prove the main result of the paper.

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2. Preliminaries Let G be a simply connected domain bounded by a simple finite rectifiable curve Γ = {t ∈ C : t = t(s), 0 ≤ s ≤  < ∞} with arc-length measure ν(t) = s. In the sequel we use the notation Γ(t, r) = Γ ∩ B(t, r),

B(t, r) = {τ ∈ C : |τ − t| < ε},

t ∈ Γ, r > 0.

Recall that a curve Γ is called Carleson if ν[Γ(t, r)] ≤ C0 r with C0 > 0 not depending on t and r. 2.1. Vekua’s generalized singular operator As is known, the theory of generalized analytic functions was developed by L. Bers and I.N. Vekua, we refer to their books [2], [26]–[27]. Generalized analytic functions of the class Ur,2 (A, B; G), r > 2, in the sense of I.N. Vekua, are regular solutions of the equation

where ∂z¯ =

1 2



∂ ∂x 1

∂z¯Φ(z) + A(z)Φ(z) + B(z)Φ(z) = 0  ∂ , A(z), B(z) ∈ Lr (G), r > 2. + i ∂y

Let f ∈ L (Γ). It is known that the integral  1 Ω1 (z, τ )f (τ ) dτ − Ω2 (z, τ )f (τ ) dτ , Φ(z) = 2πi

(2.1)

(2.2)

Γ

where Ω1 and Ω2 are the so-called basic normalized kernels of the class Ur,2 (A, B; G), is a regular solution of (2.1), see details in [26], [27]. The integral in (2.2) is called the generalized Cauchy type integral. The corresponding generalized singular integral is introduced as  1 Ω1 (t, τ )f (τ ) dτ − Ω2 (t, τ )f (τ ) dτ . (2.3) S$Γ f (t) = lim ε→0 2πi Γε

In [17] there was proved the following conventional statement: Proposition 2.1. If the singular integral S$Γ f (t) exists for almost all t ∈ Γ, then the function Φ(z) admits angular boundary values almost everywhere when z → t non-tangentially and for these boundary values the formula holds 1 (2.4) Φ± (t) = S$Γ f (t) ± f (t) 2 almost everywhere. Conversely, the almost everywhere existence of the boundary values Φ± (t) yields that of the singular integral (2.3). In [18] for the case of constant p the following statement was proved.

Vekua’s Generalized Singular Integral on Carleson Curves

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Proposition 2.2. Let Γ be a Carleson curve. The operator S$Γ is bounded in the space Lp (Γ) if r p> . (2.5) r−2 We prove a more general result for variable exponents p(t), which is even in the case of constant p is stronger than the existing result of Proposition 2.2, because we admit all the range 1 < p < ∞, avoiding the restriction in (2.5). Moreover, based on our earlier results on classical integral operators in weighted variable spaces, we are also able to prove the boundedness of the operator S$Γ in weighted spaces Lp(·) (Γ, ) with a certain class of general weights, see (2.10), including power type weights with a natural range for their exponents. All the results are obtained under the natural assumption on the variable exponent p(t), see (2.6), (2.7) below. 2.1.1. On singular integrals. It is known that the Cauchy singular integral  1 f (τ ) dτ SΓ f (t) = 2πi τ −t Γ

converges almost everywhere for any f ∈ L1 (Γ) on every rectifiable curve, see for instance, [3], p.137, Theorem 14.4). G.David [4] proved that the singular operator SΓ is bounded in the space Lp (Γ) with constant p, 1 < p < ∞, if and only if Γ is a Carleson curve. An extension of David’s result for the case of variable p(t) was made in [7], [12], [13]. 2.1.2. On convergence of S$Γ f (t). In Theorem 3.1 we show that the the integral S$Γ f converges almost everywhere in case of an arbitrary rectifiable curve and f ∈ L1 (Γ). Then in view of Proposition 2.1 we arrive at the following conclusion. Conclusion. Relation (2.4) holds almost everywhere for an arbitrary f ∈ L1 (Γ) and a rectifiable curve Γ. 2.2. On variable exponent Lebesgue spaces. We refer to [5], [15], [23], [25] for details on Lebesgue spaces Lp(·) with variable exponent, but recall some basic definitions with respect to curves on the complex plane. Let a function p(t) be defined on Γ and satisfy the conditions 1 < p∗ ≤ p(t) ≤ p∗ < ∞, t ∈ Γ and |p(t) − p(τ )| ≤

1 A , |t − τ | ≤ , t, τ ∈ Γ. 1 2 ln |t−τ |

(2.6) (2.7)

By Lp(·) (Γ, ), where (t) ≥ 0, we denote the weighted Banach space of measurable functions f : Γ → C such that ⎧ ⎫ #  # ⎨ ⎬ # (t)f (t) #p(t) # f Lp(·)(Γ,) := f p(·) = inf λ > 0 : ## dν(t) ≤ 1 < ∞. (2.8) # ⎩ ⎭ λ Γ

286

V. Kokilashvili and S. Samko We write Lp(·) (Γ) := Lp(·) (Γ, 1).

The H¨older inequality holds # # # # # f (t)g(t)dν(t)# ≤ kf Lp(·)(Γ) f  p
(·) , L (Γ) # #

1 1 +  ≡ 1, p(t) p (t)

Γ

where the constant k = p1∗ + (p1
)∗ < 2 does not depend on f and g. We deal with weights of the form m . wk (|t − tk |), tk ∈ Γ, (t) =

(2.9)

(2.10)

k=1

where wk (r) may oscillate as r → 0+ between two power functions (radial Zygmund-Bary-Stechkin type weights). The Zygmund-Bary-Stechkin class of admissible weights is defined in Subsection 2.3. In particular, the power weights m . |t − tk |βk , tk ∈ Γ, (2.11) (t) = k=1

with the condition −

1 1 < βk <  , p(tk ) p (tk )

(2.12)

are admitted. 2.3. On Zygmund-Bary-Stechkin-type weights We use the abbreviation a.i = almost increasing, a.d. = almost decreasing. Let W = {w ∈ C([0, ]) : w(0) = 0, w(x) > 0 for x > 0, w(x) is a.i.}. The numbers

 ln lim inf

mw = sup x>1

h→0

ln x



(2.13)

  ln lim sup w(hx) w(h)

w(hx) w(h)

and Mw = sup x>1

h→0

ln x

(see [20], [22], [21]), are known as as the lower and upper indices of the function w(x) (compare these indices with the Matuszewska-Orlicz indices, see [16], p. 20). We have 0 ≤ mw ≤ Mw ≤ ∞ for w ∈ W . Definition 2.3. ([1]) The Zygmund-Bary-Stechkin type class Φ0δ , 0 < δ < ∞, is defined as Φ0δ := Z 0 ∩ Zδ , where Z 0 is the class of functions w ∈ W satisfying the condition  h w(x) dx ≤ cw(h) (Z0 ) x 0 and Zδ is the class of functions w ∈ W satisfying the condition   w(h) w(x) dx ≤ c δ , (Zδ ) 1+δ h h x where c = c(w) > 0 does not depend on h ∈ (0, ].

Vekua’s Generalized Singular Integral on Carleson Curves

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The following statement is valid, see [20],[22] for δ = 1 and [6] for an arbitrary δ > 0. Theorem 2.4. Let w ∈ W . Then w ∈ Z 0 if and only if mw > 0, and w ∈ Zδ , δ > 0, if and only if Mw < δ, so that w ∈ Φ0δ ⇐⇒ 0 < mw ≤ Mw < δ.

(2.14)

Besides this, for w ∈ and any ε > 0 there exist constants c1 = c1 (ε) > 0 and c2 = c2 (ε) > 0 such that Φ0δ

c1 xMw +ε ≤ w(x) ≤ c2 xmw −ε ,

0 ≤ x ≤ .

(2.15)

is a.i.},

(2.16)

is a.d.}.

(2.17)

The following properties are also valid mw = sup{µ ∈ R1 : x−µ w(x) −ν

Mw = inf{ν ∈ R : x 1

w(x)

Note that the indices mω and Mω may be also well defined for functions w(x) positive for x > 0 which do not necessarily belong to W , for example, for functions in the class ) = {w : ∃a ∈ R1 such that wa (x) := xa w(x) ∈ W }. W Obviously, mwa = a + mw , Mwa = a + Mw . Observe that various non-trivial examples of functions in Zygmund-BaryStechkin type classes with coinciding indices may be found in [20], Section II; [19], Section 2.1, and with non-coinciding indices in [22]. In the sequel we shall also need the following technical lemma (its Euclidean version was proved in [9], for the Carleson context the proof is the same). Lemma 2.5. Let Γ be a bounded Carleson curve, the exponent p satisfy condition (2.7) and let w be any function such that there exist exponents a, b ∈ R1 and the constants c1 > 0 and c2 > 0 such that c1 ra ≤ w(r) ≤ c2 r−b , 0 ≤ r ≤  = diam (Γ). Then 1 [w(|t − t0 |)]p(t0 ) ≤ [w(|t − t0 |)]p(t) ≤ C[w(|t − t0 |)]p(t0 ) , (2.18) C where C > 1 does not depend on t, t0 ∈ Γ.

3. The main result We prove the following theorem. Theorem 3.1. I. Let Γ be a closed simple rectifiable curve. The singular integral S$Γ f exists for almost all t ∈ Γ for any f ∈ L1 (Γ). II. Let Γ be a Carleson curve and let p(t) satisfy conditions (2.6) and (2.7). Then the operator S$Γ is bounded in the space Lp(·) (Γ, ) with weight (2.10), where wk (r) are such functions that 1 r p(tk ) wk (r) ∈ Φ01 (3.1)

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) and or equivalently wk ∈ W −

1 1 < mwk ≤ Mwk <  , k = 1, 2, . . . , m. p(tk ) p (tk )

(3.2)

In particular, the weights (2.11)–(2.12) are admitted. 3.1. The required tools Everywhere in the sequel Γ is a bounded Carleson curve. Let  1 |f (τ )| dν(τ ) Mf (t) = sup r>0 ν[Γ(r, t)]

(3.3)

Γ(r,t)

be the maximal operator along Γ. In [8] the following statement was proved (see also [9] for a similar statement for bounded domains in Rn ; observe also that Theorem 3.2 for Carleson curves in the case of power weights was proved in [13], see also [10]). Theorem 3.2. Let p(t) satisfy conditions (2.6), (2.7). The operator M is bounded in Lp(·) (Γ, ) with the weight (2.10), where wk (r) satisfy (3.1)–(3.2). The following theorem was proved in [13], [10] in the case of power weights and is similarly extended to the case of Zygmund-Bary-Stechkin-type weights, taking into account Theorem 3.2. Theorem 3.3. Let p(t) satisfy conditions (2.6), (2.7) on Γ. The singular operator SΓ is bounded in Lp(·) (Γ, ) with the weight (2.10), where wk (r) satisfy (3.1)–(3.2). 3.2. Potentials over rectifiable curves Let  f (τ ) I α f (t) = dν(τ ), |τ − t|α

0 < α < 1.

(3.4)

Γ

Observe that the behavior of the potential Iα f (t) along an arbitrary rectifiable curve Γ is not quite trivial. Thus the potential Iα f (t) of a bounded function may prove to be an unbounded function. W.E. Sewell [24] in his investigations on approximation in the complex plane specially singled out the class Vα of rectifiable curves Γ along which the integral in (3.4) converges uniformly in t, so that  dν(τ ) sup < ∞; (3.5) |τ − t|α t∈Γ Γ

for Γ ∈ Vα . Observe that the condition sup ν[Γ(t, r)] ≤ Crβ

with

β>α

(3.6)

t∈Γ

is sufficient for (3.5) to be satisfied. Thus (3.5) is in particular valid on Carleson curves.

Vekua’s Generalized Singular Integral on Carleson Curves

289

Lemma 3.4. Let 0 < α < 1 and let Γ be a bounded rectifiable curve and f ∈ L1 (Γ). The integral in (3.4) exists for almost all t ∈ Γ. When Γ is a Carleson curve, the operator Iα is bounded in the space L1 (Γ) on every Carleson curve, and on any rectifiable curve with property (3.5). Proof. First we prove the almost everywhere convergence. Let s be an arc length of a point t, t = t(s), τ = t(σ). We have  Γ

f (τ )dν(τ ) = |t − τ |α

#  # # s − σ #α f [τ (σ)]dσ # # # t(s) − t(σ) # |s − σ|α ,

(3.7)

0

where  is the length of the curve. Let now Π(Γ) be the set of all points t = t(s) ∈ Γ such that for t ∈ Π(Γ) simultaneously two conditions are fulfilled:  0

f [τ (σ)]dσ |s − σ|α

exists and |t (s)| = 1.

As is known, in case of a rectifiable curve Γ, almost all points of the curve belong to Π(Γ). For a fixed t = t(s) ∈ Π(Γ) from the condition |t (s)| = 1 and the fact that Γ is assumed to have no intersections it follows that # # # s−σ # # # (3.8) # τ (s) − τ (σ) # ≤ C = C(s). Then from (3.7) we derive the almost everywhere convergence of the integral on the left-hand side. The L1 -boundedness in the case of Carleson curve is a matter of direct verification. We have    dν(t) α |I f (t)|dν(t) ≤ C |f (τ )|dν(τ ) |τ − t|α Γ

Γ

Γ

and by the help of the standard binary decomposition of Γ(t, 1) into the portions Γk (t) = {τ ∈ Γ : 2−k−1 < |t − τ | < 2−k } it is easy to check that condition (3.5) holds for an arbitrary Carleson curve.
Theorem 3.5. Let p(t) satisfy conditions (2.6), (2.7) on Γ. The operator  f (τ ) dν(τ ) α I f (t) = |τ − t|α Γ

is bounded in the space L conditions are fulfilled: F dν(τ ) < ∞, and 1) [(τ )]p
(τ )

p(·)

(Γ, ) with any weight (t) for which the following

Γ

2) the maximal operator M is bounded in the space Lp(·) (Γ, ), in particular for weights (2.10) with wk (r) satisfying conditions (3.1)–(3.2).

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Proof. The proof follows the standard lines; we refer in particular to [10], where a weighted Sobolev-type theorem was proved for Iα on Carleson curves in the case of power weights, see for instance the domination of Iα f (t) by the maximal operator in formulas (8.6) and (9.33) in paper [10], however we give the proof for the completeness of presentation. Without loss of generality we may assume that diam Γ > 1. By H¨older inequality (2.9) we obtain 8 8  8 [(·)]−1 8 f (τ ) dν(τ ) 8 8 ≤ kf Lp(·)(Γ,) 8 |τ − t|α | · −t|α 8Lp
(·) (Γ\Γ(t,1)) Γ\Γ(t,1)

8 8 ≤ kf Lp(·)(Γ,) 8−1 8Lp
(·) (Γ) ≤ Cf Lp(·)(Γ,) since

8 −1 8 8 8 p
(·) < ∞ L (Γ)

 ⇐⇒

(3.9)

dν(τ ) < ∞; [(τ )]p
(τ )

Γ

the existence of the last integral in the case of weights (2.10) with wk (r) satisfying conditions (3.1)–(3.2), follows from Lemma 2.5, inequalities (2.15) and condition (3.2). As regards the integral over Γ(t, 1), it may be dominated by the maximal function in the usual way. Indeed, let γk (t) = Γ(t, 2−k )\Γ(t, 2−k−1 ). Then   ∞  ∞ |f (τ )| dν(τ )
|f (τ )| dν(τ )
α(k+1) = ≤ 2 |f (τ )| dν(τ ) |τ − t|α |t − τ |α k=0γ (t) k

Γ(t,1)

≤C

∞


k=0

Γ(t,2−k )

2−(1−α)k M f (t) = C1 M f (t).

k=0

Then the application of Theorem 3.2 completes the proof of Theorem 3.5.




Note that in [14] it was proved that in the case of an infinite curve and constant p, the Sobolev theorem for the potential type operator holds if and only if Γ is a Carleson curve. For the extension of the Sobolev theorem on Carleson curves to the case of weighted spaces with variable p(t) we refer to papers [11], [10], [12]. 3.3. Proof of Theorem 3.1: reduction to the classical singular operator The proof is based on the observation that the functions Ω1 (z, t) and Ω2 (z, t) have the following structure Ω1 (z, t) =

m1 (z, t) 1 + t−z |t − z|α

and Ω2 (z, t) =

m2 (z, t) |t − z|α

(3.10)

where α = 2r ∈ (0, 1) and the functions m1 (z, t) and m2 (z, t) are continuous and bounded when t runs Γ and z runs a bounded domain, see [26], [27]. Consequently,

Vekua’s Generalized Singular Integral on Carleson Curves

291

for the generalized singular integral S$Γ f (t) we have S$Γ f (t) = SΓ f (t) + Iα f (t), where Iα f (t) =

1 2πi

 Γ

1 m1 (t, τ )f (τ ) dτ − |τ − t|α 2πi

 Γ

m2 (t, τ )f (τ ) dτ . |τ − t|α

(3.11)

(3.12)

Therefore, the question of convergence or boundedness of S$Γ f (t) is reduced to that of the singular integral SΓ f (t) and the integral with a weak singularity. The almost everywhere convergence of the first term on the right-hand side of (3.11) is known, see Subsection 2.1.1, while that of the second term was proved in Theorem 3.5. Similarly the boundedness of these terms in the space Lp(·) (Γ, ) under the assumptions of Theorem 3.1 follows from Theorems 3.3 and 3.5, which completes the proof of Theorem 3.1. Acknowledgments This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr.06-1000017-8792. The first author was supported also by GNSF grant ST0630-010.

References [1] N.K. Bary and S.B. Stechkin. Best approximations and differential properties of two conjugate functions (in Russian). Proceedings of Moscow Math. Soc., 5:483–522, 1956. [2] L. Bers. Theory of pseudo-analytic functions. Institute for Mathematics and Mechanics, New York University, New York, 1953. 191 pages. [3] A. B¨ ottcher and Yu. Karlovich. Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Basel, Boston, Berlin: Birkh¨ auser Verlag, 1997. 397 pages. [4] G. David. Op´erateurs int´egraux singuliers sur certaines courbes du plan complexe. ´ Ann. Sci. Ecole Norm. Sup. (4), 17(1):157–189, 1984. [5] X. Fan and D. Zhao. On the spaces Lp(x) (Ω) and W m,p(x) (Ω). J. Math. Anal. Appl., 263(2):424–446, 2001. [6] N.K. Karapetiants and N.G. Samko. Weighted theorems on fractional integrals in the generalized H¨ older spaces H0ω () via the indices mω and Mω . Fract. Calc. Appl. Anal., 7(4):437–458, 2004. [7] V. Kokilashvili, V. Paatashvili, and Samko S. Boundedness in Lebesgue spaces with variable exponent of the Cauchy singular operators on Carleson curves. In Ya. Erusalimsky, I. Gohberg, S. Grudsky, V. Rabinovich, and N. Vasilevski, editors, “Operator Theory: Advances and Applications”, dedicated to 70th birthday of Prof. I.B.Simonenko, volume 170, pages 167–186. Birkh¨ auser Verlag, Basel, 2006. [8] V. Kokilashvili, N. Samko, and S. Samko. Singular operators in variable spaces Lp(·) (Ω, ) with oscillating weights. Math. Nachrichten. to appear.

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[9] V. Kokilashvili, N. Samko, and S. Samko. The maximal operator in variable spaces Lp(·) (Ω, ). Georgian Math. J., 13(1):109–125, 2006. [10] V. Kokilashvili and S. Samko. Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent. Acta mathematica Sinica, page (submitted). [11] V. Kokilashvili and S. Samko. Sobolev theorem for potentials on Carleson curves in variable Lebesgue spaces. Mem. Differential Equations Math. Phys., 33:157–158, 2004. [12] V. Kokilashvili and S. Samko. Weighted Boundedness in Lebesgue spaces with Variable Exponent of Classical Operators on Carleson Curves. Proc. A. Razmadze Math. Inst., 138:106–110, 2005. [13] V. Kokilashvili and S. Samko. Boundedness in Lebesgue spaces with variable exponent of maximal, singular and potential operators. Izvestija VUZov. SeveroKavkazskii region. Estestvennie nauki, Special issue “Pseudodifferential equations and some problems of mathematical physics”, dedicated to 70th birthday of Prof. I.B. Simonenko, pages 152–158, 2006. [14] V.M. Kokilashvili. Fractional integrals on curves. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 95:56–70, 1990. [15] O. Kov´ acˇik and J. R´ akosnˇik. On spaces Lp(x) and W k,p(x) . Czechoslovak Math. J., 41(116):592–618, 1991. [16] L. Maligranda. Indices and interpolation. Dissertationes Math. (Rozprawy Mat.), 234:49, 1985. [17] K.M. Musaev. Some boundary properties of generalized analytic functions. Dokl. Akad. Nauk SSSR, 181:1335–1338, 1968. [18] K.M. Musaev. Boundedness of the Cauchy singular integral in a class of generalized analytic functions. Izv. Akad. Nauk Azerba˘ıdzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk, 7(6):3–8, 1986. [19] N.G. Samko. Criterion of Fredholmness of singular operators with piece-wise continuous coefficients in the generalized H¨ older spaces with weight. page 363. Proceedings of IWOTA 2000, Setembro 12–15, Faro, Portugal, Birkh¨ auser, In: “Operator Theory: Advances and Applications”, v. 142. [20] N.G. Samko. Singular integral operators in weighted spaces with generalized H¨older condition. Proc. A. Razmadze Math. Inst, 120:107–134, 1999. [21] N.G. Samko. On compactness of Integral Operators with a Generalized Weak Singularity in Weighted Spaces of Continuous Functions with a Given Continuity Modulus. Proc. A. Razmadze Math. Inst, 136:91, 2004. [22] N.G. Samko. On non-equilibrated almost monotonic functions of the Zygmund-BaryStechkin class. Real Anal. Exch., 30(2):727–745, 2005. [23] S.G. Samko. Differentiation and integration of variable order and the spaces Lp(x) . Proceed. of Intern. Conference “Operator Theory and Complex and Hypercomplex Analysis”, 12–17 December 1994, Mexico City, Mexico, Contemp. Math., Vol. 212, 203-219, 1998. [24] W.E. Sewell. Generalized derivatives and approximation by polynomials. Trans. Amer. Math. Soc., 41(1):84–123, 1937.

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[25] I.I. Sharapudinov. The topology of the space Lp(t) ([0, 1]) (Russian). Mat. Zametki, 26(4):613–632, 1979. [26] I.N. Vekua. Generalized analytic functions (Russian). Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959. [27] I.N. Vekua. Generalized analytic functions. Pergamon Press, London, 1962. Vakhtang Kokilashvili Black Sea University and A. Razmadze Mathematical Institute Tbilisi, Georgia e-mail: [email protected] Stefan Samko University of Algarve Portugal e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 295–308 c 2008 Birkh¨
auser Verlag Basel/Switzerland

On Homotopical Non-invertibility of C ∗-extensions Vladimir Manuilov Abstract. We have presented recently an example of a C ∗ -extension, which is not invertible in the semigroup of homotopy classes of C ∗ -extensions. Here we reveal the cause for existence of homotopy non-invertible C ∗ -extensions: it is related to non-exact C ∗ -algebras and to possibility to distinguish different tensor C ∗ -norms by K-theory. We construct a special C ∗ -algebra, K-theory of which hosts an obstruction for homotopical non-invertibility, and show that this obstruction for our example does not vanish. Mathematics Subject Classification (2000). Primary 46L80; Secondary 19K33 46L05. Keywords. C ∗ -algebra, C ∗ -extension, homotopy.

1. Introduction A C ∗ -extension is a short exact sequence /E /B 0

/A

/0

∗

(1)

of C -algebras. The classical Brown–Douglas–Fillmore theory [4] classifies C ∗ extensions of A by B, i.e., different E’s in (1), up to stable unitary equivalence modulo split C ∗ -extensions and results in a functor Ext(A, B), which turns out to be an abelian semigroup in the case when B is stable, i.e., B ∼ = B ⊗ K, where K denotes the C ∗ -algebra of compact operators. The Brown–Douglas–Fillmore theory works nicely for C ∗ -extensions of nuclear C ∗ -algebras because of the two features: 1. Ext(A, B) is a group, i.e., each C ∗ -extension is invertible; 2. Ext(A, B) is homotopy invariant. Both of these do not hold in general! Research was partially supported by RFFI grant 05-01-00923.

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The first example of a non-invertible C ∗ -extension was constructed by J. Anderson [1], then a lot of other examples followed, [19, 20, 12, 17, 7]. Moreover, the example by E. Kirchberg, [12], gives a non-invertible (hence non-trivial) C ∗ extension of a contractible (i.e., homotopy equivalent to 0) C ∗ -algebra, thus showing that Ext is not homotopy invariant in general. Besides the stable unitary equivalence, there is another natural equivalence relation for C ∗ -extensions – the homotopy equivalence. It gives another functor, Exth (A, B), of homotopy classes of C ∗ -extensions. As the homotopy equivalence is weaker than the stable unitary equivalence modulo split C ∗ -extensions, so the identity map gives rise to a surjective semigroup homomorphism Ext(A, B) → Exth (A, B). Since Exth is patently homotopy invariant, it was interesting to know, if non-invertibility of C ∗ -extensions persists on the homotopy level, in other words, if Exth is a group. In [16] we, jointly with K. Thomsen, provided the first example of a non-invertible element of Exth . Recall that A. Connes and N. Higson, the founders of E-theory, have expressed in [5] their opinion that E-theory is the quotient, modulo homotopy, of the theory of C ∗ -extensions. But E-theory always has a group structure, so the Connes–Higson construction, which defines a map Exth (A, B) → E1 (A, B), cannot be faithful. So the relation between two theories is more complicated. Our conjecture is that E-theory coincides with the group of invertible elements of Exth (A, B). The aim of this paper is to reveal the cause for existence of homotopy noninvertible C ∗ -extensions (more exactly, one of the causes, the only one known to us yet) and to get a better understanding of the methods of [16]. It was first understood by S. Wassermann [21] that non-invertibility (in Ext) of C ∗ -extensions can be revealed by using non-exact C ∗ -algebras. Recall that a C ∗ -algebra D is exact if, after taking the minimal tensor C ∗ -product of D by the C ∗ -algebras from (1), the sequence 0

/ B ⊗min D

/ E ⊗min D

/ A ⊗min D

/0

is exact as well. Wassermann’s idea was that if the C ∗ -norm on the algebraic tensor product A * D inherited from E ⊗min D/B ⊗min D is strictly greater than the minimal tensor C ∗ -norm then this prevents (1) from being invertible. This is not enough for homotopy non-invertibility, but here the following idea of G. Skandalis (cf. [10]) can help: the difference between tensor C ∗ -norms on A*D can sometimes be detected on the level of K-theory. The next ingredient is to find an appropriate C ∗ -algebra D and then to construct an ideal RB,D in the corona C ∗ -algebra Q(B⊗min D) with two properties: firstly, RB,D should have a non-trivial K0 -group and, secondly, the images, under the Busby invariant of the extension (1), of all elements of E ⊗min D/B ⊗min D that vanish in A⊗min D, should lie in RB,D . Then, in order to produce a homotopy non-invertible C ∗ -extension, it suffices to find a projection in E ⊗min D/B ⊗min D that vanishes in A ⊗min D and which is mapped by the Busby invariant of the extension (1) into a projection in RB,D , the K-theory class of which is non-trivial.

On Homotopical Non-invertibility of C ∗ -extensions

297

2. Basic definitions For a C ∗ -algebra B, let M (B) denote the multiplier C ∗ -algebra of B, which can be defined as the maximal C ∗ -algebra that contains B as an essential ideal. Let Q(B) = M (B)/B be the corresponding corona C ∗ -algebra and let q : M (B) → Q(B) be the quotient ∗-homomorphism. Recall that any extension (1) gives rise in a natural way to a ∗-homomorphism E → M (B), which passes to quotients: A → Q(B). The latter is called the Busby invariant of the extension (1). It is well known that, up to a rather fine form of equivalence, a C ∗ -extension can be recovered from its Busby invariant, so we will not distinguish C ∗ -extensions and their Busby invariants. Let IB = C[0, 1] ⊗ B (we use the notation ⊗ for the C ∗ -algebra tensor product when it is unique, i.e., when there is only one C ∗ -norm on the algebraic tensor product). The evaluation map evs : IB → B, s ∈ [0, 1], gives rise to the map ev I s : Q(IB) → Q(B). Two extensions of A by B are called homotopic if there I 1 ◦ Φ coincide exists a ∗-homomorphism Φ : A → Q(IB) such that ev I 0 ◦ Φ and ev with the Busby invariants of the two C ∗ -extensions. The set Exth (A, B) is defined as the set of homotopy classes of C ∗ -extensions of A by B. If B is stable, B ∼ = B ⊗ K, then the set Exth (A, B) has a natural semigroup structure. Stability of B implies existence of two elements V1 , V2 ∈ M (B) with the properties V1∗ V1 = 1M(B) ,

V2∗ V2 = 1M(B) ,

V1 V1∗ + V2 V2∗ = 1M(B) .

Let τ, τ  : A → Q(B) be the Busby invariants of two extensions. Then the sum of these two extensions is defined to have the Busby invariant given by τ$(a) = Adv1 (τ (a)) + Adv2 (τ  (a)), a ∈ A, where vi = q(Vi ) ∈ Q(B), i = 1, 2, and Adv (x) = vxv ∗ . It is well known that this definition, up to unitary equivalence (hence up to homotopy), does not depend on choice of V1 and V2 . More detailed information on C ∗ -extensions can be found in [3]. Besides the minimal and the maximal tensor C ∗ -norm on algebraic tensor products we use one more tensor C ∗ -norm, the one defined by the extension (1). Assume that there is a unique tensor C ∗ -norm on the algebraic tensor product B * D. Then, thanks to the exactness of the maximal tensor product, E ⊗min D/B ⊗ D is a quotient of A ⊗max D = E ⊗max D/B ⊗ D. On the other hand, A ⊗min D is the quotient of E ⊗min D/B ⊗ D. Therefore A * D is a dense subspace in E ⊗min D/B ⊗ D. We denote the norm on A* B inherited from E ⊗min D/B ⊗ D by  · E . Since this norm is a cross-norm, we may view E ⊗min D/B ⊗ D as a tensor product of A and D and write E ⊗min D/B ⊗ D = A ⊗E D. If  · α and  · β are tensor C ∗ -norms on A * D and if xα ≥ xβ for any x ∈ A * D then we denote the canonical surjective ∗-homomorphism by qα,β : A ⊗α D → A ⊗β D. For more information on tensor products of C ∗ -algebras we refer to [22].

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3. A canonical ∗-homomorphism induced by extensions

 Let Dk , k ∈ N, be a sequence of nuclear unital C ∗ -algebras and let k∈N Dk be a  C ∗ -algebra of norm-bounded sequences (d1 , d2 , . . .), d k ∈ Dk . Denote by pk : k∈N Dk → Dk the canonical projection. Let D ⊂ k∈N Dk be a unital C ∗ -algebra such that (pk )|D is surjective for each k ∈ N. To shorten notation we denote the restriction (pk )|D also by pk . We also assume that there is only one tensor C ∗ -norm on the algebraic tensor product B * D. The case we are most interested is when the C ∗ -algebras Dk = Mnk are the matrix algebras of increasing dimension nk and the ‘big’ C ∗ -algebra D is not exact. Denote by JA,D the kernel of the canonical surjection qmax,min : A ⊗max D

/ A ⊗min D.

Let τ : A → Q(B) be the Busby invariant of the C ∗ -extension (1) and let i : Q(B) → Q(B ⊗ D) and j : D → Q(B ⊗ D) be the canonical ∗-homomorphisms. Then i ◦ τ (A) and j(D) commute in Q(B ⊗ D), hence there is a well-defined ∗-homomorphism / Q(B ⊗ D) τD : A ⊗max D due to the universal property of the maximal tensor product of C ∗ -algebras. Another description of the map τD is given by composing the ∗-homomorphism qmax,E : A ⊗max D → A ⊗E D with the Busby invariant of the C ∗ -extension 0

/ B⊗D

/ E ⊗min D

/ A ⊗E D

/ 0.

Consider the ∗-homomorphism idA ⊗pk : A ⊗max D

/ A ⊗ Dk .

Lemma 1. Let x ∈ A ⊗max D. Then x ∈ JA,D if and only if idA ⊗pk (x) = 0 for all k ∈ N.  Proof. This follows from injectivity of the canonical map A ⊗min D → k∈N (A ⊗ Dk ) and from commutativity of the diagram / A ⊗min D A ⊗max DT TTTT TTTT TTTT  TT* k∈N idA ⊗pk   (A ⊗ Dk ). k∈N
Let χ : Q(B ⊗ D)

/ M (B ⊗ D)

be a continuous selfadjoint homogeneous section satisfying χ(y) < 2y for any y ∈ Q(B ⊗ D) (such sections exist by the Bartle–Graves theorem, [2, 14]). Since the map idB ⊗pk : B ⊗ D → B ⊗ Dk is surjective, it extends to a surjective ∗-homomorphism idB ⊗pk : M (B ⊗ D)

/ M (B ⊗ Dk ).

(2)

On Homotopical Non-invertibility of C ∗ -extensions

299

Lemma 2. If x ∈ JA,D then idB ⊗pk ◦ χ ◦ τD (x) ∈ B ⊗ Dk for any k ∈ N. Proof. Let / M (B ⊗ Dk ), k ∈ N,

χk : Q(B ⊗ Dk )

be continuous selfadjoint homogeneous sections satisfying χk (z) < 2z for any z ∈ Q(B ⊗ Dk ). Denote by I idDk : A ⊗ Dk τ⊗

/ Q(B ⊗ Dk )

the ∗-homomorphism obtained by composing τ ⊗ idDk : A ⊗ Dk → Q(B) ⊗ Dk with the canonical embedding Q(B) ⊗ Dk ⊂ Q(B ⊗ Dk ). The ∗-homomorphism (2) passes to quotients: id B ⊗pk : Q(B ⊗ D)

/ Q(B ⊗ Dk )

and by checking on simple tensors one finds that  id B ⊗pk ◦ τD = (idB ⊗pk ) ◦ (idA ⊗pk ), which implies that idB ⊗pk ◦ χ ◦ τD (x) − χk ◦ (id B ⊗pk ) ◦ (idA ⊗pk )(x) ∈ B ⊗ Dk

(3)

for any k ∈ N and for any x ∈ A ⊗max D. By Lemma 1, if x ∈ JA,D then idA ⊗pk (x) = 0 for any k ∈ N, so due to homogeneity of χk it follows from (3)
that idB ⊗pk ◦ χ ◦ τD (x) ∈ B ⊗ Dk for any k ∈ N.

4. Yet another short exact sequence Consider the ∗-homomorphism  P = k∈N idB ⊗pk : M (B ⊗ D) /



M (B ⊗ Dk ) (4)  k∈N B ⊗ Dk ⊂ k∈N M (B ⊗

k∈N



and denote by NB,D the preimage of P of the ideal Dk ), .  B ⊗ Dk ⊂ M (B ⊗ D). NB,D = P −1 k∈N

Since B ⊗ D ⊂ NB,D is an ideal in M (B ⊗ D), is it an ideal in NB,D as well. Denote the corresponding quotient C ∗ -algebra NB,D /B ⊗ D by RB,D . Lemma 3. RB,D is an ideal in Q(B ⊗ D).   Proof. First notice that k∈N B ⊗ Dk is an ideal in k∈N M (B ⊗ Dk ). Notice also that the map P (4) is injective. This implies that NB,D is an ideal in M (B ⊗ D),
hence RB,D is an ideal in Q(B ⊗ D). Thus we get a short exact sequence 0

/ RB,D

/ Q(B ⊗ D)

/ M (B ⊗ D)/NB,D

/ 0.

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Proposition 4. For any ∗-homomorphism τ : A → Q(B) one has τD (JA,D ) ⊂ RB,D . In other words, any C ∗ -extension (1) gives rise to a morphism of extensions 0

/ JA,D

/ A ⊗max D

0

 / RB,D

 / Q(B ⊗ D)

τD

/ A ⊗min D

/0

 / M (B ⊗ D)/NB,D

/ 0.

Proof. It follows from Lemma 2 that . . idB ⊗pk ◦ χ ◦ τD (x) ∈ k∈N

k∈N

B ⊗ Dk

for any x ∈ JA,D , so, by definition of NB,D , one has χ◦τD (x) ∈ NB,D ⊂ M (B⊗D). The quotient map q : M (B ⊗ D) → Q(B ⊗ D) is the left inverse for χ, q ◦ χ = idQ(B⊗D) , hence τD (x) = q(χ ◦ τD (x)) ∈ q(NB,D ) = RB,D .




The ideal RB,D is useful because of its non-trivial K-theory.

5. Some K-theory calculations

 Let ν = {nk }k∈N be an increasing sequence of positive integers. Let Z∞ = k∈N Z be the abelian group of all (not necessarily bounded) infinite sequences of integers and let Zν ⊂ Z∞ be a subgroup of all sequences (mk )k∈N , mk ∈ Z, satisfying |mk | < ∞. k∈N nk  From now on we set Dk = Mnk and D = k∈N Dk . We also assume that B is either K or IK = K ⊗ C[0, 1]. sup

Proposition 5. For B and D as above, 1. One has K0 (B ⊗ D) ∼ = Zν and K1 (B ⊗ D) = 0; 2. There is a split surjection T : K0 (NB,D ) → Z∞ such that (a) it maps the positive cone K0 (NB,D )+ of K0 (NB,D ) into the positive cone (Z∞ )+ of Z∞ ; (b) the composition T ◦ i∗ coincides with the isomorphism K0 (B ⊗ D) ∼ = Zν , where i∗ : K0 (B ⊗ D) → K0 (NB,D ) is induced by the inclusion B ⊗ D ⊂ NB,D . Proof. The K-groups of B ⊗ D are easy to calculate due to stability of K-theory: one has K∗ (B ⊗ D) ∼ = K∗ (D). The latter was calculated, e.g., in [6]. So let us turn to the group K0 (NB,D ). : D → Dk . and let Let ik : Dk → D be the natural right-inverse of pk  ∞ V1 , V2 , V3 , . . . be a sequence of isometries in M (B) such that k=1 Vk Vk∗ = 1 with convergence in the strict topology. Then ϕk = AdVk ⊗ik : B ⊗ Dk → B ⊗ D is a ∗-homomorphism. Let qk = Vk Vk∗ ⊗ ik (1Dk ) ∈ M (B ⊗ D).

On Homotopical Non-invertibility of C ∗ -extensions

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Lemma 6. Each ϕk is quasi-unital, in particular, ϕk (B ⊗ Dk ) B ⊗ D = qk B ⊗ D. Proof. The inclusion ⊆ is obvious. To prove the reversed inclusion it suffices to show that qk (b ⊗ d) ∈ ϕk (B ⊗ Dk ) B ⊗ D for every b ∈ B and every d = (d1 , d2 , d3 , . . . ) ∈ D. Write b = b1 b2 , b1 , b2 ∈ B, and dk = xy, x, y ∈ Dk . Then qk (b ⊗ d) = Vk Vk∗ b ⊗ (0, 0, 0, . . . , 0, dk , 0, . . . ) = ϕk (Vk∗ b1 ⊗ x) (Vk b2 ⊗ (0, 0, . . . , 0, y, 0, 0, . . . )) ∈ ϕk (B ⊗ Dk ) B ⊗ D.




Since B ⊗ Dk and B ⊗ D are both σ-unital there is, for each k, a ∗-homomorphism ϕk : M (B ⊗ Dk ) → M (B ⊗ D) which extends ϕk and is strictly continuous on the unit ball (see, e.g., [13], Corollary 1.1.15). ∞ Lemma 7. For each x = (x1 , x2 , x3 , . . . ) ∈ k=1 M (B ⊗ Dk ), the sum
∞ ϕk (xk ) k=1

converges in the strict topology of M (B ⊗ D). Proof. Let z ∈ B ⊗ D and  > 0 be arbitrary. It suffices to show that there is a K ∈ N such that 8 8 8
M 8 8 8 ϕ (x ) z 8 j=N j j 8 ≤ x when M > N ≥ K. Note first that ϕk (1Dk ) = qk so that ϕi (xi ) ϕj (xj ) = ϕj (xj ) ϕi (xi ) = 0 when i = j. It follows that 8 8 8
M 8 8 8 8 j=N ϕj (xj )8 ≤ x n for all M > N . Set Pn = j=1 Vj Vj∗ and note, by checking on simple tensors, that limn→∞ (Pn ⊗ 1D ) z − z = 0. Choose K ∈ N such that (PK ⊗ 1D ) z − z ≤ . Then 8 8
8 8
8 8 M 8 8 M 8 8 8 ϕj (xj ) z 8 ≤ 8 ϕj (xj ) PK z 8 8 + x = x 8 j=N

j=N

when M > N ≥ K.




It follows from Lemma 7 that we can define a ∗-homomorphism
∞  / M (B ⊗ D) such that Φ(m) = ϕ (mk ) Φ: ∞ k=1 M (B ⊗ Dk ) k=1 k ∞ where m = (m1 , m2 , m3 , . . . ) ∈ k=1 M (B ⊗ Dk ). Lemma 8. One has   1. P ◦ Φ(m) = AdV1 ⊗1D1 (m1 ), AdV2 ⊗1D2 (m2 ), AdV3 ⊗1D3 (m3 ), . . . for all m = (m1 , m2 , m3 , . . . ), where P is the map defined by (4); 2. Φ k∈N B ⊗ Dk ⊂ NB,D .

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Proof. To prove the first assertion, it suffices to show that idB ⊗pk ◦ Φ (m1 , m2 , m3 , . . . ) = AdVk ⊗1Dk (mk ) for each k. To this end let x ∈ B ⊗ Dk . Then idB ⊗pk ◦ Φ (m1 , m2 , m3 , . . . ) x = idB ⊗pk (Φ (m1 , m2 , m3 , . . . ) (idB ⊗ik ) (x)) = idB ⊗pk (ϕk (mk ) (idB ⊗ik ( x)) = idB ⊗pk ◦ ϕk (mk ) x = (∗) When b ∈ B, d ∈ Dk we find that idB ⊗pk ◦ ϕk (b ⊗ d) = (idB ⊗pk ) ◦ ϕk (b ⊗ d) = Vk bVk∗ ⊗ d. It follows then by strict continuity of both AdVk ⊗1Dk and idB ⊗pk ◦ ϕk that (∗) = AdVk ⊗1Dk (mk ) x, completing the proof of the first assertion of Lemma 8. The second assertion is obvious.
∞ Lemma 9. The map P∗ : K∗ (NB,D ) → K∗ ( k=1 B ⊗ Dk ) is a split surjection. Proof. Set Wk = Vk ⊗ 1Dk ∈ M (B ⊗ Dk ). It follows from ∞Lemma 8 that it suffices ∞ to show that the endomorphism ψ : k=1 B ⊗ Dk → k=1 B ⊗ Dk given by ψ (x1 , x2 , x3 , . . . ) = (AdW1 (x), AdW2 (x), AdW3 (x), . . . ) induces the identity map on K-theory. The latter follows from standard arguments ∞by using that ψ = AdW , where W = (W1 , W2 , W3 , . . . ) is an isometry in M ( k=1 B ⊗ Dk ).
∞ ∞ Lemma 10. The groups K0 ( k=1 B ⊗ Dk ) and k=1 K0 (B ⊗ Dk ) are isomorphic under the obvious map. Proof. Let

 ι : K0 ( ∞ k=1 B ⊗ Dk )

/ ∞ K0 (B ⊗ Dk ) k=1

be the map we are considering. ∞ Surjectivity: Let (xk ) ∈ k=1 K0 (B ⊗ Dk ). Since B is stable there are projections pk , qk ∈ B ⊗ Dk such that xk = [pk ] − [qk ]

 for all k. Then p = (p1 , p2 , p3 , . . . ) , q = (q1 , q2 , . . . ) are projections in ∞ k=1 B ⊗Dk and ι ([p] − [q]) = (xk ). ∞ ∞ Injectivity: Let x, y ∈ K0 ( k=1 B ⊗ Dk ) be such that ι(x) = ι(y). Note that It follows k=1 B ⊗ Dk has an (uncountable) approximate unit of projections. ∞   therefore that there is m ∈ N and projections p, q, p , q ∈ M ( B ⊗ Dk ) = m k=1 ∞ M (B ⊗ D ) such that m k k=1 x = [p] − [q],

y = [p ] − [q  ].

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(cf. Proposition 5.5.5 in [3]). Write p = (p1 , p2 , p3 , . . . ) , q = (q1 , q2 , q3 , . . . ) , p = (p1 , p2 , p3 , . . . ) , q  = (q1 , q2 , q3 , . . . ), where pk , qk , pk , qk ∈ Mm (B ⊗ Dk ) . Then [pk ] − [qk ] = [pk ] − [qk ] for each k since ι(x) = ι(y). Since B is stable there are projections rk ∈ B ⊗ Dk such that pk ⊕ qk ⊕ rk = Vk Vk∗ and Vk∗ Vk = pk ⊕ qk ⊕ rk in M2m+1 (B ⊗ D k ) for some partial isometry Vk ∈ M2m+1 (B ∞⊗ Dk ). Set r = B ⊗ D , V = (V , V , . . . ) ∈ M ( (r1 , r2 , r3 , . . . ) ∈ ∞ k 1 2 2m+1 k=1 k=1 B ⊗ Dk ) and note that p ⊕ q  ⊕ r = V V ∗ , V ∗ V = p ⊕ q ⊕ r ∞ in M2m+1 ( k=1 B ⊗ Dk ).
Note that our proof holds not only for our B, but for a general stable C ∗ algebra with an approximate unit of projections. It does not hold in complete generality, cf. [6]. Now we can finish the proof of Proposition 5. Let trk be the standard trace on B ⊗ Dk normalized by trk (e) = 1, where e ∈ B ⊗ Dk is a minimal projection. As isomorphism (trk )∗ : K0 (B ⊗ Dk ) → Z, the composition ∞this trace definesan ∞ ∞ is also an isomorphism. Put k=1 (trk )∗ ◦ ι : K0 ( k=1 B ⊗ Dk ) → Z .∞ (trk )∗ ◦ ι ◦ P∗ . T = k=1

Then, by Lemmas 9 and 10, T is split surjective. It is obvious that T respects positivity and that its restriction onto K0 (B ⊗ D) is an isomorphism onto Zν .
Corollary 11. The quotient map TI : K0 (RB,D ) → Z∞ /Zν induced by T is split surjection and maps the positive cone K0 (RB,D )+ of K0 (RB,D ) to the positive cone (Z∞ /Zν )+ of Z∞ /Zν . Proof. Consider the six-term exact sequence K0 (B ⊗ D) O

/ K0 (NB,D )

∂

K1 (RB,D ) o

... o

/ K0 (RB,D )

(5)

 K1 (B ⊗ D)

By Proposition 5, the map K0 (B ⊗ D) → K0 (NB,D ) induced by the inclusion is injective, hence the boundary map ∂ is zero. One also has K1 (B ⊗ D) = 0. Therefore, the sequence (5) is a short exact sequence of K0 -groups, hence K0 (RB,D ) = K0 (NB,D )/Zν is mapped surjectively onto Z∞ /Zν . Positivity of TI is obvious.
Thus we see that K0 (RB,D ) is non-trivial. Our argument also shows that the cases B = K and B = IK give the same K0 -groups. More exactly, let evs : IK → K be the evaluation map at the point s ∈ [0, 1]. It induces the maps evs : NIK,D → NK,D and ev I s : RIK,D → RK,D .

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Corollary 12. The composition of (ev I s )∗ : K0 (RIK,D ) → K0 (RK,D ) with the quotient map TI : K0 (RK,D ) → Z∞ /Zν is an isomorphism independent of s. Remark 13. The positive cone (Z∞ /Zν )+ satisfies the following property: if ξ, η ∈ (Z∞ /Zν )+ and ξ = 0 then ξ + η = 0.

6. Homotopical non-invertibility: a sufficient condition The key ingredient to produce homotopically non-invertible extensions is existence of a non-trivial projection p ∈ JA,D . Theorem 14. Let τ : A → Q(K) be a ∗-homomorphism. Suppose there exists a projection p ∈ JA,D ⊂ A ⊗max D such that the class [τD (p)] ∈ K0 (RK,D )+ is mapped by TI to a non-trivial element in (Z∞ /Zν )+ . Then the class of τ in Exth (A, K) is not invertible. Proof. Suppose that τ is homotopically invertible. Then there exists an extension τ  : A → Q(K) such that their sum τ$ : A → Q(K) given by τ$(a) = v1 τ (a)v1∗ + v2 τ  (a)v2∗ , a ∈ A, is homotopic to the zero ∗-homomorphism via a homotopy Φ : A → Q(IK). Then τ$D : A ⊗max D → Q(K ⊗ D) is also homotopic to zero via the homotopy ΦD : A ⊗max D → Q(IK ⊗ D) and ΦD (p) ∈ RIK,D ⊂ Q(IK ⊗ D). Then I 1 ◦ ΦD (p)] = TI[$ τD (p)], 0 = TI[ev I 0 ◦ ΦD (p)] = TI[ev hence  (p)] = 0 TI[τD (p)] + TI[τD  I in Z∞ /Zν . But T [τD (p)] is non-negative and TI[τD (p)] is both non-negative and non-trivial, hence we get a contradiction, cf. Remark 13.
So, in order to produce examples of homotopically non-invertible C ∗ -extensions, one has to find a projection satisfying the conditions of Theorem 14. We do that in the next section.

7. Homotopical non-invertibility of some C ∗ -extensions related to property T groups Let G be an infinite countable discrete group with the property T of Kazhdan, [11, 8], which means that the trivial representation is isolated in the space of all representations of G. We assume that G has an infinite sequence πk , k ∈ N, of inequivalent finite-dimensional irreducible unitary representations. Denote by nk the dimension of the representation πk and fix the sequence ν = (nk )k∈N . Let π ¯k be the contragredient representation of πk . The latter means that there is an antilinear involution J on the Hilbert space H πk of the representation πk and π ¯k = Jπk J. Put Dk = B(Hπ¯k ) = Mnk and D = k∈N Dk . Let C ∗ (G) be the (full) group C ∗ -algebra of G. Then the representation π ¯k defines a ∗-homomorphism C ∗ (G) → Dk .

On Homotopical Non-invertibility of C ∗ -extensions

305

By [18], C ∗ (G) contains a copy of C generated by the minimal projection p0 corresponding to the trivial representation of G, as a direct summand. We identify elements of G with their images in C ∗ (G). Let S ⊂ G be a finite symmetric set of generators of G (recall that countable property T groups are always finitely generated, cf. [8], Theorem 1.10). Put 1
g ∈ C ∗ (G). (6) x= g∈S |S| Since S is symmetric (i.e., for each g ∈ S, S contains g −1 ), x is selfadjoint. So, Sp(x) ⊂ [−1, 1]. Property T means that there is a spectral gap between 1 ∈ Sp x and the remaining part of the spectrum: there exists some δ > 0 such that Sp(x) ⊂ [−1, 1 − δ] ∪ {1}. The projection p0 is the spectral projection of x corresponding to the point 1 [9]. Let E be the C ∗ -algebra on the Hilbert space H = ⊕k∈N Hπk generated by all operators of the form ⊕k∈N πk (g), g ∈ G, and by all compact operators K(H) on H. Put A = E/K. Then the C ∗ -extension 0 → K → E → A → 0 is the C ∗ -extension considered by S. Wassermann in [20]. We need the following modification of this C ∗ -extension [16]. k Fix a sequence µ = (mk )k∈N of positive integers with limk→∞ m nk = ∞. Let mk πk be the direct sum of mk copies of the representation πk . Define Eµ as the C ∗ -algebra generated by all operators of the form ⊕k∈N πkmk (g), g ∈ G, and by all compact operators on Hµ = ⊕k∈N Hπmkk . Note that Eµ /K ∼ = A, because both E/K and Eµ /K can be obtained from the group ring C[G] by completing it with respect to the same C ∗ -algebra seminorm  ·  = lim supk→∞ πk (·). If we denote the inclusion Eµ ⊂ B(Hµ ) by τ µ then, passing to the quotients, the composition τ µ = q ◦ τ µ : A → Q(K) is the Busby invariant of the extension 0

/K

/ Eµ

/A

/ 0.

(7)

Let σ : C ∗ (G) → Eµ be the ∗-homomorphism defined by σ(g) = ⊕k∈N πkmk (g), g ∈ G. Composing it with the quotient map, we get a surjective ∗-homomorphism ¯k : C ∗ (G) → D. σ I : C ∗ (G) → A. Consider also another ∗-homomorphism ⊕k∈N π Let ∆ : G → G × G be the diagonal homomorphism. It induces the ∗homomorphism ∆ : C ∗ (G) → C ∗ (G) ⊗max C ∗ (G). Composing it with the map ¯k ) : C ∗ (G) ⊗max C ∗ (G) σ I ⊗ (⊕k∈N π

/ A ⊗max D

we obtain a ∗-homomorphism ¯k )) : C ∗ (G) ∆ ◦ (I σ ⊗ (⊕k∈N π

/ A ⊗max D.

Put ¯k ))(p0 ) ∈ A ⊗max D. p = ∆ ◦ (I σ ⊗ (⊕k∈N π Lemma 15. The projection p satisfies the following conditions: 1. p ∈ JA,D ; µ 2. TI[τD (p)] = [µ + Zν ] = 0 in Z∞ /Zν .

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Proof. To show that p ∈ JA,D one has to check that qmax,min (p) = 0 in A ⊗min D. The latter was proved by S. Wassermann in [20], cf. [16]. µ µ Let us calculate TI[τD (p)]. The map τD factorizes through the Busby invariant A ⊗ Eµ D

/ Q(K ⊗ D)

of the C ∗ -extension 0

/ K⊗D

/ Eµ ⊗min D

/ A ⊗ Eµ D

/ 0,

which is obtained from the inclusion Eµ ⊂ M (K ⊗ D) by passing to quotients. Therefore, it suffices to calculate T [¯ p] ∈ Z∞ , where p¯ = (σ ⊗ (⊕k∈N π ¯k ))(∆(p0 )) ∈ Eµ ⊗min D is the projection corresponding to the trivial representation of G. Consider the map idEµ ⊗pk : Eµ ⊗min D → Eµ ⊗ Dk . Then T [¯ p] = (Tk )k∈N , p)). By the spectral theorem, idEµ ⊗pk (¯ p) equals the where Tk = tr(idEµ ⊗pk (¯ spectral projection of the image 1
σ(g) ⊗ π ¯k (g) ∈ B(Hµ ) ⊗ Dk g∈S S of x (6) corresponding to the (still isolated) point 1. So, it remains to calculate the dimensions of the trivial representation as a subrepresentation of σ ⊗ π ¯k . But σ = ⊕k∈N πkmk and it is known, [20], that the representation πl ⊗¯ πk does not contain ¯k contains exactly one one-dimensional the trivial representation if l = k and πk ⊗ π µ (p)] = [µ + Zν ] = 0.
trivial representation. Hence, Tk = mk and TI[τD Corollary 16. The C ∗ -extension (7) is not invertible in Exth (A, K). Corollary 17. The semigroup Exth (A, K) contains a copy of the semigroup (Z ∞ /Zν )+ .

8. Concluding remarks Remark 18. We have seen that the extensions Eµ and Eµ
define different elements in Exth (A, K) when the sequences µ and µ have (very) different growth. It would be interesting to obtain conditions on µ and µ equivalent to homotopy equivalence of these extensions. We also would like to know if Wassermann’s example itself (without modification) is non-trivial up to homotopy. Remark 19. It is not necessary to have a projection with the properties as in Theorem 14 in an ideal of A ⊗max D. A smaller tensor product suffices. Let  · α be a tensor C ∗ -norm on A * D such that  · E ≤  · α for any extension E of A. Then one can use a projection in A ⊗α D satisfying the properties of Lemma 15 to prove homotopical non-invertibility of C ∗ -extensions of A in the same way. Using this tensor C ∗ -product one can hope to prove homotopical non-invertibility of similar C ∗ -extensions arising from property τ groups ([15]; this property is weaker than the property T and means that the trivial representation is isolated from

On Homotopical Non-invertibility of C ∗ -extensions

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finite-dimensional representations only), when the projection corresponding to the trivial representation does not lift to a projection in A ⊗max D, but may lift to a projection in A ⊗α D. Acknowledgement I am grateful to K. Thomsen for his fruitful long-term collaboration and hospitality during my visits to the ˚ Arhus University. I also appreciate hospitality of Isaac Newton Institute, Cambridge, and of the Organizing Committee of WOAT 2006, Lisbon.

References [1] J. Anderson, A C ∗ -algebra A for which Ext(A) is not a group. Ann. Math. 107 (1978), 455–458. [2] R.G. Bartle, L.M. Graves, Mappings between function spaces. Trans. Amer. Math. Soc. 72 (1952), 400–413. [3] B. Blackadar. K-theory for Operator Algebras. MSRI Publ. 5, Springer-Verlag, 1986. [4] L.G. Brown, R.G. Douglas, P.A. Fillmore, Extensions of C ∗ -algebras and Khomology. Ann. Math. 105 (1977), 265–324. [5] A. Connes, N. Higson. D´eformations, morphismes asymptotiques et K-th´eorie bivariante. C. R. Acad. Sci. Paris S´er. I Math. 311 (1990), 101–106. [6] M. Dadarlat and S. Eilers, On the classification of nuclear C ∗ -algebras. Proc. London Math. Soc. (3) 85 (2002), 168–210. ∗ [7] U. Haagerup, S. Thorbjornsen, A new application of random matrices: Ext(Cred (F2 )) is not a group. Ann. Math. 162 (2005), 711–775. [8] P. de la Harpe, A. Valette, La propri´ et´e (T) de Kazhdan pour les groupes localement compacts. Ast´erisque, 175, 1989. [9] P. de la Harpe, A.G. Robertson, A. Valette, On the spectrum of the sum of generators for a finitely generated group. Israel J. Math. 81 (1993), 65–96. [10] N. Higson, V. Lafforgue, G. Skandalis, Counterexamples to the Baum–Connes conjecture. Geom. and Funct. Anal. 12 (2002), 330–354. [11] D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups. Funkts. Anal. Prilozh. 1 (1967), No. 1, 71–74 (in Russian). English translation: Funct. Anal. Appl. 1 (1967), 63–65. [12] E. Kirchberg, On semi-split extensions, tensor products and exactness of C ∗ -algebras. Invent. Math. 112 (1993), 449–489. [13] K. Knudsen-Jensen, K. Thomsen. Elements of KK-theory. Birkh¨ auser, 1991. [14] T. Loring, Almost multiplicative maps between C ∗ -algebras. Operator Algebras and Quantum Field Theory, Rome, 1996, 111–122. [15] A. Lubotzky, Discrete groups, expanding graphs and invariant measures. Birkh¨ auser, 1994. [16] V. Manuilov, K. Thomsen, On the lack of inverses to C ∗ -extensions related to property T groups. Canad. Math. Bull. 50 (2007), to appear.

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[17] N. Ozawa, An application of expanders to B(l2 ) ⊗ B(l2 ). J. Funct. Anal. 198 (2003), 499–510. [18] A. Valette, Minimal projections, integrable representations and property T. Arch. Math. 43 (1984), 397–406. [19] S. Wassermann, Tensor products of free group C ∗ -algebras. Bull. London Math. Soc. 22 (1990), 375–380. [20] S. Wassermann, C ∗ -algebras associated with groups with Kazhdan’s property T. Ann. Math. 134 (1991), 423–431. [21] S. Wassermann, Liftings in C ∗ -algebras: a counterexample. Bull. London Math. Soc. 9 (1976), 201–202. [22] S. Wassermann, Exact C ∗ -algebras and related topics. Lecture Notes Series 19, Seoul National Univ., 1994. Vladimir Manuilov Dept. of Mechanics and Mathematics Moscow State University Leninskie Gory, Moscow 119992 Russia and Harbin Institute of Technology P. R. China e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 309–320 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Galois-fixed Points and K-theory for GL(n) S´ergio Mendes To the memory of Professor Jos´e de Sousa Ramos Abstract. Let F be a nonarchimedean local field and let G = GL(n) = GL(n, F). Let E/F be a finite Galois extension. We use the Hasse-Herbrand function ψE/F to identify the K-theory groups of the reduced C ∗ -algebra Cr∗ GL(n, F) with the Galois-fixed points of the K-theory groups of the reduced C ∗ -algebra Cr∗ GL(n, E). Mathematics Subject Classification (2000). Primary 22E50; Secondary 46L80. Keywords. Local field, general linear group, reduced group C ∗ -algebra, Ktheory.

Introduction Let F be a local nonarchimedean field and let G = GL(n) = GL(n, F). Let Cr∗ (G) denote the reduced C ∗ -algebra of G. The noncommutative C ∗ -algebra Cr∗ (G) is strongly Morita equivalent to the commutative C ∗ -algebra C0 (Irrt (G)) where Irrt (G) denotes the tempered dual of G, see [11]. As a consequence of this, there is an isomorphism of K-groups K∗ C ∗ (G) ∼ = K ∗ Irrt (G). r

Let E/F be a finite Galois extension of F . Base change [1] determines a map bE/F : Irrt GL(n, F) −→ Irrt GL(n, E). The base change map allows one to see the tempered representations of GL(n, F) as a subset of the tempered representations of GL(n, E). The Galois group Gal(E/F ) acts on Irrt GL(n, E) π σ (x) = π(σ −1 (x)) for all σ ∈ Gal(E/F ). If E/F is cyclic the action is completely determined by a generator σ of Gal(E/F ). The tempered representations of GL(n, E) fixed under the action of Gal(E/F ), i.e., such that π ∼ = π σ for all σ in Gal(E/F ), are precisely those obtained from GL(n, F)

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under base change, see [1, Theorem 6.2, p.51]. From now on we call such points the Galois-fixed points. Therefore, we have an identification Gal(E/F )

Irrt GL(n, E)

∼ = Irrt GL(n, F).

Now, since K ∗ is a functor Gal(E/F ) acts on K ∗ Irrt GL(n, E) and it seems natural that, at the level of the K-theory groups, K ∗ Irrt GL(n, F) should identify with the Galois-fixed points of K ∗ Irrt GL(n, E), which we denote by K ∗ Irrt GL(n, E)

Gal(E/F )

.

In this paper we concentrate on the cases of GL(1) and GL(2). Specifically, we use the Hasse-Herbrand function to identify the K-theory groups K ∗ Irrt GL(n, E)

Gal(E/F )

∼ = K ∗ Irrt GL(n, F)

for n = 1, 2, when E/F is unramified or tamely ramified.

1. Background and notation Let F be a nonarchimedean local field of characteristic 0. Then F is a finite extension of Qp . The prime p is the residue characteristic of F . Let oF , pF , F , kF = oF /pF , qF and |.|F denote, respectively, the ring of integers, the prime ideal, a uniformizing parameter, the residue field, the cardinality of kF and the absolute value on F normalized so that |x|F = q −υF (x) where υF denotes a valuation on F . n n 0 Let UF = o× F and UF = 1 + pF for a positive integer n. Put UF = UF . × The group F is a Hausdorff topological group with a filtration by compact open subgroups F × ⊃ UF ⊃ UF1 ⊃ UF2 ⊃ UF3 ⊃ · · · ⊃ {1}. The collection {UFn : n ≥ 0} is a neighborhood basis of the identity. There is an isomorphism of topological groups [5, p.13] F× ∼ =< F > ×UF . It follows that the dual group is O F× ∼ = T × U/ F

(1.1)

where T is the unit circle. We use the terminology of [14] and define a quasicharacter of F × to be a continuous homomorphism χ : F × → C× . If χ is unitary then it is called a character. From (1.1), χ can be written as a product χ(x) = zχυF (x) .χ0 (x

−υ(x) ) F

where zχ = χ( F ) ∈ T and χ0 lies in U/ F. Since χ0 is continuous, there exist  ∈ N0 such that χ0 is trivial on UF . We call the least  = χ such that χ0|UF = 1

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311

the conductor of χ. In particular, if χ = 0 then χ is trivial on UF and is called an unramified character. We write χ(x) = z υF (x) for an unramified character. To each character χ of F × we associate a pair of parameters (zχ , χ ) ∈ T × N0 , where zχ = χ( F ) and χ is the conductor of χ. These parameters do not  completely determine the character χ. In fact, UF /UFχ is a finite cyclic group of   −1 order (qF − 1)qFχ , see [9, p.12], and so is the dual group UF /UFχ . We need an enumeration of the countable set U/ F . To each natural number / ν we attach the finite set of all characters χ0 ∈ UF for which χ = ν. This enumeration is not canonical. Fixing an enumeration one can write the dual group of F × as a countable disjoint union P × ∼ O Tχ . (1.2) F = χ

We now recall some results about ramification theory. Let E/F be a finite Galois extension and write G = Gal(E/F ). The ith ramification group of E/F is defined to be Gi = {σ ∈ G : σx − x ∈ pi+1 E , x ∈ oE }. for all integers i ≥ −1. They form a decreasing sequence of groups G−1 = G ⊃ G0 ⊃ G1 ⊃ · · · ⊃ Gi ⊃ Gi+1 ⊃ · · · . One can extend the definition of Gi to a continuous parameter t ∈ [−1, +∞[: if i−1 < t ≤ i then we define Gt = Gi . We define a real function ϕ : [−1, +∞[→ R by  u dt . ϕE/F (u) = 0 (G0 : Gt ) ϕE/F is a step function and is a homeomorphism of the interval [−1, +∞[ into itself. The inverse ψE/F = ϕ−1 E/F is called the Hasse-Herbrand function, for which we collect some properties [12, Prop. 13, p. 73]: (i) ψE/F is continuous, piecewise linear, increasing and convex; (ii) ψE/F (0) = 0; (iii) ψE/F (N) ⊂ N. Recall that an extension E/F is unramified if [E : F ] = [kE : kF ] = f (E/F ). The number f = f (E/F ) is called the residue degree of E/F . The extension is totally ramified if f (E/F ) = 1 and is tamely ramified if p e(E/F ) where p = ch(kF ) > 1. The ramification degree is the integer e = e(E/F ) such that e F . The numbers e = e(E/F ) and f = f (E/F ) are related by [E : F ] = ef . E = It follows that e(E/F ) = 1 and f (E/F ) = [E : F ] if and only if E/F is unramified. Example. If E/F is unramified then ψE/F (x) = x. If E/F is tamely ramified then ψE/F (x) = ex, where e = e(E/F ) is the ramification degree.

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We end this section by recalling the notion of Weil group and some representation theory results. Let F be a separable algebraic closure of F and let F ur be the maximal unramified extension of F on F /F . There is a projection map d : Gal(F /F ) → Gal(F ur /F ). I the profinite completion of Z. The group The group Gal(F ur /F ) is isomorphic to Z, I Z is dense in Z: the isomorphism takes the geometric Frobenius automorphism ΦF ∈ Gal(F ur /F ) to the generator +1 of Z. The Weil group WF of F is by definition the pre-image of ΦF in Gal(F /F ). We thus have a surjective map, still denoted by d, d : WF → Z. The pre-image of 0 is the inertia group IF . Therefore, we have a short exact sequence 1 → IF → WF → Z → 0. The topology of WF is dictated by the short exact sequence of topological groups: IF is given the profinite topology induced from Gal(F /F ) and Z is given the discrete topology. The Weil group is a locally compact group with maximal compact subgroup IF . The map WF → Gal(F /F ) is a continuous homomorphism with dense image. See [13] for a detailed account of the Weil group for local fields. Given a topological group G we denote Gab the quotient Gab = G/Gc , where c G is the closure of the commutator subgroup of G. Gab is the maximal abelian Hausdorff quotient of G and is called the abelianization of G. The central result of local class field theory is Artin Reciprocity [10]: ArtE/F : F × /NE/F E × ∼ = Gal(E/F )ab where NE/F : E × → F × is the norm map. The Weil group is particularly suitable for class field theory and Artin reciprocity is now an isomorphism of topological groups ∼ W ab . ArtF : F × = F

The Weil-Deligne group (or simply L-group) of a nonarchimedean field is the group LF = WF × SU(2). A Langlands parameter (or L-parameter ) is a continuous homomorphism φ : LF → GL(n, C) (GL(n, C) is given the discrete topology) such that φ(ΦF ) is semisimple, where ΦF is a geometric Frobenius in WF . Two L-parameters are equivalent if they are conjugate under GL(n, C). The set of equivalence classes of L-parameters is denoted Φ(GL(n, F)). An L-parameter is tempered if φ(WF ) is bounded [2, §10.3]. A representation of G = GL(n, F) on a complex vector V is smooth if the stabilizer of each vector in V is an open subgroup of G. The set of equivalence

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313

classes of irreducible smooth representations of G is the smooth dual IrrGL(n, F) of G. We will use the local Langlands correspondence for GL(n) [7, 8]: FL

: Φ(GL(n, F)) → IrrGL(n, F).

The image of all the tempered L-parameters under F L is called the tempered dual of GL(n, F) and is denoted by Irrt GL(n, F). Given a representation ρ of a group G on a vector space V , a matrix coefficient of ρ is a function f : G → C such that f (g) = ρ(g)v, ω where v ∈ V , ω ∈ V ∗ (V ∗ denotes the dual space of V ). The inner product is given by the duality between V and V ∗ . A representation is called cuspidal if and only if the support of every matrix coefficient is compact modulo the centre of G. Let E/F be a finite Galois extension of F . Then WE is a subgroup of WF . Base change is defined by restriction of L-parameter from LF to LE . Langlands functoriality predicts the existence of a base change map bE/F : Irrt GL(n, F) → Irrt GL(n, E). In the next section we will give a description of this map for GL(1) and GL(2).

2. Galois-fixed points Let E/F be a finite Galois extension. For simplicity we will restrict to cyclic extensions. We will denote by σ a generator of Gal(E/F ). The Galois group Gal(E/F ) acts on tempered representations of GL(n, E). The Galois-fixed points of Irrt GL(n, E) are precisely the tempered representations of GL(n, E) obtained from GL(n, F) under base change, see [6]. 2.1. Galois-fixed points for GL(1) For GL(1) base change is easy to describe. For each character χF of F × we get a character χE of E × by composing with the norm map NE/F : E × → F × . We call χE the base change of χF and denote χE = bE/F (χF ) = χF ◦ NE/F . Proposition 2.1. [6] The characters of E × of the form χE = χF ◦ NE/F , where χF is a character of F × , are precisely the Galois-fixed points. For each character χ of E × the Galois group Gal(E/F ) acts on the associated parameters: (zχ , χ )σ = (zχσ , χσ ) ∈ T × N0 . The Galois-fixed points are given by (zχE , χE ), where χE = χF ◦ NE/F and χF is a character of F × . Proposition 2.2. Let E/F be a finite separable extension, either unramified or tamely ramified. Then ψ (ν) NE/F (UE E/F ) = UFν for all ν ≥ 0, where ψE/F is the Hasse-Herbrand function.

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Proof. If E/F is unramified it follows from [12, Proposition 1, p. 81]. Now, suppose E/F is tamely ramified and let F0 /F be the maximal unramified subextension of F in E/F . Then E/F0 is a totally tamely ramified extension. Since G1 = {1}, we also have Gψ(ν) = {1}, for all ν ≥ 0. By [12, Corollary 4, p. 93] we have ψ(ν)

NE/F0 (UE

) = UFν0 .

Since F0 /F is unramified we have NF0 /F (UFν0 ) = UFν . By transitivity of the norm map it follows that ψ(ν)

NE/F (UE

ψ(ν)

) = NF0 /F (NE/F0 (UE

)) = NF0 /F (UFν0 ) = UFν .


Theorem 2.3. Let E/F be a finite cyclic Galois extension, either unramified or tamely ramified. Let χF be a character of F × and let χE = χF ◦ NE/F . Then zχE = zχf F and χE = ψE/F (χF ) where f = f (E/F ) is the residue degree and ψE/F is the Hasse-Herbrand function. Proof. The equality zχE = zχf F follows from [9, Lemma 2.5]. From Proposition 2.2 we have that ψ

NE/F (UE E/F

(ν)

) = UFν

whenever E/F is unramified or tamely ramified. We only have to check that if χ χE = χF ◦ NE/F and NE/F (UEm ) = UF F then m is the conductor χE of χE . We have χ χE (UEm ) = χF ◦ NE/F (UEm ) = χF (UF F ) = 1. If r is an integer such that 0 < r < m then UEm ⊂ UEr and  χF

NE/F (UEm ) = UF

⊂ NE/F (UEr ) = UFs

for some s with χF > s. Then, by definition of conductor, χF ◦ NE/F (UEr ) = χF (UFs ) = 1, and we conclude that m = χE . By Proposition 2.2, m = ψE/F (χF ) and the result follows.
2.2. Galois-fixed points for GL(2) The smooth irreducible representations of GL(2, F) arise essentially in two different ways: induced representations and cuspidal representations. The tempered irreducible representations are the only contributors for the Ktheory of the C ∗ -algebra Cr∗ GL(2, F), see [11]. We recall that if (π, V ) is tempered then it has a unitary central character ωπ . Now, we give a more detailed description of the tempered spectrum Irrt GL(2, F).

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315

• Principal series. Let χ1 and χ2 be two quasicharacters of F × . Let π(χ1 , χ2 ) denote the (parabolically) induced representation iG,B (χ1 ⊗ χ2 ), where G is the group GL(2, F) and B is the Borel subgroup. It is well known that π(χ1 , χ2 ) is ±1 irreducible if and only if χ1 χ−1 2 = |.|F . Moreover, if χ1 and χ2 are unitary then π(χ1 , χ2 ) is called a unitary principal series representation and is tempered. The only equivalence relation between principal series representations is given by π(χ1 , χ2 )  π(χ2 , χ1 ). Thus, the irreducible unitary principal series of GL(2, F) are tempered and are indexed by unordered pairs {χ1 , χ2 } of characters of F × . • Subquotients of principal series. Let χ be a character of F × . The principal 1 −1 series iG,B (χ|.|F 2 ⊗ χ|.|F2 ) has a unique irreducible subquotient, the twist of the −1

1

Steinberg representation: χ⊗St(2) = Q(χ|.|F 2 , χ|.|F2 ). When χ is unitary, χ⊗St(2) is tempered. • Cuspidal representations. Every irreducible representation which is not parabolically induced is a cuspidal representation. A cuspidal representation is tempered provided that it has unitary central character. From now on, all the representations will be tempered and in particular will have unitary central character. Base change Let E/F be a finite Galois extension. We now describe the base change map bE/F : Irrt GL(2, F) → Irrt GL(2, E) for tempered representations of GL(2). In what follows, χ, χ1 , χ2 denote (unitary) characters of F × and NE/F : E × → F × is the norm map. • Let π(χ1 , χ2 ) be a tempered principal series representation of GL(2, F). Then bE/F (π(χ1 , χ2 )) = π(χ1 ◦ NE/F , χ2 ◦ NE/F ) is a tempered principal series representation of GL(2, E). • Let χ be a character of F × and let π = χ ⊗ StF (2) be the twist of the Steinberg representation. The corresponding L-parameter is φF = χ ⊗ spin(1/2) (here, we regard χ as a quasicharacter of WF via class field theory). spin(1/2) denotes the (2j + 1)-dimensional irreducible representation of SU(2). Base change for L-parameters is given by restriction. Accordingly, φF |LE = χ|WE ⊗ spin(1/2) = φE . Applying the Langlands correspondence we conclude that bE/F (F L(φF )) = E L(φF |LE ) bE/F (χ ⊗ StF (2)) = E L(φE ) = χ ◦ NE/F ⊗ StE (2). It remains to study the cuspidal part of the tempered dual.

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Let Φ0 (GL(2, F)) be the set of equivalence classes of irreducible 2-dimensional smooth (complex) representations of WF . Denote by Irr0 GL(2, F) the subset of Irrt GL(2, F) consisting of equivalence classes of irreducible cuspidal representations of GL(2, F). The local Langlands correspondence gives a bijection [4, p. 219] FL

: Φ0 (GL(2, F)) → Irr0 GL(2, F).

The cuspidal representations are parameterized by admissible pairs [4, p. 124]. We recall the definition of an admissible pair. Definition 2.4. Let K/F be a quadratic extension and let ξ be a quasicharacter of K × . The pair (K/F, ξ) is called admissible if (1) ξ does not factor through the norm map NK/F : K × → F × and, 1 does factor through NK/F , then K/F is unramified. (2) if ξ|UK Denote the set of F -isomorphism classes of admissible pairs (K/F, ξ) by P2 (F ). According to [4, p.215], the map P2 (F ) (K/F, ξ)

−→ Φ0 (GL(2, F)) −→ IndK/F ξ

(2.1)

is a canonical bijection, where we see ξ as a quasicharacter of WK via the class field theory isomorphism WK ∼ = K × and IndK/F is the functor of induction from representations of WK to representations of WF . We use the theory of Bushnell and Henniart [3, 4] to describe the base change of cuspidal representations. This description, however, is uncomplete for we restrict to a part of the cuspidal spectrum. We assume that the ground field F has odd residue characteristic. Cuspidal representations are parameterized by admissible pairs (K/F, ξ), where K/F is quadratic and ξ is a quasicharacter of K × . We further require the admissible pair (K/F, ξ) to have unitary character ξ. This ensures that the associated cuspidal representation has unitary central character. Every quadratic extension K/F is either unramified of tamely totally ramified. Up to F -isomorphism, for F with odd residue characteristic p there are only three such extensions: one unramified and two tamely totally ramified. Therefore, the cuspidal spectrum decomposes as follows: P Φ0 (GL(2, F)) = Φur (GL(2, F)) Φtr (GL(2, F)) where Φur (GL(2, F)) (resp., Φtr (GL(2, F))) denote the cuspidal representations of GL(2, F) parameterized by admissible pairs (K/F, ξ) with K/F unramified (resp., tamely totally ramified). For Φtr (GL(2, F)) the base change map was established by Bushnell and Henniart [3]. We restrict to unramified base change E/F , F with odd residue characteristic and to cuspidal representations Φtr (GL(2, F)). Now, let E/F be a finite cyclic Galois extension, unramified with odd degree. We require that [E : F ] is odd to prevent the tame extensions K/F to factor through E/F (i.e., F ⊂ K ⊂ E) in which case we could no longer guarantee

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317

the base change of a cuspidal representation of GL(2, F) to remain a cuspidal representation of GL(2, E). According to [3, Theorem 4.6, p. 704], base change is given by : bE/F : Φtr (GL(2, F)) → Φtr (GL(2, E)), bE/F (σ) = σE where (K/F, ξ) is an admissible pair with ξ unitary, and where σ = IndK/F ξ, σE = IndKE/E ξE and ξE = ξ ◦ NKE/E . This concludes the description of the base change map for GL(2).

3. K-theory In this section we show that, given an unramified or tamely ramified extension E/F , cyclic and Galois, we have an identification at the level of K-theory groups K ∗ Irrt GL(n, E)Gal(E/F ) ∼ = K ∗ Irrt GL(n, F) when n = 1, 2. The result can be extended easily to GL(n), n > 2, for certain classes of representations (e.g., for principal series representations). 3.1. GL(1) The unitary dual of GL(1, F) is a countable disjoint union of circles and so has the structure of a locally compact Hausdorff space P Irrt GL(1, F) ∼ Tχ (3.1) = with χ ∈ U/ F . Here Tχ denotes one copy of the unit circle T. Each K-group is a countably generated free abelian group: Q Q Zχ ∼ Zχ K ∗ (Irrt GL(1, F)) ∼ = = with χ ∈ U/ F and ∗ = 0, 1. χ ∈ N0 denotes the conductor of χ. Theorem 3.1. Let E/F be a finite cyclic Galois extension, either ramified of tamely ramified. Then the K-theory groups K ∗ Irrt GL(1, F) can be identified with the subset of Galois-fixed points: K ∗ Irrt GL(1, E)Gal(E/F ) . Proof. Given a character χF of F × let χE denote the character χE = χF ◦NE/F of E × , where NE/F is the norm map. The characters χE are the Galois-fixed points for the action of Gal(E/F ) on Irrt GL(1, E). It follows from Proposition 2.2 and Theorem 2.3 that Q Q Gal(E/F ) ∼ Zψ ( ) K ∗ Irrt GL(1, E) Z = = χE

E/F

χF

/F U

∼ = K ∗ Irrt GL(1, F) where ψE/F is the Hasse-Herbrand function.




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3.2. GL(2) According to [11], the tempered dual of GL(2) is in bijection with a locally compact Hausdorff space. This space is a countably disjoint union of circles and symmetric product of circles. Let us describe the components of this disjoint union. Recall that the dual of F × is a countable disjoint union of unitary circles, indexed by conductors of characters of F × , see (1.2). The cuspidal representations σ of GL(2) arrange themselves in the tempered spectrum Irrt GL(2) as a disjoint union of unitary circles. There are countably many disjoint circles, parameterized by admissible pairs (K/F, ξ), with ξ unitary. The unitary twists of the Steinberg representation χ ⊗ St(2) contribute to the tempered spectrum with a countable disjoint union of unitary circles, indexed by characters χ0 = χ|UF of the group of units UF . Finally, the unitary principal series arrange themselves in the tempered spectrum as a countable union of symmetric products of circles Sym2 (T) = T2 /S2 , where S2 denotes the cyclic group with two elements. Recall that the unitary principal series are parameterized by unordered pairs {χ1 , χ2 } of unitary characters of F × and that π(χ1 , χ2 )  π(χ2 , χ1 ). Therefore, the principal series part of the tempered spectrum is a countable disjoint union of symmetric products of circles. The symmetric product Sym2 (T) has the homotopy type of a single circle, see [9, Lemma 4.2, p.10]. From the discussion above the K-theory of Irrt GL(2, F) can be computed easily. The main result can now be stated and proved. Theorem 3.2. (i) Let E/F be a finite cyclic Galois extension, either ramified of tamely ramified. Let Irr0 GL(2, F) denote the cuspidal part of the tempered dual. Then the K-theory groups K ∗ (Irrt GL(2, F)\Irr0 GL(2, F)) can be identified with the subset of Galois-fixed points: K ∗ (Irrt GL(2, E)\Irr0 GL(2, E))Gal(E/F ) . (ii) Suppose F has odd residue characteristic. Let E/F be a finite cyclic Galois extension, unramified with odd degree. Then the K-theory groups K ∗ Irrtr GL(2, F) can be identified with the subset of Galois-fixed points: K ∗ Irrtr GL(2, E)

Gal(E/F )

t

.

0

Proof. (i) An element from Irr GL(2, F)\Irr GL(2, F)) is either a tempered principal series or a unitary twist of the Steinberg representation. Let χF , ηF , ζF be characters of F × and let χE , ηE , ζE denote the usual composition with the norm map. The representations χE ⊗ St(2) and π(ηE , ζE ) are the Galois-fixed points for the action of Gal(E/F ) on the tempered spectrum Irrt GL(2, E)\Irr0 GL(2, E). It follows from Theorem 3.1 that the Galois-fixed points K ∗ (Irrt GL(2, E)\Irr0 GL(2, E))Gal(E/F )

Galois-fixed Points and K-theory for GL(n) are given by the direct sums Q

Q

ZχE and

319

Z{ηE ,ζE }

where {, } denote unordered pairs. From Theorem 3.1, we have Q Q ZχE = ZψE/F (χF ) /F U

and

Q

Z{ηE ,ζE } =

Q

Z{ψE/F (ηF ),ψE/F (ζF )} ,

/F U

where ψE/F is the Hasse-Herbrand function. These direct sums can be identified with the K-groups K ∗ (Irrt GL(2, F)\Irr0 GL(2, F)). (ii) Now, let E/F be a finite cyclic Galois extension, unramified of odd degree. From [9, Theorem 6.2, p. 18], we have Q Q Gal(E/F ) ∼ K ∗ (Irrtr GL(2, E) Z(KE/E,ψKE/K (ξ )) Z(KE/E,ξE ) = = =

Q

/F U

Z(K/F,ξ ) ∼ = K ∗ Irrtr GL(2, F)

/F U

where ξE = ξ ◦ NKE/K and ψKE/K is the Hasse-Herbrand function.




Acknowledgments The author wishes to acknowledge the helpful comments and suggestions of the anonymous referee. Thanks must also be given to Paul Broussous for his help on many occasions. This paper draws on the author’s PhD research, conducted at the University of Manchester under the supervision of Prof Roger Plymen, and financed by the Funda¸c˜ao para a Ciˆencia e Tecnologia, Terceiro Quadro Comunit´ario de Apoio, SFRH/BD/10161/2002.

References [1] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Studies 120, Princeton University Press, Princeton 1989. [2] A. Borel, Automorphic L-functions. In Automorphic Forms, Representations, and L-Functions, Proc. Sympos. Pure Math. 33 (1979), 27–61. [3] C.J. Bushnell and G. Henniart, The essentially tame local Langlands correspondence, I, J. Amer. Math. Soc. 18 (2005), 685–710. [4] C.J. Bushnell and G. Henniart G., The Local Langlands Conjecture for GL(2), Springer-Verlag, Berlin 2006. [5] I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, 2nd ed., AMS, Providence 2002.

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[6] P. G´erardin and J.P. Labesse, The solution of a base change problem for GL(2). In Automorphic Forms, Representations, and L-Functions, Proc. Sympos. Pure Math. 33 (1979) part 2, 115–133. [7] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139(2000), 439–455. [8] M. Harris and R. Taylor, On the Geometry and Cohomology of Some Simple Shimura Varieties, Ann. Math. Study 151, Princeton University Press 2001. [9] S. Mendes and R.J. Plymen, Base change and K-theory for GL(n), J. Noncommut. Geom. 1(2007), 311–331. [10] J. Neukirch, Algebraic Number Theory, Springer-Verlag, Berlin, 1999. [11] R.J. Plymen, The reduced C ∗ -algebra of the p-adic group GL(n). J. Functional Analysis 72 (1987), 1–12. [12] J.-P. Serre, Local Fields. Springer-Verlag, New York, 1979. [13] J. Tate, Number theoretic background. In Automorphic Forms, Representations, and L-Functions, Proc. Sympos. Pure Math. 33 (1979) part 2, 3–26. [14] A. Weil, Basic Number Theory. Classics in Math., Springer-Verlag, 1995. S´ergio Mendes ISCTE Av. das For¸cas Armadas 1649-026, Lisbon Portugal e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 321–346 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory Kalle M. Mikkola and Ilya M. Spitkovsky Abstract. The spectral or canonical factorization of matrix- or operatorvalued function F defined on the imaginary axis is defined as F = Y ∗ X, where Y ±1 , X ±1 are H∞ (bounded and holomorphic on {Re z > 0}), or, more generally, Y ±1 , X ±1 belong to some weighted strong H2 space. It is well known that the invertibility of the corresponding Toeplitz operator P+ F P+ is necessary for this factorization to exist, where P+ : L2 → H2 is the orthogonal projection. When F is positive, this condition is also sufficient for the factors to be H∞ . In the general (indefinite) case, this is not so. However, if F is smooth enough, then the H∞ canonical factorization does exist even in the indefinite case; we give a solution assuming that F is the Fourier transform of a measure with no singular continuous part. If the (Popov function determined by the) transfer function of a control system has a canonical factorization, then a well-posed optimal state feedback exists for the corresponding control problem. Conversely, a well-posed optimal state feedback determines a canonical factorization of the transfer function. We generalize this to unstable systems, i.e., to transfer # functions that are holomorphic and bounded on some right half-plane {z # Re z > r}. Then we show that if the generalized Popov Toeplitz operator is uniformly positive, then the canonical factorization exists (the stable case is well known). However, the results on the regularity of the factors and in the nonpositive case remain very few – we explain them and the remaining open problems. Mathematics Subject Classification (2000). Primary 47A68; Secondary 49N05. Keywords. Spectral factorization, proper J-canonical factorization, unstable canonical factorization, regularity, regular well-posed linear systems.

The first author was supported by Magnus Ehrnrooth Foundation. The second author was supported in part by NSF Grant DMS-0456625.

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1. Introduction The spectral or canonical factorization problem arises in many different areas, such as Wiener-Hopf problems, and it has been studied for decades; see [GKS03] [CG81] [LS87]. In control theory such problems are frequently encountered, and the properties of the spectral factors lead to a solution that has many desirable properties from the control point of view. Therefore, the applications of spectral factorization to control problems are numerous. Below we briefly explain the control-theoretic motivation of this article. When the (control or) input operator of a system is bounded and the system is stabilizable and detectable, then lumped-system methods can be readily extended to distributed (or infinite-dimensional) systems to show the existence of a wellposed optimal state-feedback to standard control problems. As a by-product, the spectral factorization of the (Popov function determined by the) transfer function is obtained. For more general systems, often spectral factorization has been used to establish the existence of such feedback, particularly for positive-definite quadratic cost-minimization problems in the stable case [WW97] [Sta97] [CW99]. In this article we present and extend the factorization results known for indefinite stable case and positive unstable case problems and present some open problems. For the readers’ convenience, we list here the notation and terminology in the order of appearance, followed by the corresponding page numbers. 2 Section 1: H∞ 322, C+ 322, G H∞ 322, iR 323, B 323, L∞ strong 323, H 323, ˆ 323, + π/ + 323, π+ 323, R+ 323, Cω 324, U, X, Y 324. Section 2: π− 326, R− 326, τ t 326, TI 326, TIC 326, Dˆ 326, G 326, spectral 1 1 factorization 326, δ 327, MTI 327, MTId 327, MTIL 327, MTIC 327, MTICL 327, MTICd 327. Section 3: D 332, discrete time 332, 2 332, π + 332, H2 (D; Cn×n ) 332. ∞ Section 4: L2c 335, L2ω 335, TICω 335, TIC∞ 335, H∞ ω 335, proper 335, H∞ 335, proper J-canonical factorization 336. Section 5: stabilizable 337, w.r.c.f. 338, normalized 338, UR 338, proper J-canonical factorization 338, U 339, J-coercive 339, generalized Popov Toeplitz operator 339, realization 340. Now we explain the simplest possible case. To find a formula for the minimizing control of a stable system with transfer function G, one often writes the “Popov function” F := |G|2 of the system as the square of a boundedly invertible transfer function, using, e.g., the following well-known result: Proposition 1.1 (Positive scalar spectral factorization). Let F ∈ L∞ (iR). Then F = |f |2 a.e. on iR for some f ∈ G H∞ (C+ ) iff there is  > 0 such that F ≥  a.e. on iR. By H∞ (C+ ) we denote the space# of bounded holomorphic scalar functions on the right half-plane, C+ := {z ∈ C # Re z > 0}, by G H∞ we denote the subset

Factorization and Control

323

of invertible elements (f, f −1 ∈ H∞ ). Thus, an essentially bounded measurable scalar-valued function F on the imaginary axis iR can be written as the squared norm of (the boundary trace of) an invertible element of H∞ if and only if the function F is bounded from below on the imaginary axis iR. Proposition 1.1 is also true for matrix-valued functions (with F = f ∗ f on iR) and even for strong-L∞ functions with values in B(U), the space of bounded linear operators over a Hilbert space U, by Proposition 2.3 below. Moreover, certain smoothness properties of F are inherited by f . To present the indefinite (and infinite-dimensional) formulation of the above spectral factorization result, we must add a smoothness assumption, to be made more precise in Section 2 (definitions are given below the proposition): Proposition 1.2 (Indefinite spectral factorization). Let F ∈ L∞ strong (iR; B(U)) be ∞ ikT / sufficiently “smooth” (e.g., let F = F1 + k=−∞ e Ek , where F1 ∈ L1 (R; B(U)),  k Ek B(U) < ∞, and T > 0). Then F = f ∗ g for some f, g ∈ G H∞ (C+ ; B(U)) iff the Toeplitz operator 2 + TIF : u → π/ + F u is boundedly invertible on H (C ; U). Moreover, the smoothness of F is inherited by f and g. (See Section 2 for details, proof and uniqueness: if f, g are as#above, then all solutions are given by F = (E −∗ f )∗ (Eg), E ∈ GB(U) := {E ∈ B(U) # E −1 ∈ B(U)}.) This proposition has already been essentially known in the case Ek = 0 for k = 0 [GL73a], as well as in the case F1 = 0 and dim U < ∞. The latter case, via an obvious change of variables (described for example in [BKS02]), reduces to the classical setting of the matrix functions from the Wiener algebra on the unit circle, see [CG81] [LS87] [GKS03]. In the positive case F ≥ 0, the proposition was established in [CW99].) ∞ for each u0 ∈ U. By u ∈ H2 (C+ ; U) we By F ∈ L∞ strong we mean that F u0 ∈ L mean that u is U-valued and holomorphic on C+ and uH2 := supr>0 u(r+i·)2 < F∞ ∞. By Fˆ1 we mean the Laplace transform Fˆ1 (s) := −∞ e−st F1 (t) dt. By π/ + we denote the orthogonal projection L2 (iR; U) → H2 (C+ ; U) (note that π/ v I = π O + + v (v ∈ L2 (R; U)), where π+ is the orthogonal projection L2 (R; U) → L2 (R+ ; U), R+ := [0, +∞)). In the equality F = f ∗ g we refer to the boundary functions of f and g on iR (such functions exist for any f, g ∈ H∞ , and in our case they are continuous). For F ≥ 0, the equivalence claim in the above proposition reduces to the (generalized) positive result mentioned above, because then F ≥ I iff TIF is invertible (and necessarily f ∗ = g ∗ E ∗ E for some E ∈ GB(U) [RR85]). However, in the indefinite case, the above proposition is not true without a smoothness assumption, by Theorem 3.3 below. Moreover, the smoothness result in Proposition 1.2 is important in the applications even in the positive case. The canonical factorization means the same as the above spectral factorization except that the requirements on f ±1 and g ±1 are slightly relaxed (e.g., they are only required to be weighted (strong) H2 ) [CG81] [LS87]. On the other hand, in spectral factorization one sometimes allows f (not g) to be noninvertible. All such

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factorizations and their “unstable” counterparts are covered in the unstable formulation presented later below, without loosing the properties that are important in control theory. The above factorizations are typically applied to the “Popov function” F := G∗ JG, JwhereK G is the transfer function of a stable control system and often J = I 0 or J = I0 −I . The factors are then used to determine a formula for the (J-)optimal control (a minimizing or a minimax input, depending on the problem), which can thus be shown to have the required smoothness and stability properties. These properties also allow one to write the factors in terms of the solution of an algebraic Riccati equation. Moreover, a solution of this form is always given by some “well-posed state-feedback” determined by the Riccati equation (alternatively, by the factors). See, e.g., [Win89], [WW97], [CW99], [Sta98], [Mik02] or Example 5.10. Analogous comments apply to the unstable factorization explained below. Conversely, a J-optimal control determines a canonical factorization. The optimal feedback is “well posed” iff the factors and # their inverses are # proper, i.e., bounded on some right half-plane C+ ω := {z ∈ C Re z > ω}. We shall use that additional assumption in our definitions of factorizations in Section 2–5. This is the only exception that we make to the standard definition of the canonical factorization. However, even without that requirement, everything stated in this article would be true mutatis mutandis (with ill-posed state-feedback allowed). The same also applies to discrete-time control problems (with the unit disc taking the role of the right half-plane), where properness becomes redundant. In the unstable case, the definition of the corresponding factorization is a bit more system-theoretic. We give a rough formulation of our unstable main results in Proposition 1.3 below (to be made precise later). Throughout this article, U, X and Y denote complex Hilbert spaces of arbitrary dimensions. Proposition 1.3 (Unstable J-canonical factorization). Let ω ∈ R, J = J ∗ ∈ GB(Y), and G ∈ H∞ (C+ ω ; B(U, Y)). Assume that G is sufficiently smooth (see Sections 5–6) or that J ≥ 0. If G is J-coercive and stabilizable, then G = hH −1 , where h and H are “almost H∞ ”, h∗ Jh equals a constant in B(U) (as a multiplication operator on the −1 is bounded on the domain of the part smooth elements of L2 (iR; U)) and H π/ +H 2 of G in L (iR; U) (see Definition 5.2 for details). Moreover, the smoothness of G is inherited by h and H to a certain degree. (In the matrix-valued case our “almost H∞ ” is equivalent to the Cayley transform of the H2 space over the unit disc. More exact definitions and details are delayed to Section 5.) The most important new contribution of the above theorem is the case J ≥ 0; known smoothness assumptions (for general J) are rather strong, so that part remains mostly an open problem. If ω = 0, then G has a boundary function, denoted again by G, which lies in ∗ ∗ L∞ strong (iR; B(U, Y)). In this case, any spectral factorization f g of G JG determines −1 a J-canonical factorization H := g , h := GH of G. However, the J-canonical

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factorization is a strict generalization of the spectral factorization even in the stable case (it covers unstable factors of a stable function and even singular cost functions; see Lemma 4.1), and yet it has the optimal control properties described above. In the sections to come, we provide some details and known and new results on the above problems (propositions). Much still remains open [WW97] [Sta98] [Mik02] [Mik06b]: Open problems. The following open problems have arisen in control theory: 1) Find further sufficient smoothness conditions for the equivalence in Proposition 1.2. 2) Find conditions under which f is regular (i.e., it has a limit at +∞), or even more smooth. 3) Solve the corresponding unstable canonical factorization problems (see Sections 5–6). Although the stable positive case of Proposition 1.1 is well known and we solve the unstable positive case in Theorem 5.7, the above regularity problem “2” is very useful even in these two settings. The most important problem would be ˆ L + G0 , where G0 ∈ B(U, Y) to establish the factorization in the case where G = G −ω· 1 and e GL ∈ L (R+ ; B(U, Y)), and to show that the factor H (and hence also h) is of the same form in Proposition 1.3. Further possible interesting assumptions and desirable conclusions are explained in Section 6. In Section 2 we present in more detail the stable spectral factorization problem, known results and open problems. We also extend indefinite spectral factorization to convolutions with measures having no singular continuous part. This convolution operator class is popular in, e.g., control theory. In Section 3 we show by an example that the factors need not be bounded nor even proper, even if the factorization problem is the one needed for the standard H∞ control problem. We also treat discrete-time factorization problems (on the unit disc). In Section 4 we present the stable form of the J-canonical problem; this is the generalization of spectral factorization that is needed for control problems, but the two coincide if the latter exists (which happens when, e.g., the system is exponentially detectable). In Section 5 we present the general (= possibly unstable) J-canonical factorization problem. Then we solve it in the uniformly positive case and explain in further detail the relations to control theory. In Section 6 we go through the open problems, possible ways of solving them, reasonable assumptions and known results, including new ones. Most of this article can be read without any knowledge of systems and control theory, to which we refer in some motivating remarks only.

2. Stable spectral factorization We shall now formulate the theory behind Propositions 1.1 and 1.2. More detailed proofs and additional results and information can be found in [Mik02, Chapters

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2–5], particularly in Section 5.2, but most of the results are older, as explained in the notes to this section. To be able to formulate our assumptions and to avoid the technicalities with L∞ strong , we do this in “time domain” instead of its “Laplace transform”, the frequency domain. Indeed, the H∞ functions C+ → B(X) correspond 1-1 to time-invariant, causal operators on L2 (R+ ; X), and similarly, the L∞ strong functions (equivalence classes) iR → B(X) correspond to time-invariant operators on L2 (R; X). Therefore, we shall present the theory in terms of such operators. By π+ f we denote the restriction of a function f to R+ := [0, +∞), by π− f to R− := (−∞, 0), and we set (τ t f )(r) := f (r + t). Definition 2.1 (TI, TIC). We define TI(U, Y) to be the (closed) subspace of operators E ∈ B(L2 (R; U); L2 (R; Y)) that are time-invariant, i.e., τ t E = E τ t for all t ∈ R. We define TIC(U, Y) to be the (closed) subspace of operators D ∈ TI(U, Y) that are causal, i.e., π− Dπ+ = 0, or, equivalently, Dπ+ L2 ⊂ π+ L2 .1 2 2 In TI and TIC we use F the B(L , L ) norm and the adjoint E f, gL2 = f, E gL2 , where f, g := R f g¯ dm. As mentioned above, TIC can also be identified with H∞ : ∗

ˆ For each D ∈ TIC(U, Y) there is a unique Proposition 2.2 (Transfer functions D D). ∞ + ˆ function D ∈ H (C ; B(U, Y)), called the transfer function (or symbol or Laplace / =D Iu transform) of D, such that Du ˆ on C+ for all u ∈ L2 (R+ ; U). The mapping I is an isometric isomorphism onto. D → D I denotes Similarly, TI(U, Y) corresponds to L∞ strong (iR; B(U, Y)). The symbol u the Laplace Transform of u : R → U:  e−st u(t) dt (2.1) u I(s) := R

for s such that e−s· u is an L1 function. Thus, for u ∈ L2 (R+ ; U), the transform converges absolutely for s ∈ C+ . As usual, we extend the definition of u I (a.e.) to the imaginary axis by using the Plancherel transform (which coincides with the nontangential boundary limit a.e.). Thus, u → u I becomes an isomorphism of L2 (R+ ; U) onto H2 (C+ ; U). By G# we denote the subset of invertible operators; e.g., GTIC(U, Y) := {X ∈ TIC(U, Y) # X −1 ∈ TIC(Y, U)} (where X X −1 = IY and X −1 X = IU ). We use the term spectral factorization for a factorization E = Y ∗ X with X , Y ∈ GTIC(U) (i.e., Xˆ , Yˆ ∈ G H∞ (C+ ; B(U))), when E = TI(U). The term canonical factorization is usually used as a generalization of the above, where Xˆ ±1 and Yˆ ±1 are only required to be in certain weighted H2 (in the matrix-valued case). 1 The maps in TIC(U, Y) can be isometrically identified with causal Toeplitz operators (π Dπ ), + + which are exactly the “time-invariant causal” operators L2 (R+ ; U) → L2 (R+ ; Y).

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In the (uniformly) positive case, a spectral factorization always exists: Proposition 2.3 (Positive spectral factorization of TIC). Let E ∈ TI(U). Then E = X ∗ X for some X ∈ GTIC(U) iff E ≥ I for some  > 0, or equivalently, iff π+ E π+ ≥ π+ . (This can be derived from [RR85, pp. 50–54]. See [Sta97, Lemma 18(ii)] or [Mik02, pp. 142–143] for details. Naturally, “E ≥ I on L2 ” means that E u, uL2 ≥ Iu, uL2 for all u ∈ L2 , where I is the identity operator.) By the isomorphisms mentioned above, this proposition is equivalent to the direct generalization of Proposition 1.1: we have Eˆ ≥ I iff Xˆ ∗ Xˆ = Eˆ for some Xˆ ∈ H∞ . The “only if” part of Proposition 1.2 is simple, as shown in (a) below. We also list further useful known properties of the spectral factorization. Lemma 2.4 (SpF ⇒ ∃ ∃Toeplitz−1 ). Let E = Y ∗ X for some X , Y ∈ GTIC(U). Then the following hold: (a) The Toeplitz operator π+ E π+ has the inverse π+ X −1 π+ Y −∗ π+ . (b) (Uniqueness) If E = W ∗ Z , where Z , W ∈ GTIC(U), then Z = EX and W = E −∗ Y for some E ∈ GB(U). (c) If E = E ∗ , then E = X ∗ EX for some E = E ∗ ∈ GB(U). To generalize Proposition 1.2, we next define subclasses of TI that provide sufficient regularity for the spectral factorization even in the indefinite case. By MTI (“M” for measure) we denote the convolutions with measures that consist of an L1 function (absolutely continuous part) plus a discrete part (atoms): 1

MTI, MTId , MTIL ). We defineMTI(U, Y) to be the space of opDefinition 2.5 (MTI, 2 erators E ∈ TI(U, Y) of the form E u = (f + ∞ k=0 Ek δtk ) ∗ u (u ∈ L (R; U)), i.e.,  ∞ ∞
(E u)(t) = Ek u(t − tk ) + f (t − r)u(r) dr, (2.2) −∞

k=0

where f ∈ L1 (R; B(U, Y)), Ek ∈ B(U, Y), tk ∈ R for all k, and the “uniform total variation norm” ∞
E MTI := f L1 + Ek B (2.3) k=0

is finite. Here δt ∗ := δ0 (· − t)∗ = τ −t is the delay of time t ∈ R. If f = 0, we write E ∈ MTId (“d” for discrete). We say that E ∈ MTI belongs 1 to the Wiener class (E ∈ MTIL ) if tk = 0 for all k, i.e., if E has no delays. For the causal versions (f = π+ f and tk ≥ 0 for all k) of these spaces, we add the letter C at the end: 1

1

MTIC := MTI∩TIC, MTICL := MTIL ∩TIC, and MTICd = MTId ∩TIC. (2.4)

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Thus, operators E ∈ MTIL (U, Y) are the ones of form E u = f ∗ u + Eu, 1 E ∈ B(U, Y). Obviously, MTI(U), MTIL (U) and MTId (U) are Banach algebras. These classes are also closed under inverses (in TI). 1 The class MTIL (or its Laplace transform) provides sufficient smoothness for Proposition 1.2. The same holds for general MTI maps at least in the finitedimensional case: 1

Theorem 2.6 (MTI SpF). Let A stand for MTIL or for MTId (or for MTI and assume that dim U < ∞). Let E ∈ A(U). Then the Toeplitz operator π+ E π+ ∈ B(L2 (R+ ; U)) is invertible iff E has a spectral factorization E = Y ∗ X , where X , Y ∈ GTIC(U). Moreover, if X , Y are as above, then X

±1

,Y

±1

(2.5)

∈ A(U).

Since here Eˆ, Xˆ ±1 , Yˆ ±1 are continuous on C+ , we could easily write the above result in the frequency domain (with Eˆ(ir) = Yˆ (ir)∗ Xˆ (ir) for all r ∈ R). The assumption dim U < ∞ can be removed at least when E ≥ 0 or the delays of E are equally spaced, i.e., when there is T > 0 such that tk is a multiple of T for every k ∈ N (also such a property is inherited by X ±1 and Y ±1 ). Other properties preserved in the factorization include “exponential stability” #(both in the TI sense and in the MTI sense) and the smallest group containing {tk # k ∈ Z}; the details are given in [Mik02, Section 5.2]. As Theorem 3.3 below shows, A cannot be replaced by TIC (in Theorem 2.6) unless we allow for X , Y ∈ TIC, as in Proposition 3.1 below. Proof of Theorem 2.6. “If” follows from Lemma 2.4(a), so we assume that π+ E π+ is invertible. The case A = MTId follows from [BKS02, Theorem 5.16], so we are 1 left with MTIL and MTI. 1 1◦ Case A = MTIL . Let C stand for the Cayley transform s → 1−s 1+s . One 1

can show that CLMTIL , the Cayley transform of the Laplace transform (symbol) 1 of MTIL is a “decomposable algebra”. Most of this is proved in [GL73b, Theorem 4.3] and [CG81, pp. 62–63] for the scalar case, and the general case is analogous. 1 The rational functions are dense in CLMTIL , and functions holomorphic on a 1 neighborhood of ∂D are contained in CLMTIL . It follows from [Gri93, Theorem 5] that the class contains its inverses. The invertibility of the Toeplitz operator of E implies the invertibility of the discrete-time Toeplitz operator of C Eˆ (in particular, it is Fredholm). This and the 1 above properties of CLMTIL allow us to apply [GL73a, Theorem 2] to establish a 1 factorization in CLMTIL . The invertibility of the Toeplitz operator implies that the indices of the factorization are zero, i.e., that it is a spectral factorization 1 (inverse Cayley and Laplace transform it back to MTIL ). ◦ 2 Case A = MTI. First use the MTId case and Lemma 2.7(a) to observe that the discrete part Ed of E has a spectral factorization W ∗ Z . Since L1 ∗ is

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an ideal of MTI, we have W −∗ (f ∗)Z −1 = g∗ for some g ∈ L1 (R; B(U)). Therefore, W −∗ E Z −1 = W −∗ (Ed + f ∗)Z −1 = I + g∗. One easily verifies that also 1 π+ W −∗ E Z −1 π+ is invertible, hence I +g∗ = Y ∗ X for some X , Y ∈ MTICL (U), by 1◦ above. Thus, E = (Y W )∗ (X Z ). Remark: If there is T > 0 such that tk = nk T, nk ∈ Z for all k, then we can reduce the MTId case to factorization over 1 . Using the methods of 1◦ , this “equallyspaced delays” case can be settled without assuming that dim U < ∞.
The above proof was based on the fact that the invertibility of the Toeplitz operator of E ∈ MTI implies that of its discrete part (hence allowing us to factor 1 the discrete (MTId ) and Wiener class (MTIL ) parts separately): ∞ Lemma 2.7 (MTI Toeplitz). Let E ∈ MTI(U). Set Ed := k=0 Ek δtk , so that E = Ed + f ∗, where Ek , tk , f are as in (2.2). Then the following hold: (a) If E ∈ MTI(U, Y) and π+ E π+ ∈ GB(L2 ), then π+ Ed π+ ∈ GB(L2 ). (b) If E ∈ MTI(U) and E ≥ 0, then Ed ≥ 0. (Here GB(L2 ) := GB(L2 (R+ ; U), L2 (R+ ; Y)). Naturally, it can be nonempty only if dim U = dim Y, so that essentially U = Y in (a).) Proof. (a) We will prove that if the Toeplitz operator TE := π+ E π+ with E ∈ MTI is coercive, i.e., TE u ≥ u for all u ∈ π+ L2 , then so is TEd (with the same  > 0). Claim (a) follows from this, because T is invertible iff T and T ∗ are coercive. Define F ∈ MTId and f ∈ L1 by F := Ed and f ∗ := E − F (so that E = F u + f ∗ u for all u ∈ L2 ). Let δ > 0 be arbitrary. Let also u be an arbitrary unit vector in L2 (R+ ; U). By the Riemann-Lebesgue Lemma there is Tδ > 0 such that T > Tδ =⇒ fI(·)I u(· − iT )2 < δ.

(2.6)

Because Fˆ (i·) is almost periodic ([Mik02, Lemma C.1.2(h2)]), there is T > Tδ such that Fˆ (it) − Fˆ (i(t − T )) < δ for all t ∈ R. Therefore (recall that iT · π/ I := πO / u2 = u2 = 1 and +u + u, hence π +  = π+  = 1; note also that e iT · I(· − iT )) L(e u) = u π+ F u2 = π+ eiT · F u2  = π/ + (F u)(· − iT )2

iT · = π/ F u)2 + L(e

ˆ = π/ u(· − iT )2 + F (· − iT )I

ˆ I(· − iT )2 − π/ ˆ ˆ u(· − iT )2 ≥ π/ +F u + [F (· − iT ) − F ]I ˆ I(· − iT )2 − δI ≥ π/ u2 +F u ˆI(· − iT )2 − π/ II(· − iT )2 − δ ≥ π/ +E u +f u ˆI(· − iT )2 − δ − δ ≥ π/ +E u

= π+ E eiT · u2 − 2δ

= π+ E π+ eiT · u2 − 2δ

=  − 2δ.

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√ (A pedantic reader is invited to add the missing 2π’s. Note also that Fˆ u I(· −  iT ) := Fˆ (·)I u(· − iT ) etc. (not (F u)(· − iT )).) Because δ > 0 was arbitrary, this implies that π+ F u2 ≥  = u2 . (b) The proof is similar to that of (a) and hence omitted (see [Mik02, Lemma 5.2.3(c)]).
In the positive case, Proposition 2.3 guarantees us the existence of a spectral factorization. However, it is important to know that the factors share the MTI smoothness of E . That is the new contribution of the next theorem (which follows from Theorem 2.6 and Proposition 2.3 except for the fact that the assumption dim U < ∞ is no longer needed): Theorem 2.8 (Positive MTI SpF). Let A be one of the classes TI, MTI, MTId , 1 MTIL . Let E ∈ A(U). Then E ≥ I for some  > 0 iff E has a spectral factorization E = X ∗ X , where X ∈ GTIC(U).

(2.7)

Moreover, if X is as above, then X ±1 ∈ A(U). Note that if E ≥ 0, then E ≥ I for some  > 0 iff E is invertible; or equivalently, iff π+ E π+ is invertible. [Sta98] Proof of Theorem 2.8. Since Proposition 2.3 covers “if” and the case A = TI, we assume that E ≥ I for some  > 0 and that A ⊂ MTI. Then E = Z ∗ Z for some Z ∈ GTIC, by Proposition 2.3. By Lemma 2.4(a), π+ E π+ is invertible, and Z = LX ∈ A for some L ∈ GB(U), where E = Y ∗ X is the factorization given by Theorem 2.6, if dim U < ∞. The reference to [BKS02] in the proof of Theorem 2.6 required that dim U < ∞. To overcome this, use instead [BR86, Theorem I], which covers the case where I − E TI < 1; for general E ≥ I, we have I − E /E TI TI < 1.
As mentioned above and shown in Section 3, the invertibility of the Popov Toeplitz operator does not always (in the non-positive case) guarantee the existence of a (stably invertible) spectral factorization. Therefore, we are left with several open problems: Remark 2.9 (Open problems). From the control-theoretic point of view, there are still open problems even in this stable case. In the positive case (D ∗ JD ≥ I), we want to find further assumptions of D that would guarantee at least the regularity of X . The same holds in the indefinite case, where we are also interested in further assumptions that would guarantee that X ∈ GTIC (or at least X ∈ GTIC∞ , see Definition 4.2). Thus, the open problems are mostly the same as in the unstable case (Section 6).

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Notes for Section 2 More details on the results and concepts in this section can be found in [Mik02]. Proposition 2.2 and Lemma 2.4 are well known [Sta98]. 1

The factorization results for MTICL are due to Israel Gohberg and Yuri Leiterer, the MTId results and their history can be found in [BKS02]. The results for MTI were obtained in [Mik02, Section 5.2] using those mentioned above. As such they are new, but Theorem 2.8 was already proved in [Win89], under the assumptions that dim U < ∞ and tk = kT for each k, for some T > 0 (“equallyspaced delays”). The basic properties of the MTI classes are treated in [Mik02, Sections 2.6, 4.1 and D.1], but parts of them can be found in numerous sources. Theorem 2.6 holds, mutatis mutandis, also for the class of H∞ maps of the I where D ∈ B(U, Y) and fIu0 ∈ H2 (C+ ; Y) (u0 ∈ U) for some ω < 0, form Dˆ = D + f, ω by [Mik02, Theorem 8.4.9]. The application of spectral factorization to optimal control theory is widespread. Examples close to our point of view include [Win89], [Sta97], [WW97], [CW99], [Sta98] and [Mik02]. As shown in [Mik02, Section 9.14] (or [Mik07b]), also unstable canonical factors could be applied to the same problems. Such factors have already been applied in other areas. We present such factorization in the following sections and extend it to cover unstable (not strong-L∞ ) factorands or non-H∞ transfer functions. ˆ∗ ˆ If U is nonseparable, then elements of L∞ strong , e.g., X X and E , are considered equal iff Xˆ ∗ Xˆ u, v = E u, væ on iR for every u, v ∈ H2 . If U is separable, then this is equivalent to Xˆ (ir)∗ Xˆ (ir) = Eˆ(ir) being true for a.e. r ∈ R. Further details can be found in [Mik02, Chapter 3] or in [Mik08].

3. Unstable factors and discrete-time setting The spectral or canonical factorization theory has a discrete-time analogue, which we briefly present in this section. We also present an example where the factor is unstable (Xˆ ∈ H∞ ) even though the Toeplitz operator of the factorand is invertible. Some proofs can be more easily handled in the discrete-time setting, but the continuous-time well-posedness criterion and applications make the continuoustime setting more natural in the other sections. In control theory, we are usually only interested in the spectral factorization of the Popov operator E := D ∗ JD (for some “input/output map” D ∈ TIC(U, Y), J = J ∗ ∈ B(Y)), which naturally arises F ∞ in cost-minimizing or minimax control problems. (Here we refer to the cost 0 y(t), Jy(t) dt, where y is the output of a realization of D; cf. Example 5.10.) However, any self-adjoint E ∈ TI(U) can be written in this form (choose ∗ 2 r2 > E  and X ∈ GTIC(U) J X Ksuch that X X =Jr−II 0−K E (Proposition 2.3), and then set Y := U × U, D := rI ∈ TIC(U, Y), J := 0 I ).

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# + Set D := {z ∈ C # |z| < 1}. The Cayley transform s → 1−s 1+s maps C → D and iR ∪ {∞} → ∂D, bijectively. Through this map one can move the spectral factorization problem from iR to the unit circle ∂D, as explained below. This latter discrete-time setting is analogous to the continuous-time setting explained U), L2 (iR; U) above: just replace H∞ (C+ ; ∗) by H∞ (D; ∗), L2 (R± ; U) by 2 (Z± ; 2 n by L (∂D; U), the Laplace transform by the Z-transform u I(z) := n un z etc. 2 Here Z+ := N := {0, 1, 2, . . .}, Z− := {−1, −2, −3, . . .}, and  (Z+ ; U) stands for square-summable functions Z+ → U. For clarity, we denote the orthogonal projection 2 (Z; U) → 2 (N; U) by π + and the corresponding orthogonal projection F 2π iθ 2 + . Here u2 L2 (∂D; U) → H2 (D; U) by π/ H2 := supr<1 0 u(re ) dθ. In this discrete-time setting, the following result is well known. Proposition 3.1 (H2 canonical factorization). Let Dˆ ∈ H∞ (D; B(Cn , Y)) and J = J ∗ ∈ B(Y). Then the discrete-time Toeplitz operator π + D ∗ JDπ + is (boundedly) invertible on 2 (N; Cn ) iff Dˆ ∗ J Dˆ = Xˆ ∗ S Xˆ for some Xˆ ∈ GH2 (D; Cn×n ), S = S ∗ ∈ GB(Cn×n ) such that X −1 π + X is bounded on E2 (N; Cn ). Moreover, we can D 0 for some m ≤ n. choose X and S so that, in addition, S = I0m −In−m (See [CG81, Theorem VII.3.2 & Proposition VIII.4.3] or [LS87] for the proof and details. All such canonical factorizations are parameterized by XˆE∗ SE XˆE , where XˆE = E Xˆ , SE = E −∗ SE ∗ for some E ∈ GCn×n .) A similar result also holds in the general case (with U in place of Cn ); the details are given in [Mik02, p. 542]. If Dˆ ∗ J Dˆ is positive, then Xˆ ∈ G H∞ , by Proposition 2.3, but that is not the case in general, by Example 3.2 below. Example 3.2. (Unbounded SpF on the unit disc) Let J := diag(1, 1, −2). By Proposition 2.3 (and the Cayley transform), there is h ∈ G H∞ (D) such that |h| equals √ 1/2 on the left hemicircle and 3/2 on the right hemicircle. Set ⎡ ⎤ ih h (3.1) Dˆ := ⎣h(−·) h(−·)⎦ ∈ H∞ (D; C3×2 ). Then 0 1   1 f ∗ ˆ ˆ ˆ ∈ L∞ (∂D; C2×2 ), (3.2) E := D J D = ¯ f −1 where f ∈ L∞ (∂D; C) assumes exactly two values, fr := 14 − i 34 on the right hemicircle and fl := 34 − i 14 on the left hemicircle. −1 ˆ Therefore, Eˆ11 = 1 > 0 and Eˆ22 − Eˆ21 Eˆ11 E12 = −1−|f |2 ≤ − for some  > 0. ˆ The Schur decomposition shows that E is invertible on H2 , hence, by Proposition 3.1, there exists a unique (modulo a unit constant) Xˆ ∈ GH2 (∂D; C2×2 ) such J 0 K Xˆ = Eˆ a.e. on ∂D. that Xˆ ∗ I0 −I Let Er and El be the two values of Eˆ. Since the eigenvalues of Er−1 El are not positive (not even real), we have Xˆ ∈ G H∞ (D; C2×2 ) (see [LS87]); actually, Xˆ , Xˆ −1 are unbounded at ±i but holomorphic elsewhere on D.

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Note. These Dˆ and J correspond to the standard full-information H∞ control ˆ problem setting. √ (We can also have J equal to diag(1, 1, −1) by multiplying D by diag(1, 1, − 2).) Proof. One easily verifies that the eigenvalues of   −1 fr − fl −1 − fr f¯l (3.3) Er−1 El = f¯l − f¯r −1 − f¯r fl 1 + |fr |2 9 are given by λ± := Re t ± (Re t)2 − 4(|t|2 + |s|2 ), where t := −1 − fr f¯l , s := fr − fl , hence these values are not real. If we had Xˆ ∈ G H∞ , then the factorization would exist in all Lp spaces; however, it does not exist for p = 2π/ arg λ± (since λ± are not real; see [LS87] for details). We conclude that there is z0 ∈ ∂D such that Xˆ or Xˆ −1 is not essentially bounded on any neighborhood of z0 (because ∂D is compact). Since Xˆ ∗ = D E D E 0 0 and Eˆ−1 Xˆ ∗ 10 −1 = Xˆ −1 a.e. on ∂D, neither of Xˆ and Xˆ −1 can EˆXˆ −1 10 −1 be bounded on any neighborhood of z0 (note that Eˆ−1 ∈ L∞ ). The local holomorphicity of Eˆ is inherited by Xˆ (see [LS87] or [Mik02, Lemma 9.15.5]), hence z0 = ±i; by symmetry (we have Eˆ(s) = Eˆ(−s) for s ∈ ∂D, hence Xˆ (s) = Xˆ (−s) for s ∈ D, by the Poisson Formula), Xˆ must be essentially unbounded on each neighborhood of both +i and −i. (Similarly, h is locally holomorphic on ∂D \ {±i}, so no “a.e.” is needed in this example.)
Through the Cayley transform, we obtain a version of Proposition 3.1 for Dˆ ∈ H (C+ ; B(Cn , Y)), with the exception that the Cayley transform of H2 (D; Cn×n ) equals the “weighted H2 space” of functions Xˆ : C+ → Cn×n ) such that (1 + 2 + n×n ). This follows from the fact that the Toeplitz operas)−1 Xˆ ( 1−s 1+s ) ∈ H (C ; C tor π+ E π+ is invertible iff the discrete-time Toeplitz operator π + F π + is invertible, + ˆ∗ ˆ ˆ∗ ˆ where Fˆ (z) := Eˆ( 1−z 1+z ). Also the factorization D J D = X S X in this C version is often called a canonical factorization, and that is what we do in Theorem 3.3 below. Proposition 3.1 and similar results have applications in the discrete-time control theory analogous to their above counterparts in continuous-time theory. The main difference is that in discrete time well-posedness and mostly also regularity become redundant, hence we are usually left just with the question whether the factor Xˆ and its inverse Xˆ −1 are bounded (i.e., Xˆ , Xˆ −1 ∈ H∞ (D; ∗)). A partial solution to this is that if Dˆ is holomorphic on a neighborhood of the closed unit disc D, then so is Xˆ . Next we record an important continuous-time version of Example 3.2. We observe that (even for the Popov function Eˆ of the “standard full-information H∞ control problem”) the canonical factorization of (the continuous-time variant of) Proposition 3.1 may be (a) proper but unbounded, and (b) not even proper. ∞

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Theorem 3.3 (Unbounded factor X ). There exists D ∈ TIC(C2D, C3 ) with E the fol0 lowing properties: The map E := D ∗ JD ∈ TI(C2 ), where J = I2×2 0 −1 , has an invertible Toeplitz operator π + E π + . Moreover, D is “minimax J-coercive”, i.e., −1 T11 ≥ I and T22 − T21 T11 T12 ≤ −I for some  > 0, where T := π+ E π+ . NeverE D −1 2 0 Xˆ is the canonical factorization of ∈ TIC(C ), where Xˆ ∗ 10 −1 theless, X , X Eˆ ∈ L∞ (iR; B(C2 )). Moreover, we can, in addition, choose D so that any one of the following holds: (a) Xˆ and Xˆ −1 are both bounded on any smaller right half-plane C+ ω, ω > 0 ˆ (and holomorphic at infinity, hence regular, and so is D); ˆ converges (b) Xˆ and Xˆ −1 are both unbounded on any right half-plane (and D(s) uniformly as Re s → +∞). Proof. We obtain (a) by taking the Cayley transform of Example 3.2. If we rotate ˆ Eˆ, Xˆ by 90 degrees before transforming, then one discontinuity is moved to D, ˆ −1 (and then Cayley transformed to ∞), so that we obtain (b) (replace D(s) by −s ˆ ∗ ˆ ˆ ˆ e D(s) to make D(+∞) = 0 while having D J D unchanged on iR).
D E D E 0 E = 1 0 .) (Note that Xˆ is unique modulo E ∈ GB(C2 ) such that E ∗ 10 −1 0 −1 Even this canonical factorization (with possibly unbounded Xˆ and Xˆ −1 ) can be used to solve the H∞ optimal control problem as long as Xˆ , Xˆ −1 are bounded on a right half-plane (then the feedback loop remains well posed although unstable) – if not, then the optimal state feedback loop becomes ill posed. This is the difference between (a) and (b) above. Indeed, as shown in [Mik02, Example 11.3.7(a)] (using the option (b) above), there is a strongly stable well-posed linear system (“Salamon–Weiss semigroup control system”) whose transfer function is Dˆ and for which there exists a H∞ suboptimal control but the (J-optimal) “minimax control” cannot be written in (well-posed) state-feedback form. That cannot happen when the transfer function is rational or when the system is exponentially stable and has a bounded input operator. A third sufficient condition is that the input map D is in MTIC, by Theorem 2.6. In [Mik02, Example 11.3.7(b)] (which uses the option (a) above), the situation is similar except that the minimax control can be produced by state feedback, which in turn can be computed from the solution of the corresponding Riccati equation (due to the regularity of the factors). However, in that case the feedback loop is unstable, unlike in the three cases mentioned in the above paragraph.

4. Stable J -canonical factorization As we saw in the previous section, the spectral factorization is not sufficient to cover all the situations corresponding to the well-posed optimal state feedback. Therefore, it is often replaced by a more general factorization. To that end, in

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this section we define the “proper J-canonical factorization” (for D ∈ TIC; see Section 5 for unstable D). It differs from the “canonical factorization” of the previous section in the sense that the factor and its inverse have to be proper, thus guaranteeing well-posedness in applications. We first note that a spectral factorization D ∗ JD = X ∗ SX can also be written as an “inner-outer” factorization D = N M −1 : Lemma 4.1. Let D ∈ TIC(U, Y), J = J ∗ ∈ B(Y). (a) If D ∗ JD = X ∗ Y , where Y , X , X −1 ∈ TIC(U), then M := X −1 and N := DM satisfy M , M −1 , N ∈ TIC, D = N M −1 , and N ∗ JN = S for some S = S ∗ ∈ B(U). (b) Conversely, if D = N M −1 , M , M −1 , N ∈ TIC, and N ∗ JN = S ∈ B(U), then X := M −1 and Y := SX satisfy D ∗ JD = X ∗ Y and Y , X , X −1 ∈ TIC(U). (Note that here Y need not be invertible. If D ∗ JD ≥ I, then we can have S = I, by Proposition 2.3, hence then N is “J-inner”.) Proof. (a) Now S := N ∗ JN = X −∗ X ∗ Y X −1 = Y X −1 ∈ TIC. But S = S ∗ , hence S ∈ B(U), by [Sta97, Lemma 6]. (b) This is trivial.
Unfortunately, such a factorization need not always exist even if the Popov Toeplitz operator π+ D ∗ JDπ+ is invertible, by Theorem 3.3. Nevertheless, through the Cayley transform we obtain a factorization where Xˆ ±1 are weighted H2 in certain sense. For this to correspond to a well-posed state-feedback in control applications, we must have Xˆ ±1 bounded on some right half-plane C+ ω , i.e., X ±1 ∈ TICω , for some ω ∈ R, where TICω stands for time-invariant causal maps L2ω → L2ω : # Definition 4.2 (TICω , TIC∞ , L2c , L2ω ). Let ω ∈ R. Set# L2c := {u ∈ L2 # u has a compact support}, L2ω (R; U) := eω· L2 (R; U) := {eω· f # f ∈ L2 (R; U)}. We define TICω (U, Y) to be the (closed) subspace of operators D ∈ B(L2ω (R; U); L2ω (R; Y)) for which τ t D = Dτ t for all t ∈ R and π− Dπ+ = 0. We set TIC∞ := ∪ω∈R TICω . By identifying maps that coincide on the (dense) subset of functions having a compact support, one can show that # TICω (U, Y) ⊂ TICα (U, Y) whenever ω < α. Moreover, TICω (U, Y) = {eω· De−ω· # D ∈ TIC(U, Y)}. As in Proposition 2.2, we ∞ + ˆ observe that TICω corresponds to H∞ ω (U, Y) := H (Cω ; B(U, Y)) (through D → D), hence TIC∞ corresponds isometrically to the set of transfer functions bounded on some right half-plane. Recall that such functions are called proper; they are ∞ denoted by H∞ ∞ := ∪ω∈R Hω (we identify analytic functions on a half-plane with their restrictions). Note also that L2ω equals the L2 space with the weight function e−2ω· . As explained above, the “canonical factorization” of Proposition 3.1 exists iff the Popov Toeplitz operator π+ D ∗ JDπ+ is invertible, and this factorization leads

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to well-posed optimal state feedback in control applications iff the factor and its inverse are proper. We formulate this “proper canonical factorization” below. Our definition is a generalization of the “(J, S)-inner-outer” factorization of Lemma 4.1. When D ∈ TIC, it can be defined as follows: Definition 4.3. Let D ∈ TIC(U, Y), J = J ∗ ∈ B(Y). We call D = N M −1 a proper J-canonical factorization if M ∈ GTIC∞ , N = DM , M π+ M −1 is bounded on L2 , and there exists S ∈ B(U) such that N ∗ JN = S on L2c . By “N ∗ JN = S on L2c ” we mean that N u, JN vL2 = u, SvL2 (u, v ∈ It obviously follows that S = S ∗ (and N ∗ JN = S if N ∈ TIC). By “M π+ M −1 is bounded on L2 ” we mean that M π+ M −1 (originally defined, e.g., on L2ω for some ω ≥ 0) coincides with an element of B(L2 ). It follows 2 2 2 N L2c ⊂ L2 . By the closed-graph theorem, the map that J N K M2Lc ⊂ L , hence also 2 M : L ([−T, T ]; U) → L is continuous for any T > 0. The condition “N ∗ JN = S on L2c ” in Definition 4.3 can be rewritten in the “standard form”: L2c ).

Lemma 4.4. Let N ∈ TIC∞ (U, Y), J ∈ B(Y), S ∈ B(U) and N L2c ⊂ L2 . Then N ∗ JN = S on L2c iff



v0 = u0 , Sv0 a.e. on iR for every u0 , v0 ∈ U. N u0 , J N

(4.1)

If dim U < ∞ then a boundary function Nˆ in the weighted L2 (iR; B(U, Y)) exists (this is not so in general [Mik06a, Section 4]), and hence then (4.1) can be written as “Nˆ∗ J Nˆ = S a.e. on iR”, or equivalently, as Dˆ ∗ J Dˆ = Xˆ ∗ S Xˆ , where X := M −1 . Thus, Definition 4.3 coincides with the standard definition of a canonical factorization (if S ∈ GB(U)) with the exception that we require that Xˆ ±1 is bounded on some right half-plane; cf. Proposition 3.1 and the comments below it. Proof of Lemma 4.4. By the Plancherel Theorem, (4.1) holds iff N u, JN vL2 = u, SvL2 whenever u = f u0 , v = gu0 for some characteristic functions f and g of finite-measurable subsets of R+ (because the boundary function of NˆfIu0 = fINˆu0

at ir equals fI(ir) times the boundary function N u0 of Nˆu0 ). By time-invariance, linearity and density, this then holds for all u, v ∈ L2c (R+ ; U).
See Propositions 5.8 and 5.9 for the uniqueness, existence and importance of the proper J-canonical factorization. If u ∈ L2 ([−T, 0]; U), then M u = (M π− M −1 )π− M u ∈ L2 , because M u ∈ L2ω for some ω ≥ 0, π− L2ω ⊂ L2 , and M π− M −1 = I − M π+ M −1 ∈ B(L2 ). Use translation-invariance for general u ∈ L2c .

2 Proof:

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Notes for Section 4 Lemma 4.1 is close to [Sta98, Lemma 4.6], where X ±1 ∈ TIC(U), S ±1 ∈ B(U). In the standard (matrix-valued) definition of the factorization, the condition “N ∗ JN = S on L2c ” is usually replaced by “Eˆ = Xˆ ∗ S Xˆ ” (where Eˆ = Dˆ ∗ J Dˆ in our setting; cf. Lemma 4.4) or similar, and also non-self-adjoint factorands are treated. Moreover, the requirement M ∈ GTIC∞ is usually replaced by the assumption that Mˆ is holomorphic on C+ ; the boundedness of M π+ M −1 then leads to Mˆ and Xˆ being weighted-H2 . However, in this operator-valued setting, and also from the point of view of applications, our definition seems more appropriate.

5. Unstable canonical factorization In some control problems the input/output map D or the transfer function Dˆ of the control system is stable (Dˆ ∈ H∞ ). Even in unstable settings, the transfer function must at least be “stabilizable”; otherwise typical control problems have no solutions. We define stabilizability in Lemma 5.1 below, where five equivalent characterizations are given. One of them involves “coprime factorizations” (w.r.c.f.’s), which take the role of the “inner-outer factorizations” of Lemma 4.1 in the general case. In this section we also generalize “proper J-canonical factorization” to unstable maps. We then show that a positive factorization problem has a solution iff the map involved is stabilizable. We then end this section by outlining some control-theoretic applications of proper J-canonical factorization. Throughout this section we assume that D ∈ TIC∞ (U, Y) and J = J ∗ ∈ B(Y) (outside Example 5.10). For such a general D the formulation of a proper Jcanonical factorization is slightly different from Definition 4.3 (although equivalent when D ∈ TIC := TIC0 ). As noted below that definition, there cannot be such a factorization unless D satisfies the condition (ii) below. Here we give several equivalent characterizations of this important “stabilizability” condition (see further below for the definitions of (iii) and (v)). Lemma 5.1 (Stabilizable D). The following are equivalent for any D ∈ TIC∞ (U, Y): (i) D = N M −1 for some N , M ∈ TIC (such thatJ M K−1 ∈ TIC∞ (U)). 2 2 (ii) D = N M −1 where N , M , M −1 ∈ TIC∞ and N M Lc ⊂ L . (iii) D has a normalized w.r.c.f. (iv) There exists ω ∈ R such that for each v ∈ L2ω (R− ; U) there exists u ∈ L2 (R+ ; U) such that π+ D(v + u) ∈ L2 . (v) D has a stabilizable realization. If (i) (hence also (ii)–(v)) holds, then we call D stabilizable. If D = N M −1 is a w.r.c.f., then every w.r.c.f. of D is given by D = (N V )(M V )−1 (V ∈ GTIC(U)). The proof is given in [Mik07a]; part (v) is explained in, e.g., [Mik02], [Sta05] or [Mik07a].

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A factorization J K of the2 form (i) is 2called a w.r.c.f. 2(a weakly coprime right factorization) Dif N M E u ∈ L =⇒ u ∈ L for every u ∈ Lloc (R+ ; U). (By [Mik07a], ˆ N this holds iff Mˆ f ∈ H∞ =⇒ f ∈ H∞ for every f ∈ H∞ ∞ (U). This is often J N K∗ J N K ˆ ˆ written as gcd(N , M ) = I.) It is called normalized if M = I on L2 . If M Nˆ, Mˆ ∈ H∞ are rational, then they are weakly coprime iff they have no common zeros on C+ ∪ {∞}. Below we shall see that in this unstable case w.r.c.f.’s take the role of the “inner-outer factorizations” of Lemma 4.1. Without stabilizability, no typical control problems for D have any solutions. Obviously, any D ∈ TIC is stabilizable (take M = I). ˆ Example. The function D(s) = (s − 1)−1/2 satisfies D ∈ TIC∞ (C) but it cannot be stabilizable, because it is not meromorphic at 1 (indeed, by (i) we should have Dˆ = Nˆ/Mˆ, where Nˆ, Mˆ ∈ H∞ (C)). 2 In Definition 4.3 we required M π+ M −1 to be bounded # on L 2. When D is 2 2 2# not bounded on L , we must replace L by UR := {u ∈ L Du ∈ L }, where we use the natural extension of D defined below. This space UJR will precisely K 2be more 2 defined below. For clarity, we also make the requirement N L ⊂ L explicit in c M the following direct generalization of Definition 4.3. −1 J-canonical factorization of D if Definition 5.2. We call D = N J NMK 2 a proper M ∈ GTIC∞ (U), N = DM , M Lc ⊂ L2 , N ∗ JN = S on L2c and M π+ M −1 is bounded UR → UR .

Given any stabilizable D, we define UR ⊂ L2 (R; U) to be the closure of the # space {u ∈ L2ω (R; U) # uUR < ∞} (for some ω ∈ R such that D ∈ TICω ), where u2UR := u22 + Du22 . We extend D to UR by continuity. Obviously, Definitions and K 4.3 are equivalent when D ∈ TIC (below Definitions 5.2 we observed that J5.2 N L2 ⊂ L2 there too). c M We list below some obvious basic properties of UR : Remark 5.3. If D = N1 M1−1 is a [normalized] w.r.c.f., then M1 is an [isometric] isomorphism ofJL2K(R; U) onto UR . Therefore, UR is independent of ω. The map DI is isometric UR → L2 . If D = N M −1 is a proper J-canonical factorization, then M L2c ⊂ UR . Thus, UR = M1 L2 (R; U) is an equivalent definition of UR (for any w.r.c.f. D = N1 M1−1 ). In Theorem 5.7 we shall show that in the “uniformly positive case”, the Jcanonical factorization is always a w.r.c.f. (with N ∗ JN ∈ B(U)). (This does not hold in general, by Theorem 3.3.) From Remark 5.3 we conclude the converse: Corollary 5.4. A w.r.c.f. D = N M −1 is a proper J-canonical factorization iff N ∗ JN ∈ B(U). The proper J-canonical factorization problem for D can reduced to the stable case (to that of N1 ∈ TIC):

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Lemma 5.5. If D = N1 M1−1 is a w.r.c.f., then D has a proper J-canonical factorization iff N1 has a proper J-canonical factorization. Indeed, then N1 = N2 M2−1 is a proper J-canonical factorization for N1 iff D = N2 (M1 M2 )−1 is a proper J-canonical factorization for D. Proof of Lemma 5.5. If N1 = N2 M2−1 is a proper J-canonical factorization of N1 , then M1 M2 π+ M2−1 M1−1 is bounded UR → UR , by Remark 5.3. If D = N M −1 is a proper J-canonical factorization of D and we set M2 := M1−1 M , then N1 = DM1 = N M2−2 and M2 π+ M2−1 = M1−1 M π+ M −1 M1 is bounded L2 → L2 , by Remark 5.3. The other properties are straight-forward (see also Lemma 5.1).
If D ∈ TIC and the Popov Toeplitz operator π+ D ∗ JDπ+ is uniformly positive (≥ I), then D ∗ JD has a spectral factorization X ∗ X (X ∈ GTIC), by Proposition 2.3. In Theorem 5.7 we shall generalize this result to the unstable case where D ∈ TIC. To present that result, we start by defining the corresponding concepts. # Definition 5.6. We equip U := {u ∈ L2 (R+ ; U) # Du ∈ L2 } with the norm u2 := U

u22 + Du22 . We call D J-coercive if D ∗ JD ∈ GB(U), positively J-coercive if, in addition, D ∗ JD ≥ 0.

Obviously, U becomes a Hilbert space and U = UR ∩ L2 (R+ ; U). If D ∈ TIC, then U = L2 (R+ ; U) with an equivalent norm. Therefore, we call D ∗ JD ∈ B(U) the generalized Popov Toeplitz operator (since it equals π+ D ∗ JDπ+ ∈ B(L2 (R+ ; U)) when D ∈ TIC). Thus, J-coercivity means the invertibility of the generalized Popov Toeplitz operator. Note that D is positively J-coercive iff Du, JDuL2 ≥ u2U for some  > 0. J K Example. A map of the “LQR” form D := DI1 (for any D1 ∈ TIC∞ ) is positively I-coercive, although it need not be stabilizable. Now we can generalize Proposition 2.3 to the unstable case: Theorem 5.7. Assume that D ∈ TIC∞ (U, Y) is positively J-coercive. Then D has a proper J-canonical factorization iff D is stabilizable. Moreover, the proper Jcanonical factorization is a w.r.c.f., and we can have S = I. In particular, for this factorization, say D = N M −1 , we have N , M ∈ TIC. See also Corollary 5.4. Proof of Theorem 5.7. By Lemma 5.1, we may assume that D is stabilizable and has a normalized w.r.c.f. D = N1 M1−1 . Obviously, M1 ∈ GB(L2 (R+ ; U), U). From this we conclude that D being positively J-coercive means that π+ N1∗ JN1 π+ ∈ GB(L2 (R+ ; U)), i.e., that N1 is positively J-coercive. By Proposition 2.3 and Lemma 4.1(a), N1∗ JN1 = X ∗ X , where X ∈ GTIC(U), and N1 = N X is a proper J-canonical factorization, hence so is D = N M −1 , where M := M1 X −1 , N := N1 X −1 by Lemma 5.5. Obviously, also N M −1 is a w.r.c.f.
Some indefinite existence results are indicated in Remark 6.4.

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The rest of this section is devoted to the reasons and applications of our definition of proper J-canonical factorization. An important motivation is Theorem 5.7 above. Another one is that this definition is a direct generalization of the case where Dˆ ∈ H∞ is matrix-valued; see, e.g., [CG81] or [LS87], except that we require properness (Mˆ±1 ∈ H∞ ∞ ), which is necessary for the state-feedback to be well posed. (However, everything in this article would remain true if we would remove the properness requirements and allow state-feedback to be improper. Moreover, one can establish an analogous discrete-time theory, and there properness is redundant.) However, the most important justification of this definition is that it is exactly what one needs in the optimization of control systems by (well-posed) state feedback. To briefly illustrate this, we cite below two propositions from [Mik07b] (in fact, the former one is [Mik02, Theorem 9.14.3] slightly modified). These propositions are not applied in this article. The reader interested in the exact definitions of stabilization, Riccati equations or J-optimal state feedback may consult [Mik02] or [Mik06b] (or [Mik07b]). A realization of D means a system (cf. Example 5.10) whose transfer function ˆ Every proper (or H∞ ) function has a Well-Posed Linear System realization is D. ∞ (see, e.g., [Sal89] or [Sta05]; the input and output operators may be unbounded). Proposition 5.8. Let Σ be a realization of D. Assume that D is J-coercive. Then the following are equivalent: (i) Σ is output-stabilizable and D has a proper J-canonical factorization. (ii) There exists a J-optimal state-feedback pair for Σ. (iii) The integral Riccati equation for Σ and J has a U-stabilizing solution. We mention that there is also a constructive connection between the factorization and optimal state feedback [Mik07b] [Mik02]. Proposition 5.9. Assume that D is J-coercive. Then the following are equivalent: (i) D has a proper J-canonical factorization N M −1 . (ii) Some realization of D has a J-optimal state-feedback pair. (iii) D is stabilizable and every stabilizable realization of D has a J-optimal statefeedback pair. Moreover, if (i) holds, then all proper J-canonical factorizations of D are given by D = (N E)(M E)−1 (E ∈ GB(U)), and necessarily S ∈ GB(U). Naturally, without J-coercivity (e.g., when J = 0) the J-canonical factorization may be highly nonunique. Certain conditions on the realization Σ are known to guarantee (i)–(iii) of Proposition 5.8. In quadratic minimization or H∞ control problems, these conditions typically lead to natural extensions of classical solutions, as shown in [Mik02, Chapters 9–12] in a fairly general setting. Such assumptions include having a bounded input operator or a smoothing semigroup (see, e.g., [CZ95], [vK93], [LT00], [CW06] or [Mik02, Section 9.2]). They usually also allow one to use the standard Riccati equation instead of the integral one mentioned in Proposition 5.8.

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To illustrate realizations, state-feedback and optimization for readers not familiar with control theory, we finish this section by presenting the following simple example. Perhaps the most common problem in control theory is to stabilize a system and to minimize the so-called LQR cost function, through state feedback. Theorem 5.7 guarantees the existence of a solution in the state-feedback form, as explained in the example below. Further details are given in [Mik06b] and in [Mik02]. Example 5.10. (a) Let A generate a strongly continuous semigroup on X, and let B ∈ B(U, X), C ∈ B(X, Y) and D ∈ B(U, Y). For the system x˙ = Ax + Bu, y = Cx + Du, x(0) = x0 , assume that for each initial state x0 ∈ X there exists an input u ∈ L2 (R+ ; U) such that y ∈ L2 . (We call y the output.) J K Then the map D ∈ TIC∞ (U, Y × U) determined by D := DI1 , Dˆ1 := D + C(· − A)−1 B is stabilizable and I-coercive, and the cost function  ∞   2 2 y(t)2Y + u(t)2U dt = Du, IDuL2 (5.1) y2 + u2 = 0

has a unique minimizing input u ∈ L2 (R+ ; U) for each initial state x0 ∈ X. Moreover, this input is given by the state-feedback u(t) = F x(t) (t ≥ 0) for some F ∈ B(X, U). The feedback operator F is unique and it can be determined from the standard Riccati equation PBB ∗ P = A∗ P + PA + C ∗ C, P ≥ 0, F = −B ∗ P (slightly modified if D = 0 or J = I). [CZ95] Set Mˆ := I + F (s − A − BF )−1 B, Nˆ := D + C(s − A − BF )−1 B. Then M and N are the closed-loop maps from an external disturbance to the feedback ˆ ˆ ∈ H∞ and they are weakly coprime [Mik06b] loop to u and y. Moreover, M J N, N K [Mik07a]. Furthermore, D = M M −1 is an I-optimal factorization. Conversely, given a proper I-canonical factorization of D we can compute F . (b) Even if we allow B and C to be moderately unbounded, we get results very similar to those in (a), although then the state-feedback operator F may become unbounded and the Riccati equations may become very difficult to solve. (The latter depeneds on how regular the I-canonical factors are.) Moreover, we can use also the more general (positively J-coercive) cost function Du, JD instead of (5.1)= Du, IDuL2 , and D does not have to be of the form [ ∗I ]. (c) In H∞ optimal control J 0 K problems one often uses the indefinite cost function Du, JDu, where J = I0 −I . If B and C are unbounded, then the “J-optimal state-feedback” (that achieves the (minimax) saddle point of the cost function) might be ill posed, by Theorem 3.3(b) (and [Mik02, Example 11.3.7(a)]). (Here Dˆ1 is the transfer function of the system determined by A, B, C, D and Dˆ is the transfer function of the system determined by A, B, [ C0 ] , [ D I ].)

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Notes for Section 5 Some special cases of J-coercivity (such as the invertibility of the standard Popov Toeplitz operator in the stable case) have been used for decades in optimal control problems. Many applications and some historical remarks on the finite-dimensional setting can be found in [IOW99]. The term “J-coercive” is from [Sta98], where a special case was treated. Our generalization is from [Mik02], where (in Chapters 10 and 11, particularly in Section 10.3) it is shown that this covers all classical (non-singular) assumptions and leads to a solution of standard minimization and H∞ problems. Propositions 5.8 and 5.9 are from [Mik07b], but the latter can be deduced from the former (and known results; the uniqueness from [Mik06b, Lemma A.5]), and the former is almost [Mik02, Theorem 8.4.3], where also (iii) is explained. Under some regularity, the integral Riccati equation is equivalent to the standard algebraic Riccati equation. See, e.g., [FLT88], [Win89], [WW97], [CW99], [Sta97], [Sta98] [LT00] or [Mik02] for Example 5.10(b).

6. Open problems & optional assumptions In this section we shall explain in detail the open problems stated in the introduction. That is, we shall specify some possible interesting sufficient assumptions for Propositions 1.2 and 1.3 and explain the regularity and stability properties that one might wish to get for the J-canonical factor M and its inverse M −1 (or for X −1 and X in the stable case of Section 2; cf. Lemma 4.1). We also explain how the problem could be reduced to a coprime factorization problem. Throughout this section we assume that D ∈ TIC∞ (U, Y) (i.e., that Dˆ is holomorphic and bounded on some right half-plane with values in B(U, Y)) and that J = J ∗ ∈ B(Y). In the indefinite case (even stable, by Theorem 3.3(b)), one must assume 1 certain regularity of D (such as D ∈ MTICL ) to guarantee the existence of a proper J-optimal factorization. In the best case we can also guarantee some regularity for the factors (N , M ). Further details are given in Properties 6.2, after which we provide additional information on the assumptions and motivation and a list of known results. However, we first formulate the perhaps most important open special case of J-optimal factorization as a conjecture: Conjecture 6.1. Assume that D ∈ B(U, Y), ω ∈ R and e−ω· f (·) ∈ L1 (R+ ; B(U, Y)). Set Du := Du + f ∗ u (u ∈ L2ω (R; U)). If D is stabilizable and J-coercive, then D has a J-optimal factorization D = N M −1 . Moreover, then N and M are of the same form as D. Obviously, D ∈ TICω (U, Y). For ω = 0 the conjecture is true (use Theorem 2.6 and Lemma 4.1(a)).

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As the case ω = 0 shows, the conjecture might be true even if we allow for delays in D. On the other hand, even the case with the stronger assumption e−ω· f (·) ∈ Lp (R+ ; B(U, Y)) would be interesting, at least if true for all p > 1. A weaker sufficient assumption on D would of course be even more useful (e.g., Lpstrong in place of Lp ). Indeed, in typical control problems, one can often show that the I/O map is stabilizable and J-coercive. The problem is to show that the “J-canonical factorization” is proper (i.e., that M , M −1 ∈ TIC∞ ; see Definitions 4.3 and 5.2; this is always so in the positive case by Theorem 5.7, but not in general) and that the factors (N and) M and M −1 are regular – most preferably highly regular. We explain below two possible interpretations for “regularity” and list the typical assumptions that are often met in applications. Properties 6.2. In many applications, the following smoothness properties may be assumed on D and are desirable for M and M −1 . ˆ (1) (Regularity) lims→+∞, s∈R D(s) exists (or even exists uniformly for s ∈ C when Re s → +∞). (2) Dˆ is the Laplace transform of some weighted Lp function (e.g., Du = eω· f ∗ u, f ∈ Lp (R+ ; B(U, Y))) [plus delays], for p = 1 (or for other p ∈ (0, 2)). (Note that (2.) implies (1.).) Also the stability of the factors (N , M ∈ TIC; perhaps even M −1 ∈ TIC if D ∈ TIC) is highly desirable but not necessary for all applications. By Lemma 5.1, we must assume that D is stabilizable. Also J-coercivity and/or the following properties may be assumed: (α) (Positive case) Du, JDu ≥ 0 ∀u ∈ U. (β) (Stable case) D ∈ TIC. (γ) dim U < ∞ (and dim Y < ∞). Regularity (1.) of the factors, even if true in weak sense, allows one to use the extension of the standard algebraic Riccati equation theory developed in, e.g., [Sta97], [WW97] and [Mik02]. Any smoothness results would also provide useful information on the optimal feedback. In most applications the transfer functions are scalar- or matrix-valued, hence (γ) can be assumed if necessary, but also the operator-valued setting has become increasingly popular in some physical applications. Even in the positive case, not many sufficient conditions for factor regularity are known: Remark 6.3 (Nonnegative case D ∗ JD 0). Assume that D is stabilizable and positively J-coercive. Then a proper J-canonical factorization of D exists; it is characterized in Theorem 5.7. Also for this factorization, any regularity results on the factors would be very useful for the reasons explained J K above. In general, regularity is not preserved even if D ∈ TIC(C, C) or D = DI1 ∈ TIC(C, C2 ), D is positively I-coercive, and J = I (see [WW97, Example 11.5] or [Mik02, Proposition 9.13.1(c1)]).

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From Theorem 3.3(b) we observe that (type (1.)) regularity is not sufficient for the existence of a proper J-canonical factorization in the stable, indefinite J-coercive case. Several sufficient conditions in the stable case were presented in Section 2, but in the indefinite unstable case not much is known (cf. the realizationbased conditions mentioned below Proposition 5.9). The factorization in that case, particularly Conjecture 6.1, is a major open problem. We also mention the following coprime factorization problem, which is an alternative way of finding J-canonical factorizations: Remark 6.4. By Lemmata 5.5 and 5.1, the J-canonical factorization problem can always be reduced to the stable case. Thus, if D is J-coercive and has a w.r.c.f. D = N1 M1−1 such that N1 satisfies the assumptions of, e.g., Theorem 2.6, then we obtain a J-canonical factorization of D (see Lemma 4.1(a)). One sufficient condition for this is that D lies in the Callier–Desoer class [CD80]. Then, actually, we would get a coprime factorization D = N M −1 in the same class, by [Mik02, Theorem 8.4.9(β)] (and [CD80] or [VSF82, Theorem 2.1]), which also provides similar results for some other classes. Also other kind of sufficient conditions for such reducibility would be interesting. Thus, results on [weakly] right coprime factorization within a suitable class of operators lead to results on proper J-canonical factorization.

References [BKS02] Albrecht B¨ ottcher, Yuri I. Karlovich, and Ilya M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Operator Theory: Advances and Applications, vol. 131, Birkh¨ auser Verlag, Basel, 2002. [BR86]

R.G. Babadzhanyan and V.S. Rabinovich, Factorization of almost periodic operator functions, Differential and integral equations and complex analysis (Russian), Kalmytsk. Gos. Univ., Elista, 1986, pp. 13–22, i.

[CD80]

Frank M. Callier and Charles A. Desoer, Stabilization, tracking and disturbance rejection in multivariable convolution systems, Ann. Soc. Sci. Bruxelles S´er. I 94 (1980), no. 1, 7–51.

[CG81]

Kevin F. Clancey and Israel Gohberg, Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, vol. 3, Birkh¨ auser-Verlag, Basel Boston Berlin, 1981.

[CW99] Frank M. Callier and Joseph J. Winkin, The spectral factorization problem for multivariable distributed parameter systems, Integral Equations Operator Theory 34 (1999), no. 3, 270–292. [CW06] Ruth F. Curtain and George Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim. (2006). [CZ95]

Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Springer-Verlag, New York, 1995.

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[FLT88] Franco Flandoli, Irena Lasiecka, and Roberto Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler–Bernoulli boundary control problems, Ann. Mat. Pura Appl. 153 (1988), 307–382. [GKS03] Israel Gohberg, Marinus A. Kaashoek, and Ilya M. Spitkovsky, An overview of matrix factorization theory and operator applications, Factorization and integrable systems (Faro, 2000), Oper. Theory Adv. Appl., vol. 141, Birkh¨auser, Basel, 2003, pp. 1–102. [GL73a] Israel C. Gohberg and Yuri Laiterer, A criterion for factorization of operatorfunctions with respect to a contour, Soviet Math. Dokl. 14 (1973), 425–429. [GL73b]

, The factorization of operator-functions relative to a contour. III. Factorization in algebras. (Russian), Math. Nachr. 55 (1973), 33–61.

[Gri93]

Gustaf Gripenberg, Stability of Volterra equations with measure kernels in Banach spaces, J. Math. Anal. Appl. 178 (1993), 156–164.

[IOW99] Vlad Ionescu, Cristian Oar˘ a, and Martin Weiss, Generalized Riccati theory and robust control. a Popov function approach, John Wiley, New York, London, 1999. [LS87]

Georgii S. Litvinchuk and Ilia M. Spitkovskii, Factorization of measurable matrix functions, Operator Theory: Advances and Applications, vol. 25, Birkh¨ auser Verlag, Basel, 1987, Translated from the Russian by Bernd Luderer, with a foreword by Bernd Silbermann.

[LT00]

Irena Lasiecka and Roberto Triggiani, Control theory for partial differential equations: continuous and approximation theories. I abstract parabolic systems, Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000.

[Mik02] Kalle M. Mikkola, Infinite-dimensional linear systems, optimal control and algebraic Riccati equations, Doctoral dissertation, technical report A452, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 2002, www.math.hut.fi/˜kmikkola/research/. [Mik06a]

, Characterization of transfer functions of Pritchard-Salamon or other realizations with a bounded input or output operator, Integral Equations Operator Theory 54 (2006), no. 3, 427–440.

[Mik06b]

, State-feedback stabilization of well-posed linear systems, Integral Equations Operator Theory 55 (2006), no. 2, 249–271.

[Mik07a]

, Weakly coprime factorization and state-feedback stabilization of discrete-time systems, submitted, 2007.

[Mik07b]

, Indefinite optimal control and well-posed minimizing control, in preparation, 2007.

[Mik08]

, Fourier multipliers for L2 functions with values in nonseparable Hilbert spaces and operator-valued H p boundary functions, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, to appear.

[RR85]

Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford University Press, New York, Oxford, 1985.

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Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory 21 (1989), 147–164.

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[Sta97]

Olof J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc. 349 (1997), 3679–3715. [Sta98] , Feedback representations of critical controls for well-posed linear systems, Internat. J. Robust Nonlinear Control 8 (1998), 1189–1217. [Sta05] , Well-Posed Linear Systems, Encyclopedia Math. Appl., vol. 103, Cambridge University Press, Cambridge, 2005. [vK93] Bert van Keulen, H∞ -control for distributed parameter systems: A state space approach, Birkh¨ auser Verlag, Basel Boston Berlin, 1993. [VSF82] Mathukumalli Vidyasagar, Hans Schneider, and Bruce A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Autom. Control 27 (1982), no. 5, 880–894. [Win89] Joseph Winkin, Spectral factorization and feedback control for infinite-dimensional control systems, Doctoral dissertation, Facult´ees Universitaires NotreDame de la Paix ` a Namur, 1989. [WW97] Martin Weiss and George Weiss, Optimal control of stable weakly regular linear systems, Math. Control Signals Systems 10 (1997), 287–330. Kalle M. Mikkola Helsinki University of Technology, Institute of Mathematics P.O. Box 1100; FIN-02015 HUT, Finland http://www.math.hut.fi/˜kmikkola/ e-mail: [email protected] Ilya M. Spitkovsky College of William and Mary, Department of Mathematics P.O. Box 8795; Williamsburg, VA 23187-8795 U.S.A. http://www.math.wm.edu/˜ilya/ e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 347–353 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Compact Linear Operators Between Probabilistic Normed Spaces Kourosh Nourouzi Abstract. A pair (X, N ) is said to be a probabilistic normed space if X is a real vector space, N is a mapping from X into the set of all distribution functions ( for x ∈ X, the distribution function N (x) is denoted by Nx , and Nx (t) is the value Nx at t ∈ R ) satisfying the following conditions: (N1) Nx (0) = 0, (N2) Nx (t) = 1 for all t > 0 iff x = 0, t ) for all α ∈ R\{0}, (N3) Nαx (t) = Nx ( |α| (N4) Nx+y (s + t) ≥ min{Nx (s), Ny (t)} for all x, y ∈ X, and s, t ∈ R0 + . In this article, we study compact linear operators between probabilistic normed spaces. Mathematics Subject Classification (2000). Primary 47S50; Secondary 46S50. Keywords. Compact operator, P N -space, bounded set, N -compact set, weakly bounded operator, strongly bounded operator.

1. Introduction and preliminaries An interesting and important generalization of the notion of metric space was introduced by Menger [5] under the name of statistical metric space, which is now called probabilistic metric space. Menger showed how one could replace a numerical distance between points p and q by a distribution function Fpq whose value Fpq (x) at the real number x is interpreted as the probability that the distance between p and q is less than x. For more results of probabilistic analysis we refer [5, 6, 7]. An important family of probabilistic metric spaces are probabilistic normed ˇ spaces (briefly, P N -spaces) that were introduced by Serstnev [8] in 1962. Since then in many related articles in a straightforward modification they were also called fuzzy normed spaces. The theory of probabilistic normed spaces is important as a generalization of deterministic results of linear normed spaces. This research was in part supported by a grant from IPM (No. 86470033).

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In the sequel, we shall adopt the usual terminology of the theory of probabilistic normed spaces as in [1, 2, 7]. Throughout the paper, we assume that X is a real vector space, R+ is the set of all positive real numbers, and the symbol ∧ will denote infimum. By a distribution function f : R → R0 + we mean that it is non-decreasing and left-continuous with inf t∈R f (t) = 0, and supt∈R f (t) = 1. Definition 1.1. A mapping N from X into the set of all distribution functions is called an N -norm on X if the following conditions are satisfied for all x, y ∈ X, and c ∈ R: (N1) Nx (0) = 0, (N2) Nx (t) = 1 for all t ∈ R+ if and only if x = 0, t ), for all t ∈ R+ and c = 0, (N3) Ncx (t) = Nx ( |c| (N4) Nx+y (s + t) ≥ min{Nx (s), Ny (t)}, for all s, t ∈ R. Then (X, N ) is said to be a P N -space. Theorem 1.2. [3] Let (X, N ) be a P N -space, and that (N5) Nx (t) > 0, for all t ∈ R+ implies that x = 0. If α ∈ (0, 1), define xα = ∧{t ∈ R+ : Nx (t) ≥ α}. Then { α : α ∈ (0, 1)} is an ascending family of norms on X which is called the family of α–norms on X corresponding to the probabilistic norm N on X. Definition 1.3. Let (X, N ) be a P N -space, and {xn } be a sequence in X. The N sequence {xn } is said to be N -convergent to x ∈ X and denoted by xn −→ x or xn → x if limn→∞ Nxn −x (t) = 1, for all t > 0. Definition 1.4. [4] Let T : (X, N ) → (Y, N  ) be a linear operator, where (X, N ) and (Y, N  ) are P N -spaces. The operator T is called: 1. Weakly continuous at z ∈ X if for any  > 0, and α ∈ (0, 1) there exists δ > 0 such that for all x ∈ X if Nx−z (δ) ≥ α, then NT (x)−T (z) () ≥ α. The operator T is said to be weakly continuous on X if T is weakly continuous at each point of X. 2. Weakly bounded on X if for every α ∈ (0, 1) there exists mα > 0 such that if Nx ( mtα ) ≥ α, then NT (x) (t) ≥ α, for all x ∈ X and t ∈ R. We will denote the set of all weakly bounded operators from (X, N ) into (Y, N  ) by WB(X, Y ). It is proved that WB(X, Y ) is a vector space 3. Strongly continuous at z ∈ X if for every  > 0 there exists δ > 0 such that NT (x)−T (z) () ≥ Nx−z (δ), for all x ∈ X. If T is strongly continuous at each point of X, then T is said to be strongly continuous on X. 4. Strongly bounded on X if there exists a positive real number M such that NT (x) (t) ≥ Nx ( Mt ), for all x ∈ X and t ∈ R. Theorem 1.5. [4] Let (X, N ) and (Y, N  ) be two P N -spaces and T : (X, N ) → (Y, N  ) a linear operator. Then T is weakly (strongly) continuous if and only if T is weakly (strongly) bounded.

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The following condition of an N -norm will be useful: (N6)

The function Nx (·) is a continuous function on R and strictly increasing on {t : 0 < Nx (t) < 1}, for every nonzero element x.

Theorem 1.6. [4] Let (X, N ) and (Y, N  ) be two P N -spaces satisfying (N5) and (N6). The linear operator T : (X, N ) → (Y, N  ) is weakly bounded if and only if T is bounded with respect to α-norms corresponding to N and N  . Theorem 1.7. [4] Let T : (X, N ) → (Y, N  ) be a linear operator, where (X, N ) and (Y, N  ) are P N -spaces. If T is strongly continuous on X, then T is sequentially continuous, that is for any sequence {xn } in X with xn → x implies that T (xn ) → T (x). Definition 1.8. [4] A subset B of a P N -space X is said to be N -bounded if there are t > 0 and 0 < r < 1 such that Nx (t) > 1 − r, for all x ∈ B. Definition 1.9. The N -closure of a subset E of a P N -space (X, N ) is denoted by E and defined by the set of all x ∈ X such that there is a sequence {xn } of elements of E with xn → x. We say that E is N -closed if E = E. Definition 1.10. A subset B in a P N -space (X, N ) is called N -compact if each sequence of elements of B has an N -convergent subsequence.

2. Compact operators In this section we aim to introduce the compact operators between P N -spaces and give some properties of them. Definition 2.1. Let (X, N ) and (Y, N  ) be two P N -spaces. A linear operator T : (X, N ) → (Y, N  ) is called a compact operator if for every N -bounded subset M of X the subset T (M ) of Y is relatively compact, that is the N -closure of T (M ) is an N -compact set. Example. Let (X,  · 1 ) and (Y,  · 2 ) be two normed spaces, and T : X → Y be a compact operator. Then it is easy to see that T : (X, N1 ) → (Y, N2 ) is a compact operator, where N1 and N2 are the standard probabilistic norms induced by the norms  · 1 and  · 2 , respectively, i.e.,  t t>0 t+xi Nix (t) = 0 t ≤ 0, for i = 1, 2. Theorem 2.2. Let T : (X, N ) → (Y, N  ) be a linear operator. Then T is compact if and only if it maps every N -bounded sequence {xn } in X onto a sequence {T (xn )} in Y which has an N -convergent subsequence.

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Proof. Suppose that T is a compact operator, and {xn } is an N -bounded sequence in (X, N ). The N -closure of {T (xn ) : n ∈ N} is an N -compact set. So {T (xn )} has an N -convergent subsequence by definition. Conversely, let A be an N -bounded subset of (X, N ) and {xn } be a sequence in the N -closure of T (A). For given  > 0, n ∈ N and t > 0, there exists {yn } in T (A) such that Nx n −yn ( 2t ) > 1 − . Let yn = T (zn ), where zn ∈ A. Since A is an N -bounded set, so is {zn }. On the other hand, T (zn ) has an N -convergent subsequence {ynk } = {T (znk )}. Let ynk → y for some y ∈ Y . Hence Ny n −y ( 2t ) > 1 − , for all nk > n0 . We have k

t t ), Ny n −y ( )} > 1 − , k 2 2 for all nk > n0 . Now {xnk } is an N -convergent subsequence of {xn }. Thus the N -closure of T (A) is an N -compact set.
Nx n

k

−y (t)

≥ min{Nx n

k

−ynk (

Lemma 2.3. Let (X, N ) be a P N -space satisfying (N5) and {xn } be a sequence in N X. Then xn −→ x if and only if limn→∞ xn − xα = 0, for all α ∈ (0, 1). N

Proof. Suppose that xn −→ x. Choose α ∈ (0, 1) and t > 0. There exists k ∈ N such that Nxn −x (t) > 1 − α, for all n ≥ k. It follows that xn − x1−α ≤ t, for all n ≥ k. Thus xn − x1−α → 0. Conversely, let xn − xα → 0, for every α ∈ (0, 1). Fix α ∈ (0, 1), and t > 0. There exists k ∈ N such that ∧{r > 0 : N (xn − x, r) ≥ 1 − α} < t, for all n ≥ k. Hence for every n ≥ k there is 0 < tn < t such that N (xn − x, tn ) ≥ 1 − α. It implies that Nxn −x (t) ≥ 1 − α, for all n ≥ k, i.e., N
xn −→ x. For a P N -space (X, N ), consider the subset Bα [x, r] = {y ∈ X : Nx−y (r) ≥ α}, where x ∈ X, α ∈ (0, 1), and r > 0. Theorem 2.4. Let (X, N ) be a P N -space satisfying (N5) and Nx (·) is a continuous function on R for a fixed element x ∈ X. Then X is of finite dimension if and only if Bα [x, r] is an N -compact set in X, for each α ∈ (0, 1), and r > 0. Proof. Let Aα [x, r] = {y ∈ X : x − yα ≤ r}, where α ∈ (0, 1) and r > 0. We first show that Bα [x, r] = Aα [x, r]. If y ∈ Bα [x, r], then Nx−y (r) ≥ α. Since x − yα ≤ r, y ∈ Aα [x, r]. Now if y ∈ Aα [x, r], then x − yα ≤ r, or ∧{t > 0 : Nx−y (t) ≥ α} ≤ r. If ∧{t > 0 : Nx−y (t) ≥ α} < r, there exists s such that 0 < s < r and Nx−y (s) ≥ α. Thus y ∈ Bα [x, r]. If ∧{t > 0 : Nx−y (t) ≥ α} = r, there is {tn } such that tn → r, and Nx−y (tn ) ≥ α. By the continuity of Nx (·) we obtain Nx−y (r) = limn→∞ Nx−y (tn ) ≥ α. Hence y ∈ Bα [x, r]. Consequently Aα [x, r] = Bα [x, r]. Suppose now that dim X < ∞, x ∈ X, and r > 0. Choose the sequence {xn } in Bα [x, r]. It is clear that Aα [x, r] is a compact subset of (X,  · α ). Hence there ·α

is a subsequence {xnk } of {xn } and v ∈ Aα [x, r] such that xnk −→ v. Because in ·β

finite-dimensional spaces all norms are equivalent, xnk −→ v, for all β ∈ (0, 1).

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N

Thus by Lemma 2.3 we obtain xnk −→ v. Since Bα [x, r] = Aα [x, r], we have v ∈ Bα [x, r]. Conversely, let Bα [x, r] be compact. To show that X is finite-dimensional, it suffices to prove that Aα [x, r] is compact with respect to α-norm. Choose a sequence {xn } of Aα [x, r]. Since Bα [x, r] is compact, it has an N -convergent subsequence {xnk }. Lemma 2.3 implies that {xnk } is convergent under  · α . Thus Aα [x, r] is a compact set in normed space (X,  · α ). This shows that X is of finite dimension.
Lemma 2.5. Let T : (X, N ) → (Y, N  ) be a compact operator, where (X, N ) and (Y, N  ) are P N -spaces satisfying (N5). Then T : (X,  · 1α ) → (Y,  · 2α ) is an ordinary compact operator for all α ∈ (0, 1). Proof. We show that for each bounded sequence {xn } in (X,  · 1α ), the sequence {T (xn )} has a convergent subsequence in (Y,  · 2α ). Let {xn } be a bounded sequence in (X,  · 1α ). There exists M > 0 such that xn 1α < M for all n ∈ N. By the definition of  · α , for every n there exists tn > 0 such that tn < M , and N (xn , tn ) ≥ α. Because N is non-decreasing, α ≤ Nxn (tn ) ≤ Nxn (M ). Hence Nxn (M ) ≥ α, for all n, that is {xn } is N -bounded. Thus {T (xn )} has an N convergent subsequence {T (xnk )}. By Lemma 2.3, {T (xnk )} is convergent under
 · 2α . Theorem 2.6. Let (X, N ) and (Y, N  ) be two P N -spaces satisfying (N5) and (N6). Then a) Every compact linear operator T : (X, N ) → (Y, N  ) is weakly continuous. b) If dim X = ∞ then the identity operator I : (X, N ) → (X, N ) is not a compact operator. Proof. a) Choose α ∈ (0, 1). Let  · 1α and  · 2α are α-norms on X and Y corresponding to the N -norms N and N  , respectively. By Lemma 2.5, T : (X,  · 1α ) → (Y,  · 2α ) is a compact operator and therefore there exists Mα > 0 such that T (x)2α ≤ Mα x1α . Hence T is weakly bounded by Theorem 1.6. Now Theorem 1.5 implies that T is weakly continuous. b) The identity operator I maps Bα [0, 1] to itself. Suppose on the contrary that I is a compact operator. Then B α [0, 1] is compact, for all α ∈ (0, 1). Now B α [0, 1] ⊆ Aα [0, 1] = Bα [0, 1] implies that Bα [0, 1] is closed and so compact. Thus X is finite dimensional by Theorem 2.4, which is a contradiction.
Theorem 2.7. Let (X, N ) and (Y, N  ) be two P N -spaces. Then the set of all compact linear operators from X into Y is a linear subspace of WB(X, Y ). Proof. Suppose that T1 and T2 are compact linear operators from X into Y . Let {xn } be any N -bounded sequence in X. The sequence {T1 (xn )} has an N -

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convergent subsequence {T1 (xnk )}. The sequence {T2 (xnk )} also has an N -convergent subsequence {T2 (zn )}. Hence {T1 (zn )} and {T2 (zn )} are N -convergent sequences. Let T1 (zn ) → u, and T2 (zn ) → v. If t > 0, we have t t  lim N(T (t) ≥ lim min{NT 1 (zn )−u ( ), NT 2 (zn )−v ( )}. 1 +T2 )(zn )−u−v n→∞ 2 2  Thus limn→∞ N(T (t) = 1, for all t > 0 . This implies that T1 + 1 +T2 )(zn )−u−v  T2 is a compact operator. Now if T (xnk ) → y, then limn→∞ NαT (t) = 1 (xn )−αy n→∞

k

t ) = 1, for all α ∈ R \ {0}, and t > 0. Hence αT1 is also a limn→∞ NT 1 (xn )−y ( |α| k compact operator.


Theorem 2.8. Let (X, N ) be a P N -space, T : (X, N ) → (X, N ) be a compact linear operator, and S : (X, N ) → (X, N ) be a strongly continuous linear operator. Then ST and T S are compact operators. Proof. Let {xn } be any N -bounded sequence in X. Then {T (xn )} has an N convergent subsequence {T (xnk )}. Let limn→∞ T (xnk ) = y. Since S is strongly continuous, by Theorem 1.7 we have S(T (xnk )) → S(y). Hence ST (xn ) has an N -convergent subsequence. This proves ST is compact. Now to show that T S is compact, choose any N -bounded sequence {xn }. There exist t0 > 0 and r0 ∈ (0, 1) such that Nxn (t0 ) > 1 − r0 for all n ≥ 1. By Theorem 1.5 we conclude that the operator S is a strongly bounded operator. Thus there is M > 0 such that  (t0 M ) > 1 − r0 , for all n. It follows that {S(xn )} is N -bounded sequence in NS(x n) S(X). Because T is compact, {T (S(xn ))} has an N -convergent subsequence. This completes the proof.
Lemma 2.9. Let (X, N ) be a P N -space satisfying (N5), Nx (·) be a continuous function on R and dim X < ∞. Then each N -bounded sequence {xn } in (X, N ) has an N -convergent subsequence. Proof. Let {xn } be an N -bounded sequence in (X, N ). There are t0 > 0 and r0 ∈ (0, 1) such that N (xn , t0 ) > 1 − r0 , for all n ∈ N. Hence {xn } ∈ B1−r0 [0, t0 ]. By Theorem 2.4, B1−r0 [0, t0 ] is an N -compact set, so {xn } has an N -convergent subsequence.
Theorem 2.10. [4] Let (X, N ) and (Y, N  ) be two P N -spaces satisfying (N5) and (N6). If T : (X, N ) → (Y, N  ) is a linear operator where dim X < ∞, then T is weakly continuous. Theorem 2.11. Let (X, N ) and (Y, N  ) be two P N -spaces satisfying (N5) and N  (x, ·) is continuous function on R, and T : (X, N ) → (Y, N  ) a linear operator. i) If T is weakly bounded and dim T (X) < ∞, then T is a compact operator. ii) If (X, N ) and (Y, N  ) satisfying (N6) and dim X < ∞, then T is a compact operator.

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Proof. i) Let {xn } be an N -bounded sequence of (X, N ). There are t0 > 0 and r0 ∈ (0, 1) such that N (xn , t0 ) > (1 − r0 ) for all n ∈ N. Since T is weakly bounded, there is M1−r0 > 0 such that for all n, t0 ) ≥ 1 − r0 . Nxn (t0 ) ≥ 1 − r0 =⇒ NT (xn ) ( M1−r0 It follows that {T (xn )} is an N -bounded sequence in T (X). Since dim T (X) < ∞, the sequence {T (xn )} has an N -convergent subsequence by Lemma 2.9. Hence T is compact. ii) Theorem 2.10 implies that T is weakly continuous. We also imply by Theorem 1.5 that T is weakly bounded. Since dim T (X) < ∞, by the part (i) we conclude that T is a compact operator.


References [1] C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space. Aequationes Math. 46 (1993) 91-98. [2] C. Alsina, B. Schweizer and A. Sklar, Continuity properties of probabilistic norms. J. Math. Anal. Appl. 208 (1997) 446-452. [3] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11, 3 (2003), 687–705. [4] T. Bag, S.K. Samanta, Fuzzy bounded linear operators. Fuzzy Sets Syst. 151 (2005), 513–547. [5] K. Menger, Statistical metrics. Proc. Nat. Acad. Sci. USA 28 (1942), 535–537. [6] B. Schweizer, Sklar A., Statistical metric spaces. Pacific J. Math. 10 (1960), 313–334. [7] B. Schweizer, Sklar A., Probabilistic metric spaces. New York, Amsterdam, Oxford: North-Holland 1983. ˇ [8] A.N. Serstnev, On the motion of a random normed space. Dokl. Akad. Nauk SSSR 149 (1963) 280-283, English translation in Soviet Math. Dokl. 4 (1963) 388-390. Kourosh Nourouzi Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran e-mail: [email protected] and Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O.Box 19395-5746, Tehran, Iran e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 355–384 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schr¨ odinger Operators Vladimir S. Rabinovich and Steffen Roch Abstract. The aim of this paper is to study relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on Lp (Rn ) perturbed by singular potentials. Our approach to this problem is via the limit operators method. This method associates with each band-dominated operator A a family op(A) of so-called limit operators which reflect the properties of A at infinity. Consider the compactification of Rn by the “infinitely distant” sphere S n−1 . Then the set op(A) can be written as the union of its components opηω (A) where ω runs through the points of S n−1 and where opηω (A) collects all limit operators of A which reflect the properties of A if one tends to infinity “in the direction of ω”. Set spηω A := ∪Ah ∈opηω (A) sp Ah . We show that the distance of an eigenvalue λ ∈ / spess A to spηω A determines the exponential decay of the λ-eigenfunctions of A in the direction of ω. We apply these results to estimate the exponential decay of eigenfunctions of electro-magnetic Schr¨ odinger operators for a large class of electric potentials, in particular, for multiparticle Schr¨ odinger operators and periodic Schr¨ odinger operators perturbed by slowly oscillating at infinity potentials. Mathematics Subject Classification (2000). Primary 47G30; Secondary 35J10, 35P05, 35S05, 47A53, 47N20. Keywords. Pseudodifferential operators, limit operators, essential spectra, exponential decay, Schr¨ odinger operators.

This work has been partially supported by CONACYT project 43432 and DFG Grant 444 MEX112/2/05.

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1. Introduction The main aim of this paper is to study the relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators perturbed by singular potentials. We are going to attack this problem by the limit operators method. This method was already used earlier to describe the location of essential spectra of perturbed pseudodifferential operators, which has found applications to electro-magnetic Schr¨ odinger operators, square-root KleinGordon operators and Dirac operators under quite general assumptions on the behavior of magnetic and electric potentials at infinity. By means of the limit operators method, also a simple and transparent proof of the celebrated Hunziker, van Winter, Zjislin theorem for multi-particle Hamiltonians has been obtained. In [28, 30], the limit operators method was applied to study the location of the essential spectrum of discrete Schr¨ odinger operators. The basic idea of the limit operators method is as follows. For 1 < p < ∞, we consider the Banach algebras Ap of all band-dominated operators on the Lebesgue spaces Lp (Rn ) or lp (Zn ). For each operator A ∈ Ap , there is an associated family of so-called limit operators which are defined by sequences h tending to infinity. We denote this family by op(A) and call it the operator spectrum of A. The results of [31, 32] (see also [33] for a comprehensive account) show that  spess A = spAh (1.1) Ah ∈op(A)

for a large class of band-dominated operators on Rn or Zn . Since the limit operators of an operator are more simple objects than the operator itself, identity (1.1) provides an effective tool to study the essential spectra for large classes of operators. For instance, differential and pseudodifferential operators of order m ∈ R belong to the algebra Ap after multiplication by the operator (I − ∆)−m/2 of order reduction. It should be noted that formulas similar to (1.1) have been obtained independently in [17] by means of admissible geometric methods. We also refer to the papers [14, 13, 21, 4] and the references therein where C ∗ -algebra techniques have been employed to study essential spectra of Schr¨ odinger operators. The methods of [17, 14, 13, 21, 4] are applicable only for self-adjoint or normal operators acting on Hilbert spaces, whereas the limit operators approach allows one to consider non self-adjoint operators on Lp -type spaces, for example Schr¨ odinger operators with complex potentials. ˜ n be the compactification of Rn homeomorphic to the closed unit ball Let R n n ˜ n \ Rn is homeomorphic to the unit sphere S n−1 . We B ⊂ R . Its boundary R n ˜ denote the point in R \ Rn which corresponds to ω ∈ S n−1 by ηω , and we write opηω (A) for the set of all limit operators of A defined by sequences h tending to ηω in the topology of the compactification. The set  spAh (1.2) spηω A := Ah ∈opηω (A)

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can be considered as the local essential spectrum of A at the point ηω . Note that  spηω A. spess A = ω∈S n−1

In this paper we are going to show that the “distance” of an eigenvalue λ ∈ / spess A to the local essential spectrum spηω A determines the exponential decay of the λeigenfunction uλ of A in the direction of ηω . We apply this result to estimate the exponential decay of eigenfunctions for a large class of electro-magnetic Schr¨ odinger operators with electric potentials, in particular, for multi-particle Schr¨ odinger operators and periodic Schr¨ odinger operators perturbed by slowly oscillating potentials. We plan to examine further applications of the limit operators method to exponential decay estimates of eigenfunctions of other operators important in mathematical physics (including Dirac, Pauli, Klein-Gordon and Maxwell operators) in a forthcoming paper. Exponential decay estimates are subject of an intensive research. We would like to mention the work of Agmon [1, 2] where estimates of eigenfunctions of second-order elliptic operators have been obtained in terms of a special (Agmon)metric. In contrast to Agmon’s approach, we use pseudodifferential operators with analytical symbols and limit operators. Pseudodifferential operators with analytical symbols were employed earlier for exponential estimates of solutions of pseudodifferential equations in [19, 26, 28, 27, 25, 20] and for the tunnel effect in [23, 24, 25]. The paper is organized as follows. In Section 2 we collect some auxiliary material on the Banach algebra Ap (Rn ) of the band-dominated operators on Lp (Rn ) and on a related Wiener algebra Wp (Rn ), and we recall a criterion from [31, 32, 33] for operators in these algebras to be Fredholm on Lp (Rn ). Then we apply this criterion to derive the equality (1.1) on which all further considerations are based. In Section 3 we employ this equality to study the essential spectrum of elliptic pseum dodifferential operators in the class OP S1, 0 which are perturbed by measurable potentials. Section 4 is devoted to applications of limit operators to exponential estimates of solutions of pseudodifferential equations with analytical symbols perturbed by singular potentials. In Section 5 we consider the electro-magnetic Schr¨ odinger operator (Hu)(x) := (i∂xj − aj (x))ρjk (x)(i∂xk − ak (x))u(x) + Φ(x)u(x),

x ∈ Rn ,

as an unbounded operator on L2 (Rn ) with domain H 2 (Rn ) where Rn is equipped with a Riemann metric ρjk satisfying inf

x∈Rn , ω∈S n−1

ρjk (x)ω j ω k > 0.

(1.3)

Here we use the Einstein summation convention, and ρjk (x) is the tensor inverse to ρjk (x). We suppose that the functions ρjk and ak are infinitely differentiable and bounded with all of their derivatives and that they are slowly oscillating at infinity. Under certain conditions on the electric potential, we prove that the limit operator

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of H defined by a sequence g = (gm ) is unitarily equivalent to the operator (Hg u)(x) := −ρjk g ∂xj ∂xk u(x) + Φg (x)u(x), where the numbers ρjk g are independent of x and Φg (x) is the limit of the Φ(x+gm ) in the sense that s-limm→∞ Φ(x + gm )(I − ∆)−1 = Φg (x)(I − ∆)−1 with s-lim referring to the strong limit. Formula (1.1) then implies  spess H = sp Hg .

(1.4)

Hg ∈op(H)

Set µH (ω) :=

inf

Hg ∈opηω (H)

sp Hg ,

let g be a sequence tending to ηω , and assume that ρjk =: ρjk g ω depends on ω only. The following theorem settles an exponential decay estimate for the λeigenfunctions uλ of the operator H. Theorem 1.1. Let the eigenfunction uλ of H correspond to an eigenvalue λ<µ ˆ H :=

inf

ω∈S n−1

µH (ω).

Then ev˜ uλ ∈ H 2 (Rn ) where v˜ is a C ∞ -function on Rn which coincides with the function x → l(x/|x|)|x| outside a (small) neighborhood of the origin and where l : S n−1 → R+ is an arbitrary smooth positive function such that ? µH (ω) − λ for all ω ∈ S n−1 . l(ω) < (1.5) (ρω ω, ω)  Here, (x, y) = nj=1 xj yj refers to the standard inner product on Rn . We apply Theorem 1.1 to estimate eigenfunctions associated with points in the discrete spectrum of multi-particle Schr¨ odinger operators and of periodic Schr¨ odinger operators perturbed by slowly oscillating potentials. In particular we odinger operator H, and will obtain an estimate of µH (ω) for a multi-particle Schr¨ then we apply Theorem 1.1 to get the exponential decay estimates for eigenfunctions of H corresponding to eigenvalues λ < µ ˆ H . Different forms of exponential decay estimates for eigenfunctions related with the discrete spectrum of multiparticle Schr¨ odinger operators were obtained by Agmon [1], see also [10] and [11]. We consider also periodic Schr¨ odinger operators perturbed by a slowly oscillating potential, that is, operators HΦ+Ψ = −∆+(Φ+Ψ)I where Φ is a continuous Zn -periodic function and Ψ is a continuous function slowly oscillating at infinity. It is well known that the spectrum of a Zn -periodic Schr¨ odinger operator has a band structure with at most countably many spectral gaps, and that the discrete

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spectrum of these operators is empty (see, for instance, [35], XIII.16, and the excellent survey by Kuchment [15]). We prove that if [a, b] is a spectral band of the periodic operator HΦ then [a + mΨ , b + MΨ ] with mΨ := lim inf Ψ(x), x→∞

MΨ := lim sup Ψ(x) x→∞

is a spectral band in the essential spectrum of the perturbed operator HΦ+Ψ . Similarly, if (α, β) is a spectral gap of H + ΦI, then (α + MΨ , β + mΨ ) is a gap in the essential spectrum of HΦ+Ψ . Hence, if (α + MΨ , β + mΨ ) is not empty, then this interval can contain points of the discrete spectrum of HΦ+Ψ . Note that the discrete spectrum of periodic Schr¨ odinger operator perturbed by decreasing potentials has been studied in [5, 6, 7]. We derive exponential estimates for the eigenfunctions uλ of HΦ+Ψ with respect to eigenvalues λ located on the left of the essential spectrum of HΦ+Ψ and for eigenvalues λ in the gaps (α + MΨ , β + mΨ ) of the essential spectrum of HΦ+Ψ .

2. Local invertibility at infinity and Fredholmness of band-dominated operators on Rn In what follows we thoroughly suppose that p ∈ (1, ∞), and we use the following standard notations. For each Banach space X, let L(X) stand for the Banach algebra of all bounded linear operators on X and K(X) for the associated ideal of the compact operators. For h ∈ Rn , consider the shift operator (Vh u)(x) := u(x−h) which acts as an isometry on Lp (Rn ). Further, let Cbu (Rn ) refer to the C ∗ -algebra of all bounded and uniformly continuous functions on Rn , C0 (Rn ) to its subalgebra consisting of all functions with compact support, SO(Rn ) to the subalgebra of Cbu (Rn ) of all functions a which are slowly oscillating in the sense that lim sup |a(x + y) − a(x)| = 0

x→∞ y∈K

for every compact K ⊂ Rn , Cb∞ (Rn ) to the space of all infinitely differentiable functions which are bounded together with all of their derivatives, and finally SO∞ (Rn ) to the intersection Cb∞ (Rn ) ∩ SO(Rn ). Definition 2.1. An operator A ∈ L(Lp (Rn )) is band-dominated if, for every function ϕ ∈ Cbu (Rn ), lim ϕt A − Aϕt IL(Lp (Rn )) = 0 t→0

where ϕt (x) := ϕ(t1 x1 , . . . , tn xn ) for t = (t1 , . . . , tn ) ∈ Rn . The set of all band-dominated operators is a Banach subalgebra of L(Lp (Rn )) which we denote by Ap (Rn ). This algebra is inverse closed in L(Lp (Rn )), and it contains the ideal K(Lp (Rn )). Let Ip (Rn ) denote the set of all operators A ∈ L(Lp (Rn )) such that lim ϕt AL(Lp (Rn )) = lim Aϕt IL(Lp (Rn )) = 0

t→0

t→0

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for every function ϕ ∈ Cbu (Rn ) with ϕ(0) = 0. One easily checks that Ip (Rn ) ⊂ Ap (Rn ) and that Ip (Rn ) is a closed two sided ideal of Ap (Rn ). For R > 0, let PR denote the operator of multiplication by the characteristic function of the ball BR := {x ∈ Rn : |x| < R}, which acts as a projection on Lp (Rn ). Definition 2.2. Let A ∈ L(Lp (Rn )), and let h = (hm ) be a sequence tending to infinity. The linear operator Ah is called a limit operator of A defined by the sequence h if, for every R > 0, lim (Ah − V−hm AVhm )PR L(Lp (Rn ))

m→∞

= lim PR (Ah − V−hm AVhm )L(Lp (Rn )) = 0. m→∞

(2.1)

Evidently, every operator A has at most one limit operator with respect to a given sequence h, which justifies the notation Ah . It is immediate from the definition that Ah is a bounded linear operator on Lp (Rn ) and that Ah  ≤ A. We denote the set of all limit operators of A by op(A). An operator A ∈ L(Lp (Rn )) is called rich if every sequence h tending to infinity has a subsequence g for which the limit operator Ag exists. The set of all rich operators in Ap (Rn ) is a closed subalgebra of Ap (Rn ) which we denote by A$p (Rn ). Theorem 2.3. Let A ∈ A$p (Rn ). Then the coset A + Ip (Rn ) is invertible in the quotient algebra Ap (Rn )/Ip (Rn ) if and only if all limit operators Ah ∈ op(A) are invertible and if the norms of their inverses are uniformly bounded, sup Ah ∈op(A)

A−1 h  < ∞.

(2.2)

Note that the invertibility of the coset A + Ip (Rn ) in Ap (Rn )/Ip (Rn ) is equivalent to the existence of an operator R ∈ Ap (Rn ) such that RA − I, AR − I ∈ Ip (Rn ). An operator A ∈ Ap (Rn ) with this property is also said to be locally invertible at infinity on Lp (Rn ). Let χ0 denote the characteristic function of the semi-open cube I0 := (0, 1]n and set χα (x) := χ0 (x − α) for α ∈ Zn . Definition 2.4. Let Wp (Rn ) stand for the set of all operators A ∈ L(Lp (Rn )) with
sup χ0 V−α AVα−γ χ0 IL(Lp (Rn )) AWp (Rn ) := γ∈Zn α∈Z

=




γ∈Zn

n

sup χα Aχα−γ IL(Lp (Rn )) < ∞.

α∈Zn

Provided with the operations inherited from L(Lp (Rn )) and with the above defined norm, the set Wp (Rn ) becomes a Banach algebra, the so-called Wiener algebra. The importance of the Wiener algebra lies in the fact that the boundedness condition (2.2) in Theorem 2.3 is redundant for rich operators in Wp (Rn ).

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Theorem 2.5. Let A ∈ Wp$ (Rn ) := Wp (Rn )∩A$p (Rn ). Then A is locally invertible at infinity on Lp (Rn ) if and only if all limit operators of A are invertible on Lp (Rn ). The local invertibility at infinity coincides with the common Fredholm property for operators which are locally compact. Definition 2.6. An operator Q ∈ L(Lp (Rn )) is locally compact if aQ and QaI are compact operators on Lp (Rn ) for every function a ∈ C0 (Rn ). We denote the set of all locally compact operators by LC p (Rn ). Proposition 2.7. LC p (Rn ) ∩ Ap (Rn ) is a closed ideal of Ap (Rn ). Proof. Let Q ∈ LC p (Rn ) ∩ Ap (Rn ), A ∈ Ap (Rn ), and a ∈ C0 (Rn ). Then aQA is clearly a compact operator. We show that the operator QAaI is compact, too. Let ϕ ∈ C0∞ (Rn ) be a function with ϕ(x) = 1 if |x| ≤ 1 and ϕ(x) = 0 if |x| ≥ 2, and set ϕR (x) := ϕ(x/R) for R > 0. Choose R0 such that ϕR a = a for all R > R0 . Since A ∈ Ap (Rn ), QAaI = QAϕR aI = QϕR AaI + TR

for R > R0

where the TR are operators with limR→∞ TR L(Lp (Rn )) = 0. Since the operators AϕR I are compact for every R > R0 , the operator QAaI is compact.
Note that the coset A + (LC p (Rn ) ∩ Ap (Rn )) is invertible in the quotient algebra Ap (Rn )/(LC p (Rn ) ∩ Ap (Rn )) if and only if there exist an operator R ∈ Ap (Rn ) and operators Q1 , Q2 ∈ LC p (Rn ) ∩ Ap (Rn ) such that RA = I + Q1 ,

AR = I + Q2 .

(2.3)

Theorem 2.8. The operator A ∈ Ap (R ) is Fredholm on L (R ) if and only if the following conditions hold: (i) A is invertible modulo the ideal LC p (Rn ) ∩ Ap (Rn ) of Ap (Rn ), (ii) A is invertible modulo the ideal Ip (Rn ) of Ap (Rn ). n

p

n

Proof. Let conditions (i) and (ii) be fulfilled. Then there are operators R , R in Ap (Rn ) such that AR = I + Q2 , (2.4) R A = I + Q1 , where Then

Qj

AR = I + Q2 (2.5) R A = I + Q1 , n  ∈ LC p (R ) ∩ Ap (R ) and Qj ∈ Ip (Rn ). Set R := R + R − R AR . n

RA − I

= R A + R A − R AR A − I = (I − R A) (R A − I) = −Q1 Q1 .

In the same way one gets The operators

Q1 Q1

AR − I = −Q2 Q2 .   and Q2 Q2 are compact. Indeed, since Q1 lim Q1 Q1 − ϕR Q1 Q1 L(Lp(Rn )) = 0. R→∞

∈ Ip (Rn ),

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V.S. Rabinovich and S. Roch

Moreover, since Q1 ∈ Ap (Rn ), lim Q1 Q1 − Q1 ϕR Q1 L(Lp(Rn )) = 0.

R→∞ LC p (Rn )

∩ Ap (Rn ), the operator Q1 ϕR Q1 is compact. Hence, Since, finally, Q1 ∈ being the norm limit of compact operators, the operator Q1 Q1 is compact. Analogously, the compactness of Q2 Q2 follows. Thus, A is a Fredholm operator, and R is its inverse modulo compact operators. Conversely, let A ∈ Ap (Rn ) be a Fredholm operator. Then there are operators R ∈ Ap (Rn ) and T1 , T2 ∈ K(Lp (Rn )) such that RA = I + T1 ,

AR = I + T2 .

(2.6)

Because of K(L (R )) ⊂ LC p (R ) ∩ Ap (R ) and K(L (R )) ⊂ Ip (R ), the equalities (2.6) imply conditions (i) and (ii).
p

n

n

n

p

n

n

Theorems 2.5 and 2.8 have the following consequences. Theorem 2.9. Let A ∈ Wp$ (Rn ). Then A is a Fredholm operator on Lp (Rn ) if and only if (i) A is invertible modulo the ideal LC p (Rn ) ∩ Ap (Rn ) of Ap (Rn ), (ii) all limit operators of A are invertible. Theorem 2.10. Let A ∈ Wp$ (Rn ), and let condition (i) of Theorem 2.9 be satisfied. Then  sp Ah . (2.7) spess A = Ah ∈op(A)

˜n

Let R denote the compactification of Rn homeomorphic to the unit ball ˜ n \ Rn be the infinitely distant point B := {x ∈ Rn : |x| ≤ 1}, and let ηω ∈ R n−1 . We denote by opηω (A) the set of the which corresponds to the point ω ∈ S limit operators of A which correspond to sequences h tending to ηω in the topology ˜ n . Then, under the conditions of Theorem 2.10, equality (2.7) can be written as of R   sp Ah . (2.8) spess A = n

ω∈S n−1 Ah ∈opηω (A)

3. Applications to pseudodifferential operators 3.1. Basics We start with recalling some facts about pseudodifferential operators. Standard references are [39, 36, 38]. We will use the following standard notations. The n-tuple α := (α1 , · · · , αn ), αj ∈ N ∪ {0}, is a multi-index, |α| := α1 + . . .+ αn is its length, ∂xα := ∂xα11 . . . ∂xαnn is the operator of αth partial derivative, and ξ := (1 + |ξ|2 )1/2 . We say that a C ∞ -function a on Rn × Rn is a symbol in the H¨ormander class m S1, 0 if
# β α # #∂x ∂ξ a(x, ξ)# ξ−m+|α| < ∞ sup |a|N := |α|+|β|≤N

(x, ξ)∈Rn ×Rn

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for every N ∈ N ∪ {0}. The class of all pseudodifferential operators   −n (Op(a)u)(x) = (2π) a(x, ξ) ei(x−y, ξ) u(y) dy dξ, u ∈ C0∞ (Rn ), Rn

Rn

m m m with symbols in S1, 0 is denoted by OP S1, 0 . For m ∈ R, we use the notation D m to refer to the pseudodifferential operator with symbol a(x, ξ) = ξ . It is well known (see, for instance, [38], Chap. VI) that pseudodifferential 0 p n operators with symbols in S1, 0 are bounded on L (R ) for p ∈ (1, ∞) and there are a constant Cp and a number N ∈ N such that

Op(a)L(Lp (Rn )) ≤ Cp |a|N .

(3.1)

We will also have to work with pseudodifferential operators with double symbols a on Rn × Rn × Rn which satisfy the estimates # β γ α # #∂ ∂ ∂ a(x, y, ξ)# ≤ Cαβγ ξm−|α| x y ξ for every choice of multi-indices α, β, γ. We denote the class of these symbols by m m S1, 0, 0 and write OP S1, 0, 0 for the associated class of pseudodifferential operators with double symbols. The latter act on C0∞ (Rn ) via   a(x, y, ξ) ei(x−y, ξ) u(y) dy dξ. (3.2) (Opd (a)u)(x) = (2π)−n Rn

Rn

⊂ More precisely, every pseudodifferential One knows that operator Opd (a) with double symbol can be viewed as a common pseudodifferential operator, Op(b), with   b(x, ξ) := (2π)−n a(x, x + y, ξ + η) e−i(y, η) dy dη (3.3) m OP S1, 0, 0

m OP S1, 0.

Rn

Rn

m S1, 0.

a symbol in The double integral in (3.3) is understood as an oscillatory integral. An important example of a pseudodifferential operator with double symbol m m in S1, 0, 0 is the Weyl quantization operator OpW (a) with a ∈ S1, 0 which acts on C0∞ (Rn ) by     x+y −n , ξ ei(x−y, ξ) u(y) dy dξ. (OpW (a)u)(x) := (2π) a 2 Rn Rn 0 n Theorem 3.1. OP S1, 0 ⊂ Wp (R ) for every p ∈ (1, ∞). m ∞ n Proof. Let A = Op(a) ∈ OP S1, 0 . Then A acts on C0 (R ) by  (Au)(x) = kA (x, x − y) u(y) dy,

(3.4)

and the kernel function kA ∈ C ∞ (Rn × (Rn \ {0})) satisfies the estimates # β α # #∂x ∂z kA (x, z)# ≤ C|a|M |z|−n−m−|α|−N

(3.5)

Rn

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V.S. Rabinovich and S. Roch

for all multi-indices α and β and for all N ≥ 0 such that n + m + |α| + N > 0. The constant C > 0 in (3.5) is independent of A, whereas M depends on n, m, α, β and N ([38], p. 241). We set κγ (A) := sup χ0 V−α AVα−γ χ0 IL(Lp (Rn )) . α∈Zn

Hence, for the operator A in (3.4), κ0 (A) ≤ AL(Lp (Rn )) ≤ C|a|M , whereas for γ = 0,   κγ (A) ≤




|kA (x, x − y + γ)|p dy I0

1/p

p/p


dx

I0

with 1/p + 1/p = 1. Applying estimate (3.5) with α = β = 0, m = 0 and N = 1 we obtain κγ (A) ≤ C|a|M |γ|−n−1 for γ = 0. Hence,


γ∈Z

sup χ0 V−α AVα−γ χ0 IL(Lp (Rn )) ≤

n n α∈Z




κγ (A) ≤ C|a|M

γ∈Zn

for some M ∈ N.




0 0 ˜n ˜n Definition 3.2. Let S1, 0 (R ) (resp. S1, 0, 0 (R )) denote the class of all symbols in 0 0 n ˜n S1, 0 (resp. in S1, 0, 0 ) which can be extended to a continuous function on R × R n n n m ˜n ˜ (resp. on R × R × R ). We further say that a belongs to S1, 0 (R ) (resp. to m −m 0 0 ˜n ˜n ˜n S1, lies in S1, 0, 0 (R )) if the function aξ 0 (R ) (resp. in S1, 0, 0 (R )). 0 $ n ˜n Proposition 3.3. OP S1, 0 (R ) ⊆ Wp (R ) for every p ∈ (1, ∞). 0 ˜n Proof. Let a ∈ S1, 0 (R ), and let h = (hj ) be a sequence tending to infinity. Then

V−hj Op(a)Vhj = Op(a(x + hj , ξ)). For every compact subset K of Rn , the sequence of the functions a(x + hj , ξ), ˜n j ∈ N, is uniformly bounded and equicontinuous on the compact subset K × R ˜ n . By the Arzel`a-Ascoli theorem, there are a subsequence g = (gj ) of h of Rn × R ˜ n ) such that and a function ag ∈ C ∞ (Rn × R # β α # #∂ ∂ a(x + gj , ξ) − ∂ β ∂ α ag (x, ξ)# = 0 lim sup (3.6) x ξ x ξ j→∞ (x, ξ)∈K×Rn

0 for all multi-indices α, β. The limit (3.6) implies that ag ∈ S1, 0 and that Op(ag ) is 0 ˜n the limit operator of Op(a) with respect to the sequence g. Hence, OP S1, 0 (R ) ⊆ $ n Wp (R ).


Essential Spectra of ΨDO

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0 n ˜n Let a ∈ S1, 0, 0 (R ) and let h : N → Z be a sequence tending to infinity. Then there exists a subsequence g = (gj ) of h such that # β α # #∂x ∂ξ a(x + gj , y + gj , ξ) − ∂xβ ∂ξα ag (x, y, ξ)(x, ξ)# = 0 sup lim j→∞ (x, y, ξ)∈K1 ×K2 ×Rn

(3.7) for every compact set K1 × K2 ⊂ Rn × Rn and all multi-indices α, β. One can prove as above that Opd (ag ) is the limit operator of Opd (a) with respect to g. 3.2. Essential spectra of elliptic pseudodifferential operators of zero order m A pseudodifferential operator Opd (a) ∈ OP S1, 0, 0 is said to be uniformly elliptic n on R if inf n |a(x, y, ξ)| > 0. (3.8) lim n R→∞ (x, y)∈R ×R , |ξ|≥R

0 p n Theorem 3.4. An operator Opd (a) ∈ OP S1, 0, 0 is Fredholm on L (R ) if and only if it is (i) uniformly elliptic on Rn , and (ii) locally invertible at infinity.

Proof. Let the ellipticity condition (3.8) hold. It is easy to show that then there 0 is a double symbol b ∈ S1, 0, 0 such that Opd (b) Opd (a) = I + Opd (t1 ),

Opd (a) Opd (b) = I + Opd (t2 )

−1 and are therefore where the operators Opd (t1 ) and Opd (t2 ) belong to OP S1,0,0 p n locally compact on L (R ). Hence, by Theorem 2.9, Opd (a) is a Fredholm operator on Lp (Rn ). Conversely, it is well known that the Fredholmness of an operator Opd (a) 0 p n n in OP S1, 0, 0 implies its uniform ellipticity (3.8). Since K(L (R )) ⊂ Ip (R ), the Fredholmness of Opd (a) also implies the local invertibility at infinity of Opd (a).


The previous results imply the following. 0 ˜n Theorem 3.5. Let A = Op(a) ∈ OP S1, 0, 0 (R ) be a uniformly elliptic pseudodifferential operator. Then A is a Fredholm operator on Lp (Rn ) for p ∈ (1, ∞) if and only if all limit operators of A are invertible. Moreover,  spess (A : Lp (Rn ) → Lp (Rn )) = sp (Ah : Lp (Rn ) → Lp (Rn )). (3.9) Ah ∈op(A)

3.3. Essential spectra of perturbed elliptic pseudodifferential operators of nonzero order Here we consider a class of pseudodifferential operators of order m ≥ 0 perturbed by a singular potential Φ, i.e., operators of the form A = B + ΦI where ˜ n ); (A) B = Opd (b) ∈ OP S m and bξ−m ∈ S 0 (R 1, 0, 0

1, 0, 0

(B) the operator ΦD−m is locally compact on Lp (Rn ) and belongs to Wp$ (Rn ).

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V.S. Rabinovich and S. Roch

We let H m, p (Rn ) denote the Sobolev space with norm uH m, p (Rn ) := Dm uLp(Rn ) and consider A as a bounded operator from H m, p (Rn ) into Lp (Rn ). Theorem 3.4 implies the following. Theorem 3.6. Let the operator A = Opd (b) + ΦI satisfy conditions (A) and (B). Then A, considered as acting from H m, p (Rn ) to Lp (Rn ), is a Fredholm operator if and only if (i) the operator Opd (b) is uniformly elliptic on Rn , and (ii) all limit operators of the operator AD−m are invertible on Lp (Rn ). Let the operator A satisfy conditions (A) and (B), and let m ≥ 0. We say that a complex number λ belongs to spp A resp. to sppess A if the operator A − λI : H m, p (Rn ) → Lp (Rn ) is not invertible resp. not Fredholm. Let A := Opd (b) + ΦI be a pseudodifferential operator of order m ≥ 0 and let A˜h be a limit operator of A˜ := AD−m with respect to a sequence h. Then we call Ah := A˜h Dm the limit operator of A defined by the sequence h. Note that Ah is a bounded operator operator from H m, p (Rn ) to Lp (Rn ). We write op(A) for the set of all limit operators of A defined in this way. The following is a consequence of Theorem 3.6. Theorem 3.7. Let the operator A = Opd (b) + ΦI satisfy conditions (A) and (B), and let the operator Opd (b) be uniformly elliptic on Rn . Then  spp Ah . (3.10) sppess A = Ah ∈op(A)

Proposition 3.8. Let A := Opd (b) + ΦI again satisfy conditions (A) and (B), and let m > 0. Moreover, suppose that the operator A is uniformly elliptic. Then A can be viewed of as an unbounded closed operator on L2 (Rn ) with domain H m, 2 (Rn ) =: H 2 (Rn ), and the spectrum spA and the essential spectrum spess A of A, considered as an unbounded operator on L2 (Rn ) coincide with the corresponding spectra sp2 A and sp2ess A of A, considered as a bounded operator acting from H m (Rn ) to L2 (Rn ). For the proof see [3], pages 27–32.




The next proposition states some sufficient conditions for a potential Φ to be subject of condition (B). In case p = 2, similar results have been obtained [9], p. 62–71. Proposition 3.9. (i) If m > 0 and Φ ∈ L∞ (Rn ), then the operator ΦD−m belongs to the Wiener algebra Wp (Rn ) and is locally compact. (ii) If m < n and Φ ∈ Ln/m (Rn ), then the operator ΦD−m belongs to Wp (Rn ) and is compact on Lp (Rn ). (iii) If m > n and Φ ∈ Lp (Rn ), then the operator ΦD−m belongs to Wp (Rn ) and is compact on Lp (Rn ).

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Proof. (i) One has κγ (ΦD−m ) ≤ ΦL∞ (Rn ) γ−n−1 , which implies ΦD−m ∈ Wp (Rn ). The local compactness of this operator follows since m is positive. (ii) The operator D−m acts boundedly from Lp (Rn ) into Lq (Rn ) ∩ Lp (Rn ) if 1 1 m −m is an Lp -Fourier multiplier for every p ∈ (1, ∞). Write q = p − n . Indeed, ξ   ξ−m = |ξ|−m |ξ|m ξ−m . The operator |ξ|m ξ−m is an Lp -Fourier multiplier, too, whereas Op(|ξ|−m ) is a Riesz potential which is bounded as an operator from Lp (Rn ) into Lq (Rn ) by the Hardy-Littlewood-Sobolev theorem (see, for instance, [38], Chap. VIII). The generalized H¨older inequality uvLs (Rn ) ≤ uLp(Rn ) vLq (Rn ) holding if

1 s

=

1 p

1 q

+

implies ΦD−m L(Lp (Rn )) ≤ Cp ΦLn/m(Rn ) .

Further, for x ∈ I0 , −m

χ0 V−α ΦD

(3.11)

 Vα−γ χ0 v(x) = Φ(x + α) I0

kD−m (x − y + γ) v(y) dy.

In case γ = 0 we conclude from the equality Φ(· + α)Ln/m (Rn ) = ΦLn/m(Rn ) and from estimate (3.11) that χ0 V−α ΦD−m Vα χ0 IL(Lp (Rn )) ≤ Cp ΦLn/m(Rn ) . Let now γ = 0. Then, for u ∈ Lp (Rn ), χ0 V−α ΦD−m Vα−γ χ0 uLp(Rn ) ≤ ΦLn/m(Rn ) χ0 V−α D−m Vα−γ χ0 uLq (Rn ) where

1 p

=

1 q

+

m n.

Further, with

1 p

+

1 p


= 1, one has

χ0 V−α D−m Vα−γ χ0 uLq (Rn )    q
 q1 p # #p
#kD−m (x − y + γ)# dy ≤ dx uLp(Rn ) . I0

I0

Applying estimates (3.5) and (3.11) we obtain sup χ0 V−α ΦD−m Vα−γ χ0 IL(Lp (Rn )) ≤ C0 ΦLn/m(Rn ) γ−n−1 .

α∈Zn

Hence, Φ D−m ∈ Wp (Rn ). In order to prove the compactness of ΦD−m , we choose a sequence of functions Φk in C0∞ (Rn ) such that limk→∞ Φ − Φk Ln/m(Rn ) = 0. Employing estimate (3.11) again we find lim ΦD−m − Φk D−m L(Lp (Rn )) = 0.

k→∞

Since the operators Φk D−m are compact on Lp (Rn ), so is ΦD−m .

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(iii) Let m > n. Then the function kD−m is continuous at the point 0, and it satisfies estimate (3.5). We claim that then D−m is a bounded operator from Lp (Rn ) into Lp (Rn ) ∩ L∞ (Rn ). For u ∈ Lp (Rn ), let  v(x) := kD−m (x − y + γ) u(y) dy. Rn

By the H¨older inequality, |v(x)| ≤ kD−m Lp
(Rn ) uLp(Rn ) . Thus, if Φ ∈ Lp (Rn ), then ΦD−m L(Lp (Rn )) ≤ Cp ΦLp (Rn ) , verifying the claim. The compactness of ΦD−m on Lp (Rn ) for p < ∞ can be shown as in (ii).


4. Exponential estimates for perturbed pseudodifferential equations 4.1. Weights Let w be a positive measurable function on Rn which we call a weight. We denote by Lp (Rn , w) the space of all measurable functions u on Rn for which uLp(Rn , w) := wuLp (Rn ) < ∞. Let D be a convex domain in Rn which contains the point 0 ∈ Rn . In what follows we consider weights of the form w(x) := exp v(x) where v is a function satisfying ∂xj v ∈ Cb∞ (Rn ) for j = 1, . . . , n and (∇v)(x) ∈ D for every x ∈ Rn . We denote the class of these weights by R(D) and write Rsl (D) for the class of all weights in R(D) such that lim ∂ 2 v(x) x→∞ xi xj

=0

(4.1)

for 1 ≤ i, j ≤ n. The weights in Rsl (D) are called slowly oscillating. Let l : S n−1 → R be a positive C ∞ -function. We associate with l the weight wl (x) := exp vl (x)

where vl (x) := l(x/|x|) |x|

and consider the open star-like domain Ωl := {x ∈ Rn : x = tl(ω)ω, t ∈ [0, 1), ω ∈ S n−1 }. One can show that vl (x) = max (x, y) for all x ∈ Rn where (x, y) =

n j=1

y∈Ωl

xj yj is the scalar product on Rn .

(4.2)

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369

Proposition 4.1. (i) The function vl is positively homogeneous, that is vl (tx) = tvl (x) for all t > 0 and x ∈ Rn . Moreover, vl ∈ C ∞ (Rn \ {0}). (ii) (∇vl )(x) ∈ Ωl for every x ∈ Rn and (∇vl )(ω) = l(ω)ω ∈ ∂Ωl for every point ω ∈ S n−1 . Proof. Statement (i) is evident. To prove (ii), fix x ∈ Rn . The continuous function y → (x, y) attains its maximum over Ωl at some point ξ(x) ∈ Ωl . Thus, vl (x) = (x, ξ(x)). In particular, ξ(x) is a stationary point of the function y → (x, y) on Ωl , that is, ∂(x,∂yξ(x)) = 0 for i = 1, . . . , n, whence i n
∂(x, ξ(x)) ∂ξi (x) ∂vl (x) = ξj (x) + = ξj (x). ∂xj ∂yi ∂xj i=1

This shows that (∇vl )(x) = ξ(x) ∈ Ωl for every x ∈ Rn . It is further evident that ξ is a homogeneous function of order zero. Hence, (∇vl )(ω) = ξ(ω) = l(ω)ω.
Let v˜l be a C ∞ -function on Rn which coincides with vl outside a certain neighborhood of the origin. Then the weight w ˜l := exp v˜l belongs to the class Rsl (D). Moreover, lim (∇˜ vl )(x) = (∇vl )(ω) = l(ω)ω. (4.3) x→ηω

4.2. ΨDO with analytical symbols m Let D ⊂ Rn be a convex domain. We say that a symbol a belongs to S1, 0, 0 (D) n if the function ξ → a(x, y, ξ) extends analytically into the tube domain R + iD and if |∂xβ ∂yβ ∂ξα a(x, y, ξ + iη)| ≤ Cαβ ξm−|α| for all triples (x, y, ξ + iη) ∈ Rn × Rn × (Rn + iD). For proofs of the following propositions see Section 4.5 in [33] and page 308 in [12], respectively. m Proposition 4.2. Let a be a symbol in S1, 0, 0 (D) and w a weight in R(D). Then

w−1 Opd (a)wI = Opd (a(x, y, ξ + iθw (x, y)))

(4.4)

m is a pseudodifferential operator in OP S1, 0, 0 , and  1 (∇v)((1 − t)x + ty) dt. θw (x, y) = 0

Proposition 4.3. Let Banach spaces X1 and Y1 be densely embedded into Banach spaces X2 and Y2 , respectively. Further let A : X2 → Y2 and A|X1 : X1 → Y1 be Fredholm operators with the same index, ind (A : X2 → Y2 ) = ind (A|X1 : X1 → Y1 ). Then every solution u ∈ X2 of the equation Au = f with f ∈ Y1 belongs already to X1 .

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Theorem 4.4. Let w be a weight in R(D) with limx→∞ w(x) = ∞, and let A := Opd (b) + ΦI be an operator which satisfies the following conditions: ˜ n ), and conditions (A) and (B) hold; (i) b ∈ S m (D), bξ−m ∈ S 0 (R 1, 0, 0

1, 0, 0

(ii) Opd (b) is an elliptic operator on Rn ; (iii) for every t ∈ [−1, 1], all limit operators Ahtw : H m, p (Rn ) → Lp (Rn ) of the operator Atw := Opd (b(x, y, ξ + itθw (x, y)) + ΦI are invertible.

If u is a function in H m, p (Rn , w−1 ) for which Au ∈ Lp (Rn , w), then u belongs to H m, p (Rn , w). Proof. Note that A : H m, p (Rn , wt ) → Lp (Rn , wt ) is a Fredholm operator if and only if w−t Awt I : H m, p (Rn ) → Lp (Rn ) is a Fredholm operator, and that the Fredholm indices of these operators coincide. The conditions of the theorem guarantee that w−t Awt I : H m, p (Rn ) → Lp (Rn ) is a Fredholm operator for every t ∈ [−1, 1]. The representation (4.4) and the estimate (3.1) for the norm of pseudodifferential operators imply that the mapping [−1, 1]  t → w−t Awt I : H m, p (Rn ) → Lp (Rn ) is norm continuous. Thus, the Fredholm index of the operators w−t Awt I : H m, p (Rn ) → Lp (Rn ) is independent of t ∈ [−1, 1], and so is the index of the operators A : H m, p (Rn , wt ) → Lp (Rn , wt ). Hence, ind (A : H m, p (Rn , w−1 ) → Lp (Rn , w−1 )) = ind (A : H m, p (Rn , w) → Lp (Rn , w)). Moreover, the embedding of H m, p (Rn , w) into H m, p (Rn , w−1 ) is dense. Proposition 4.3 implies that all solutions of the equation Au = f with f ∈ Lp (Rn , w), which a priori lie in H m, p (Rn , w−1 ), already belong to H m, p (Rn , w).
Corollary 4.5. Let the conditions of Theorem 4.4 hold for every t ∈ [0, 1]. If u ∈ Lp (Rn ) is a solution of the equation Au = 0 then u ∈ H m, p (Rn , w). For a proof of the corollary, note that the uniform ellipticity of A implies that every solution u ∈ Lp (Rn ) of the equation Au = 0 already belongs to H m, p (Rn ). The next theorem follows from Corollary 4.5 if one takes into account that lim ∇vl (x + gk ) = l(ω)ω

gk →ηω

for every sequence g tending to ηω . Write sppdis A for the discrete spectrum of A.

Essential Spectra of ΨDO

371

m Theorem 4.6. Let the operator A = Opd (b) + ΦI be such that b ∈ S1, 0, 0 (D), −m 0 n ˜ ), and Opd (b) is uniformly elliptic. Let further conditions (A) bξ ∈ S1, 0, 0 (R and (B) be satisfied. Moreover, let Opd (bg ) + Φg I be a limit operator of A defined by the sequence g, and let l be a positive C ∞ -function on the unit sphere S n−1 . If λ ∈ sppdis A, but

λ∈ / spp (Opd (bg (x, y, ξ + itl(ω)ω)) + Φg I) for every ω ∈ S n−1 , every t ∈ [0, 1] and every sequence g tending to ηω , then every x λ-eigenfunction of A lies in H m, p (Rn , el( |x| )|x| ). 4.3. ΨDO with slowly oscillating symbols Here we introduce a class of pseudodifferential operators for which the statements of Theorems 4.4 – 4.6 can be formulated more explicitly. We say that a double m symbol a ∈ S1, 0, 0 is slowly oscillating if lim

sup

x→∞ ξ∈Rn , y∈K

|∂xβ ∂ξα a(x, x + y, ξ)|ξ−m = 0

for all multi-indices α, β = 0 and for every compact subset K of Rn . We denote m m the class of these symbols by SO1, 0, 0 and write OP SO1, 0, 0 for the corresponding class of pseudodifferential operators. Further we set ˜ n ) := OP SOm ∩ OP S m (R ˜ n ). OP SOm (R 1, 0, 0

1, 0, 0

1, 0, 0

We consider perturbed pseudodifferential operators of the form A = Opd (b) + ΦI m ˜n where Opd (b) ∈ OP SO1, 0, 0 (R ) for some m > 0 and where Φ := Φ1 + Φ2 with −m a compact operator and Φ2 ∈ SO(Rn ). Φ1 D Let h : N → Zn be a sequence tending to infinity. There are a subsequence g of h and a function bg on Rn such that

b(x + gk , y + gk , ξ) → bg (ξ) ∞

(4.5)

in the topology of the space C (R × R × R ) (see [33], p. 228, for details). Since the values bg (ξ) of the limit function are independent of x and y, the limit m ˜n operators of Opd (b) ∈ OP SO1, 0, 0 (R ) have symbols which are independent of x and y, too. Similarly, the limit operators of the potential Φ are operators of multiplication by the constants n

n

n

Φg2 := lim Φ2 (x + gk ) = lim Φ2 (gk ). k→∞

(4.6)

k→∞

Thus, all limit operators Ag := Op(bg (ξ)) + Φg2 of A are invariant with respect to shifts. One can prove that then the spectrum of the limit operator Ag , considered as an unbounded operator on Lp (Rn ) with domain H m, p (Rn ), is independent of p ∈ (1, ∞) and that spAg = {λ ∈ C : λ = bg (ξ) + Φg2 , ξ ∈ Rn }. This implies the following result.

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m ˜n Theorem 4.7. Let A = Opd (b)+ΦI where Opd (b) ∈ OP SO1, 0, 0 (R ) for some m > −m being compact on 0 is a uniformly elliptic operator, Φ := Φ1 + Φ2 with Φ1 D Lp (Rn ) and Φ2 ∈ SO(Rn ). Then the essential spectrum of A, the latter considered as an unbounded operator on Lp (Rn ) with domain H m, p (Rn ), is independent of p ∈ (1, ∞), and  spess A = {λ ∈ C : λ = bg (ξ) + Φg2 , ξ ∈ Rn } (4.7) Ag ∈op(A)

where the constant functions Φg2 are defined by (4.6). Our next goal are estimates of the exponential decay of eigenfunctions of pseudodifferential operators with slowly oscillating symbols acting on spaces with slowly oscillating weights. m m −m Theorem 4.8. Let A = Opd (b) + ΦI where b ∈ SO1, ∈ 0, 0 ∩ S1, 0, 0 (D), bξ 0 n ˜ ), and Opd (b) is a uniformly elliptic operator. Let further w = exp v be S1, 0, 0 (R a weight in Rsl (D). Finally, let λ ∈ sppdis A and uλ be a λ-eigenfunction of A. If g inf |bg (ξ + itθw ) + Φg2 − λ| > 0

ξ∈Rn

for every sequence g tending to infinity such that the limits (4.5), (4.6), and g θw := lim (∇v)(x + gk ) k→∞

exist, then uλ lies in Lp (Rn , wl ). Let l be a positive C ∞ -function on the unit sphere S n−1 such that Ωl ⊂ D, x and let wl := exp vl where vl (x) = l( |x| )|x|. We modify wl in a neighborhood of the origin to obtain a smooth weight w ˜l belonging to Rsl (D). Then Theorem 4.8 implies the following result. m m −m Theorem 4.9. Let A = Opd (b) + ΦI where b ∈ SO1, ∈ 0, 0 ∩ S1, 0, 0 (D), bξ p 0 n ˜ ), and Opd (b) is uniformly elliptic. Let λ ∈ sp A, and let uλ be a λS1, 0, 0 (R dis eigenfunction of A. For every t ∈ [0, 1], every ω ∈ S n−1 , and for every sequence g tending to ηω such that the limits (4.5) and (4.6) exist, let

inf |bg (ξ + itl(ω)ω) + Φg2 − λ| > 0.

ξ∈Rn

Then uλ ∈ Lp (Rn , w ˜l ).

5. Schr¨ odinger operators 5.1. Essential spectra In this concluding section we will thoroughly have p = 2. Hence, the subscript p is often omitted. Consider the electro-magnetic Schr¨odinger operator H := (i∂xj − aj )ρjk (i∂xk − ak ) + ΦI

Essential Spectra of ΨDO

373

on Rn equipped with the Riemann metric ρjk such that inf

x∈Rn , ω∈S n−1

ρjk (x)ω j ω k > 0.

(5.1)

Here we use again the Einstein summation convention, and ρjk (x) refers to the tensor inverse to ρjk (x). Condition (5.1) implies the uniform ellipticity of H. We further suppose that (I) the entries ρjk of the metric tensor are real-valued functions in SO∞ (Rn ), (II) the magnetic potential (a1 , a2 , . . . , an ) is a real-valued vector function with components aj in SO∞ (Rn ), (III) the electric potential Φ is a complex-valued function such that Φ(I − ∆)−1 is a locally compact operator on L2 (Rn ) which belongs to W $ (Rn ). We consider H as a closed unbounded operator on L2 (Rn ) with domain H 2 (Rn ). Theorem 3.7 then implies  spHg (5.2) spess H = Hg ∈op(H)

where g g Hg = (i∂xj − agj I)ρjk g (i∂xk − ak ) + Φ I, jk ρjk g := lim ρ (gl ) ∈ R, l→∞

agk := lim ak (gl ) ∈ R, l→∞

(5.3)

and where Φg I is the limit operator of ΦI with respect to the sequence g. g For ag = (ag1 , . . . , agn ) ∈ Rn , set (Tag u)(x) := ei(a , x) u(x). The so-defined operator Tag is a unitary operator on L2 (Rn ). Consequently, the limit operator Hg is unitarily equivalent to the operator jk g jk g  Tag Hg Ta−1 g = −∂xj ρg ∂xk + Φ I = −ρg ∂xj ∂xk + Φ I =: Hg

(5.4)

with zero magnetic potential. This implies that  spess H = sp Hg Hg ∈op(H)

and that the essential spectrum of H is independent of the magnetic field. Let Φ = Φ1 + Φ2 where Φ2 (I − ∆)−1 is a compact operator on L2 (Rn ) g and where Φ1 ∈ SO(Rn ). Then the limit operators Hg = −ρjk g ∂xj ∂xk + Φ1 I are operators with constant coefficients, whence  g n (5.5) spess H = {λ ∈ R : λ = ρjk g ξj ξk + Φ1 , ξ ∈ R }. g

Let ΓΦ1 ⊂ C denote the set of all partial limits at infinity of the function Φ1 . It turns out that ΓΦ1 is a closed connected set of the complex plane. Identity (5.5) implies the following.

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Theorem 5.1. (i) spess H = ΓΦ1 + [0, ∞) ⊆ C. (ii) If the function Φ1 is real-valued, then spess H = [lim inf Φ1 (x), +∞). x→∞

5.2. Exponential estimates of eigenfunctions of the discrete spectrum Let conditions (I), (II), (III) hold and assume that the electric potential Φ is real-valued. We will also suppose that the Schr¨ odinger operator H as well as its limit operators are self-adjoint (see, for instance, [9] and Chapter X in [34]). From Proposition 3.9 we conclude that the limit operator (wt Hw−t )g of t w Hw−t is unitarily equivalent to the operator g g 2 g 2 jk g := −ρjk Htw g ∂xj ∂xk − t |(∇v) |ρg − ρg (∇v)j ∂xk + Φ I

(5.6)

where (∇v)g := lim (∇v)(gk ) k→∞

(∈ Rn )

(5.7)

and g g g g |(∇v)g |2ρg := ρjk g (∇v)j (∇v)k = (ρg (∇v) , (∇v) )

for ρg :=

(5.8)

n (ρjk g )j, k=1 .

Theorem 5.2. Let conditions (I), (II), (III) hold and let Φ = Φ1 + Φ2 where Φ1 ∈ SO(Rn ) and Φ2 (I − ∆)−1 is a compact operator on L2 (Rn ). Further let w = exp v be a weight in Rsl such that ; |(∇v)g |ρg < Φg1 − λ (5.9) for every sequence g tending to infinity. If λ is an eigenvalue of H such that λ < mΦ1 := lim inf Φ1 (x), x→∞

then every λ-eigenfunction of H belongs to H 2 (Rn , w). Proof. The essential spectrum of H is the interval [mΦ1 , ∞). Thus, λ ∈ spdis H. g is an operator with constant Since Φ1 is slowly oscillating, the limit operator Htw coefficients which is, hence, unitarily equivalent to the operator of multiplication by its symbol g g jk jk g 2 g 2 O H tw (ξ) := −ρg ξj ξk − iρg (∇v)j ξk + Φ − t |(∇v) |ρg .

From (5.6), (5.8) and (5.9) we obtain that there is an ε > 0 such that g g g 2 O R(H tw (ξ) − λ) ≥ |(∇v) |ρg + Φ1 − λ ≥ ε.

(5.10)

g g This estimate implies that λ ∈ / spHtw for every limit operator Htw of Htw and for every t ∈ [0, 1]. Thus, the assertion follows from Corollary 4.5.


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Next we introduce a function on S n−1 which describes the local distribution of the essential spectrum of the operator H by µH (ω) :=

inf

inf

Hg ∈opηω (H) {u∈H 2 (Rn ):uL2 (Rn ) =1}

(Hg u, u)

(5.11)

where Hg is given by (5.4). We further set µ ˆH :=

inf

ω∈S n−1

µH (ω).

The self-adjointness of Hg implies that sp Hg ⊆ [µH (ω), ∞), whence spess H ⊆ [ˆ µH , +∞). Theorem 5.3. Let conditions (I), (II), (III) hold. Moreover assume that the operator H and all limit operators of H are self-adjoint. Assume also that the n−1 n limits limx→ηω ρjk (x) =: ρjk and set ρω := (ρjk ω exist for every ω ∈ S ω )j, k=1 . ∞ n−1 for which Let finally d be a positive C -function on S ? µH (ω) − λ (5.12) d(ω) < (ρω ω, ω) for all ω ∈ S n−1 . If λ is an eigenvalue of H with λ < µ ˆH , then every associ˜d ) where wd (x) := exp vd (x) = ated λ-eigenfunction of H belongs to H 2 (Rn , w x exp(d( |x| )|x|). Proof. For every sequence g tending to ηω and for every u ∈ H 2 (Rn ), g − λI)u, u)L2 (Rn ) R ((Htw d

≥ (Hg − (λ + |(∇vd )(ω)|2ρω )I)u, u)L2 (Rn ) ≥ (µH (ω) − λ − d2 (ω)(ρω ω, ω))uL2(Rn ) .

In view of condition (5.12) there exists an ε > 0 such that g − λI)u, u)L2 (Rn ) ≥ εu2L2 (Rn ) R ((Htw d

(5.13)

for every u ∈ H 2 (Rn ). Estimate (5.13) implies g − λI)uL2 (Rn ) ≥ εuL2(Rn ) (Htw d

(5.14)

g − λI)∗ uL2 (Rn ) ≥ εuL2 (Rn ) (Htw d

(5.15)

and for every u ∈ H (R ). From estimate (5.14) we conclude that 2

n

g − λI : H 2 (Rn ) → L2 (Rn ) Htw d g R(Htw d

(5.16)

− λI) is closed. Indeed, for is an injective operator and that its range g 2 − λI)u → f where u ∈ H (Rn ). By estimate (5.14), the closedness, let (Htw n n d g 2 n − λI is (un ) is a Cauchy sequence in L (R ). Let u denote its limit. Since Htw d g 2 n a closed operator, one has u ∈ H (R ) and (Htwd − λI)u = f . Hence, f is in g g − λI. Similarly, estimate (5.15) implies that ker(Htw − λI)∗ ∩ the range of Htw d d g 2 n ∗ H (R ) = {0}. The uniform ellipticity of the operator (Htwd − λI) implies that

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g every solution u ∈ L2 (Rn ) of the equation (Htw − λI)∗ u = 0 belongs to H 2 (Rn ). d g ∗ Thus, ker(Htwd − λI) = {0}. Further, g g g R(Htw − λI) = R(Htw − λI) = (ker(Htw − λI)∗ )⊥ = L2 (Rn ). d d d

Hence, the operator (5.16) is bijective and possesses a bounded inverse (defined on g for every sequence all of L2 (Rn )). This implies that λ is a resolvent point of Htw d ˜d ).
g. In view of Corollary 4.5, every λ-eigenfunction of H belongs to H 2 (Rn , w Theorem 5.3 associates to each eigenvalue λ(< µ ˆH ) and to the distribution function µH a weight wd (x) = exp(d(x/|x|)|x|) such that uλ ∈ H 2 (Rn , w ˜d ). If only the infimum µ ˆH of the essential spectrum of H is available, then one still gets a rough “isotropic” exponential decay estimate of the λ-eigenfunction uλ , namely uλ ∈ H 2 (Rn , w ˜d ) where ? µ ˆH − λ d|n| . and 0 < d < wd (x) = e supω∈S n−1 (ρω ω, ω) 5.3. Multi-particle Schr¨ odinger operators We consider an atomic type system of N + 1 particles with coordinates xi ∈ R3 , 0 ≤ i ≤ N , and interacting real-valued potentials ϕij , 0 ≤ i < j ≤ N , defined on R3 . The particle x0 (considered as the “nucleus”) has infinite mass and is fixed at x0 = 0. We assume that the functions ϕij are subject to the following conditions: (i) every function ϕij is the sum of a function ϕij ∈ L2 (R3 ) and a function ϕij such that ϕij (I − ∆)−1 is a compact operator on L2 (R3 ); (ii) ϕij ≥ 0 for 0 ≤ i < j ≤ N . The configuration space of the system consists of the product of N copies of R3 which we identify with R3N and denote by X. The generic point in X is written as x = (x1 , . . . , xN ) where the xi = (xi1 , xi2 , xi3 ) are the coordinates of the particles. The Schr¨odinger operator of the system is the elliptic operator H on X defined by H=−

N N

1 ∆i − ϕ0j (xj ) + 2m i i=1 j=1

N


ϕij (xi − xj )

(5.17)

1≤i<j≤N

where ∆i are usual Laplace operator on R3 with respect to the variable xi and the mi are positive numbers representing the mass of the particles. We consider potentials Φ which are functions on X of the form
Φ(x) = Φij (x) 0≤i<j≤N



where

if i = 0, 1 ≤ j ≤ N, ϕ0j (xj ) ϕij (xi − xj ) if 1 ≤ i < j ≤ N. Conditions (i), (ii) provide the self-adjointness of H as an unbounded operator on L2 (R3N ) with domain H 2 (R3N ) (see, for instance, [34], Chap. X). Moreover, in view of Proposition 3.9, Φ(I − ∆)−1 ∈ W $ (R3N ). Φij (x) =

Essential Spectra of ΨDO

377

Let r ≤ N . For each r-tuple (i1 , . . . , ir ) of integers 1 ≤ i1 < . . . < ir ≤ N , we let Σ(i1 , ..., ir ) stand for the set of all ω ∈ S 3N −1 with ω k = 0 if k ∈ {i1 , . . . , ir } / {i1 , . . . , ir }. Further set and ω k = 0 for k ∈  
1 Ai1 , ..., ir := − ∆j + ϕ0j (xj ) 2mj 1≤j≤N, j =i1 , ..., ir
ϕij (xi − xj ). + 1≤i<j≤N, i =i1 , ..., ir , j =i1 , ... ir

Recall the definition of the local distribution function µH (ω) from (5.11). Proposition 5.4. (i) If 1 ≤ r ≤ N − 1 and ω ∈ Σ(i1 , ..., ir ) , then µH (ω) = inf sp Ai1 , i2 , ..., ir .

(5.18)

and µH (ω) ≤ 0. (ii) If r = N and ω ∈ Σ(1, ..., N ) , then µH (ω) = 0. 3N which tends to ηω . Proof. (i) Let h = (hm ) = (h1m , . . . , hN m ) be a sequence in Z We distinguish two cases.

Case (a): The coordinate sequences (him1 ), . . . (himr ) tend to infinity, whereas all other coordinate sequences (hjm ) with j ∈ / {i1 , . . . , ir } remain bounded. Denote by Li1 , ..., ir the set of all pairs (iµ , iν ) with iµ , iν ∈ {i1 , . . . , ir } for i i which the sequence (hmµ − himν ) is bounded. We can assume that limm→∞ (hmµ − iν / Li1 , ..., ir (otherwise we pass to a suitable subsehm ) = ∞ for all pairs (iµ , iν ) ∈ quence of h). Then there is a subsequence of h, for which the limit operator of H exists and such that this limit operator is unitarily equivalent to the self-adjoint operator

1 Hh := − ∆j − ϕ0j (xj ) 2mj 1≤j≤N 1≤j≤N, j =i1 , ..., ir
ϕij (xi − xj ) + 1≤i<j≤N, i =i1 , ..., ir , j =i1 , ..., ir

+




ϕij (xi − xj ).

(i, j)∈Li1 , ..., ir

The non-negativity of the potentials ϕij implies that the spectrum of Hh is contained in the spectrum of the operator

1 Hh := − ∆j − ϕ0j (xj ) 2mj 1≤j≤N 1≤j≤N, j =i1 , ..., ir
ϕij (xi − xj ). + 1≤i<j≤N, i =i1 , ..., ir , j =i1 , ..., ir

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˜ g of Hh which For the reverse inclusion, note that there exists a limit operator H  ˜ equals Hh . Theorem 3.7 implies that sp Hh ⊇ spess Hh ⊇ sp Hg = sp Hh . Hence, the spectra of Hh and Hh coincide. Case (b): Again we assume that the coordinate sequences (him1 ), . . . (himr ) tend to infinity, but now we also allow that the sequences (hjm1 ), . . . (hjml ) tend to infinity for some indices j1 , . . . , jl ∈ / {i1 , . . . , ir }. Since the potentials ϕij (xj ) are non-negative, the inclusion sp Hh ⊆ sp Hh follows as in case (a). Applying the Fourier transform with respect to the variables xk , k = i1 , . . . , ir we get that Hh is unitarily equivalent to the operator of multiplication by the operator-valued function which is defined at (ξi1 , . . . , ξir ) ∈ R3r by / (ξi , . . . , ξi ) := H 1 r h

1 2 1 2 ξ + ...+ ξ + Ai1 , ..., ir . 2mi1 i1 2mir ir

(5.19)

It follows from (5.19) that sp Hh = sp Hh = [inf sp Ai1 , ..., ir , +∞)  the semifor 1 ≤ r ≤ N − 1. Note that the spectrum of the operator  Hh contains 1 ∆ of Hh the axis [0, +∞). Indeed, there exists a limit operator − 1≤j≤q 2m j j spectrum of which is [0, +∞), and by Theorem 3.7, sp Hh ⊇ spess Hh ⊃ [0, +∞). Thus, µH (ω) = inf sp Ai1 , ..., ir ≤ 0. 3N Let h = (hm ) = (h1m , . . . , hN which tends to ηω for some m ) be a sequence in Z point ω ∈ Σ(1, ..., N ) . Then each of the sequences (h1m ), . . . , (hN m ) tends to infinity. Let U be the set of all pairs (j, k) for which the sequence (hjm − hkm ) is bounded. Then there exists a subsequence of h for which the limit operator of H exists and is unitarily equivalent to the operator

1 BU := − ∆j + ϕij (xi − xj ). 2mj 1≤j≤N

(i, j)∈U

Again, the non-negativity of the potentials ϕij implies that sp BU ⊆ [0, +∞). For the reverse inclusion, note that there exists a limit operator of BU which is equal  1 to − 1≤j≤q 2m ∆j and which, thus, has the interval [0, +∞) as its spectrum. j Hence, sp BU = [0, +∞), implying that µH (ω) = 0.
As a corollary we obtain the Hunziker, van Winter, Zjislin theorem (see, e.g., [8], 3.3.3) on the location of the essential spectrum of multi-particle Schr¨ odinger operators. Theorem 5.5 (HWZ theorem). The infimum µ ˆH of the essential spectrum of the multi-particle Schr¨ odinger operator H is µ ˆH =

inf

1≤i1 <...
inf sp Ai1 , ..., ir .

Essential Spectra of ΨDO

379

The following theorem verifies the exponential decay of the eigenfunctions of the Hamiltonian H. It is a straightforward consequence of Theorem 5.3 and Proposition 5.4. Theorem 5.6. Let λ < µ ˆH < 0 be an eigenvalue of H and l : S n−1 → R be a ∞ positive C -function such that ? µH (ω) − λ l(ω) < (ρω, ω)  1 1 1 1 , ..., , ..., , ..., 2m1 2m1 2mN 2mN is a block-diagonal matrix with each value occurring 3 times, and µH (ω) is given ˜l ) by (5.11) and (5.18). Then each λ-eigenfunction uλ of H belongs to H 2 (Rn , w where wl (x) = el(x/|x|)|x|. 

where

ρ = diag

5.4. Perturbed periodic Schr¨ odinger operators Here we are going to examine Schr¨odinger operators HΦ = −∆ + ΦI with a real-valued, continuous and Zn -periodic function Φ, called the electric potential. The operator HΦ is a self-adjoint operator on L2 (Rn ) with domain H 2 (Rn ). We will also consider different kinds of perturbations of the operator HΦ . We will need the following proposition.  Proposition 5.7. Let P := |α|≤m aα Dα be a uniformly elliptic differential operator with continuous and Zn -periodic coefficients aα , which we consider as an unbounded closed operator on L2 (Rn ) with domain H 2 (Rn ). Then (i) spP g = spP for every limit operator P g ∈ op(P ), (ii) spess P = spP . Proof. First we prove (i) and (ii) for the operator P , thought of as a bounded operator from H m (Rn ) to L2 (Rn ). For (i), let the sequence g define the limit operator P g of P . By the periodicity of the coefficients aα , lim P g − V−gk AVgk H 2 (Rn )→L2 (Rn ) = 0.

k→∞

This equality implies that if λ ∈ / sp2 P g , then λ ∈ / sp2 P . Hence, sp2 P ⊆ sp2 P g . 2 2 2 g 2 2 g But sp P ⊇ spess P ⊇ sp P , whence sp P = sp P . Assertion (ii) now follows from (i) and from the identity  sp2ess P = sp2 P g . P g ∈op(A)

The corresponding statements (i) and (ii) for P as an unbounded operator follow from the discussion before Theorem 3.7.


380

V.S. Rabinovich and S. Roch Let us start with the unperturbed periodic operator HΦ := −∆ + ΦI = P (x, D).

This operator can be treated by means of the Floquet transform
(F f )(x, k) := Vα (f e−i(x, k) ) for x, k ∈ Rn . α∈Zn

The function Ff is Zn -periodic with respect to the first variable x and satisfies a cyclic condition with respect to the second variable k (also known as the quasiimpulse), (F f )(x, k + γ) = e−i(γ, x) (F f )(x, k) for all γ ∈ 2πZn and x, k ∈ Rn . The Floquet transform acts as a unitary operator from L2 (Rn ) to L2 (T∗ , L2 (T)) with T∗ := Rn /2πZn referring to the so-called Brillouin zone and T := Rn /Zn , and its inverse is given by  1 −1 v(x, k)ei(x, k) dk (F v)(x) = (2π)n [0, 2π]n where the function v(x, k) ∈ L2 (T∗ , L2 (T)) is continued to a Zn -periodic function with respect to the variable x ∈ Rn . A straightforward calculation yields F P (x, D)F −1 = P (x, Dx + k). Hence, the operator P (x, D), considered as an unbounded operator on L2 (Rn ), is unitarily equivalent to the “direct integral”  P (x, Dx + k) dk T∗

of unbounded self-adjoint operators P (x, Dx + k) on L2 (T) with domain C 2 (T). The operators P (x, Dx + k) have a discrete and real spectrum for every k ∈ T∗ . For k ∈ T∗ , let λj (k), j ≥ 1, refer to the eigenvalues of the operator P (x, Dx + k). It is a basic property of spectra of periodic Schr¨ odinger operators that spHΦ = spess HΦ =

∞ 

[νj (HΦ ), µj (HΦ )]

j=1

where νj (HΦ ) :=

min

k∈[0, 2π]n

λj (k),

µj (HΦ ) :=

max

k∈[0, 2π]n

λj (k)

(5.20)

and νj (HΦ ) → ∞ as j → ∞ (see [35], Section XIII.16, and [15]). Hence, the spectrum (= the essential spectrum) of a periodic Schr¨ odinger operator is the union of the intervals [νj (HΦ ), µj (HΦ )], which are called the spectral bands of HΦ . The existence of spectral gaps, i.e., of intervals which do not intersect any of the spectral bands is what makes a crystal a semi-conductor. Note that there is also a semi-infinite spectral gap (−∞, ν(HΦ )), where ν(HΦ ) := inf 1≤j≤∞ νj (HΦ ) denotes the infimum of the spectrum of HΦ .

Essential Spectra of ΨDO

381

Next we consider potentials Φ + Ψ where Φ is as above and Ψ is a measurable function on Rn such that HΦ+Ψ is a self-adjoint operator and Ψ(I − ∆)−1 is compact on L2 (Rn ). Then, clearly, spess HΦ+Ψ = spess HΦ , but points of the discrete spectrum of HΦ+Ψ can appear in spectral gaps of the operator HΦ . The following is a corollary of Theorem 5.3. Theorem 5.8. Let λ ∈ (−∞, ν(HΦ )) be an eigenvalue of HΦ+Ψ and uλ an associated eigenfunction. Then uλ ∈ H 2 (Rn , w ˜d ) for every positive number d < ν(H) − λ where wd (x) = ed|x| . Finally we consider potentials Φ + Ψ + Ω where Φ and Ψ are as above and Ω is a real-valued function in SO(Rn ). The Schr¨odinger operator HΦ+Ψ+Ω is selfadjoint. For a ∈ L∞ (Rn ) and ω ∈ S n−1 , set ma (ω) := lim inf a(x), x→ηω

ma :=

inf

ω∈S n−1

Ma (ω) := lim sup a(x), x→ηω

ma (ω),

Ma :=

inf

ω∈S n−1

Ma (ω).

Theorem 3.7 implies the following description of the essential spectrum of the perturbed periodic Schr¨ odinger operator. Note that spess HΦ+Ψ+Ω = spess HΦ+Ω . Theorem 5.9. Under the above conditions, spess HΦ+Ψ+Ω =

∞ 

[νj (Φ) + mΩ , µj (Φ) + MΩ ].

j=1

Corollary 5.10. Let (µj (Φ), νj+1 (Φ)) be a spectral gap of the operator HΦ . If νj+1 (Φ) − µj (Φ) > MΩ − mΩ , then the interval (µj (Φ) + MΩ , νj+1 (Φ) + mΩ ) is a spectral gap in the essential spectrum of HΦ . In the opposite case, if νj+1 (Φ) − µj (Φ) ≤ MΩ − mΩ , then the spectral gap (µj (Φ), νj+1 (Φ)) of the unperturbed operator HΦ is contained in the spectrum of the perturbed operator HΦ+Ψ+Ω . (Thus, the perturbation Ω closes the spectral gap (µj (Φ), νj+1 (Φ)).) The following is a direct consequence of Theorem 5.3. Theorem 5.11. Let λ < ν(HΦ ) + mΩ ) be an eigenvalue of HΦ+Ψ+Ω located below the essential spectrum. Further, let d : S n−1 → R be a positive smooth function and d(ω) < ν(HΦ ) + mΩ (ω) − λ for every ω ∈ S n−1 . ˜d ). Then each λ-eigenfunction of HΦ+Ψ+Ω belongs to H 2 (Rn , w

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The problem of the exponential estimates for eigenfunctions of perturbed periodic operators is more involved if the eigenvalues are located in the spectral gaps (µj , νj+1 ) of the unperturbed periodic operator. Consider the operator HΦ+Ψ . The limit operators of the operator e−tw˜d (x) HΦ+Ψ etw˜d (x) with respect to a sequence g tending to ηω are of the form Agω, t, d := (i∇ + itd(ω)ω)2 + Φg I where lim sup |Φ(x + gk ) − Φg (x)| = 0

k→∞ x∈Rn

due to the periodicity of Φ. From Proposition 5.7 we infer that the spectrum of Agω, t, d equals the spectrum of the operator Aω, t, d := (i∇ + itd(ω)ω)2 + ΦI. Hence, Corollary 4.5 yields the following result. Theorem 5.12. Let λ be an eigenvalue of the operator HΦ+Ψ which belongs to the spectral gap (µj (Φ), νj+1 (Φ)) of the periodic operator HΦ . Further, let d : S n−1 → R be a positive smooth function such that inf

t∈[0, 1], ω∈S n−1

|λ − spAω, t, d | > 0.

Then each λ-eigenfunction of HΦ+Ψ belongs to H 2 (Rn , w ˜d ).

References [1] S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Princeton University Press, Princeton, 1982. [2] S. Agmon, Spectral properties of Schr¨ odinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4(1975), 151–218. [3] M.S. Agranovich, Elliptic Operators on Closed Manifolds. Modern Problems of Mathematics 63, Partial Differential Equations 6, VINITI, 1990 (Russian). [4] W. Amerin, M. M˘ antoiu, R. Purice, Propagation properties for Schr¨ odinger operators affiliated with certain C ∗ -algebras. Ann. H. Poincar´e In-t 6(2002), 3, 1215–1232. [5] M.Sh. Birman, The discrete spectrum of the periodic Schr¨ odinger operator perturbed by a decreasing potential. Algebra i Analiz 8(1996), 3-20. Russian, Engl. transl.: St. Petersburg Math. J. 8(1997), 1–14. [6] M.Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schr¨ odinger operator I. Regular perturbations. In: Boundary Value Problems, Schr¨ odinger Operators, Deformation Quantization, Math. Top. 8, Akademie-Verlag, Berlin 1995, p. 334–352. [7] M.Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schr¨ odinger operator II. Nonregular perturbations. Algebra i Analiz 9(1997), 62–89. Russian, Engl. transl.: St. Petersburg Math. J. 9(1998), 1073–1095.

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[8] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin, Heidelberg, New York 1987. [9] E.B. Davies, Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics 42, Cambridge University Press, Cambridge 1995. [10] R. Froese, I. Herbst, Exponential bound and absence of positive eigenvalue for N-body Schr¨ odinger operators. Comm. Math. Phys. 87(1982), 429–447. [11] R. Froese, I. Herbst, M. Hoffman-Ostenhof, T. Hoffman-Ostenhof, L2 -exponential lower bound of the solutions of the Schr¨ odinger equation. Comm. Math. Phys. 87(1982), 265–286. [12] I. Gohberg, I. Feldman, Convolution Equations and Projection Methods for Their Solution. Nauka, Moskva 1971. Russian, Engl. transl.: Amer. Math. Soc. Transl. Math. Monographs 41, Providence, R.I., 1974. [13] V. Georgescu, A. Iftimovici, Crossed Products of C ∗ -Algebras and Spectral Analysis of Quantum Hamiltonians. Comm. Math. Phys. 228(2002), 519–560. [14] V. Georgescu, A. Iftimovici, Localization at infinity and essential spectrum of quantum Hamiltonians. arXiv:math-ph/0506051V1, June 20, 2005. [15] P. Kuchment, On some spectral problems of mathematical physics. In: Partial Differential Equations and Inverse Problems, C. Conca, R. Manasevich, G. Uhlmann, M. S. Vogelius (Editors), Contemp. Math. 362, Amer. Math. Soc. 2004. [16] S. Lang, Real and Functional Analysis. Graduate Texts in Mathematics 142, Springer, New York 1993 (third ed.). [17] Y. Last, B. Simon, The essential spectrum of Schr¨ odinger, Jacobi, and CMV operators. Preprint 304 at http://www.math.caltech.edu/people/biblio.html [18] P.D. Lax, Functional Analysis. Wiley-Interscience, 2002. [19] Ya.A. Luckiy, V.S. Rabinovich, Pseudodifferential operators on spaces of functions of exponential behavior at infinity. Funct. Anal. Prilozh. 4(1977), 79–80. [20] M. M˘ antoiu, Weighted estimations from a conjugate operator. Letter in Math. Physics 51(2000), 17–35. [21] M. M˘ antoiu, C ∗ -algebras, dynamical systems at infinity and the essential spectrum of generalized Schr¨ odinger operators. J. Reine Angew. Math. 550(2002), 211–229. [22] M. M˘ antoiu, R. Purice, A priori decay for eigenfunctions of perturbed Periodic Schr¨ odinger operators. Preprint Universit´e de Gen`eve, UGVA-DPT 2000/02-1071. [23] A. Martinez, Microlocal exponential estimates and application to tunnelling. In: Microlocal Analysis and Spectral Theory, L. Rodino (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences Vol. 490, 1996, p. 349–376. [24] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer, New York 2002. [25] S. Nakamura, Agmon-type exponential decay estimates for pseudodifferential operators. J. Math. Sci. Univ. Tokyo 5(1998), 693–712. [26] V.S. Rabinovich, Pseudodifferential operators with analytic symbols and some of its applications. Linear Topological Spaces and Complex Analysis 2, Metu-T¨ ubitak, Ankara 1995, p. 79–98.

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[27] V. Rabinovich, Pseudodifferential operators with analytic symbols and estimates for eigenfunctions of Schr¨ odinger operators. Z. f. Anal. Anwend. (J. Anal. Appl.) 21(2002), 2, 351–370. [28] V.S. Rabinovich, On the essential spectrum of electromagnetic Schr¨ odinger operators. In: Contemp. Math. 382, Amer. Math. Soc. 2005, p. 331–342. [29] V.S. Rabinovich, Essential spectrum of perturbed pseudodifferential operators. Applications to the Schr¨ odinger, Klein-Gordon, and Dirac operators. Russian J. Math. Phys. 12(2005), 1, 62–80. [30] V.S. Rabinovich, S. Roch, The essential spectrum of Schr¨ odinger operators on lattices. J. Phys. A: Math. Gen. 39(2006), 8377–8394. [31] V.S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators. Integral Eq. Oper. Theory 30(1998), 4, 452–495. [32] V.S. Rabinovich, S. Roch, B. Silbermann, Band-dominated operators with operatorvalued coefficients, their Fredholm properties and finite sections. Integral Eq. Oper. Theory 40(2001), 3, 342–381. [33] V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory. Operator Theory: Adv. and Appl. 150, Birkh¨ auser, Basel, Boston, Berlin 2004. [34] M. Reed, B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Selfadjointness. Academic Press, New York, San Francisco, London 1975. [35] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York, San Francisco, London 1978. [36] M.A. Shubin, Pseudodifferential Operators and Spectral Theory. Springer, New York 2001 (second ed.). [37] B. Simon, Semiclassical analysis of low lying eigenvalues II. Tunnelling. Ann. Math. 120(1984), 89–118. [38] E.M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, New Jersey, 1993. [39] M.E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey, 1981. Vladimir S. Rabinovich Instituto Politechnico National ESIME-Zacatenco, Ed.1, 2-do piso Av. IPN, Mexico, D.F., 07738 Mexico e-mail: [email protected] Steffen Roch Technische Universit¨ at Darmstadt Fachbereich Mathematik Schlossgartenstrasse 7 64289 Darmstadt Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 385–391 c 2008 Birkh¨
auser Verlag Basel/Switzerland

On Finite Sections of Band-dominated Operators Vladimir S. Rabinovich, Steffen Roch and Bernd Silbermann Abstract. In an earlier paper we showed that the sequence of the finite sections Pn APn of a band-dominated operator A on lp (Z) is stable if and only if the operator A is invertible, every limit operator of the sequence (Pn APn ) is invertible, and if the norms of the inverses of the limit operators are uniformly bounded. The purpose of this short note is to show that the uniform boundedness condition is redundant. Mathematics Subject Classification (2000). Primary 47N40; Secondary 47L40, 65J10. Keywords. Band-dominated operators, finite sections, stability, limit operators.

1. Introduction Let 1 < p < ∞. We will work Banach space lp (Z+ ) of all sequences (xn )∞ n=0  on the p of complex numbers with |xn | < ∞. We provide this space with its standard basis which consists of all sequences ei := (0, . . . , 0, 1, 0, . . .) with the 1 standing at the ith position. Every bounded linear operator on lp (Z+ ) admits a matrix representation (aij )i,j∈Z+ with respect to the standard basis. We call an operator A ∈ L(lp (Z+ )) a band operator if the associated matrix is a band matrix, i.e., if there is a k such that aij = 0 whenever |i − j| ≥ k. The operator A is said to be band-dominated if it is the norm limit of a sequence of band operators. Let n ∈ N. The nth finite section of an operator A ∈ L(lp (Z+ )) with matrix representation (aij )i,j∈Z+ is the n × n-matrix (aij )n−1 i,j=0 . We identify this matrix with the operator Pn APn where Pn is the projection Pn : lp (Z+ ) → lp (Z+ ),

(x0 , x1 , . . .) → (x0 , . . . , xn−1 , 0, 0, . . .).

This work has been partially supported by CONACYT project 43432 and DFG Grant 444 MEX112/2/05.

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V.S. Rabinovich, S. Roch and B. Silbermann

The sequence (Pn APn ) of the finite sections of A is said to be stable if there is an n0 such that the operators Pn APn : im Pn → im Pn are invertible for every n ≥ n0 and if the norms of their inverses are uniformly bounded. There is an intimate relation between stability of sequences and Fredholmness of operators. For, we associate to the sequence A = (Pn APn ) the block diagonal operator (1.1) Op (A) := diag (P1 AP1 , P2 AP2 , P3 AP3 , . . .) considered as acting on lp (Z+ ) = im P1 ⊕im P2 ⊕im P3 ⊕· · · . It is an easy exercise to show that the sequence A is stable if and only if the operator Op (A) is a Fredholm operator on lp (Z+ ), i.e., if its kernel and its cokernel have finite dimension. In general, the equivalence between stability and Fredholmness seems to be of less use. But if we start with the sequence A = (Pn APn ) of the finite sections method of a band-dominated operator A, then we end up with a band-dominated operator Op (A) on lp (Z+ ) again. But for band-dominated operators on lp (Z+ ), there is a general Fredholm criterion which expresses the Fredholm property of a banddominated operator in terms of its limit operators. To state this result, we need a few notations. It will be convenient to work on the Banach space lp (Z) of the two-sided infinite sequences. The space lp (Z+ ) can be considered as a closed subspace of lp (Z) in a natural way. We let P denote the projection P : lp (Z) → lp (Z),

(xn ) → (. . . , 0, 0, x0 , x1 , x2 , . . .)

and write Q for the complementary projection I − P . Usually we will identify an operator A on lp (Z+ ) with the operator P AP acting on lp (Z). For every m ∈ Z, we introduce the shift operator Um : lp (Z) → lp (Z),

(xn ) → (xn−m ).

Let further H stand for the set of all sequences h : N → N which tend to infinity. An operator Ah ∈ L(lp (Z)) is called a limit operator of A ∈ L(lp (Z+ )) with respect to the sequence h ∈ H if U−h(n) P AP Uh(n) tends ∗ -strongly to Ah as n → ∞. Here, ∗ -strong convergence means strong convergence of the sequence itself and of its adjoint sequence. Notice that every operator can possess at most one limit operator with respect to a given sequence h ∈ H. The set σop (A) of all limit operators of a given operator A is the operator spectrum of A. Notice further that every sequence h : N → N which tends to infinity has a strongly monotonically increasing subsequence, g say, and that the existence of the limit operator Ah implies the existence of Ag and the identity Ah = Ag . Thus, it is sufficient to consider limit operators with respect to strongly monotonically increasing sequences. It is not hard to see that every limit operator of a Fredholm operator is invertible. A basic result of [2] claims that the operator spectrum of a band-dominated operator is rich enough in order to guarantee the reverse implications. Here is a

On Finite Sections of Band-dominated Operators

387

summary of the results from [2] needed in what follows. A comprehensive treatment of this topic is in [4]; see also the references mentioned there. Theorem 1.1. Let A ∈ L(lp (Z+ )) be a band-dominated operator. Then (a) every sequence h ∈ H possesses a subsequence g such that the limit operator Ag exists. (b) the operator A is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded. An elegant proof which also works for band-dominated operators on other discrete groups than Z is due to Roe [6]. Thus, and by the above mentioned equivalence between stability of the sequence A and Fredholmness of the associated operator Op (A), one will get a stability criterion for A by computing all limit operators of Op (A). This computation has been carried out in [2, 3, 5], see also Chapter 6 in [4]. Here is the result. Theorem 1.2. Let A ∈ L(lp (Z+ )) be a band-dominated operator. Then the finite sections sequence (Pn APn )n≥1 is stable if and only if the operator P AP + Q and all operators QAh Q + P with Ah ∈ σop (A) are invertible on lp (Z), and if the norms of their inverses are uniformly bounded. The goal of this note is to show that the uniform boundedness condition in Theorem 1.2 can be removed.

2. Main result For our goal, we will need a subsequence version of Theorem 1.2. We choose and fix a strongly monotonically increasing sequence η : N → N. Further, we write Hη for the set of all (infinite) subsequences of η and σop,η (A) for the collection of all limit operators of A with respect to subsequences of η. Then we have the following version of Theorem 1.2. Theorem 2.1. Let A ∈ L(lp (Z+ )) be a band-dominated operator and η : N → N a strongly monotonically increasing sequence. Then the sequence (Pη(n) APη(n) )n≥1 is stable if and only if the operator P AP + Q and all operators QAh Q + P

with

Ah ∈ σop,η (A)

are invertible on lp (Z), and if the norms of their inverses are uniformly bounded. Thus, instead of all limit operators of A with respect to monotonically increasing sequences h, only those with respect to subsequences of η are involved. The following proof of Theorem 2.1 is an adaptation of the proof of Theorem 1.2 given in [5].

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Proof. Let A ∈ L(lp (Z+ )) be a band-dominated operator, set Aη := (Pη(n) APη(n) ), and associate with the sequence Aη the block diagonal operator Op (Aη ) := =

diag (Pη(1) APη(1) , Pη(2) APη(2) , Pη(3) APη(3) , . . .) ∞
Uµ(n) Pη(n) APη(n) U−µ(n) n=1

acting on l (Z ) where µ(1) := 0 and µ(n) := η(1) + · · · + η(n − 1) for n ≥ 2, and where the series converges in the strong operator topology. It is still true that Op (Aη ) is a band-dominated operator on lp (Z+ ) and that the sequence Aη is stable if and only if the operator Op (Aη ) is Fredholm. Let h ∈ H be a sequence which tends to infinity and for which the limit operator Op (Aη )h exists. We call numbers of the form η(1) + η(2) + · · · + η(n) η-triangular and distinguish between two cases: Either there is a subsequence g of h such that the distance from g(n) to the set of all η-triangular numbers tends to infinity as n → ∞, or there are a k ∈ Z and a subsequence g of h such that g(n) + k is η-triangular for all n. The figure illustrates the shifted operator U−g(n) Op (Aη )Ug(n) in the neighborhood of its 00-entry (marked by 0). p

+

!1 !1 l(n)

l(n)

P(n) AP(n)

P(n) AP(n) 0k k

0

P(n+1) AP(n+1)

Case 1

P(n+1) AP(n+1)

Case 2

In the first case, we let ∆n denote the largest η-triangular number which is less than g(n). Then l(n) := g(n) − ∆n defines a sequence l which tends to infinity, and the limit operator Op (Aη )h = Op (Aη )g of Op (Aη ) coincides with the limit operator Al of A. Let now g be a subsequence of h such that each g(n) + k is η-triangular for some integer k. Then the sequence l defined by l(n) := g(n) + k tends to infinity, the limit operator of Op (Aη ) with respect to the sequence l exists, and Op (Aη )l = U−k Op (Aη )g Uk .

On Finite Sections of Band-dominated Operators

389

Let d(n) be the (uniquely determined) positive integer such that l(n) = η(1) + η(2) + · · · + η(d(n)). The sequence d is strongly monotonically increasing. Thus, the sequence η ◦ d is a subsequence of η and tends to infinity. Without loss we can assume that the limit operator of A with respect to the sequence η ◦ d exists (otherwise we pass to a suitable subsequence of d and, hence, of l and g). Then Op (Aη )h

=

Op (Aη )g

= =

Uk Op (Aη )l U−k Uk (QAη◦d Q + P AP )U−k .

Thus, each limit operator of Op (Aη ) is either a limit operator of A or of the form Uk (QAη◦d Q + P AP )U−k

with k ∈ Z and Aη◦d ∈ σop,η (A).

(2.1)

Next we are going to show that, conversely, each limit operator of A and each operator of the form (2.1) appears as a limit operator of Op (Aη ). Let Al be a limit operator of A with respect to a sequence l ∈ H. Choose a strongly monotonically increasing sequence d : N → N such that η(d(n) + 1) − l(n) → ∞ and set h(n) := (η(1) + η(2) + · · · + η(d(n))) + l(n). Then h ∈ H, the limit operator Op (Aη )h exists, and it is equal to Al . Let now d : N → N be a strongly monotonically increasing sequence such that the limit operator Aη◦d of A exists, and let k ∈ Z. Consider h(n) := (η(1) + η(2) + · · · + η(d(n))) + k. Again, h ∈ H, the limit operator Op (Aη )h exists, and now this limit operator is equal to U−k (QAη◦d Q + P AP )Uk . Thus, σop (Op (Aη )) = σop (A) ∪ {U−k (QAh Q + P AP )Uk : k ∈ Z, Ah ∈ σop,η (A)}. This equality shows that the conditions of the theorem are necessary. They are also sufficient since the invertibility of A implies those of all limit operators of A, and if both A and QAh Q + P are invertible then the operator U−k (QAh Q + P AP )Uk is invertible for every integer k.
Corollary 2.2. Let A ∈ L(lp (Z+ )) be a band-dominated operator, and let η : N → N be a strongly monotonically increasing sequence for which the limit operator Aη exists. Then the sequence (Pη(n) APη(n) )n≥1 is stable if and only if the operators P AP + Q and QAη Q + P are invertible on lp (Z). Indeed, under the conditions of the corollary, the set σop,η (A) is a singleton. Here is the announced main result of the present paper.

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Theorem 2.3. Let A ∈ L(lp (Z+ )) be a band-dominated operator and η : N → N a strongly monotonically increasing sequence. Then the sequence (Pη(n) APη(n) )n≥1 is stable if and only if the operator P AP + Q and all operators QAh Q + P

with

Ah ∈ σop,η (A)

are invertible on lp (Z). Proof. The necessity of invertibility of the mentioned operators follows from Theorem 2.1. Conversely, let P AP + Q and all operators QAh Q + P with Ah ∈ σop,η (A) be invertible on lp (Z). Contrary to what we want to show, assume that the sequence Aη = (Pη(n) APη(n) ) fails to be stable. Then there is a strongly monotonically increasing sequence d : N → N such that (Pη(d(n)) APη(d(n)) )−1  ≥ n

for all n ∈ N

where we agree upon writing A−1 n  = ∞ if the matrix An fails to be invertible. Thus, no subsequence of the sequence Aη◦d is stable. Let g be a subsequence of η ◦ d for which the limit operator Ag exists. (The existence of a sequence g with these properties follows from Theorem 1.1 (a).) Then Ag ∈ σop,η (A), and the operators P AP + Q and QAg Q + P are invertible by hypothesis. Corollary 2.2 implies that the subsequence Ag of Aη◦d is stable. Contradiction.
There is also a version of Theorem 2.3 for band-dominated operators on lp (Z) which we will briefly sketch. For n ∈ N, consider the projections  x(m) if − n ≤ m < n p p Rn : l (Z) → l (Z), (Rn x)(m) := 0 otherwise. The finite sections sequence for an operator A on lp (Z) is the sequence (Rn ARn ) where the Rn ARn are viewed of as operators on im Rn , provided with the norm induced by the norm on L(lp (Z)). The stability of the sequence (Rn ARn ) as well as the notion of a band-dominated operator on lp (Z) are defined as above, with obvious modifications. Let again η : N → N be a strongly monotonically increasing sequence. In σ+,η (A) and σ−,η (A), we collect all limit operators of A with respect to subsequences of η and of −η, tending to +∞ and −∞, respectively. The following can be proved in the same vein as Theorem 2.3. Theorem 2.4. Let A ∈ L(lp (Z)) be a band-dominated operator. Then the finite sections sequence (Rn ARn )n≥1 is stable if and only if the operator A, all operators QAh Q + P

with

Ah ∈ σ+ (A)

P Ah P + Q

with

Ah ∈ σ− (A)

and all operators p

are invertible on l (Z).

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391

We would like to mention that the stability of the finite sections sequence for band-dominated operators on l∞ can be studied as well. This involves some technical subtleties (when working with adjoint sequences, for instance), but it is easier with respect to the concern of the present paper: Indeed, for p = ∞, the uniform boundedness condition in Theorem 1.1 (b) is already redundant. For much more on this topic, we refer to the recent textbook [1]. It remains an open question whether the uniform boundedness condition in Theorem 1.1 (b) is redundant for p ∈ (1, ∞) or at least for p = 2.

References [1] M. Lindner, Infinite Matrices and their Finite Sections. An Introduction to the Limit Operator Method. Birkh¨ auser, Basel, Boston, Berlin 2006. [2] V.S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators. Integral Equations Oper. Theory 30(1998), 4, 452–495. [3] V.S. Rabinovich, S. Roch, B. Silbermann, Algebras of approximation sequences: Finite sections of band-dominated operators. Acta Appl. Math. 65(2001), 315–332. [4] V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory. Operator Theory: Adv. and Appl. 150, Birkh¨ auser Verlag, Basel, Boston, Berlin 2004. [5] S. Roch, Finite sections of band-dominated operators. Preprint 2355 TU Darmstadt, July 2004, 98 p., to appear in Memoirs Amer. Math. Soc. [6] J. Roe, Band-dominated Fredholm operators on discrete groups. Integral Equations Oper. Theory 51(2005), 3, 411–416. Vladimir S. Rabinovich Instituto Politechnico National ESIME-Zacatenco, Ed.1, 2-do piso Av. IPN, Mexico, D.F., 07738 Mexico e-mail: [email protected] Steffen Roch Technische Universit¨ at Darmstadt Fachbereich Mathematik Schlossgartenstrasse 7 D-64289 Darmstadt, Germany e-mail: [email protected] Bernd Silbermann Technische Universit¨ at Chemnitz Fakult¨ at f¨ ur Mathematik D-09107 Chemnitz, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 393–416 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Characterization of the Range of One-dimensional Fractional Integration in the Space with Variable Exponent Humberto Rafeiro and Stefan Samko Abstract. Within the frameworks of weighted Lebesgue spaces with variable exponent, we give a characterization of the range of the one-dimensional Riemann-Liouville fractional integral operator in terms of convergence of the corresponding hypersingular integrals. We also show that this range coincides with the weighted Sobolev-type space Lα,p(·) [(a, b), ]. Mathematics Subject Classification (2000). Primary 46E30; Secondary 47B38. Keywords. Fractional integrals, Riesz potentials, Bessel potentials, variable exponent spaces, Marchaud fractional derivative.

1. Introduction Recently the spaces I α [Lp(·) (Rn )] and B α [Lp(·) (Rn )] of Riesz and Bessel potential spaces were studied within the frameworks of variable exponents p(·) in papers [1] and [2] in the case of the whole space Rn . In particular, the following characterization of the space of Bessel potentials was obtained in [2]: 0 B α [Lp(·) (Rn )] = Lp(·) (Rn ) I α [Lp(·) (Rn )] = {f ∈ Lp(·) (Rn ) : Dα f ∈ Lp(·) (Rn )}, (1.1) where Dα f is the Riesz fractional derivative. A similar characterization for potentials over a domain in Rn remains an open question even in the case of constant p. For an analogue of the Riesz derivative adjusted for domains in Rn we refer to [14]. In this paper we solve such a problem of characterization in the one-dimensional case n = 1. We study the range of fractional integrals over the space Lp(·) (Ω, ) with variable exponent p(·) and a power type weight , where Ω = (a, b) is a finite or infinite interval. We obtain a characterization of this range in terms of convergence of the corresponding Marchaud derivatives and show that this range may be also obtained as the restriction on Ω of Bessel potentials with densities in

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Lp(·) (Ω, ). We refer to [19], p. 229–232, for such results in the non-weighted case and constant p. Note that an increasing interest to the variable exponent Lebesgue spaces Lp(·) observed last years was caused by possible applications (elasticity theory, fluid mechanics, differential equations, see for example [15]). We refer to papers [20] and [13] for basics on the Lebesgue spaces with variable exponents and to the surveys [6], [10], [18] on harmonic analysis in such spaces. One of the breakthrough results obtained for variable p(x) was the statement on the boundedness of the HardyLittlewood maximal operator in the generalized Lebesgue space Lp(·) under certain conditions on p(x), see [4] and the further development in the above survey papers. The importance of the boundedness of the maximal operator is known due to the fact that many convolution operators occurred in applications may be dominated by the maximal operator. This tool is also used in this paper. J K Let 0 < α < 1 and x ∈ (a, b). We study the ranges I α Lp(·) [(a, b), ] of the Riemann-Liouville fractional integration operators  x  b 1 1 ϕ(t) dt ϕ(t) dt α α (Ia+ ϕ)(x) = , (I ϕ)(x) = (1.2) b− 1−α Γ(α) a (x − t) Γ(α) x (t − x)1−α over weighted Lebesgue spaces Lp(·) [(a, b), ] with variable exponent p(x). We show that the ranges of operators (1.2) coincide (Theorem 4.5) under natural assumptions and obtain necessary and sufficient conditions for a function f to belong to this range (Theorem 4.4). Finally we show that this range coincides with the Sobolev type space Lα,p(·) [(a, b), ] (Theorem 4.15). When developing necessary tools for the proof, we also obtain results of independent interest for Hardy-type operators (Theorems 3.4, 3.8) and for singular operators with fixed singularity (Theorem 3.6). A non-weighted result of a type of Theorem 4.15 for variable exponents was obtained in [1] and [2] for the Riesz potential operator in the case of the whole space Ω = Rn . We deal not with the Riesz potential operator, but with the fractional α integration operator Ia+ which has the unilateral nature. However the main novelty in comparison with [1] and [2] is not only in a different nature of the operator or admission of the weight, but in the fact that the case of a domain in Rn , when we may have an essential influence of the boundary, is more difficult. We show how it is possible to characterize this range in the one-dimensional case with Ω = (a, b), −∞ < a < b ≤ ∞. In comparison with [1] and [2], the results obtained in this paper require different terms and methods. Notation |Ω| is the Lebesgue measure of a set Ω ⊆ Rn , B(x, r) = {y ∈ Rn : |x − y| < r};  is a weight, i.e., an a.e. finite and positive function; P(Ω) and P1 (Ω), see (2.1)–(2.2); w-Lip (Ω), see (2.3); w-Lipx0 (Ω), see (3.2); M is the maximal operator, see (2.7); P (Ω) is the set of exponents p ∈ P(Ω) such that M is bounded in Lp(·) (Ω, ).

Characterization of the Range of One-dimensional . . .

395

2. Preliminaries 2.1. On spaces Lp(·) with variable exponents Although our main results concern the one-dimensional case n = 1, some auxiliary statements below are given for the multidimensional case. We refer to [13], [16] for details on variable Lebesgue spaces over domains in Rn , but give some necessary definitions. For a measurable function p : Ω → [1, ∞), where Ω ⊂ Rn is an open set, we put p+ = p+ (Ω) := ess sup p(x)

and

p− = p− (Ω) := ess inf p(x). x∈Ω

x∈Ω

In the sequel we use the notation P(Ω) := {p ∈ L∞ (Ω) : 1 < p− ≤ p(x) ≤ p+ < ∞}

(2.1)

P1 (Ω) := {p ∈ L∞ (Ω) : 1 ≤ p− ≤ p(x) ≤ p+ < ∞}.

(2.2)

and The generalized Lebesgue space Lp(·) (Ω) with variable exponent is introduced as the set of functions f on Ω for which  Ip (f ) := |f (x)|p(x) dx < ∞. Ω

By w-Lip (Ω), for bounded Ω, we denote the class of exponents p ∈ L∞ (Ω) satisfying the log-condition |p(x) − p(y)| ≤

C , − ln |x − y|

|x − y| ≤

By p (x) we denote the conjugate exponent: The weighted Lebesgue space L on Ω functions f for which

p(·)

1 p(x)

1 , x, y ∈ Ω. 2 +

1 p
(x)

(2.3)

≡ 1.

(Ω, ) is defined as the set of all measurable

    f f Lp(·)(Ω,) = f Lp(·)(Ω) = inf λ > 0 : Ip ≤ 1 < ∞. λ

In [11] the following theorem was proved. Theorem 2.1. Let p ∈ P1 (Rn ). The class C0∞ (Rn ) is dense in the space Lp(·) (Rn , ) with an a.e. positive weight  if [(x)]p(x) ∈ L1loc (Rn ).

(2.4) R

Lemma 2.2. Let Ω be a bounded domain in Rn and p ∈ P1 (Ω) w-Lip (Ω). There exists an extensionRp$(x) of p(x) to the whole space Rn such that p$(x) ≡ p(x) for x ∈ Ω, p$ ∈ P(Rn ) w-Lip (Rn ), p$(x) is constant outside some large fixed ball and p$− (Rn ) = p− (Ω);

p$+ (Rn ) = p+ (Ω).

(2.5)

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Proof. It is known in general that any continuous function defined on an arbitrary closed set in Rn may be extended to the whole space Rn with preservation of its continuity modulus, see [21], Ch. 6, Section 2. This extension p$ may be realized in such a way that (2.5) is valid, see for example, [5], Theorem 4.2. To get an extension constant outside some ball, it suffices to arrange a new extension in the form D  x E x p$∗ (x) = η p$(x) + 1 − η C R R where C is any constant such that p− (Ω) ≤ C ≤ p+ (Ω) and η(x) is any C0∞ function with support in the ball |x| < 2 and equal identically to 1 in the ball |x| < 1, and R is sufficiently large so that Ω ⊆ {x ∈ Rn : |x| ≤ R}. (Then p$∗ (x) ≡ p(x) for x ∈ Ω and p$∗ (x) ≡ C for |x| ≥ 2R).
Everywhere in the sequel, when Ω is unbounded, we assume that there exists the limit p(∞) := lim p(x). In the case p(x) ≡ const beyond some big ball, we x→∞

use the notation p(x) ≡ p∞ (= p(∞)), |x| > R. In case of unbounded domains we will also use the decay condition |p(x) − p(∞)| ≤

C , ln(1 + |x|)

x ∈ Ω.

2.2. On maximal and convolution operators in Lp(·) Let  1 (Mϕ)(x) = sup |ϕ(y)|dy R r>0 |B(x, r)| B(x,r) Ω

(2.6)

(2.7)

be the Hardy-Littlewood maximal operator. The following theorem for the weight (x) = (1 + |x|)γ

m .

|x − xk |βk ,

xk ∈ Ω, k = 1, 2, . . . , m

(2.8)

k=1

was in particular proved in [12] when Ω is bounded and in [9], when Ω is unbounded. R Theorem 2.3. Let p ∈ P(Ω) w-Lip (Ω) and  be weight of form (2.8). I) When Ω is bounded, the maximal operator is bounded in Lp(·) (Ω, ) if and only if n n < βk <  , k = 1, 2, . . . , m. (2.9) − p(xk ) p (xk ) II) Let Ω be unbounded and p be constant outside some ball of large radius R > 0: p(x) ≡ p∞ , |x| > R. The maximal operator is bounded in Lp(·) (Ω, ) if and only if condition (2.9) and the condition
n n − <γ+ βk <  p∞ p∞ m

k=1

are satisfied.

(2.10)

Characterization of the Range of One-dimensional . . .

397

By P (Ω) we denote the set os exponents p ∈ P(Ω) such that M is bounded in Lp(·) (Ω, ). Let Ω = Rn . For dilatations 1 Kε f (x) = n ε



 k Rn

x−y ε

 f (y)dy

the following weighted statement is valid. p
(·)

Theorem 2.4. Let  be a weight, −1 ∈ Lloc , p ∈ P1 (Rn ) and k(x) be an integrable  sup |k(y)|dx < ∞. Then

function on Rn with A :=

# # # # # sup Kε f (x)# ≤ A(Mf )(x) # ε>0 #

i) so that ii)

Rn |y|≥|x|

8 8 8 8 8 sup Kε f (x)8 8 ε>0 8

for all

≤ C1 f Lp(·)(Rn ,)

Lp(·) (Rn ,)

in the case p(·) ∈ P (Rn ). If in addition (2.4), then also iii)

f ∈ Lp(·) (Rn , ),

 k(y)dy = 1 and (x) satisfies condition Rn

Kε f (x) → f

as ε → 0 in Lp(·) (Rn , ) and almost everywhere. Proof. For the non-weighted case the statement of the theorem is known, see [4]. Statement i) can be proved exactly as in [8] since the step functions are dense in Lp(·) (Rn , ); statement ii) is an immediate consequence of i). To prove iii), observe that C0∞ (Rn ) is dense in the space Lp(·) (Rn , ) by Theorem 2.1. So splitting f = f1 + fε , where f1 ∈ C0∞ (Rn ) and fε Lp(·) (Rn ,) < ε, we have Kε f − f Lp(·)(Rn ,) For I2,ε we obtain 8 8 8 8 I2,ε ≤ 8Kε fε 8

≤ =

Lp(·) (Rn ,)

Kε f1 − f1 Lp(·) (Rn ,) + Kε fε − fε Lp(·) (Rn ,) I1,ε + I2,ε .

+ fε Lp(·) (Rn ,) ≤ Cfε Lp(·) (Rn ,) ≤ Cε.

(2.11)

a.e.

The a.e. convergence Kε f1 −−→ f1 as ε → 0 with f1 ∈ C0∞ (Rn ) is obvious, see [21]. The boundedness of the maximal operator implies that Ip [(Mf )] < ∞. Using i) and Lebesgue dominated convergence theorem we have that limε→0 Ip [(Kε f1 − ε→0

f1 )] = 0, thus showing that I1,ε −−−→ 0. Thus we have convergence in Lp(·) (Rn , )norm and a.e. convergence.


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By Theorem 2.4, the boundedness in Lp(·) (Rn , ) of the maximal operator guarantees the boundedness of convolution operators  Af (x) = k(y)f (x − y)dy Rn

whose kernels k(x) have decreasing integrable dominants. However, the boundedness of the maximal operator requires in general the local log-condition (2.3). Meanwhile, for rather “nice” kernels k(x) this condition may be avoided. Namely, in [7] the following result was obtained. C n Theorem 2.5. Let k(y) satisfy the estimate |k(y)| ≤ (1+|y|) for some ν, y ∈ R   1 1 ν > n 1 − p(∞) + q(∞) . Then the convolution operator A is bounded from the

space Lp(·) (Rn ) to the space Lq(·) (Rn ) under the only assumption that the exponents p, q ∈ P1 (Rn ) satisfy decay condition (2.6) and q(∞) ≥ p(∞). 2.3. Boundedness of potential and singular operators in weighted Lp(·) -spaces The following result is known, see [12], where Theorem 2.6 was stated for the single power weight; its validity for a finite product of power weights is reduced to the case of a single weight by the standard introduction of the unity partition. For the completeness of presentation we give details of such a reduction in Appendix, see Section 5. Theorem 2.6. Let Ω ⊂ Rn be Ra bounded domain, let α(x) ∈ L∞ (Ω) and ess inf Ω α(x) > 0, let p ∈ P(Ω) w-Lip (Ω) and let  be weight of form (2.8) with xk ∈ Ω. Under condition (2.9) the operator  f (y) Iα(·) f (x) = (x) dy n−α(x) Ω (y)|x − y| is bounded in the space Lp(·) (Ω). The following theorem on the boundedness of the singular operator 1 Sϕ(t) = π

b a

ϕ(t) dt , t−x

x ∈ (a, b)

was proved in [11]. R Theorem 2.7. Let −∞ < a < b < ∞ and let p ∈ P(a, b) w-Lip (a, b). The operator S is bounded in the space Lp(·) [(a, b), ], where  is weight (2.8) with xk ∈ [a, b], k = 1, 2, . . . , m, if and only if −

1 1 < βk <  , p(xk ) p (xk )

k = 1, 2, . . . , m.

Characterization of the Range of One-dimensional . . .

399

3. Hardy-type inequalities in variable exponent setting 3.1. Definition and assumptions Let now n = 1 and Ω = [a, b], where −∞ < a < b ≤ ∞, and consider the space Lp(·) [(a, b), ] with the weight  when b < ∞ |x − a|µ(x) |b − x|ν(x) (x) = , (3.1) when b = ∞ |x − a|µ(x) (1 + |x|)ν(x) where the exponents µ(x), ν(x) are bounded functions which have finite limits µ(a) = lim µ(x), ν(b) = lim ν(x). We need the following notation for the class of x→a x→b exponents. Definition 3.1. Let Ω = (a, b), where −∞ ≤ a < b ≤ ∞ and let x0 ∈ [a, b]. By w-Lipx0 (Ω) we denote the class & % A 1 ∞ , (3.2) w− Lipx0 (Ω) = µ ∈ L (Ω) : |µ(x) − µ(x0 )| ≤ , |x − x0 | ≤ 1 2 ln |x−x 0| in case x0 = ∞, and

 w− Lip∞ (Ω) = µ ∈ L∞ (Ω) : |µ(x) − µ(∞)| ≤

For µ ∈ w-Lipa (a, b)

R

A ln(2 + |x|)

 .

(3.3)

w-Lipb (a, b) with −∞ < a < b < ∞ one has

|b − x|ν(x) ≈ |x − a|µ(a) |b − x|ν(b) . R R Similarly, for µ ∈ w-Lipa (R1 ) w-Lipb (R1 ) w-Lip∞ (R1 ) |x − a|

µ(x)

|x − a|µ(x) |b − x|ν(x) ≈ |x − a|µ(a) |b − x|ν(b) (1 + |x|)µ(∞)+ν(∞)−µ(a)−ν(b) .

(3.4)

(3.5)

Remark 3.2. From Theorem 2.1 it is easy to derive that the class C0∞ ((a, b)) of infinitely differentiable functions with support in (a, b), −∞ < a < b < ∞ is dense in the space Lp(·) [(a, b), ] with the weight (3.1), if p ∈ P1 (a, b) and µ(a)p(a) > −1, ν(b)p(b) > −1. Everywhere in the sequel we assume that p(x) ≡ p∞ = const for large |x| ≥ R

in the case b = ∞.

(3.6)

3.2. Hardy inequalities The following proposition was proved in [7] (see Theorem 3.3 there). R R Proposition 3.3. Let p, r ∈ P1 (R1+ ) w-Lip0 (R1+ ) w-Lip∞ (R1+ ), p(0) = r(0) and p(∞) = r(∞). (3.7) R 1 and α, β ∈ w-Lip∞ (R+ ), The Hardy operators  x  ∞ f (y) f (y) β(x) dy and H f (x) = x dy (3.8) Hα(·) f (x) = xα(x)−1 β(·) α(y) β(y)+1 y 0 y x w-Lip0 (R1+ )

400

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are bounded from the space Lp(·) (R1+ ) into Lr(·) (R1+ ), if α(0) <

1 p (0)

,

α(∞) <

1

and

p (∞)

β(0) > −

1 , p(0)

β(∞) < −

1 . p(∞)

We need the following weighted statement derived from Proposition 3.3. R R Theorem 3.4. Let p, r ∈ P1 (R1+ ) w-Lip0 (R1 ) w-Lip∞ (R1+ ) and condition (3.7) µ(x) ν(x) be satisfied, R let 0 < b 1< ∞ and (x) =1 xR |x − b| 1 , x > 0, where µ ∈ w1 Lip0 (R+ ) w-Lip∞ (R+ ), ν ∈ w-Lipb (R+ ) w-Lip∞ (R+ ) and −

1 1 < ν(b) <  . r(b) p (b)

(3.9)

R Let also α, β ∈ w-Lip0 (R1+ ) w-Lip∞ (R1+ ). Then the Hardy-type inequalities 8 8 x 8 f (y) 8 8 8 α(x)−1 dy 8 ≤ Cf Lp(·)(R1+ ,) (3.10) 8x 8 y α(y) 8 r(·) 1 L

0

and

8 8 ∞ 8 8 f (y) 8 8 β(x) dy 8 8x 8 y β(y)+1 8

(R+ ,)

≤ Cf Lp(·)(R1+ ,)

(3.11)

Lr(·) (R1+ ,)

x

are valid if α(0) + µ(0) <

1 p (0)

,

α(∞) + µ(∞) + ν(∞) <

1 p (∞)

(3.12)

and −

1 < β(0) + µ(0), p(0)

−

1 < β(∞) + µ(∞) + ν(∞), p(∞)

(3.13)

respectively. Proof. For (3.10) we have to show that the operator  x f (y) dy Bf (x) = xα(x)+µ(x)−1 |x − b|ν(x) α(y)+µ(y) |y − b|ν(y) 0 y is bounded from Lp(·) (R1+ ) to Lr(·) (R1+ ). We have Bf Lr(·) (R1+ ) ≤ Bf Lr(·) (0, b ) + Bf Lr(·) ( b ,2b) + Bf Lr(·)(2b,+∞) . 2

2

F x f (y) For 0 < x < 2b , we have |Bf (x)| ≤ Cxα(0)+µ(0)−1 0 yα(0)+µ(0) dy so that Bf Lr(·) (0, b ) is covered by Proposition 3.3. For 2b ≤ x ≤ 2b we have 2  b   2b 2 |f (y)| |f (y)| ν(b) dy + dy , |Bf (x)| ≤ C|x − b| α(0)+µ(0) b |y − b|ν(b) 0 y 2

Characterization of the Range of One-dimensional . . .

401

whereJbothKthe integrals are finite by the H¨older inequality, and |x − b|ν(b) ∈ Lr(·) 2b , 2b . Finally, when x > 2b, we get  b  2b 2 |f (y)| dy |f (y)| dy α(∞)+µ(∞)+ν(∞)−1 + + |Bf (x)| ≤ Cx α(0)+µ(0) b y |y − b|ν(b) 0 2   x |f (y)| dy + α(∞)+µ(∞)+ν(∞) 2b y where the first two integrals are finite by the H¨older inequality and xα(∞)+µ(∞)+ν(∞)−1 ∈ Lr(·) (2b, ∞), while the last term is dominated by xα(∞)+µ(∞)+ν(∞)−1

Fx

|f (y)| dy , 0 y α(∞)+µ(∞)+ν(∞)

which is covered by Proposition 3.3. Similarly one can prove inequality (3.11). Remark 3.5. Theorem 3.4 is also valid for the weight (x) = xµ(x)


m 

|x − bk |νk (x) ,

k=1

where 0 < b1 < b2 < · · · < bm < ∞, under natural modification, i.e., 1 1 < νk (bk ) <  , k = 1, . . . , m; − r(bk ) p (bk ) m m

1 1 α(∞) + µ(∞) + ; − < β(∞) + µ(∞) + νk (∞) <  νk (∞). p (∞) p(∞) k=1

k=1

3.3. On singular operators with fixed singularity R R Theorem 3.6. Let p, r ∈ P1 (R1+ ) w-Lip0 (R1+ ) w-Lip∞ (R1+ ) and condition (3.7) µ(x) ν(x) , x > 0, where µ ∈ wbe satisfied, R let 0 < b 1< ∞ and (x) = x 1 |x R− b| 1 Lip0 (R+ ) R w-Lip∞ (R+ ) and ν ∈ w-Lipb (R+ ) w-Lip∞ (R1+ ). Let also β ∈ wLip0 (R1+ ) w-Lip∞ (R1+ ). Then the operator  ∞ ϕ(t) dt (3.14) H β(·) ϕ(x) := xβ(x) β(t) (x + t) t 0 is bounded from the space Lp(·) (R1+ , ) into Lr(·) (R1+ , ), if −

1 1 < ν(b) <  , r(b) p (b)

and −

−

1 1 < β(0) + µ(0) <  p(0) p (0)

1 1 < β(∞) + µ(∞) + ν(∞) <  . p(∞) p (∞)

(3.15)

(3.16)

Proof. Since (3.17) H β f (x) ≤ Hβ f (x) + Hβ f (x) where H and Hβ are the Hardy operators (3.8), Theorem 3.6 immediately follows from Theorem 3.4.
β

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H. Rafeiro and S. Samko

F   β Remark 3.7. In the case of a similar operator H β ϕ(x) =: 0 xt ϕ(t) x+t dt, 0 < x <  < ∞ on a finite interval, Theorem 3.6 is valid without condition (3.16). Note that a weaker version of Theorem 3.6 for a finite interval was proved in [12]. 3.4. On a Hardy-Littlewood inequality The following extension of the Hardy-Littlewood inequality to the case of variable exponents is valid. In the case where the exponent p is constant, this inequality is well known, being due to Hardy and Littlewood, see for instance [19], p. 104–106 (we take this opportunity to note that there are misprints on p. 104 in formulas (5.45)–(5.46): there should be x−αp instead of xαp ). In the case of variable p, an inequality of Hardy type for the multidimensional Riesz-type potentials over bounded domains in Rn with the weight |x − x0 |β was proved in [17] in the case n . We admit infinite intervals (a, b) and thanks 0 < α < n, α − p(xn0 ) < β < p
(x 0) to the unilateral structure of the Riemann-Liouville integral an  we can consider  1 arbitrary α > 0 and the weight exponents in the interval − p(x1 0 ) , p
(x . 0) R Theorem 3.8. Let α > 0, −∞ < a < b < ∞, p ∈ P(a, b) w-Lip (a, b) and  be weight of form (3.1) with µ ∈ w-Lipa (a, b), ν ∈ w-Lipb (a, b). Then 8 8 8 8 x 8 8 1 ϕ(t) dt 8 8 ≤ C ϕLp(·) [(a,b),] (3.18) 8 (x − a)α 8 1−α (x − t) 8 8 p(·) a

L

under the conditions: 1 1 < µ(a) <  , − p(a) p (a)

[(a,b),]

−

1 1 < ν(b) <  . p(b) p (b)

(3.19)

Inequality (3.18) is also valid in the case b = ∞, if additionally p(x) satisfies assumption (3.6), µ, ν ∈ w-Lip∞ (a, ∞) and the second condition in (3.19) is replaced by 1 1 − < ν(∞) + µ(∞) <  . (3.20) p∞ p∞ Proof. The proof follows the principal idea in [17], but uses the unilateral nature of the one-dimensional integration. Let a = 0 for simplicity. We continue ϕ(t) as zero beyond the interval (0, b) and have    x 1 1 ϕ(t) dt x−t ϕ(t) dt, x > 0, (3.21) = L xα (x − t)1−α x x 

R1

0

ξ , ξ ∈ [0, 1] . The right-hand side in (3.21) may be extended 0, ξ∈ / [0, 1] for all x ∈ R1 , if in the denominator we replace x by |x|. Then we have    1 x−t ϕ(t) dt ≤ CMϕ(x), (3.22) L |x| |x|

where L(ξ) =

α−1

R1

Characterization of the Range of One-dimensional . . .

403

where the domination by the maximal operator is possible by the pointwise inequality i) of Theorem 2.4. Then from (3.21) and (3.22) we obtain 8 8 8 8 x 8 1 8 ϕ(t) dt 8 8 ≤ CMϕLp(·) $ (R1 ,∗ ) 8 xα 8 1−α (x − t) 8 8 p(·) 0

L

[(0,b),]

where p$(x) is an extension of p(x) from (a, b) to R1 provided by Lemma 2.2. An extension ∗ (x) of the weight may be taken, according to (3.4)–(3.5), as  b < ∞, |x − a|µ(a) |b − x|ν(b) , ∗ (x) = , (3.23) |x − a|µ(a) (1 + |x|)µ(∞)−µ(a)+ν(∞) , b = ∞ With this extension, the maximal operator is bounded in the space Lp$(·) (R1 , ∗ ) by Theorem 2.3 and we arrive at (3.18).


4. On fractional integrals and derivatives in Lp(·) [(a, b), ] 4.1. On Marchaud derivative The Marchaud fractional derivative ([19], p. 200) Dα a+ f =

f (x) α + α Γ(1 − α)(x − a) Γ(1 − α)

 a

x

f (x) − f (t) dt, (x − t)1+α

(4.1)

of order 0 < α < 1, for “not so nice” functions f (x) is understood as  x−ε f (x) α f (x) − f (t) α lim + dt, ε > 0, lim Da+,ε f = ε→0 Γ(1 − α)(x − a)α Γ(1 − α) ε→0 a (x − t)1+α where f (x) is assumed to be continued as zero beyond the interval [a, b]. It is known ([19], p. 200) that Dα a+,ε f = where



f (x) α Aε f (x), + Γ(1 − α)(x − a)α Γ(1 − α)

f (x) − f (t) dt for a + ε ≤ x ≤ b, (x − t)1+α a   f (x) 1 1 for a ≤ x ≤ a + ε. − Aε f (x) = α εα (x − a)α Aε f (x) =

(4.2)

x−ε

(4.3) (4.4)

Lemma 4.1. Let −∞ < a < b < ∞, α > 0, let p ∈ P(a, b) and  be weight of form (3.1) with µ ∈ w-Lipa (a, b), ν ∈ w-Lipb (a, b). The truncated fractional p(·) [(a, b), ] for any fixed ε > 0 differentiation operator Dα a+,ε f is bounded in L under conditions (3.19). This is also valid for b = ∞, if we additionally assume that µ, ν ∈ wLip∞ (a, ∞) and p(·) ∈ P [(a, R ∞)]; for the latter inclusion, the following conditions are sufficient: p ∈ P(a, ∞) w-Lip (a, ∞), and (3.6), (3.20) and the first condition in (3.19) hold.

404

H. Rafeiro and S. Samko

Proof. After easy calculations we obtain Dα a+,ε f (x)

αχ[a+ε,b] (x) f (x) = − α Γ(1 − α)ε Γ(1 − α)

x−a 

f (x − t) dt , t1+α

x ∈ [a, b],

ε

p(·)

where the second term is bounded in L x−a F f (x−t) dt H¨older inequality we have ≤ t1+α ε

[(a, b), ]. Indeed, for x > a + ε by the 8 −1 8 1 8 p
(·) 8 where ε1+α f Lp(·) [(a,b),] ·  L (a,b)

the last factor is finite under the conditions µ(a)p (a) < 1, ν(b)p (b) < 1, in the case of finite b. In the case b = ∞ we have  x−a  f (x − t) dt ≤ k(t)f (x − t)dt (4.5) t1+α ε R with k(t) = t−1−α · χ(ε,∞) (t) and then the boundedness follows by Theorem 2.4 when p(·) ∈ P [(a, ∞)]. The sufficiency of the conditions for the latter inclusion, mentioned in the theorem, follows from (4.5) and Theorem 2.3.
Remark 4.2. The statement of Lemma 4.1 for b = ∞ in the case µ(a) = µ(∞) + ν(∞) = 0 is valid for an arbitrary p ∈ P(a, ∞) satisfying condition (2.6). This follows from (4.5) and Theorem 2.5 α 4.2. The left-hand side inverse operator to the Riemann-Liouville operator Ia+ α When considering the operator left inverse to Ia+ , we may not follow the same lines as in the known proof for the case of constant p, see [19], Section 13, since the proof there uses the p-mean continuity of the Lp space, which is no more valid in the case of variable p, see [13]. Thus we have to modify the arguments from [19] and make use of the maximal operator.

Theorem 4.3. Let −∞ < a < b < ∞, 0 < α < 1 and where p ∈ P(a, b) w-Lipb (a, b). Then

R

α f = Ia+ ϕ, ϕ ∈ Lp(·) [(a, b), ],

w-Lip (a, b) and  is weight (3.1) with µ ∈ w-Lipa (a, b), ν ∈ Dα a+ f = ϕ,

α p(·) [(a, b), ], where Dα a+ f = lim Da+,ε f with the limit in the norm of the space L ε→0

under conditions (3.19). This is also valid in the case b = ∞, if additionally µ, ν ∈ w-Lip∞ (a, b) and (3.6) and (3.20) hold. Proof. Without loss of generality we take a = 0. We need to show that lim  Dα 0+,ε f − ϕ Lp(·) [(0,b),] = 0.

ε→0

(4.6)

Characterization of the Range of One-dimensional . . . In [19], p. 227–228, there was proved the following representation ⎧ x ε ⎪ ⎪ ⎪ ⎪ ⎪ K(t)ϕ(x − εt)dt =: Aε ϕ(x), ε≤x≤b ⎪ ⎨ α α 0 D0+,ε I0+ ϕ = x ⎪ ⎪ ϕ(t) sin απ ⎪ ⎪ ⎪ dt =: Bε ϕ(x), 0 ≤ x ≤ ε, ⎪ α ⎩ πε (x − t)1−α

405

(4.7)

0

with α sin απ tα + − (t − 1)+ , K(t) = π t

 tα +

=

tα , t > 0 , 0, t < 0

(4.8)

valid for ϕ ∈ Lp , where 1 ≤ p < ∞, and therefore valid for “nice” functions. In the sequel the function ϕ(t) is assumed to be continued as zero beyond [a, b] whenever necessary, so that Aε (x) and Bε (x) are well defined on the whole line R1 . By Remark 3.2, “nice” functions are dense in Lp(·) [(a, b), ], so that to verify (4.7) on Lp(·) [(a, b), ], we only need to check the boundedness of all the operators α involved in (4.7). The operators Dα 0+,ε and I0+ are bounded by Lemma 4.1 and Theorem 2.6, respectively. The operator Aε is bounded by Theorem 2.4 (after the corresponding extension of p and  to the whole line R1 ). In the case b = ∞, it suffices to have the boundedness on any (a, N ), N < ∞, since all the operators are of Volterra type. Note that the kernel K(t) has a radial integrable decreasing majorant, so that by Theorem 2.4 |Aε ϕ(x)| ≤ CMϕ(x).

(4.9)

Representation (4.7) may be rewritten as Dα 0+,ε f (x) = χ[ε,b] (x)Aε ϕ(x) + χ[0,ε] (x)Bε ϕ(x),

x ∈ [0, b]

(4.10)

and then Dα 0+,ε f (x)−ϕ(x) = Aε ϕ(x)−ϕ(x)+ χ[0,ε] (x)[Bε ϕ(x)−Aε ϕ(x)], x ∈ [0, b]. (4.11) By Part iii) of Theorem 2.4 8 8    8 81 x−ξ 8 ϕ(ξ)dξ − ϕ(x) K → 0 (4.12) Aε ϕ−ϕLp(·)[(0,b),] ≤ 8 8 p(·) 8ε 1 ε R L $ (R1 ,∗ ) where the extension p$(x) of p(x) has been chosen according to Lemma 2.2 and the extension ∗ (x) of the weight is defined in (3.23). The condition p$ ∈ P (R1 ) of Theorem 2.4 is satisfied according to Theorem 2.3. The term χ[0,ε] (x)[Bε ϕ(x) − Aε ϕ(x)] is estimated uniformly in ε by the maximal function: χ[0,ε] (x)|Bε ϕ(x) − Aε ϕ(x)| ≤ Cχ[0,ε] (x)Mϕ(x).

(4.13)

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H. Rafeiro and S. Samko

Indeed, taking into account the inequality ε1α ≤ x1α for 0 < x < ε and the estimate obtained in (3.21)–(3.22) and estimate (4.9), we get  sin απ x ϕ(y) dy |Bε ϕ(x) − Aε ϕ(x)| ≤ + CMϕ(x) ≤ C1 Mϕ(x). πxα 0 (x − y)1−α From (4.13) we have χ[0,ε] (x)[Bε ϕ(x) − Aε ϕ(x)]Lp(·) [(0,b),] ≤ CMϕLp(·) [(0,ε),]

(4.14)

which tends to zero as ε → 0 by the boundedness of the maximal operator. It remains to conclude that from (4.11) there follows (4.6) by (4.12) and (4.14).
J K α Lp(·) [(a, b), ] of the fractional integration operator 4.3. The range Ia+ In the next theorem, we derive necessary and sufficient conditions for the representability of a function f by the fractional integral of a function in Lp(·) [(a, b), ]. R Theorem 4.4. Let −∞ < a < b < ∞ and p ∈ P(a, b) w-Lip (a, b), let  be weight of form (3.1) with µ ∈ w-Lipa (a, b), ν ∈ w-Lipb (a, b) and let conditions (3.19) be satisfied. In order that a function f (x) be representable as α ϕ f = Ia+

with

ϕ ∈ Lp(·) [(a, b), ],

it is necessary and sufficient that f ∈ Lp(·) [(a, b), ] and there exists lim Aε f (x) in ε→0

Lp(·) [(a, b), ] where Aε f (x) is the function defined in (4.3)–(4.4). This statement remains valid in the case b = ∞, if the condition f ∈ Lp(·) [(a, b), ] is replaced by f (x) ∈ Lp(·) [(a, ∞), ] (x − a)α

(4.15)

and we additionally assume that µ, ν ∈ w-Lip∞ (a, ∞) and conditions (3.6) and (3.20) hold. Proof. Necessity part is a consequence of Theorems 4.3, 3.8 and 2.6 because by (4.2) we have f (x) Aε f = Dα . a+,ε f − Γ(1 − α)(x − a)α The necessity of condition (4.15) follows from (3.18). Sufficiency part. Let a = 0 for simplicity. Given f ∈ Lp(·) [(0, b), ], we introduce the functions f (x) α Aε f + (4.16) Bε f (x) = Γ(1 − α)(x − a)α Γ(1 − α) where the limit lim Aε f (x) exists in Lp(·) [(0, b), ] by assumption. Observe that ε→0

f (x) Γ(1−α)(x−a)α

∈ Lp(·) [(a, b), ], which follows from the fact that Aε f ∈ Lp(·) [(a, b), ],

Characterization of the Range of One-dimensional . . . see (4.4). After some transformations we arrive at the representation ⎧ x ε ⎪ ⎪ ⎪ ⎪ ⎪ K(y)f (x − εy)dy, ε ≤ x ≤ b; ⎪ ⎨ α 0 I0+ Bε f (x) = x ⎪ ⎪ f (t) ⎪ sin απ ⎪ ⎪ dt, 0 ≤ x ≤ ε. ⎪ ⎩ πεα (x − t)1−α

407

(4.17)

0

similar to (4.7), see details of those transformations in [19], p. 229–230. Observe that in [19] this representation was justified for f ∈ Lp with constant p; therefore (4.17) is valid for “nice” functions dense in Lp(·) [(a, b), ] and consequently for all f ∈ Lp(·) [(a, b), ] thanks to the boundedness of all the operators involved in (4.17) (with fixed ε > 0, see the arguments in the proof of Theorem 4.3 after (4.8)) . Since {Aε f } is convergent in Lp(·) [(a, b), ] as ε → 0, then Bε f (x) converges p(·) in L [(a, b), ] to ϕ(x) ∈ Lp(·) [(a, b), ], where ϕ(x) =

L

p(·)

α f (x) lim Aε ϕ(x). + Γ(1 − α)(x − a)α Γ(1 − α) ε→0

α α ϕ. Since the operator Ia+ is continuous in We need to show that f = I0+ α [(a, b), ] by Theorem 2.6, it is sufficient to prove that f = lim I0+ Bε f . To this ε→0

end, we have to show that the right-hand side of (4.17) tends to f as ε → 0 in the norm of the space Lp(·) [(a, b), ], which is done exactly as in the proof of Theorem 4.3 thanks to the coincidence of the right-hand sides of (4.17) and (4.7).
J p(·) K α 4.4. On the interpretation of the range Ia+ L [(a, b), ] as fractional Sobolev type J p(·) K α L [(a, b), ] of the fractional In this subsection we show that the range Ia+ integration operator coincides with the fractional Sobolev space on Lα,p(·) [(a, b), ] defined as the space of restrictions of Bessel potentials onto [a, b], see Theorem 4.15. First we observe that the ranges of the left-hand sided and right-hand sided fractional integrals coincide under the appropriate assumptions. Namely, the following theorem is valid. R Theorem 4.5. Let 0 < α < 1, p ∈ P(a, b) w-Lip (a, b), −∞ < a < b < ∞, and let  be weight of form (3.1) with µ ∈ w-Lipa (a, b), ν ∈ w-Lipb (a, b). Then α α [Lp(·) [(a, b), ]] = Ib− [Lp(·) [(a, b), ]], Ia+

(4.18)

under the conditions α−

1 1 < µ(a) <  , p(a) p (a)

α−

1 1 < ν(b) <  · p(b) p (b)

(4.19)

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H. Rafeiro and S. Samko

Proof. The coincidence of the ranges stated in (4.18) under conditions (4.19) follows from the known ([19], p. 206) formulas   α α Ib− cos aπϕ + sin απra−α Sraα ϕ , ϕ = Ia+ (4.20)   −α α α α Ia+ ϕ = Ib− cos aπϕ − sin απrb Srb ϕ (4.21) where ra±α (x) = (x − a)±α and rb±α (x) = (b − x)±α . Formulas (4.20)–(4.21) being valid for ϕ ∈ C0∞ , are extended to the whole space Lp(·) [(a, b), ] by continuity of the operators involved. Indeed, the weighted singular operators ra−α Sraα , rb−α Srbα are bounded in Lp(·) [(a, b), ] by Theorem 2.7 when −1/p(a) < µ(a) − α < 1/p (a) α α and Ib− are and −1/p(b) < ν(b) − α < 1/p (b), and the fractional integrals Ia+ p(·) bounded in L [(a, b), ] by Theorem 2.6.
R Corollary 4.6. Let p ∈ P(a, b) w-Lip (a, b). In the non-weighted case, the coincidence α α Ia+ [Lp(·) (a, b)] = Ib− [Lp(·) (a, b)], (4.22) 1 1 holds if 0 < α < min{ p(a) , p(b) }. Similarly to Theorem 4.5, the following statement is proved for the whole line R1 with the help of the known relations ([19], p. 202) between the left-hand-sided and right-hand-sided fractional integrals via the singular operator. R 1 (3.6) Theorem 4.7. Let 0 < α < 1, let p ∈ P(R1 ) w-Lip (R R ) satisfy condition µ(x) ν(x) 1 1 |b−x| with µ ∈ w-Lip (R ) w-Lip (R ) and ν ∈ wand let (x) = |x−a| a ∞ R Lipb (R1 ) w-Lip∞ (R1 ). Then under conditions (4.19) and the condition 1 1 < µ(∞) + ν(∞) <  α− p∞ p∞ the following coincidence of the ranges holds α p(·) α p(·) I+ [L (R1 , )] = I− [L (R1 , )] = I α [Lp(·) (R1 , )],

where I

α

(4.23)

is the one-dimensional Riesz operator.

The space of Bessel potentials is known as the range of the Bessel operator: B α [Lp(·) (Rn )] = {f : f = B α ϕ,

ϕ ∈ Lp(·) (Rn )},

α ≥ 0,

where B is the Bessel potential operator which reduces to multiplication by  − α 1 + |ξ|2 2 in Fourier transforms; we refer to [2] for the study of these fractional type spaces with variable exponent, including the characterization of B α [Lp(·) (Rn )] in terms of convergence of some hypersingular integrals. α

Definition 4.8. For a domain Ω ⊂ Rn we define the fractional Sobolev type space Lα,p(·) (Ω) as the space of restrictions onto Ω of functions f ∈ B α [Lp$(·) (Rn )] with some extension p$(·) of p(·) from Ω to Rn : # # α,p(·) α p $(·) n # L (Ω) = B [L (R )]# (4.24) Ω

Characterization of the Range of One-dimensional . . .

409

# # and for f = Bϕ## define the norm by Ω

f Lα,p(·)(Ω) = inf ϕLp(·) $ (Rn ) where the # infimum is taken with respect to all possible ϕ in the representation # f = Bϕ## and all the extensions p$ . Ω

We need a similar weighted space. To avoid complications with extension of arbitrary weights from Ω to Rn , we restrict ourselves to the one-dimensional case and power-type weights and use their extensions in the form  when b < ∞ |x − a|µ$ (x) |b − x|ν$ (x) $(x) = , (4.25) when b = ∞ |x − a|µ$ (x) (1 + |x|)ν$ (x) Definition 4.9. Let −∞ < a < b ≤ ∞. We define the fractional Sobolev type space Lα,p(·) [(a, b), ] with weight (3.1) as # # # (4.26) Lα,p(·) [(a, b), ]) = B α [Lp$(·) (R1 , $)]# # (a,b) R where $ is an extension of form (4.25) with µ $ ∈# w-Lipa (R1 ) w-Lip∞ (R1 ), # R define the norm by ν$ ∈ w-Lipb (R1 ) w-Lip∞ (R1 ), and for f = Bϕ## (a,b)

f Lα,p(·)((a,b),) = inf ϕLp(·) $ (R1 ,$ ) where the # infimum is taken with respect to all possible ϕ in the representation # and all the extensions p$, µ $ and ν$. f = Bϕ## (a,b)

We refer to [3] for the notion of Banach function spaces and to [19], Section 26, for the notion of the Riesz fractional differentiation Dα and its truncation Dα ε, used in the following result for the Riesz and Bessel potentials. Theorem 4.10. Let X = X(Rn ) be a Banach function space, satisfying the assumptions i) C0∞ is dense in X; ii) the maximal operator M is bounded in X; iii) I α f (x) converges absolutely for almost all x for every f ∈ X and (1 + |x|)−n−α I α f (x) ∈ L1 (Rn ). Then 0 B α (X) = X I α (X) = {f ∈ X : Dα f = lim Dα ε f ∈ X}. ε→0

(4.27)

(X)

Proof. Theorem 4.10 was proved in [2] for the case of X = Lp(·) (Rn ). The analysis of the proof given in [2] shows that conditions i)–iii) are sufficient for that proof to hold within the frameworks of abstract Banach function spaces.


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As a corollary to Theorem 4.10, we obtain the following result for the case X = Lp(·) (Rn , ), important for the sequel. R Theorem 4.11. Let Ω = Rn , let p ∈ P(Rn ) w-Lip (Rn ) satisfy condition (3.6) in Rn , let  be weight of form (2.8) with the exponents satisfying condition (2.9) and the condition m
n n α− <γ+ βk <  . (4.28) p∞ p∞ k=1

Then Lα,p(·) (Rn , ) = Lp(·) (Rn , )

0

I α [Lp(·) (Rn , )]

= {f ∈ Lp(·) (Rn , ) : Dα f ∈ Lp(·) (Rn , )}.

(4.29)

Proof. Theorem 4.11 follows from Theorem 4.10. Indeed, condition i) of Theorem 4.10 for the space X = Lp(·) (Rn , ) is fulfilled by Theorem 2.1, condition ii) is satisfied by Theorem 2.3. Condition iii) is checked directly due to the known pointwise estimate  |ϕ(y)| dy α |I ϕ(x)| ≤ CMϕ(x) + C , |x − y|n−α |x−y|>1

where the second term is easily estimated by direct application of the H¨older inequality.
The next auxiliary Theorems 4.12 and 4.14 provide preliminary facts necessary for the main result of this subsection given in Theorem 4.15. R R Theorem 4.12. Let ϕ ∈ Lp(·) (R1 , ), where p ∈ P1 (R1+ ) w-Lip0 (R1 ) wR − b|ν(x) , 0 < b < ∞, where µ ∈ w-Lip0 (R1+ ) wLip∞ (R1+ ), (x) = |x|µ(x) |x R Lip∞ (R1+ ), ν ∈ w-Lipb (R1+ ) w-Lip∞ (R1+ ). Then α α ϕ = I+ ψ χ[0,∞] I+

where ψ(x) =

⎧ ⎨ ⎩

 ϕ(x) +

sin απ π

0,

0

∞

 α ϕ(−t) t dt, x x+t

(4.30)

x>0

∈ Lp(·) (R1 , ),

(4.31)

x<0

if the following conditions are satisfied: 1 1 1 1 < ν(b) <  , α − < µ(0) <  − p(b) p (b) p(0) p (0) and α−

1 1 < µ(∞) + ν(∞) <  . p(∞) p (∞)

(4.32)

(4.33)

Proof. Representation (4.30)–(4.31) is known in case p is constant and µ ≡ ν ≡ 0, see [19], p. 211. By Theorem 2.1, the set C0∞ (R1 ) is dense in the space Lp(·) (R1 , ).

Characterization of the Range of One-dimensional . . .

411

Therefore, we need to prove that the operator  ∞  α ϕ(t) t −α dt H ϕ(x) = x x +t 0 involved in (4.31) and studied in Theorem 3.6, is bounded from the space Lp(·) (R1+ , ) into the space Lp(·) (R1+ , ), where p(x) = p(−x), (x) = (−x). This boundedness does not follow formally from Theorem 3.6, because  = . However, we observe that |x − b| ∼ |x + b| for x > 2b and 0 < x < 2b . So the estimation F∞ of |(x)H −α ϕ(x)|p(x) dx is reduced to Theorem 3.6 when integrating over (0, 2b ) 0

and (2b, ∞), while for x ∈ ( 2b , 2b) the estimation is trivial by the H¨ older inequality. p(·) 1 Having proved that ψ ∈ L (R , ), we can now proceed exactly as in [19], Theorem 11.6. The main point is the interchange of integrals as in [19], possible by α Fubini’s theorem, because the double integral I0+ (H −α ϕ) is absolutely convergent, which is a matter of direct verification.
p(·) 1 |x − a|µ(x) |b − x|ν(x) with Corollary 4.13. Let R ϕ ∈ L 1 (R , ), where (x) =1 R 1 µ ∈ w-Lipa (R ) w-Lip∞ (R ) and ν ∈ w-Lipb (R ) w-Lip∞ (R1 ). Under the conditions 1 1 1 1 α− < µ(a) <  , α − < ν(b) <  (4.34) p(a) p (a) p(b) p (b)

and condition (4.33), the relation α α ϕ = I+ ψ χ(a,b) I+

holds, where ψ ∈ Lp(·) (R1 , ). Let f ∗ be the zero extension of a function f defined on (a, b) to R1 . The following theorem provides sufficient conditions for f ∗ to be representable by fractional integral on R1 if f has such a property on (a, b). To this end, we need to deal with some extension of the exponent p(x) and the weight (x) to the whole line R1 . In Theorem 4.14 we use the extension p$(x) satisfying the following conditions and p$(∞) = p$(b) = p(b), 0 p$ ∈ w− Lipb (R1 ) w− Lip∞ (R1 ),

p$ ∈ P1 (R1 )

(4.35) (4.36)

and the extension (4.25) of the weight satisfying the conditions 0 0 µ $ ∈ w− Lipa (R1 ) w− Lip∞ (R1 ) and ν$ ∈ w− Lipb (R1 ) w− Lip∞ (R1 ), (4.37) 1 . (4.38) µ $(∞) + ν$(∞) − α <  p$ (∞) α Theorem 4.14. Let −∞ < a < b < ∞ and let f (x) = Ia+ ϕ, x ∈ (a, b), with R p(·) ϕ ∈ L ((a, b), ), where p ∈ P1 (a, b) w-Lipb (a, b), (x) is weight (3.1) with

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H. Rafeiro and S. Samko

µ ∈ w-Lipa (a, b), ν ∈ w-Lipb (a, b) and 1 1 1 < ν(b) − α <  . µ(a) <  , − p (a) p(b) p (b)

(4.39)

Then α f ∗ (x) = (I+ ϕ1 )(x),

where ϕ1 (x) ∈ L

p $(·)

x ∈ R1

(R , $) is given by ⎧ x b.

(4.40)

1

(4.41)

and p$(x) and $(x) are arbitrary extensions satisfying conditions (4.35)–(4.38). Proof. The representation itself (4.40)–(4.41) is known, in the case of constant p, see [19], p. 236. Thus it is valid for C0∞ -functions. We only have to show that ϕ1 ∈ Lp$(·) (R1 , $). It suffices to show that g(x) ∈ Lp$(·) [(b, ∞), $]. It is known (see (13.33) in [19]) that g(x) has the form α   sin απ b b − τ ϕ(τ )dτ g(x) = − =: Aα (4.42) 1 ϕ. π x − b x − τ a p(·) [(a, b), ] to Lp$(·) [(b, ∞), $], it To check that the operator Aα 1 is bounded from L suffices to show that ∞ |Aψ(x)|p$(x) dx ≤ C < ∞ b

for all ψ with ψLp(·) (a,b) ≤ 1, where α  b ψ(τ )dτ b−τ µ $ (x) ν $ (x) (x − b) , x > b. Aψ(x) = (x − a) x−b (x − τ )(τ − a)µ(a) (b − τ )ν(b) a We have ∞

p $(x)

|Aψ(x)| b

2b

p $(x)

|Aψ(x)|

dx =

∞ dx +

|Aψ(x)|p$(x) dx =: U1 + U2 ,

2b

b

For U1 we obtain

# c #p$(x) # # 2b # # |ψ(τ )| dτ [ν(b)−α]p(b) # # U1 ≤ C (x − b) dx # # µ(a) # (τ − a) # a b # b #p$(x) # # 2b # # |ψ(τ )| dτ [ν(b)−α]p(b) # # +C (x − b) dx # ν(b)−α (x − τ ) ## # (b − τ ) b

c

where a < c < b. The first term here is easily estimated via ψLp(·)(a,b) by the H¨older inequality under conditions (4.39).

Characterization of the Range of One-dimensional . . .

413

The second term has the form # #  b # b−c |ψ(b − t)| dt #p$(x+b) # # [ν(b)−α]p(b) C x dx # # # 0 tν(b)−α (t + x) # 0 which is nothing else but the p$(· + b)-modular for the operator of the type (3.14)), so it is easily treated by means of Theorem 3.6 with Remark 3.7 taken into account. Finally, for U2 we have # #p$(x)  ∞ # b # dx |ψ(τ )| dτ # # U2 ≤ C # # [1+α−$ µ(∞)−$ ν (∞)]p(∞) # µ(a) (b − τ )ν(b)−α # (1 + |x|) (τ − a) 2b a where it remains to make use of the H¨older inequality.




R

Theorem 4.15. Let −∞ < a < b < ∞, p ∈ P(a, b) w-Lip (a, b) and (x) = (x − a)µ(x) (b − x)ν(x) , where µ ∈ w-Lipa (a, b), ν ∈ w-Lipb (a, b). Then D E α Ia+ Lp(·) [(a, b), ] = Lα,p(·) [(a, b), ] (4.43) under conditions (4.19). Proof. Let f ∈ Lα,p(·) [(a, b), ]. Then by the definition in (4.26) there exists an extension p$ of p and extensions µ $, ν$ of the exponents of the weight to R1 and a α,$ p(·) 1 (R , $) such that function g ∈ L f (x) = g(x) for

a ≤ x ≤ b.

α p By Theorem 4.11, g ∈ I α [Lp$(·) (R1 , $)] and consequently g ∈ I+ [L $(·) (R1 , $)] by α p(·) Theorem 4.7, which implies that f ∈ Ia+ [L (a, b), ] by Corollary 4.13. J p(·) K α L [(a, b), ] . Let f ∗ be the continuation of this Conversely, let f ∈ Ia+ function by zero beyond the interval [a, b] and let p$ be the continuation of p to R1 satisfying conditions ii)–iii) of Theorem 4.14. By Theorem 4.14 we have that f ∗ ∈ I α [Lp$(·) (R1 , )] and because f ∗ ∈ Lp$(·) (R1 , ), by Theorem 4.11 we have that
f ∗ ∈ Lα,$p(·) (R1 , ). Hence f ∈ Lα,p(·) [(a, b), ] .

5. Appendix For simplicity we prove the following technical lemma for the case of a bounded set Ω in Rn , the case of unbounded sets needs some technical modifications. We deal with the weights satisfying the condition
1 wk ∈ Lp(·) (Ω), ∈ Lp (·) (Ω), k = 1, 2, . . . , m. (5.1) wk We denote Ωk = {x ∈ Ω : wk (x) = 0} ∪ {x ∈ Ω : wk (x) = ∞}, From (5.1) it follows that |Ωk | = 0. We suppose that 0 Ωk Ωj = ∅ for all j = k.

k = 1, . . . , m.

(5.2)

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Lemma 5.1. Let Ω be an open bounded set in Rn , wk (x) be weights satisfying the assumptions (5.1) and (5.2) and let a linear operator A fulfill the conditions: i) it is bounded in the spaces Lp(·) (Ω, wk ), k = 1, . . . , m, ii) the operator χE1 AχE2 is bounded from the spaces Lp(·) (Ω,Rwk ), k = 1, . . . , m, into L∞ (Ω) for all disjoint sets E1 , E2 ⊂ Ω such that E 1 E 2 = ∅. m  wk (x). Then the operator A is bounded in the space Lp(·) (Ω, w) with w(x) = k=1

Proof. We have to prove the boundedness of the operator wA w1 in the space m  ak (x). To Lp(·) (Ω). We will make use of a corresponding partition of unity 1 = k=1

this end, we consider some neighborhoods Ek and Fk of the sets Ωk , that is, some open sets Ek and Fk such that 0 Ωk ⊂ Ek ⊂ E k ⊂ Fk ⊂ Ω and F k F j = ∅ for all k = j. (5.3) Such neighborhoods exist by assumption (5.2). We choose functions ak (x) such that ak (x) ≡ 1 on Ek and ak (x) ≡ 0 on Ω\Fk , so that ak (x)[wj (x)]±1 ≡ 0 on Ω\Fk if k = j. Then m m
w(x)
cj (y) = wk (x)bk (x) w(y) w (y) j=1 j k=1

where bk (x) and cj (y) are bounded functions supported in the same neighborhoods where the functions ak (x) and aj (y) were. Then wA

m m

ck f cj f f = bk wk A + bk wk A . w wk wj k,j=1 k=1

k=j

The first sum contains operators bounded in L to obtain the boundedness of the operators Ajk f = bk wk A

cj f , wj

p(·)

(Ω) by assumption i). It remains

j = k.

We have Ω = Ek ∪ Ej ∪ Ejk

with

Ekj = Ω\(Ek ∪ Ej ).

We denote for brevity χk = χEk , χj = χEj , χkj = χEkj . It is easily seen that cj χk ≡ 0 and bk χj ≡ 0. Taking this into account, we represent the operator Ajk = (χk +χj +χkj )Ajk (χk + 4  χj + χkj ) in the form Ajk = Bi where i=1

B1 = χkj wk bk Acj

χkj , wj

B2 = χk wk bk Acj

χkj , wj

Characterization of the Range of One-dimensional . . . B3 = χkj wk bk Acj

χj , wj

B4 = χk wk bk Acj

415

χj . wj

The operators B1 , B2 and B3 , containing the factor χjk are bounded in Lp(·) (Ω). χ (x) This follows from condition i) because χkj (x)wk (x) and wkjj (x) may be represented, whenever necessary, as χkj (x)wk (x) = u(x)wj (x)

and

v(x) χjk (x) = , wj (x) wk (x)

where u and v are bounded functions. Finally, for the operator B4 we observe that from condition ii) it follows that χ the function χk bk Acj wjj f is bounded for f ∈ Lp(·) (Ω) and then it suffices to refer to the fact that wk ∈ Lp(·) (Ω). Remark 5.2. An integral operator Af (x) =

F


K(x, y)f (y) dy satisfies condition ii)

Ω

of Lemma 5.1 if

sup

|K(x, y)| < ∞ for any ε > 0, and the weights satisfy

x,y:|x−y|≥ε

the second assumption in (5.1). Acknowledgments This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr.06-1000017-8792. Humberto Rafeiro gratefully acknowledges financial support by Funda¸c˜ ao para a Ciˆencia e Tecnologia (FCT) (Grant No. SFRH / BD / 22977 / 2005), through Programa Operacional Ciˆencia e Inova¸c˜ ao 2010 (POCI2010) of the Portuguese Government, cofinanced by the European Community Fund FSE.

References [1] A. Almeida. Inversion of the Riesz Potential Operator on Lebesgue Spaces with Variable Exponent. Frac. Calc. Appl. Anal., 6(3):311–327, 2003. [2] A. Almeida and S. Samko. Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Function Spaces and Applic., 4(2):113–144, 2006. [3] C. Bennett and R. Sharpley. Interpolation of operators., volume 129 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1988. [4] L. Diening. Maximal function on generalized Lebesgue spaces Lp(·) . Math. Inequal. Appl., 7(2):245–253, 2004. [5] L. Diening. Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and W k,p(·) . Mathem. Nachrichten, 268:31–43, 2004. [6] L. Diening, P. H¨ ast¨ o, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In “Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2, 2004, pages 38–58. Math. Inst. Acad. Sci. Czech Republick, Praha, 2005.

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[7] L. Diening and S. Samko. Hardy inequality in variable exponent Lebesgue spaces. Albert-Ludwigs-Universit¨ at Freiburg, Preprint Nr 1/2006-17.03.2006, 2006. 15 pages. [8] J. Duoandikoetxea. Fourier Analysis. Amer. Math. Soc., “Graduate Studies”, 2001. 222 pages. [9] M. Khabazi. Maximal operators in weighted Lp(x) spaces. Proc. A. Razmadze Math. Inst., 135:143–144, 2004. [10] V. Kokilashvili. On a progress in the theory of integral operators in weighted Banach function spaces. In “Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha. [11] V. Kokilashvili and S. Samko. Singular Integrals in Weighted Lebesgue Spaces with Variable Exponent. Georgian Math. J., 10(1):145–156, 2003. [12] V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted Lp(x) spaces. Revista Matem´ atica Iberoamericana, 20(2):495–517, 2004. [13] O. Kov´ ac˘ık and J. R´ akosn˘ık. On spaces Lp(x) and W k,p(x) . Czechoslovak Math. J., 41(116):592–618, 1991. [14] H. Rafeiro and S. Samko. On multidimensional analogue of Marchaud formula for fractional Riesz-type derivatives in domains in Rn . Fract. Calc. and Appl. Anal., 8(4):393–401, 2005. [15] M. R˚ uˇziˇcka. Electroreological Fluids: Modeling and Mathematical Theory. Springer, Lecture Notes in Math., 2000. vol. 1748, 176 pages. [16] S. Samko. Differentiation and integration of variable order and the spaces Lp(x) . Proceed. of Intern. Conference “Operator Theory and Complex and Hypercomplex Analysis”, 12–17 December 1994, Mexico City, Mexico, Contemp. Math., Vol. 212, 203–219, 1998. [17] S. Samko. Hardy inequality in the generalized Lebesgue spaces. Frac. Calc. and Appl. Anal, 6(4):355–362, 2003. [18] S. Samko. On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transf. and Spec. Funct, 16(5-6):461–482, 2005. [19] S. Samko, A. Kilbas, and O. Marichev. Fractional Integrals and Derivatives. Theory and Applications. London-New-York: Gordon & Breach. Sci. Publ., (Russian edition – Fractional Integrals and Derivatives and some of their Applications, Minsk: Nauka i Tekhnika, 1987.), 1993. 1012 pages. [20] I.I. Sharapudinov. The topology of the space lp(t) ([0, 1]). Mat. Zametki, 26(4):613– 632, 1979. Engl. transl. in Math. Notes. 26 (1979), no 3-4, 796–806. [21] E.M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, 1970. Humberto Rafeiro and Stefan Samko University of Algarve, Portugal e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 417–427 c 2008 Birkh¨
auser Verlag Basel/Switzerland

Orbit Representations and Circle Maps Carlos Correia Ramos, Nuno Martins and Paulo R. Pinto To the memory of Jos´e de Sousa Ramos Abstract. We yield C∗ -algebras representations on the orbit spaces from the family of interval maps f (x) = βx + α (mod 1) lifted to circle maps, in which case β ∈ N. Each orbit will encode an unitary equivalence class of an irreducible representation of: a Cuntz algebra Oβ if α = 0 and β > 1; an irrational / Q and β = 1; and a Cuntz-Krieger OAα,β whenever rotation algebra Aβ if α ∈ β > 1 and the critical point is periodic, where Aα,β is the underlying Markov transition matrix of f . Mathematics Subject Classification (2000). Primary 46L05; Secondary 37B10. Keywords. Interval maps, irrational rotation algebra, Cuntz-Krieger algebra, irreducible representations.

1. Introduction The interplay between operator algebras and discrete dynamical systems has been very fruitful since their origins. Von Neumann algebras theory has a strong relation with ergodic theory, see for example [20] and references therein or [6, 7]. After the work of Cuntz and Krieger [9] and Rieffel [19], where C*-algebras deeply related with Markov chains were found, the interplay and interest between discrete dynamical systems and C*-algebras has greatly increased. Besides interest in its own right, applications of representation theory of, e.g., Cuntz algebras to wavelets, fractals, dynamical systems, see, e.g., [3, 4, 13, 16], and quantum field theory in [1] are particularly remarkable. In the sequel of [8], we will use symbolic dynamics tools to yield further representations of Cuntz, Cuntz-Krieger and rotation algebras from interval maps. We consider the family of maps f (x) = βx + α (mod 1)

(1.1)

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defined on the unit interval I = [0, 1] with real parameters α and β. The map f can naturally be lifted to a map Φ(e2πix ) = e2πif (x) on S 1 , and the condition f (0) = f (1) implies that β ∈ N.  The generalized orbit R(x) = i∈Z f i (x) underlines again the Hilbert space attached to every point x upon we construct a representation. However, in contrast with [8], we produce not only representations of the Cuntz-Krieger algebra but also representations of the rotation algebra Aα , depending on the parameters α and β. Again the generalized orbit R(x) will encode an unitary equivalence class of irreducible representations, see Proposition 3.1 in the Cuntz algebra case, Theorem 3.4 for the rotation algebra and Theorem 3.5 in the Cuntz-Krieger case. We remark that our techniques are different from those employed in [5], where a countable family of irreducible representations of the irrational rotation algebra are yielded.

2. Preliminaries A representation of a ∗-algebra A on a complex Hilbert space H is a ∗-homomorphism π : A → B(H) into the ∗-algebra B(H) of bounded linear operators on H. Usually representations are studied up to unitary equivalence. Two representations $ are (unitarily) equivalent if there is a unitary π : A → B(H) and π $ : A → B(H) $ operator U : H →H (i.e., U is a surjective isometry) such that U π(a) = π $(a)U, for every a ∈ A. A representation π : A → B(H) of some ∗-algebra is said to be irreducible if there is no non-trivial subspace of H invariant with respect to all operators π(a) with a ∈ A. A well-known result, see, e.g., [17, Proposition 3.13.2], says that π is irreducible if and only if x ∈ B(H) : xπ(a) = π(a)x, for all a ∈ A =⇒ x = λI,

(2.1)

for some complex number λ, where I denotes the identity of A. By the very definition of commutant, (2.1) can be restated as follows: π(A) = CI. The representation is called faithful if it is injective. 2.1. Cuntz-Krieger and rotation algebras In the sequel we yield ∗-representations of two classes of C∗ -algebras. The first one is that of Cuntz-Krieger algebras. Let A = (aij ) be an n × n matrix with entries equal to 0 or 1 such that each row and column has at least one non-zero entry. The Cuntz-Krieger algebra OA associated to the matrix A is the C∗ -algebra [9] generated by (non-zero) partial isometries s1 , . . . , sn satisfying: s∗i si =

n
j=1

aij sj s∗j

(i = 1, . . . , n),

n


si s∗i = I,

(2.2)

i=1

where I denotes the identity. Recall that si is a partial isometry if and only if si s∗i si = si . The reader acquainted with the work of Cuntz and Krieger will notice

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that s∗i sj = 0 for i = j and si sj = aij si sj . Moreover, the range projections pi = si s∗i and support projections qi = s∗i si obey the following: pi pj = δij pi ,

qi qj = qj qi

and

qi sj = aij sj .

It is well known that the algebra OA is uniquely determined by the relations in (2.2), see [9, Theorem 2.13]. As a special case, we obtain the Cuntz algebra On when A is full: aij = 1 for all i and j. We now consider the second family of C*-algebras. Given θ ∈ R, as in [19], the rotation algebra Aθ is the universal C∗ -algebra generated by two unitaries u and v obeying uv = e2πiθ vu. (2.3) The C∗ -algebra Aθ is simple if and only if θ is irrational. It is clear that ∼ Aθ = An±θ , n ∈ Z. Conversely, if Aθ ∼ = Aθ
, then θ = n ± θ for some n ∈ Z (see [12] for the case θ rational and [18] for the case θ irrational). We also note that for any angle θ and k ∈ Z, we have uv k = exp(2πikθ)v k u, and hence Akθ ⊆ Aθ . If θ = pq is rational with gcd(p, q) = 1, then the irreducible representations of Aθ are all finite dimensional. Moreover, the dimension is q and the representations on Cq are given by u → zU , v → wV , where V en = en+1 , V eq−1 = e0 , U en = q ω n en , with ω = e2πip/q , {en }q−1 n=0 is the canonical orthonormal basis of C and z, w are two complex numbers of absolute value 1. If θ is irrational, the representation on l2 (Z) given by V en = en+1 and U en = e2πiθ en (where {en } is an orthonormal basis in l2 (Z)) is irreducible, but classifying all irreducible representations seems to be an impossible task, as for the Cuntz and Cuntz-Krieger algebras, see [3, 4, 8] where various families of irreducible representations are exhibited. 2.2. Piecewise monotone interval maps Let f : I → I be a piecewise monotone map of the interval I into itself, that is, there is a minimal partition of open sub-intervals of I, I = {I1 , . . . , Im } such that ∪m j=1 Ij = I and f|Ij is continuous monotone, for every j = 1, . . . , m. We define fj := f|Ij . The inverse branches are denoted by fj−1 : f (Ij ) → Ij . Let χIi be the characteristic function on the interval Ii . The following are naturally satisfied f ◦ fi−1 (x) = χf (Ii ) (x) x,

fi−1 ◦ f|Ii (x) = χIi (x) x.

Let {1,2,...,n} be the alphabet associated to some partition P = {I1 ,...,In } of open sub-intervals of I so that ∪nj=1 Ij = I, not necessarily I. The address map, is defined by ad :

n 

Ij → {1, 2, . . . , n},

ad(x) = i if x ∈ Ii .

j=1

We define Ωf := {x ∈ I : f k (x) ∈ ∪m j=1 Ij for all k = 0, 1, . . . }.

420

C.C. Ramos, N. Martins and P.R. Pinto Note that Ωf = I. The itinerary map it : Ωf → {1, 2, . . . , n}N is defined by it(x) = ad(x))ad(f (x))ad(f 2 (x)) . . .

and let Σf = it(Ωf ). Observe that the space Σf is invariant under the shift map σ : {1, 2, . . . , n}N → {1, 2, . . . , n}N defined by σ(i1 i2 . . . ) = (i2 i3 . . . ), and we have it ◦ f = σ ◦ it. We will use σ meaning in fact σ|Σf . A sequence in {1, 2, . . . , n}N is called admissible, with respect to f , if it occurs as an itinerary for some point x in I, that is, if it belongs to Σf . An admissible word is a finite sub-sequence of some admissible sequence. The set of admissible words of size k is denoted by Wk = Wk (f ). Given i1 . . . ik ∈ Wk , we define Ii1 ...ik as the set of points x in Ωf which satisfy ad(x) = i1 , . . . , ad(f k (x)) = ik . As in [8], we consider the equivalence relation on the set Ωf , Rf = {(x, y) : f n (x) = f m (y) for some n, m ∈ N0 }.

(2.4)

We write x ∼ y whenever (x, y) ∈ Rf . Consider the equivalence class Rf (x) and set Hx the Hilbert space l2 (Rf (x)) with canonical orthonormal basis {|y : y ∈ Rf (x)}, in Dirac notation. Note that Hx = Hy (are the same Hilbert spaces) whenever x ∼ y. The inner product (·, ·) is given by y|x = (|y , |z) = δy,z . For each i = 1, . . . , n, let us define an operator Si on Hx , with respect to some partition P = {I1 , . . . , In } of I, as follows: # A Si |y = χf (Ii ) (y) #fi−1 (y) . Note that χf (Ii ) (x) = 1 if and only if there is a pre-image of x in Ii . We have Si∗ |y = χIi (y) |f (y) . In fact @ A y| Si |z = y|fi−1 (z) = δy,f −1 (z) . i

On the other hand we have y| Si∗ |z = χIi (y) f (y)|z = χIi (y)δf (y),z . Since δy,f −1 (z) = χIi (y)δf (y),z we have shown that the operators Si , Si∗ are adjoint i of each other. We further remark that Si is a partial isometry: namely, Si is an isometry on its restriction to span{|y : y ∈ f (Ii )} ∩ H x and vanishes in the remaining part of Hx . For µ = µ1 . . . µk ∈ Wk we define Sµ = Sµ1 . . . Sµk . Also Sµ∗ = Sµ∗k . . . Sµ∗1 . Note that Sµ Sµ∗ |y = χIµ (y).

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3. Linear mod 1 interval maps, circle maps and representations Now, let us consider the family of maps f (x) = βx + α (mod 1)

(3.1)

such that f (0) = f (1), with β ≥ 1 and α ∈ [0, 1[. This family satisfies the conditions given in the preliminaries. There is a circle homeomorphism induced by f , as follows: Let φ : [0, 1[→ S 1 ⊂ C with φ(x) = e2πix . We set Φ : S 1 → S 1 $ with Φ(e2πix ) = e2πif (x) , where f$ is any extension of f to the interval I, obtained by choosing either f (cj ) = 0 or f (cj ) = 1, for each discontinuity point cj . The condition f (0) = f (1) implies that β = n ∈ N and the continuity of the map Φ. The behavior of the dynamical system (I, f$) (and consequently of both − (Ωf , f ) and (S 1 , Φ)) is characterized by the sequences it(f (c+ j )) and it(f (cj )), for each discontinuity point cj , see [14]. Therefore, it suffices to consider the itinerary − j of 0, since it(f (0)) = it(f (1)), and f (c+ j ) = 1, f (cj ) = 0. Let ωβ,α = {f (0)}j∈N0 be the orbit of 0. Let us consider the partition I = {I0 , . . . , In−1 }, with I0 = ]0, (1 − α)/β[ , . . . , Ij = ](j − α)/β, (j + 1 − α)/β[ , . . . , . . . In−1 = ](n − 1 − α)/β, 1[ ,

(3.2)

which is the minimal partition of monotonicity for f . The set {0, 1, . . . , n − 1} will be the alphabet. Set ξβ,α = (ξi )i∈N = itf (0) and consider the σ-invariant compact subset Σβ,α = it(I) of {0, 1, . . . , n − 1}N . Depending on the parameters α, β the orbit of 0 can be finite, in which case we obtain a Markov partition, see [14], which is a refinement of the above partition I. We denote the Markov partition by M ={J1 , . . . , Jm }. A characterization of the values of α, for which there is a Markov partition, is partially given by the following: Proposition 3.1. If the orbit ωn,α is finite then α ∈ Q. In particular α=

ξl + ξl−1 n + ξl−2 n2 + · · · + ξ1 nl−1 , with ξβ,α = (ξi )i∈N = itf (0). 1 + n + n2 + · · · + nl−1


Proof. See [14] for full details.

3.1. Case n > 1, α = 0 In this situation we have f (Ij ) = [0, 1], for every interval Ij , with j = 0, . . . , n − 1, in the partition in Eq. (3.2). See Fig. 1 for an example of a graph of f . Therefore, the operators Si , on Hx , for each i = 1, . . . , n, take the particular form: # A Si |y = #f −1 (y) . i

We have: Si∗ |y = χIi (y) |f (y) . operators Si are partial isometries and besides they satisfy the relations n These ∗ S S = 1 and Si∗ Si = 1, for i = 1, . . . , n. In fact, consider Si Si∗ acting on a i=1 i i vector |y of the canonical basis, and let k ∈ {1, . . . , n} such that y ∈ Ik . Then we have # A Si Si∗ |y = χIi (y)Si |f (y) = χIi (y) #fi−1 ◦ f (y) = χIi (y) |y .

422

C.C. Ramos, N. Martins and P.R. Pinto 1 0.8 0.6 0.4 0.2 0

0

0.2 0.4 0.6 0.8

1

Figure 1. Graph of f with α = 0 and n = 2 n n ∗ ∗ Thus i=1 Si Si |y = i=1 χIi (y) |y = χIk (y) |y = |y. Now, consider Si Si acting on a vector |y of the canonical basis. We have # # A A Si∗ Si |y = Si∗ #f −1 (y) = χIi (f −1 (y)) #f ◦ f −1 (y) = |y . i

i

i

We can now state the main result of this section. Proposition 3.2. The map πx : On → B(Hx ), given by πx (si ) = Si is a faithful irreducible representation of the Cuntz algebra On , where Hx = l2 (Rf (x)). Moreover, two such representations πx and πy are unitarily equivalent if and only if Rf (x) = Rf (y). Proof. The fact that πx is a representation of the Cuntz algebra On follows from the computation above. The remaining statements follow from [8, Theorem 7].
3.2. Case n = 1 Let n = 1 and assume α is an irrational number in ]0, 1[. Then the orbit of 0 is infinite. We consider P = {I0 , I1 }, with I0 =]0, 1 − α[, I1 =]1 − α, 1[. Thus, every orbit will be codified in the alphabet {0, 1}, via the itinerary map. The map f is invertible and we set f0 (x) = x + α, and f1 (x) = x − 1 + α. See Fig. 2 for an example of a graph of f . # A Let S0 and S1 be the operators on Hx defined by Si |y = χf (Ii ) (y) #fi−1 (y) , with i = 0, 1. We obviously have Si∗ |y = χIi (y) |f (y) , with i = 0, 1. Let V = Vx = S0∗ + S1∗ . This operator acts on the vector basis |y as V |y = |f (y) .

# A Since f is invertible, the operator V is unitary with V |y = #f −1 (y) . Let U = Ux be the diagonal operator (multiplication operator)

(3.3)

∗

U |y = φ(y) |y = e2πiy |y ,

(3.4)

Orbit Representations and Circle Maps

423

1 0.8 0.6 0.4 0.2 0

0

0.2 0.4 0.6 0.8

Figure 2. Graph of f with α = 1 −

1

√ 2 and n = 1

which is also an unitary operator, with U ∗ |y = φ(y) |y = e−2πiy |y . The von Neumann algebra generated by S0 and S1 is the same algebra as the algebra generated by p0 = S0 S0∗ , p1 = S1 S1∗ and V , since Si = pi V ∗ , with i = 0, 1. Moreover, we obtain the following result. Proposition 3.3. The operator U belongs to the von Neumann algebra generated by the operators S0 and S1 . Proof. For each µ ∈ Wk , let m(µ) be some point in Iµ ∩Rf (x). Note that if we have it(y) = (αj )∞ j=1 , for some point y ∈ Rf (x) then limj→∞ m(α1 . . . αj ) = y, and the limit is independent on the choice of m(α1 . . . αj ) ∈ Iα1 ...αj , since for each j ∈ N  there is r > j so that Iα1 ...αj ⊂ Iα1 ...αr . Let Mk = µ∈Wk e2πim(µ) Sµ Sµ∗ . We can see that limk→∞ Mk = U in the strong topology, since limk→∞ Mk v − U v = 0, for every v ∈ Hx . Therefore, U is in the von Neumann algebra generated by the operators S0 , S1 .
Now, since f is invertible, the equivalence class Rf (x) is given by a single orbit (the orbit of x), that is, Rf (x) = {f j (x) : j ∈ Z}, and Hx = l2 (Z). Theorem 3.4. Let x ∈ I. The map ρx : Aθ → B(Hx ), given by ρx (u) = Ux and ρx (v) = Vx is a faithful representation of the irrational rotation algebra Aθ = C ∗ (u, v : uv = e2πiθ vu), with θ = α. Moreover, two such representations ρx and ρy are unitarily equivalent if and only if Rf (x) = Rf (y). Proof. The definition of U = Ux and V = Vx gives U V |y = e2πi(α+y) |f (y) and V U |y = e2πiy |f (y). Therefore U V = e2πiα V U and ρx defines a (necessarily faithful) representation of the irrational rotation algebra Aα . Moreover, ρx is irreducible. Indeed, let T ∈ B(Hx ) commuting with both U and V . Since U |f j (x) = e2πi(x+jα) |f j (x), j ∈ Z, {f j (x)}j∈Z is an orthonormal basis in Hx and the eigenvalues e2πi(x+jα) are all distincts, it follows that T |f j (x) = λj |f j (x), j ∈ Z. But

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V |f j (x) = |f j+1 (x), so V T = T V gives λj = λj+1 , j ∈ Z. Therefore T is a scalar and thus ρx is irreducible. It is clear that Rf (x) = Rf (y), then ρx and ρy are unitarily equivalent. Notice that Ux and Uy have the same eigenvalues if and only if x ∼ y. Hence ρx and ρy can be unitarily equivalente only when x ∼ y.
3.3. Case n > 1, α ∈]0, 1[ We have two distinct situations to consider: the orbit of 0, ωn,α , is either finite or infinite. We will only treat the finite case, i.e., we will assume that n and α ∈]0, 1[ are such that ωn,α is a finite set. See Fig. 3 for an example of a graph of f . This happens when 0 is a periodic point or a pre-image of a periodic point. In both cases we obtain a Markov partition M = {J1 , . . . , Jm } and a transition matrix denoted by Aβ,α , see [14]. For each i = 1, . . . , m, let us consider the operator Si 1 0.8 0.6 0.4 0.2 0

0

0.2 0.4 0.6 0.8

Figure 3. Graph of f with α = 1 −

1

√ 2 and n = 2

on Hx , with respect to the Markov partition M: # A Si |y = χf (J ) (y) #f −1 (y) . i

i

Note that χf (Ji ) (x) = 1 if and only if there is a pre-image of x in Ji (also note that the operators Si are defined, in this case, with respect to the Markov partition and not with the partition I). We have Si∗ |y = χJi (y) |f (y) . The operators Si satisfy the relations m
i=1

Si Si∗ = 1,

Si∗ Si =

m


aij Sj Sj∗ ,

for i = 1, . . . , m,

j=1

where Aβ,α = (aij ) is the transition matrix defined above. In fact, consider Si Si∗ acting on a vector |y of the canonical basis, and let k ∈ {1, . . . , m} such that y ∈ Jk . Then we have # A Si Si∗ |y = χJi (y)Si |f (y) = χJi (y)χf (J ) (f (y)) #f −1 ◦ f (y) = χJi (y) |y , i

i

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425

since χIi (y)χf (Ji ) (f (y)) = χf (Ji ) (f (y)). Thus m


Si Si∗ |y =

i=1

m


χJi (y) |y = χJk (y) |y = |y .

i=1

Now, consider Si∗ Si acting on a vector |y of the canonical basis by # A S ∗ Si |y = χf (J ) (y)S ∗ #f −1 (y) = χf (J ) (y)χJi (f −1 (y)) |y . i

i

i

i

i

i

χIi (fi−1 (y))

= 1 and the condition χf (Ji ) (y) = 1 is equivalent to the Note that existence of a pre-image of y in Ji . Since y ∈ Jk this means that aik = 1. On the other hand m m

aij Sj Sj∗ |y = aij χJj (y) |y = aik |y . Thus

Si∗ Si

=

j=1

m

j=1

∗ j=1 ai,j Sj Sj .

Therefore, we have the following result:

Theorem 3.5. Let x ∈ I. Let α,n be such that the orbit of 0 is finite. The map πx : OAα,β → B(Hx ), given by πx (si ) = Si is a faithful irreducible representation of the Cuntz-Krieger algebra OAα,β . Moreover, two such representations πx and πy are unitarily equivalent if and only if Rf (x) = Rf (y). Proof. The fact that πx is a representation of the Cuntz-Krieger algebra OAα,β follows from the computation above. The remaining statements follow from [8, Theorem 7].
As in the n = 1 case, see proof of Proposition 3.3 above, the multiplication operator U |y = φ(y) |y = e2πiy |y is in the von Neumann algebra generated by the operators S1 , . . . , Sm . Proposition 3.6. The operators U, V satisfy the commutation relation U V = e2πiα V U n . Proof. We have U V |y = e2πif (y) |f (y) = e2πiα e2πiny |f (y) . On the other hand V U n |y = e2πiny |f (y) .


Therefore, U V = e2πiα V U n .

It seems interesting to look into our proofs and to analyze the C*-algebra generated by two unitaries u and v satisfying the relation uv n = zvu

(3.5)

with z ∈ S 1 . For z=1 (the “commutative” case) we get uvu−1 = v n , which is the defining relation1 of a Baumslag-Solitar group B(1, n), see [2]. One non-trivial 1 We

thank Florin Boca from University of Illinois at Urbana-Champaign for this point.

426

C.C. Ramos, N. Martins and P.R. Pinto

representation in 2 × 2 matrices is given by     n 0 1 1 u→ , v→ . 0 1 0 1 Therefore the C*-algebra generated by u and v obeying Eq. (3.5) should be regarded as a “twisted” group algebra of the untwisted z = 1 one. Acknowledgment First author acknowledges CIMA-UE for financial support. The other authors were partially supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia through the Program POCI 2010/FEDER.

References [1] M. Abe, K. Kawamura, Recursive fermion systems in Cuntz algebra, I. Commun. Math. Phys. 228 (2003), 85–101. [2] G. Baumslag, D. Solitar, Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc. 68 (1962), 199–201. [3] O. Bratteli, P.E.T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra. Mem. Amer. Math. Soc. 663 (1999), 1–89. [4] O. Bratteli, P.E.T. Jorgensen, Representation theory and numerical AF-invariants. The representations and centralizers of certain states on Od . Mem. Amer. Math. Soc. 168, no. 797, xviii+178 pp., 2004. [5] B.A. Brenken, Representations and automorphisms of the irrational rotation algebra. Pacific J. Math. 111 (1984), 257–282. [6] C. Correia Ramos, N. Martins, P.R. Pinto, J. Sousa Ramos, Orbit equivalence and von Neumann algebras for piecewise linear unimodal maps. Grazer Math. Ber. 350 (2006), 45–54. [7] C. Correia Ramos, N. Martins, P.R. Pinto, J. Sousa Ramos, Orbit equivalence and von Neumann algebras for expansive interval maps. Chaos, Solitons and Fractals 33 (2007) 109–117. [8] C. Correia Ramos, N. Martins, P.R. Pinto, J. Sousa Ramos, Cuntz-Krieger algebras representations from orbits of interval maps. Preprint. [9] J. Cuntz, W. Krieger, A class of C*-algebras and topological Markov chains Inv. Math. 56 (1980), 251–268. [10] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. 78 (1999), 401–430. [11] J. Feldman, C.E. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations. Ergodic Th. & Dynam. Sys. 9 (1989), 239–269. [12] R. Høegh, T. Skjelbred, Classification of C*-algebras admitting ergodic actions of the two-dimensional torus. J. Reine Angew. Math. 328 (1981), 1–8. [13] K. Kawamura, Representation of the Cuntz algebra O2 arising from real quadratic transformations. Preprint RIMS-1396, 2003. [14] N. Martins, J. Sousa Ramos, Cuntz-Krieger algebras arising from linear mod one transformations. Fields Inst. Commun. 31 (2002), 265–273.

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[15] N.E. Wegge-Olsen: K-Theory and C∗ -Algebras, Oxford University Press, 1993. [16] V. Ostrovskyi, Yu. Samoilenko, Introduction to the theory of representations of finitely presented *-algebras, I. Representations by bounded operators. Rev. in Math. and Math. Phys., 11, Harwood Academic Publishers, Amsterdam, iv+261 pp., 1999. [17] G.K. Pedersen, C*-Algebras and their Automorphism Groups. Academic Press, London Mathematical Society Monographs, 14, ix+416, 1979. [18] M. Pimsner, D. Voiculescu, Imbedding the irrational rotation C∗ -algebra into an AF-algebra. J. Operator Theory 4 (1980), 201–210. [19] M.A. Rieffel, C ∗ -algebras associated with irrational rotations. Pacific J. Math. 93 (1981), 415–429. [20] M. Takesaki, Theory of Operator Algebras. III. Encyclopaedia of Mathematical Sciences, 127. Operator Algebras and Non-commutative Geometry, 8. Springer-Verlag, Berlin, xxii+548 pp., 2003. Carlos Correia Ramos Departamento de Matem´ atica ´ Universidade de Evora, R. Rom˜ ao Ramalho, 59 ´ 7000-671 Evora, Portugal e-mail: [email protected] Nuno Martins and Paulo R. Pinto Centro de An´ alise Matem´ atica, Geometria e Sistemas Dinˆ amicos Departamento de Matem´ atica Instituto Superior T´ecnico, Av. Rovisco Pais 1 1049-001 Lisboa, Portugal e-mail: [email protected] e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 181, 429–437 c 2008 Birkh¨
auser Verlag Basel/Switzerland

On Generalized Spherical Fractional Integration Operators in Weighted Generalized H¨older Spaces on the Unit Sphere Natasha Samko and Boris Vakulov Abstract. For spherical convolution operators with a given power type asymptotic of their Fourier-Laplace multiplier we prove a statement on the boundedness within the framework of weighted generalized H¨older spaces on the unit sphere. The result obtained explicitly shows how spherical convolution operators under consideration improve the behavior of the continuity modulus of functions. Mathematics Subject Classification (2000). Primary 46E15; Secondary 42B15, 26A33. Keywords. spherical convolution operators, spherical potentials, indices of almost monotonic functions, Boyd-type indices, continuity modulus, generalized H¨ older spaces.

1. Introduction 

Let

k(x · σ)f (σ)dσ,

Kf (x) =

x ∈ Sn−1 ,

Sn−1

be a spherical convolution operator, and km be a spherical Fourier multiplier, corresponding to the kernel k(t). We refer for instance to [4], Ch. 6 for spherical convolutions and basics of harmonic analysis on sphere. The fractional type convolution operator  1 f (σ) α K f (x) = dσ, x ∈ Sn−1 , (1.1) γn−1 (α) |x − σ|n−1−α Sn−1

α > 0,

α = n − 1 + 2k, k = 0, 1, 2, 3, . . .

is a typical example (see [4], p. 151 for the value of the normalizing constant n−α

γn−1 (α)); the operator K α has the kernel

[2(1−t)] 2 γn−1 (α)

and its Fourier-Laplace

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N. Samko and B. Vakulov

multiplier is

 Γ m+ km (α) =  Γ m+



n−1−α 2  n−1+α 2

∼

1 mα

as

m → ∞.

(1.2)

There are known statements on the boundedness of the potential operator (1.1) within the frameworks of spaces of integrable functions, for example in Lebesgue spaces Lp (Sn−1 ) or weighted such spaces, see for instance, [4], Ch. 6 and references therein. The property of the potential K α ϕ to improve behavior of the function ϕ is in particular expressed in the Sobolev type theorem K α : Lp (Sn−1 ) → Lq (Sn−1 ) α with the “jump” 1p − 1q = n−1 . Theorems on boundedness of potential type operators K α within the frameworks of continuous functions are less known. For the generalized H¨older spaces H0ω (Sn−1 ) with a given continuity modulus ω (see their definition in Subsection 2.2), the “improvement” of the behavior of functions is expressed by theorems of the type K α : H0ω (Sn−1 ) → H0ω1 (Sn−1 ), where ω1 (h) = hα ω(h), 0 < α < 1. Such theorems including the case of weighted spaces were in particular obtained in [7], [8], [9], [10], [11], [12]. The goal of this paper is to find conditions – in terms of Fourier multiplier km – sufficient for the boundedness K : H0ω (Sn−1 , ρ)

→

1

H0ω (Sn−1 , ρ),

ω 1 (h) ≤ Cω(h)

(1.3)

for a certain class of spherical convolution operators K, generalizing the case of the potential operator K α . For simplicity we consider the case of power weight ρ(x) = |x − a|µ , a ∈ Sn−1 . We shall deal with the case when the “improvement” of ω(h) is in the above mentioned terms of multiplication by a power function: ω 1 (h) = hα ω(h) =: ωα (h),

0 < α < 1.

Note that in contrast to the case of spaces of integrable functions, the pointwise inequalities of the type |Aϕ(x)| ≤ |Bϕ(x)| between two operators A and B, do not allow to immediately derive the boundedness of the operator A in H¨ older type spaces from that of the operator B. By this reason the generalization undertaken in this paper required non-trivial efforts, in contrast to the case of spaces of integrable functions.

2. Preliminaries 2.1. On upper and lower indices of almost increasing functions Definition 2.1. A non-negative function ω on [0, ] is said to be almost increasing (or almost decreasing) if there exists a constant C ≥ 1 such that ω(t1 ) ≤ Cω(t2 ) for all t1 ≤ t2 (or t2 ≥ t1 respectively). Let W = {ω(t) ∈ C((0, ]) : ω(0) = 0, ω(t) > 0, t ∈ (0, ], ω(t) is almost increasing} .

On Generalized Spherical Fractional Integration Operators . . .

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Definition 2.2. Let ω ∈ W .The numbers ln lim inf mω = sup

h→0

t>1

ω(th) ω(h)

ln t

,

Mω = inf

t>1

ln lim sup ω(th) ω(h) h→0

ln t

introduced in such a form in [2], [3], will be referred to as the lower and upper index numbers of function ω ∈ W . For ω ∈ W we have 0 ≤ mω ≤ Mω ≤ ∞. 2.2. The spaces H0ω (Sn−1 , ρ) Let ω ∈ W and w(f, δ) =

sup x,y∈Sn−1 |x−y|<δ

|f (x) − f (y)|,

0 < δ ≤ 2.

By H ω (Sn−1 ) we denote the subspace of continuous functions on Sn−1 with the finite norm ω(f, δ) . f H ω (Sn−1 ) = f C(Sn−1) + sup 0<δ<2 ω(δ) Let ρ(x) = |x − a|µ , where a ∈ Sn−1 . The weighted space H0ω (Sn−1 , ρ) is defined as 5 6 H0ω (Sn−1 , ρ) = f : ρf ∈ H ω (Sn−1 ) and ρf |x=a = 0 . The space C 1 (Sn−1 ) used in the sequel, is understood as      x ∈ C B 12 ,2 , C 1 (S n−1 ) = f ∈ C(S n−1 ) : Dj f |x|

(2.1)

∂ϕ where Dj ϕ(x) = ∂x , j = 1, 2, . . . , n, and B 21 ,2 = {x : 12 ≤ |x| ≤ 2}. We refer to j paper [5] where various definitions of the space C 1 (S n−1 ) were considered and was shown that not all of them coincide with each other.

2.3. On spherical convolutions with a given asymptotic of the Fourier-Laplace multiplier Definition 2.3. By Wα,N with α ∈ R1 , N = 1, 2, 3, . . . we denote the class of spherical Laplace-Fourier multipliers {km }∞ m=0 , which have the following asymptotics at infinity N
  km = ci m−α−i + O m−α−N −ε , c0 = 0, (2.2) i=0

for some ε > 0. Clearly, the boundedness of the operator K with multiplier of form (2.2) from the space H0ω (Sn−1 , ρ) to the space H0ωα (Sn−1 , ρ) is reduced to that of potential type operators with the concrete Fourier-Laplace multipliers m−α−i and of the convolution operator KN corresponding to the rapidly decreasing Fourier   multiplier O m−α−N −ε .

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3. Formulation of the main result In [12] and [1] there was proved the following Theorem 3.1 for the potential operators (1.1) (first it was proved in [12] in a different form and later in [1] in the form given below). = hαω(h). The operator K α , where Theorem 3.1. Let ω ∈ W and ωα (h)  n−1  0<α< ω 1, is bounded from the space H0 S , ρ into the space H0ωα Sn−1 , ρ , ρ(x) = |x − a|µ , if Mω + α < 1

and

Mω + α < µ < n − 1 + mω .

(3.1)

We generalize this theorem and prove the following statement. µ ρ(x) = Theorem 3.2. Let 0 < α < 1. Let also ω ∈ W , ωα (h) = hα ω(h) and  n−1  |x−a| . ω A spherical convolution operator is bounded from the space H0 S , ρ into the  space H0ωα Sn−1 , ρ under conditions (3.1), if

{km }∞ m=0 ∈ Wα,N , N ≥ n + 1.

4. Proof of Theorem 3.2 4.1. A principal statement on the differentiability of the kernel Theorem 4.1. Let the multiplier km satisfy the condition c m→∞ |km | ≤ q , m where q > n. Then k  (t) has a structure

(4.1)

(t) , (4.2) 1 − t2 where (t) ∈ Lip([[−1 + δ, 1 − δ]), δ > 0 and (±1) = 0. Moreover, (t) satisfies the estimate 9 |(t)| ≤ C 1 − t2 , (4.3) so that k  (t) is a bounded function on [−1, 1]. If (4.1) is satisfied with q > n + 1, then k  (t) ∈ C([−1, 1]). k  (t) = √

Proof. It is known that by a given multiplier km the kernel k(t) may be calculated by the formula 1 K(t, ξ)(1 − ξ 2 )

k(t) = an

n−4 2

dξ,

n ≥ 3,

−1 2

where an =

2n−2 n[Γ( n 2 )] π

n+1 2

Γ( n−1 2 )

and

K(t, ξ) =

∞
m=0

m  9 km dn (m) t + iξ 1 − t2 ,

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where dn (m) =

(n + 2m − 2)(m + n − 3)! ∼ mn−2 m!(n − 2)!

as m → ∞, see [4], formula (6.9). By differentiating the series for K(t, ξ) with respect to t, we see that (4.2) is valid with 1 c(t, ξ)

(t) = an

9  n−4 1 − t2 − iξt (1 − ξ 2 ) 2 dξ

−1

where c(t, ξ) =

∞


m−1  9 mkm dn (m) t + iξ 1 − t2 .

m=1

By (4.1), the series defining the function c(t, ξ) is dominated by the series #m−1 √ ∞ ##
t + iξ 1 − t2 # . C mq−n+1 m=1 # # √ Since #t + iξ 1 − t2 # ≤ 1, this series converges absolutely and uniformly with respect to (t, ξ) ∈ [−1, 1] × [−1, 1] under condition q > n. Therefore, (t) is a continuous function. ∞  m−1 Observe that c(±1, ξ) = (±1) mkm dn (m) =: λ± does not depend on m=1

ξ. Therefore, 1 (±1) = ∓λ± ian

ξ(1 − ξ 2 )

n−4 2

dξ = 0.

(4.4)

−1

Furthermore, |c(t1 , ξ) − c(t2 , ξ)| ≤

∞
m=1

#m−1 #;  ; # # 2 2 # mkm dn (m) |t1 − t2 | + # 1 − t1 − 1 − t2 ## (4.5)

|t1 − t2 | 9 . ≤ C9 1 − t21 1 − t22

(4.6)

From (4.6) it is obvious that (t) is Lipschitzian inside the interval (−1, 1) and from (4.5) and (4.4) estimate (4.3) follows. To prove the continuity of k  (t) up to the end points t = ±1 in the case q > √ n + 1, it suffices to show that there exist the limits lim √(t) = lim (t)−(±1) . 1−t2 1−t2 t→±1

t→±1

We have (t) − (1) √ = an 1 − t2

1

−1

c(t, ξ)(1 − ξ ) 2

n−4 2

ian dξ + √ 1 − t2

1 [c(1, ξ) − c(t, ξ)]ξ(1 − ξ 2 ) −1

n−4 2

.

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N. Samko and B. Vakulov

It remains to show that there exists the limit c(1, ξ) − c(t, ξ) √ lim t→1 1 − t2 uniformly with respect to ξ ∈ [−1, 1]. We have √  m−1 ∞
t + iξ 1 − t2 −1 c(t, ξ) − c(1, ξ) √ √ = mkm dn (m) lim lim 2 2 t→1 t→1 1−t 1−t m=1 = iξ

∞


m(m − 1)km dn (m),

m=2 ∞ 

the passage to the limit being valid since the series

m(m − 1)|km |dn (m) con-

m=2

verges when q > n + 1. Similarly, the existence of the limit lim

t→1

c(t,ξ)−c(−1,ξ) √ 1−t2




is shown.

Remark 4.2. The proof of Theorem 4.1 was written for the case n ≥ 3. In case n = 2 the proof may be easily obtained in a similar way from the series representation ∞  km Tm (t) where Tm (t) = cos(m arccos t) are Chebyshev polynomials, k(t) = π1 m=0

arccos t)  √ (t) = m sin(m . if one makes use of the formula Tm 1−t2

Remark 4.3. It is well known that condition q >

n 2

in (4.1) yields

F1 −1

|k(t)|(1 −

n−3

t2 ) 2 dt < ∞ and condition q > n − 1 provides k(t) ∈ C([−1, 1]), see for example, [4], Lemma 6.3. It is natural that in Theorem 4.1 we needed the condition q > n for the information about the derivative k  (t). To get an information (4.2) about the differentiability of k(t) in the open interval (−1, 1) we had to impose the condition q > n, while for that on the closed interval [−1, 1] we needed the condition q > n+1. Corollary 4.4. If the Fourier multiplier km satisfies estimate (4.1) with q > n + 1, then the convolution operator K is bounded from the space L1 (Sn−1 ) to the space C 1 (Sn−1 ). Indeed, it suffices to ensure that the operators   ∂ x , k = 1, 2, . . . , n (Kf ) ∂xk |x| are bounded from L1 (Sn−1 ) to the space C(Sn−1 ). The latter follows from Theorem 1.2, since the operator #  #  n
# ∂ x ##  # (Kf ) = k (x · σ) σj (δjk − xj xk ) f (σ)dσ # ∂xk |x| #|x|=1 j=1 

where δjk =

Sn−1

1, j = k , has a continuous kernel. 0, j = k

On Generalized Spherical Fractional Integration Operators . . .

435

4.2. Proof itself of Theorem 3.2 Let

  Γ m + n−1−α 2  km (α) =  Γ m + n−1+α 2 be the Laplace-Fourier multiplier corresponding to the operator K. 1 may be It is known (see [6]; [4], p. 157) that the power multipliers mα+i represented in the form   N1
1 1 , (4.7) = b k (α + j) + O j,i m mα+i mα+i+N1 +1 j=i

where the constants bj,i do not depend on m, and N1 may be arbitrarily large. Taking N1 = N , we obtain for {km }∞ m=0 ∈ Wα,N :   N
1 km = , ε > 0. (4.8) aj km (α + j) + O mα+N +ε j=0 Therefore, the convolution operator K may be written in the form K=

N


aj K α+j + KN ,

(4.9)

j=0

  1 N N with the property km = O mα+N where the operator KN has the multiplier km +ε . This operator may be treated by means of Theorem 4.1. Indeed, the condition µ < n − 1 + mω in (3.1) guarantees that f L1 (Sn−1 ) ≤ Cf H0ω (Sn−1 ,ρ) .

(4.10)

The condition Mω + α < 1 implies that ωαx(x) is almost decreasing and therefore, ωα (x) ≥ Cx. Then we have the embedding gH0ωα (Sn−1 ) ≤ CgC 1 (Sn−1 ,ρ) .

(4.11)

Consequently, by (4.11), Theorem 4.1 and (4.10) we have for f = ϕρ , ϕ ∈ H0ω (Sn−1 ): 8 8 8 8 8 8 8 8 8 8 8ϕ8 8KN ϕ 8 8KN ϕ 8 8 8 ≤ C ≤ C 8 8 8 8 8 ρ 8 1 n−1 ≤ C ϕH0ω (Sn−1 ) . ρ H ωα (Sn−1 ) ρ C 1 (Sn−1 ) L (S ) 0

To state finally that

8 8 8 8 8ρKN ϕ 8 ≤ C ϕH ω (Sn−1 ) , 8 0 ρ 8H ωα (Sn−1 ) 0

it suffices to refer to the fact that ρ is a multiplier in the space H0ωα (Sn−1 ). The latter follows from the condition µ > Mω + α in (3.1). α α+N , K α+1 , 0< It remains to state that all the potential K  n−1 operators  n−1   ,...,K ωα ω α < 1, in (4.9) are bounded from H0 S , ρ to H0 S , ρ . For the potential K α this is stated in Theorem 3.1. For “better” potentials K α+j , j = 1, 2, . . . , N ,

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N. Samko and B. Vakulov

such a boundedness must be moreover valid. To justify this, we represent the potential operator K α+j in the form  Λj (x, σ) α+j f (x) = f (σ) dσ, x ∈ Sn−1 , (4.12) K |x − σ|n−1−α Sn−1

0 < α < 1,

j = 1, 2, . . . , N,

where the functions Λj (x, σ) = |x − σ| satisfy the Lipschitz condition j

|Λj (x, σ) − Λj (y, σ)| ≤ C|x − y|,

x, y ∈ Sn−1

with a constant C not depending  n x, y,σ. Then the boundedness of operators (4.12) from H0ω Sn−1 , ρ to H0ωα Sn−1 , ρ follows immediately from the following simple lemma. Lemma 4.5. Let Λ(x, σ) be any bounded on Sn−1 × Sn−1 function such that |Λ(x, σ) − Λ(y, σ)| ≤ C|x − y| with C > 0 not depending on x, y, σ ∈ Sn−1 . I. If an operator

 Kϕ(x) =

K(x, σ)ϕ(σ)dσ Sn−1

    is bounded from H ω Sn−1 to H ω1 Sn−1 , where ω and ω1 are continuity moduli, then any operator of the form  KΛ ϕ(x) = Λ(x, σ)K(x, σ)ϕ(σ)dσ  ω

Sn−1

   is also bounded from H Sn−1 to H ω1 Sn−1 .     II. If the operator K is bounded from H0ω Sn−1 to H0ω1 Sn−1 and Λ(a, σ)   is a multiplier in H0ω Sn−1 , then the operator KΛ is also bounded from  ω n−1 to H0ω1 Sn−1 . H0 S Proof. Indeed, by direct estimations one can easily obtain the following estimate for the continuity modulus of Kϕ: ω(KΛ ϕ, h) ≤ Ch · KϕC(Sn−1) + M ω(Kϕ, h) where M =

sup x,σ∈Sn−1 )

|Λ(x, σ)|. Therefore, the required boundedness for the oper-

ator Ka follows from that for the operator K, so that the part I is done. The part II is obvious.
Acknowledgment This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr. 06-1000017-8792.

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References [1] N.K. Karapetiants and N.G. Samko. Weighted theorems on fractional integrals in the generalized H¨ older spaces H0ω (ρ) via the indices mω and Mω . Fract. Calc. Appl. Anal., 7(4): 437–458, 2004. [2] N.G. Samko. On boundedness of singular operator in weighted generalized H¨older spaces H0ω (Γ, ρ) in terms of upper and lower indices of these spaces (in Russian). Deponierted in VINITI, no. 349-B91, 28p., Moscow, 1991. [3] N.G. Samko. Singular integral operators in weighted spaces with generalized H¨older condition. Proc. A. Razmadze Math. Inst, 120: 107–134, 1999. [4] S.G. Samko. Hypersingular Integrals and their Applications. London-New York: Taylor & Francis, Series “Analytical Methods and Special Functions”, vol. 5, 358 + xvii pages, 2002. [5] S.G. Samko and B.G. Vakulov. On equivalent norms in fractional order function spaces of continuous functions on the unit sphere. Fract. Calc. Appl. Anal., 3(4): 401–433, 2000. [6] B.G. Vakulov. Spherical convolution operators in the case of power decreasing of its multiplier (Russian). Deponierted in VINITI, no. 6545-83, Moscow, 1983. [7] B.G. Vakulov. Hardy-Littlewood-Sobolev theorems on potential type operators in Lp (Sn−1 , ρ). Deponierted in VINITI, no. 5435-86, 41p., Moscow, 1986. [8] B.G. Vakulov. Potential type operator on a sphere in generalized H¨older classes. Izv. Vuzov. Matematika, (11): 66–69, 1986. [9] B.G. Vakulov. Potential type operators on a sphere in generalized H¨older spaces. Deponierted in VINITI, no. 1563-86, 31p., Moscow, 1986. [10] B.G. Vakulov. Spherical operators of potential type in generalized weighted H¨older space on a sphere. Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Reg. Estestv. Nauki, (4): 5–10, 127, 1999. [11] B.G. Vakulov, N.K. Karapetiants, and L.D. Shankishvili. Spherical potentials of complex order in generalized H¨ older spaces. Izv. Nats. Akad. Nauk Armenii Mat., 36(2): 54–78 (2002), 2001. [12] B.G. Vakulov, N.K. Karapetiants, and L.D. Shankishvili. Spherical convolution operators with a power-logarithmic kernel in generalized H¨ older spaces. Izv. Vyssh. Uchebn. Zaved. Mat., (2): 3–14, 2003. Natasha Samko Universidade do Algarve, FCT Post Doc, Portugal e-mail: [email protected] Boris Vakulov Rostov State University, Russia e-mail: [email protected]