OT 59
Operator Theory: Advances and Applications Vol. 59 Editor: 1. Gohberg Tel Aviv University Ramat Aviv, Israel Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) R A. Fillmore (Halifax) C. Foias (Bloomington) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) J. A. Helton (La Jolla)
M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) J. D. Pincus (Stony Brook) M. Rosenblum (Charlottesville J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville)
Honorary and Advisory Editorial Board: P. R. Halmos (Santa Clara) T. Kato (Berkeley) P. D. Lax (New York)
M. S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)
Birkhauser Verlag
Basel Boston Berlin
Operator Theory and Complex Analysis Workshop on Operator Theory and Complex Analysis Sapporo (Japan) June 1991 Edited by T. Ando 1. Gohberg
1992
Birkhauser Verlag
Basel Boston Berlin
Editors' addresses: Prof. T. Ando Research Institute for Electronic Science Hokkaido University Sapporo 060 Japan
Prof. I. Gohberg Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv, Israel
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data Operator theory and complex analysis / Workshop on Operator Theory and Complex Analysis, Sapporo (Japan), June 1991. Ed. by T. Ando ; I. Gohberg. - Basel ; Boston ; Berlin : Birkhauser, 1992
(Operator theory ; Vol. 59) ISBN 3-7643-2824-X (Basel ...) ISBN 0-8176-2824-X (Boston ...) NE: Ando,'I§uyoshi [Hrsg.]; Workshop on Operator Theory and Complex Analysis <1991, Sapporo>; GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where copies are made for other than private use a fee is payable to »Verwertungsgesellschaft Wort<, Munich. © 1992 Birkhauser Verlag Basel, P.O. Box 133, CH-4010 Basel Printed in Germany on acid-free paper, directly from the authors' camera-ready manuscripts ISBN 3-7643-2824-X
ISBN "176-2824-X
V
Table of Contents Editorial Introduction
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ix
Scattering matrices for microschemes .
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Holomorphic operators between Krein spaces and the number . . . . . of squares of associated kernels .
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V. Adamyan
1. General expressions for the scattering matrix . 2. Continuity condition . . . . . . . . . . . References
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2 7 10
D. Alpay, A. Dijksma, J. van der Ploeg, H.S.V. de Snoo .
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. . . . . . . . . 0. Introduction . . . . . 1. Realizations of a class of Schur functions . . 2. Positive squares and injectivity . . . . . . . 3. Application of the Potapov-Ginzburg transform .
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11 11 15
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23 28
D. Alpay, H. Dym
On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains 1. Introduction . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . 3. 8(X) spaces . . . . . . . . . . . . . . . 4. Recursive extractions and the Schur algorithm 5. Np(S) spaces . . . . . . . . . . . . . . 6. Linear fractional transformations . . . . . . 7. One sided interpolation . . . . . . . . . . 8. References . . . . . . . . . . . . . . . . M. Bakonyi, H.J. Woerdeman
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The central method for positive semi-definite, contractive and strong Parrott type completion problems . . . 1. Introduction . . . . . . . . . . . . . . 2. Positive semi-definite completions . . . . 3. Contractive completions . . . . . . . . . 4. Linearly constrained contractive completions References
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. 78 . 78
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87 89 95
VI J.A. Ball, M. Rakowski
Interpolation by rational matrix functions and stability . . . of feedback systems: The 4-block case .
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Introduction . . . . . . . . . . . . . . . . . . . 1. Preliminaries . . . . . . . . . . . . . . . . . . 2. A homogeneous interpolation problem . . . . . . . 3. Interpolation problem . . . . . . . . . . . . . . 4. Parametrization of solutions . . . . . . . . . . . 5. Interpolation and internally stable feedback systems References
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. 96 . 96 100 104 109 116 131 140
H. Bart, V.E. Tsekanovskii
Matricial coupling and equivalence after extension
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Operator means and the relative operator entropy
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1. Introduction . . . . . . . 2. Coupling versus equivalence 3. Examples . . . . . . . . 4. Special classes of operators References
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143 143 145 148 153 157
J.I. Fujii 1. Introduction . . . . . . . . . . . . . . . . . 2. Origins of operator means . . . . . . . . . . 3. Operator means and operator monotone functions 4. Operator concave functions and Jensen's inequality 5. Relative operator entropy . . . . . . . . . . . .
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161 161 162 163 165 167 171
M. Fujii, T. Furuta, E. Kamei
An application of Furuta's inequality to Ando's theorem 1. Introduction . . . . . . . . . . 2. Operator functions . . . . . . . 3. Furuta's type inequalities . . . . 4. An application to Ando's theorem References
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173 173 175 176 177 179
T. Furuta
Applications of order preserving operator inequalities 0. Introduction . . . . . . . . . . . . . . . . . 1. Application to the relative operator entropy . . . 2. Application to some extended result of Ando's one References
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180 180 181 185 190
VII
I. Gohberg, M.A. Kaashoek
The band extension of the real line as a limit of discrete band extensions, I. The main limit theorem . . . . . .
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Interpolating sequences in the maximal ideal space of H°° II . .
0. Introduction . . . . . . . . I. Preliminaries and preparations II. Band extensions . . . . . . III. Continuous versus discrete . References
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191 191 193 201
205 219
K. Izuchi
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1. Introduction . 2. Condition (A2) 3. Condition (A3) 4. Condition (Al) References
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221 221 223 227 231 232
C.R. Johnson, M. Lundquist
Operator matrices with chordal inverse patterns 1. Introduction . . 2. Entry formulae . 3. Inertia formula . References
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234 234 237 243 251
P. Jonas, H. Langer, B. Textorius
Models and unitary equivalence of cyclic selfadjoint operators in Pontrjagin spaces . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The class F of linear functionals . . . . . . . . . . . . . . . . . 2. The Pontrjagin space associated with 0 E F . . . . . . . . . . . 3. Models for cyclic selfadjoint operators in Pontrjagin spaces . . . . . 4. Unitary equivalence of cyclic selfadjoint operators in Pontrjagin spaces .
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252 252 253 257 266 275 283
T. Okayasu
The von Neumann inequality and dilation theorems for contractions 1. The von Neumann inequality and strong unitary dilation 2. Canonical representation of completely contractive maps 3. An effect of generation of nuclear algebras . . . . . . . References
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285 285 287 289 290
L.A. Sakhnovich
Interpolation problems, inverse spectral problems and nonlinear equations . . . . . . .
References
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292 303
VIII
S.
Takahashi
Extended interpolation problem in finitely connected domains
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305 305 306 309 318 326
Accretive extensions and problems on the Stieltjes operator-valued functions relations . . . .
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328
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329 335
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345
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Introduction . . . . . . . . . . . . 1. Matrices and transformation formulas II. Disc Cases . . . . . . . . . . . III. Domains of finite connectivity . . .
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References
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E.R. Tsekanovskii .
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1. Accretive and sectorial extensions of the positive operators, operators of the class C(9) and their parametric representation 2. Stieltjes operator-valued functions and their realization . . 3. M.S. Livsic triangular model of the M-accretive extensions (with real spectrum) of the positive operators . . . . . . . 4. Canonical and generalized resolvents of QSC-extensions of Hermitian contractions . . . . . . . . . . . . . . . . References
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343 344
V. Vinnikov
Commuting nonselfadjoint operators and algebraic curves
1. Commuting nonselfadjoint operators and the discriminant curve 2. Determinantal representations of real plane curves . . . . . . 3. Commutative operator colligations . . . . . . . . . . . . . 4. Construction of triangular models: Finite-dimensional case . . 5. Construction of triangular models: General case . . . . . . . 6. Characteristic functions and the factorization theorem . . . . References
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348 348 350 353 355 359 364 370
P. Y. Wu
All (?) about quasinormal operators 1. Introduction . . . . . . 2. Representations . . . . 3. Spectrum and multiplicity 4. Special classes . . . . . 5. Invariant subspaces . . . 6. Commutant . . . . . . 7. Similarity . . . . . . . 8. Quasisimilarity . . . . . 9. Compact perturbation . 10. Open problems . . . . References
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Workshop Program List of Participants
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372 372 374 377 379 380 382 385 387 391 393 394
399 402
IX
EDITORIAL INTRODUCTION This volume contains the proceedings of the Workshop on Operator Theory and Com-
plex Analysis which was held at the Hokkaido University, Sapporo, Japan, June 11 to 14, 1991. This workshop preceeded the International Symposium on the Mathematical Theory
of Networks and Systems (Kobe, Japan, June 17 to 21, 1991). It was the sixth workshop
of this kind, and the first to be held in Asia. Following is a list of the five preceeding workshops with references to their proceedings:
1981 Operator Theory (Santa Monica, California, USA) 1983 Applications of Linear Operator Theory to Systems and Networks (Rehovot, Israel), OT 12
1985 Operator Theory and its Applications (Amsterdam, the Netherlands), OT 19 1987 Operator Theory and Functional Analysis (Mesa, Arizona, USA), OT 35
1989 Matrix and Operator Theory (Rotterdam, the Netherlands), OT 50 The next Workshop in this series will be on Operator Theory and Boundary Eigenvalue
Problems. It will be held at the Technical University, Vienna, Austria, July 27 to 30, 1993.
The aim of the 1991 workshop was to review recent advances in operator theory and
complex analysis and their interplay in applications to mathematical system theory and control theory.
The workshop had a main topic: extension and interpolation problems for matrix and operator valued functions. This topic appeared in complex analysis at the beginning of this
century and now is maturing in operator theory with important applications in the theory of systems and control. Other topics discussed at the workshop were operator inequalities and operator means, matrix completion problems, operators in spaces with indefinite scalar
product and nonselfadjoint operators, scattering and inverse spectral problems. This Workshop on Operator Theory and Complex Analysis was made possible through
the generous financial support of the Ministry of Education of Japan, and also of the International Information Science Foundation, the Kajima Foundation and the Japan Asso-
X
ciation of Mathematical Sciences. The organizing committee of the Mathematical Theory of Networks and Systems (MTNS) has rendered financial help for some participants of this
workshop to attend MTNS also. The Research Institute for Electronic Science, Hokkaido
University, provided most valuable administration assistance. All of this support is acknowledged with gratitude.
T. Ando, I. Gohberg August 2, 1992
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
1
SCATTERING MATRICES FOR MICROSCHEMES
Vadim Adamyan
A mathematical model for a simple microscheme is constructed on the basis of the scattering theory for a pair of different self-adjoint extensions of the same symmetric ordinary differential operator on a one-dimensional manifold, which consist of a finite number of semiinfinite straight outer lines attached to a "black box" in a form of a flat connected graph. An explicit expression for the scattering is given under a continuity condition at the graph vertices.
Contemporary technologies provide for the formation of electronic microschemes
composed of atomic clusters and conducting quasionedimensional tracks as "wires" on crystal surfaces. Electrons travelling along such "wires" behave like waves and not like particles. The explicit expressions for the functional characteristics of simple microschemes
can be obtained using the results of the mathematical scattering theory adapted for the special case when different self-adjoint extensions of the same symmetric ordinary differential operator on a one-dimensional manifold 12 are compared. The manifold S2 simulating the visual structure of a microscheme consists of m (< oo) semiinfinite straight outer lines
attached to a "black box" in a form of a flat connected graph. Different self-adjoint extensions of the mentioned symmetric operator distinguish only in boundary conditions at terminal points of outer lines. Having the scattering matrix for the pair of any self-adjoint operators of such kind with the special extension corresponding to the case of disjoint isolated outer lines one can immediately calculate by known formulae the high-frequency conductance of the "black box". In the first section of this paper the general expression for the scattering matrix for two self-adjoint extensions of the second-order differential operator in L2(11) is derived on the basis of more general results obtained in [1].
The detailed version of these results is given in the second part for the most important case when functions of the extension domain satisfy the condition of continuity
Adamyan
2
at the points of connection of outer lines to vertices of the graph. This work was initiated by some ideas from the paper [2].
1. General expressions for the scattering matrix. Remind the known concepts of the scattering theory. Let L be a Hilbert space and Ho, H be a pair of self-adjoint operators on L. Denote by Po the orthogonal projection on the absolutely continuous subspace of Ho. If the resolvent difference
(H -
zI)-1
- (Ho - zI)-1
at some (and, consequently, at any) nonreal point z is a nuclear operator, then by the Rosenblum-Kato theorem the partial isometric wave operators
W±(H, Ho) = t-foo s-lim e,Ht e-,Hot Po exist and map the absolutely continuous subspace of Ho onto the same subspace of H [3]. The scattering operator S(H, Ho) = W+(H, Ho)W-(H, Ho)
is unitary on PoL and commutes with Ho. The scattering matrix S(A), -oo < A < oo, is the scattering operator S(H, Ho) in the spectral representation of Ho. Consider the special case when H and Ho are different self-adjoint extensions of the same densely defined symmetric operator A in L with finite defect numbers (m, m). Let (e,,)' be any basis of the defect subspace ker[A* + i7]. Put
(Ho + iI)(Ho - zI)-1e,,,
1/=11 ... , m,
and introduce the matrix-function A(z),
(z) = ((Hoz + I)(Ho - zI)e,., e,), v, µ = 1, ... , M. According to the M. G. Krein resolvent formula for fixed Ho and arbitrary H m
(1)
(H -
zI)-1
= (Ho - zI)-1 - > ([A(z) + 4,V
Q]-1)µv (., e.(z))ea(z)
1
where a parameter Q is a Hermitian matrix [4]. The following parametric representation of the scattering matrix S(A) for extensions H and Ho was derived using the formula (1) in [1].
Adamyan
3
Take any decomposition of the absolutely continuous part Lo of Ho into a direct integral
Lo=j 00
®K(A)dA.
00
Without loss of generality one can assume dim K(\) < m. Let h (A) be a spectral image of the vector Poe,,. Then (2)
S(P) = I + 2iri(A2 + 1) E([A*(A + i0) +Q]-l)µ (
h. (A) )K( A) ho (A).
µ,v
Now we are going to adapt (2) for the case when a given symmetric operator A is a second-order differential operator on the above mentioned one-dimensional manifold 0 of the graph Stint and m outer semiaxes.
All self-adjoint extensions of A in L2(1) differ only by boundary conditions at the terminals (sk)i of outer lines and the corresponding connecting points (sk)in from the inside of the "black box" Stint. Take as Ho a special extension decomposing into an orthogonal sum :
m
Ho=Hnt®E®Hko; k=1
Hint
:
L2(fjint) -i L2(SZint);
HO k
(Hkf)(x)
L2(E+)
L2(E+), fi(GO) = 0.
As a consequence of the decomposition of Ho the Green function Go. (x) y), x, y E 1Z, of the operator Ho, i.e. the kernel of the resolvent (Ho - wI]-1 in L2(f) possesses the property: for any regular point w G,,,(x, y) = 0 if x E Stint and y belongs to any outer line and vice versa or if x and y are points of different outer lines. Describe the assumed properties of H nt. This operator can be considered as a self-adjoint extension of the orthogonal sum Aint ®Aii, of regular symmetric differential operators of the form
Aiv =
dPv()a +9Y()
Adamyan
4
on the corresponding segments (ribs) of the graph with continuous real-valued and p qv 0. The functions f from the A;,, domain in L2 (av, Qv ) functions p are continuously differentiable and satisfy boundary conditions
f(a-,)=f(av)=0i
f'(a-,)=f'(av)=0.
Note that every A;,, has defect indices (2N, 2N) where N is the number of the graph ribs.
Let B be the "soft" self-adjoint extension of A;v, i.e. the restriction of the conjugate operator A,,, on the subset of functions f such that f'(av) = f, (00 = 0. The operator Consider the special self-adjoint extension B = >v ®B of B can be taken as the part H ,,t in the decomposition of Ho. In this case the Green function GW(x, y) on flint X Stint coincides with the Green function E°(x, y) of B and in its turn E° (x, y) is nonzero only if "x" and "y" belong the same segments (a,,, Qv ) of 11,,,t and on these segments E° (x) y) coincides with the Green functions of B,,. Note that 6W(-, y) E L2(fl) for any regular point w and any y E Q. From the definition of the Green function Go (x) y) and its given properties it follows that the functions (Go ; (x, ,°)) i `
together with the functions (Go ; (x, a,,), Go ; (x, Qv )) N form a basis in the defect subspace M = ker[A* + ii] of A. Put 172v-1 = avi '72,, = /9
,
v = 1 ... Ni
712N+k = Sk, k = 1, ... , mi
and introduce the matrix-function r(w), F,.. (w) = GW (q, qv ),
Im w # 0, u, v = 1, ... , 2N + m.
According to the Hilbert identity for any regular points w, z of Ho and any x, y E 0 (3)
Jn
du Go. (x, u)G°(u, y)
=
w-z [G' (x, y) - G° (x, y)]. 1
As a consequence of (3) and the relation G° (x, y) = G,2,-(x, y) we have (4)
°µv(w) = Ju' +wHo][Ho = rto.,(w) - 2 [r ,,,(-i) + rY(-i) .
rlv))(x)Go:(x,rlµ)dx
Adamyan
5
Let H be an arbitrary self-adjoint extension of A in L2(SZ). The Krein resolvent formula (1) and (4) yield the following expression for the Green function G, (x, y) of H through Go. (x, y): 2N+m
(5)
G. (x, y) = GW (x) y)
- µ,v=1 ([r°(w) + Ql
(x, 1u)Go (,7.,, y),
where Q is a Hermitian matrix. Now to construct the scattering matrix S(A) for the pair H, H° notice that the parts B of H° as regular self-adjoint differential operators have discrete spectra and the parts Hk on the outer lines form the absolutely continuous component of HO. Consequently the first 2N basis vectors of the defect subspace M_ are orthogonal to the absolutely continuous subspace of H° and unlike the last m basis vectors G°;(x, '72N+k) _ G°_; (x, ,°E) belong to this subspace. The natural spectral representation of the absolutely continuous part of H0i i.e. of the orthogonal sum of the operator Hk, is the multiplication operator on the independent variable A in the space of C'"-valued functions L2(0, oo; Cm). The corresponding spectral mapping of the initial space can be defined in such way that the defect vectors turn into the vector-functions
hk(A)=
(6)
,l'
'a
1
A+iek,
A>0,
where ek E cm are the columns
e1 =
,
0
Using all above reasons we get immediately from (2) that S(.\) is the (m x m)-matrixfunction and (7)
k, 1 = 1, ... , m.
Skl(A) = bkl + 2t V L ([r0(A - i0) + Q]-1)2N+k,2N+l,
Represent the parameter Q and the matrix-function r°(w) in the block-diagonal form (8)
Q=
L
M1
WJ,
r° (w)_ [I'?t(w) 0
0
-iIm]
where W is a Hermitian (m x m)-matrix, Im is the unit matrix of order m and the matrixfunction r9 t(w) is determined by the Green function E° of the extension B as follows (9)
(rnt(w))µ = EW(*!n, riv),
Ft,v=1,...,2N.
Adamyan
6
Using (7) and (8) we get the formula: (10)
S(A) = { i I + W - M*[L + I'0,t(A - i0)]-1M } x
{V'
l
+W-M'[L+I'O,t(A-i0)]-1M}
.
This expression describes all possible scattering matrices for microschemes com-
prised of given m outer lines and a given set of N ribs. The parametric matrices W, M, L contain information on the geometrical structure of the "black box" and the boundary conditions at all vertices including the connecting points to the outer lines. Without loss of generality we can consider that all connecting points are the graph vertices. Single out now the scattering matrices for microschemes which differ only by the way the outer lines are connected to the definite inputs of the "black box", i.e. to the certain vertices of the definite graph Q. Notice that this limiting condition generally speaking leaves the Hermitian matrix W arbitrary. The matrix can now vary only as far as the subspace kerMM* remains unchanged. In what follows we will assume that this subspace and, respectively, its orthogonal complement in C2N are always invariant subspaces of the L matrix. Let Co be the orthogonal projector on the subspace ker MM* in C2N. Under the above condition and for various types of connection of outer lines to certain graph vertices only the block COLCO of the L matrix is modified. Notice that this block for a "correct" connection is always invertible. Consider now the connections for which the L matrix in (8) remains unchanged. This matrix is a parameter in the Krein formula when resolvents of B and a definite self-adjoint extension H nt of Aint are compared. Let rint(w) be the matrix-function which is determined by the Green function Ew of H nt like F() in (9) by E. According to the Krein formula of the form (5)
rint(w) = r nt(w) - rot (w)[r nt(w) + L]-1r nt(w) = L[r nt(w) + L]-1179 t(w) = L - L[r nt(w) + L]-1L. Taking into account that by the assumption that the block COLCO of the Hermitian matrix L is invertible on the subspace ker MM* C C2N, denoting by Q the corresponding inverse operator in this subspace and using (11) we can write (12)
M*[r ,,t(w) +L]-1M = M*QL[r9,(w) + L]-1LQM
= M*QM - M`Qriat(w)QM.
Adamyan
7
Inserting the last expression into (10) we find that the scattering matrix S(.1) for the connections without changing the parameter L and the subspace ker MM* has a form (13)
S(L) = { i
I'? (A - i0)M } x
L In +
x {-V' + W - Jf*fO (\ - iO)M
Y1
where the matrix parameters
W (= W - M*QM), M = QM depend on the boundary conditions at those vertices of the graph 0, which are connected with the outer lines. If there are reasons to consider that as a result of the connection of outer lines
to the graph 52 the matrices L and M are changed into M', L' so that ker(L - L') _ kerM'*M' = ker M'M, then it is natural to use the representation (14)
S(A) = {iVIIn + H(A -
i0)] [_i\/In + H(\ - i0)J -1,
H(w) = W + M'` [L + rinc(w)(Po - QL'Po)]
-1
x [Po]rino (w)Qo - I] M'.
The formula (14) can be obtained from (10) using the relation (11).
2. Continuity condition. From the physical point of view the most natural are self-adjoint extensions of A satisfying the continuity condition at the graph vertices. This condition states that all functions from the domain of any such extension possess coinciding limiting values at any vertex along all ribs incident to this vertex. Irrespective to the present problem consider now the structure of the Krein formula (5) with parameter Q for arbitrary self-adjoint extensions satisfying the continuity condition in every vertex. Take a vertex with s incident ribs, i.e. the vertex of degree s. It is convenient to enumerate the extreme points of the ribs at the vertex as i71, ... , . q,. Replacing x in Eq. (5) by qi for arbitrary y we obtain (15)
G.(rli,y) _ E Qi , ([ro (w) + Qi - 1), GW(+I.,y) N,V
It follows from the continuity condition that (16)
G.(,71,y)=G.('l2,y)=...=G,(,7e,y)
Adamyan
8
Since y and w in (15) and (16) are arbitrary it is obvious that the matrix Q in fact transforms any vector from C2N+m into a vector with equal first s components. Denote by J, the matrix of order s all components of which are unity. As Q is Hermitian, it is nothing but the following block matrix
Q=rhJ, 0
01
Q"
where h is a real constant and Q' is a Hermitian matrix of the rank 2N + m - s. Since the same procedure is valid for any vertex of arbitrary degree, the matrix with the suitable enumeration of the extreme points of the ribs and outer lines takes the block-diagonal form
(17)
h1J,1
0
0
h2J
0
0
Q=
where l is the total number of the graph vertices and s1i... , s. are corresponding degrees of the vertices. Thus the following lemma is valid. LEMMA. The parameter Q in the Krein formula (5) for extensions satisfying the continuity condition is the Hermitian matrix such that nonzero elements of every its row (column) are equal and situated at the very places where the unities of the incidence matrix of the graph 11 are.
Let the matrix Q be already reduced to the block-diagonal form (17) by a corresponding enumeration of extreme point of ribs and terminal points of outer lines. In this case the matrices W, M and COLCO of the representation (8) coincide with the diagonal matrix h1
h2
0
ID=
0
hm
where the parameters h1i ... , hm are determined by the boundary conditions in vertices to which the outer lines are connected. Using this fact and (13) we infer:
THEOREM. Let H be a self-adjoint extension of A in L2(51) satisfying the continuity condition and univalently connected with the set of parameters h1,... , hr of the corresponding matrix Q of the form (17) generating H in accordance with the Krein formula and let Ho be the special extension of A decomposing into an orthogonal sum of
Adamyan
9
the self-adjoint operators on the graph and on the outer lines. The scattering matrix S(.1) for the pair Ho, H admits the representation (18)
S(A) _ [iVIm + r(A - 10)] [_1/Im + r(A -
io)]
is the Green function of the self-adjoint extenwhere rik(w) = &(71i, 77k) and sion H nt of Ai,,t satisfying the continuity condition and determined by the same set of parameters hl, ... , h, for the same vertices like H is.
From the representation (18) it is obvious that the analytic properties of the scattering matrix S(A) are essentially determined by those of the matrix r(w) constructed by the Green function of the separated graph. For the regular differential operator the matrix (E, (77j, 'lk))1 is the meromorphic R-function. The natural problem thus arises of the partial recovery of the graph structure and the operator on it from the matrix S(A) or, equivalently, by the matrix r. In the case when the graph is reduced to a single segment this problem is the well-known problem of recovery of a regular Sturm-Liouville operator from spectra of two boundary problems. We hope to carry out the consideration of the former problem in a more general case elsewhere. In conclusion, as an example, consider an arbitrary graph with only two outer lines connected to the same vertex. In this case S(A) is the second order matrix-function but the determining matrix r(w) is degenerate and takes the form
r
E,,
where t; is the internal coordinate of the vertex of the graph tangent to the outer lines. The scattering matrix according to (18) now can be put in the usual form s(a)
t(A)
Tea)
]
,
where
r(A) =
i
2EaA_1la-i
t(A) -
A-/,u
57l`-i
are, respectively, the reflection and transition coefficients. Notice that according to the Landauer formula the resistance of the graph is given by z
R(a)
= R° It
PP
1
- Ro 4)) Ea (1, ) '
where Ro is the quantal resistance, i.e. the universal constant.
Adamyan
10
REFERENCES 1.
Adamyan, V. M.; Pavlov, B. S.: Null-range potentials and M. G. Krein's formula of generalized resolvents (in Russian), Studies on linear operators of functions. XV. Research notes of scientific seminars of the LBMI, 1986, v.149, pp. 723.
2.
3.
4.
Exner, P.; Seba, P.: A new type of quantum interference transistor, Phys. Lett. A 129:8,9 (1988), 477-480. Reed, M.; Simon, B.: Methods of modern mathematical physics. III: Scattering theory, Academic Press, New York - San Francisco - London, 1979. Krein, M. G.: On the resolvents of a Hermitian operator with the defect indices (m., m) (in Russian), Dok]. Acad. Nauk SSSR 52:8 (1946), 657-660.
Department of Theoretical Physics University of Odessa, 270100 Odessa Ukraine MSC 1991: 81U, 47A40
Operator Theory: Advances and Applications, Vol. 59 ® 1992 Birkhauser Verlag Basel
11
HOLOMORPHIC OPERATORS BETWEEN KREIN SPACES AND THE NUMBER OF SQUARES OF ASSOCIATED KERNELS
D. Alpay, A. Dijksma, J. van der Ploeg, H.S.V. de Snoo
Suppose that 0(z) is a bounded linear mapping from the Krein space a to the Kreln space 0, defined and holomorphic in a small neighborhood of z = 0. Then often 0 admits
which is
realizations as the characteristic function of an isometric, a coisometric and of a unitary If the colligations colligation in which for each case the state space is a Kreln space. satisfy
minimality
conditions
(i.e.,
are
controllable,
observable
or
closely
connected,
respectively) then the positive and negative indices of the state space can be expressed in terms of the number of positive and negative squares of certain kernels associated with 0,
depending on the kind of colligation. In this note we study the relations between the numbers of positive and negative squares of these kernels. Using the Potapov-Ginzburg transform we give
a reduction to the case where the spaces a and 6 are Hilbert spaces.
For this case these
relations has been considered in detail in [DLS1]. 0. INTRODUCTION
Let (?3,
or a and ll for short, be Krein spaces and denote by L(),@)
[...]0) and (0,
the space of bounded linear operators from tj to 0 (we write L(jY) for L(af,jY)). If TeL(l,(s3), we write Tt (EL(1t3,?3)) for the adjoint of T with respect to the indefinite inner products [.,.] and [.,.]m on the spaces tjy and 0. We say that TeL(a, l) is invertible (instead of boundedly invertible) if T ueL(6,a,).
By S(a,(s3) we denote the (generalized) Schur class of all L( ,b)
valued functions 0, defined and holomorphic on some set in the open unit disc D={zECI I z I <1}; we denote by Z(0) the domain of holomorphy of 0 in D. The class of Schur functions 0 for which 0eZ(0) will he indicated by S°(ar,0). If Z is a subset of D, we write Z*={zIzeb }. With each
0eS(a,0) we associate the function 6t defined by ®(z)=0(z)*. Clearly, 9ES(6,) ), a(6)=D(0)* and if 0ES°(j, 3), then ®eS°(af,(s3). We associate with 0 the kernels 17e(z,w)=I-0(w)`0(z)
z,wep0),
1-zoz
with values in L(R) and L((s3), respectively, and the kernel
ae(z,w)=f-0(w)0(z)`
1-wz
z,wED(0)*,
Alpay et al.
12
I-®(w)'®(z) ®(2)'-0(w)* Se(z,w)
1-wz
z-w
®(w)-0(z)
I-©(w)e(2)`
w-z
1-wz
z,wE1(®)nb(®)`, z#w,
with values in L()eft where af®b stands for the orthogonal sum of the Krein spaces a and 6. Here I is the generic notation for the identity operator on a Krein space, so in the kernels I is the identity operator on a or on 0. If we want to be more specific we write, e.g., I,3 to indicate that I is the identity operator on a. In this paper we prove theorems about the relation between the number of positive (negative) squares of the matrix kernel 5e and those of the kernels oe and ag on the diagonal of Se. We recall the following definitions. Let St be a Krein space. A kernel K(z,w) defined for z,w in some subset Z of the complex plane C with values in L(.?), like the kernels considered above, is called nonpositive (nonnegative), if K(z,w)*=K(w,z), z,wEZ, so that all matrices of the form where nEN, and are arbitrary, are hermitian, and the eigenvalues of each of these matrices are nonpositive (nonnegative, respectively). More generally, we say that K(z,w) has is positive (negative) squares if K(z,w)*=K(w,z), z,wE2i, and all hermitian matrices of the form mentioned above have at most r and at least one has exactly a positive (negative) eigenvalues. It has infinitely many positive (negative) squares if for each x at least one of these matrices has not less than K positive (negative) eigenvalues. We denote the number of positive and negative squares of K by sq+(K) and sq_(K), respectively. If, for example, sq_(K)=0, then K(z,w) is nonnegative. In the sequel we denote by ind+S and ind_.S the dimensions of the Hilbert space St. and the anti Hilbert space 52_ in a fundamental decomposition ([K(zi,zl)f;,fJ].,)i,=1i
k =.11+®R_ of S. Then ind+S2 = sq+(K) where K is the constant kernel K(z,w) = I, and the indices are independent of the chosen fundamental decomposition of St. Whenever in this paper we use the
term Pontryagin space we mean a Krein space, k2, say, for which ind_,S < oo.
The main theorems in this paper concern the relation between the values of sq+(Se) on the one hand and the values of sq+(ae) and sq+(ae) on the other hand. The most general one implies that, if icENu{0}, then sq_(Se)=ic if and only if sq_(oe)=sq_(ae)=K, and sq+(Se)=K if and only if sq+(oe)=sq+(oe)=K. To formulate this theorem we consider two fundamental decompositions R =t`f+®a_ and M=6+(D (s3_ of Of and 0, and we denote by P+ the orthogonal projections on ) onto the spaces L + and by Q+ the orthogonal projections on 0 onto the spaces (t3+. THEOREM 0.1.
Let jY and 0 be Krein spaces and let eES(1,(t3). Then:
(t) sq-(Se)
Alpay et al.
13
Of course Theorem 0.1 remains valid if the minus sign is everywhere replaced by the plus sign. Theorem 0.1 is closely related
Note that if Q_e(z) j g_ a L(R_, @_) is invertible, then ind_jY= ind_@.
to the theorem which states that if TEL(jY,@) is a contraction (i.e, rT
that the nxn matrix (1/(1-z;zj)), where z1,z2i...,z, are n different points in D,
is positive.
This follows for instance from the formula 1/(1-zw)=(27r)-lfo"(e`T-z)-l(e-`r-w)-ldr, z,WED.
In the case where a and t3 are Krein spaces such that either their positive indices ind+u and ind+@, or their negative indices ind-a and ind_tl, are finite, Theorem 0.1 can be sharpened. We
only state the version for which the last condition on the indices holds; the one for which ind+t'f and ind+@ are finite is mutatis mutandis the same. THEOREM 0.2.
Let a and @ be Pontryagin spaces,
and let /cEr'iu{0}.
Then the
following three statements are equivalent. (i) (ii)
sq-(Se) =lc.
(iii)
sq_(ag)=K and ind_ =ind_t .
sq_(ae) = K and ind_a = ind_@.
If (i)-(iii) are valid then Q_e(z)jg-eL(R_,@_)
is invertible for all, except for at most K,
points zE b(e). Moreover, if these conditions hold, then (9 can be extended to a meromorphic function on D, and the number of negative squares of each of the three kernels associated with the extended function coincides with the number of negative squares of the corresponding kernel for the function e. This theorem corresponds to the fact that if ind-a = ind($ < oo, then every contraction T E L(a, @) is also a bicontraction; see, for example, [IKL], [DR]. Theorem 0.2 implies that if ind-tj = ind_(53 < oo, then sq_(Se) = sq_(ae) = sq_(ae), which for the case where a and t3 are Hilbert spaces was already
proved in [D[S1]. The equality here means that either all three numbers sq_(Se), sq_(ae), sq_(ae) are infinite, or one of them is finite in which case all three are finite and equal. If ind_ < oo, ind_@ < oo, but ind-a # ind_@, then the equality between the numbers of negative squares of
the three kernels need not hold, as the following example shows. To avoid confusion with the signs we give a counterexample to Theorem 0.2 in which all signs are reversed. Take a = CZ. &=C and define the 1x2 matrix function eES°(jY,@) by e(z)=(kz,1), zED, where kEC and IkI <1. Then ore
is a 2x2 matrix kernel, ag is a scalar kernel and Se is a 3x3 matrix kernel. In this case we have that sq+(ae) = 0, but sq+(S9) = sq+(ae) = co
Alpay et al.
14
The proofs of Theorems 0.1 and 0.2 which we give are basically a variation of the proof of the theorem about the characterization of bicontractions in a Kreln space mentioned previously: we also make use of the Potapov-Cinzburg transform. We show in this paper that, if the kernels associated with 0 have a finite number of negative squares, then the Potapov-Ginzburg
transformation of 0 can be defined,
and,
if we denote this transformation by E, then
sq_(aE)=sq_(oe), sq_(oE)=sq_(ae) and sq_(SE)=sq_(Se). Since E is essentially a Schur function between Hilbert spaces, the proofs of Theorems 0.1 and 0.2 are obtained through a reduction of the general case to the case where Y and 0 are Hilbert spaces, which was considered in [DLS1].
Theorems 0.1 and 0.2 are related to Theorem 7.6 in the excellent lecture notes by Ando [An], which contains a detailed account of the case K = 0. To point out the connection we state the following related result. Let jy and (tS be Kreln spaces and let 0c-S(5,@). assertions are equivalent. (a) For each zED: 0(z) is a bicontraction. THEOREM 0.3.
If Z(0)=D, then the following
(d)
For each zeD: 0(z) is a contraction and Q 0(z)f0_eL()_,@_) is invertible. For each zeD: 0(z) is a contraction and P_0(z)j@_EL((t3_,j_) is invertible. sq-(ae)=0 and sq-(ac4)=0.
(e)
sq-(Se) =0.
(b)
(c)
The equivalences (a)ra(b)e*(c) are alluded to above. The implications (e)-.(d), (d)=(a) are trivial: the first one follows from the fact that the kernels ae(z,w) and og(z,w) are on the diagonal of S0(z,w) and the second follows by taking w=z in ae(z,w) and aA(z,w). To prove the implication (a)=o-(e), Ando in [An] uses complementation theory and the theory of operator ranges; see also [B2,3]. On the other hand on account of the invertibility of Q 0(z)jg_EL(U_,(93_) the
implication can be reduced to one where the spaces involved are Hilbert spaces, and then the implication can be and has been proved in different ways; see [SF] and [BR]. In this paper we are interested in the equivalence (d). .(e), which has been proved by McEnnis [Mc]. The purpose of this paper is to generalize this equivalence to the case where we allow the kernels associated with 0 E S(jY, c 3) to have negative squares. These generalizations are formulated in Theorems 0.1 and 0.2. Note that we do not require that z(0)=D.
The kernels associated with 0eS°(a,6) have appeared in a number of papers, frequently in connection with the theory of (co-)isometric and unitary colligations; of the more recent papers,
mention for instance [A], [AD1,2,3], [AG1,2], [An], [B1,2,3], [BC], [CDIS], [M] and [Y]. If ) and 0 are Hilbert spaces then 0 can be written as the
we
[DLS1,2,3],
characteristic function of (a) a closely innerconnected isometric colligation, (b) a closely outerconnected coisometric colligation, as well as of (c) a closely connected unitary
Alpay et al.
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If in all three cases we denote by St the state space of the colligation, then SF is a Kreln space with ind±(t)=sq±(oe) in case (a), ind,(S)=sq,(oe) in case (b) and ind,(S)=sq,(Se) in case (c). Proofs of these results can be found in, e.g., [DLS1]; see also Remark 1.1 in Section 1 of this paper. In the general case where ) and 6 are Krein spaces, e e S°(j, (53) can also be realized on a neighborhood of 0 as the characteristic function of a minimal unitary or (co-) colligation.
isometric colligation. In this case the state space of the colligation is uniquely determined by 9 up to weak isomorphisms and so its positive and negative indices are uniquely determined by 9, cf. Section 1 below.
We briefly describe the contents of the three sections of this paper.
In Section
1
(REALIZATIONS OF A CLASS OF SCHUR FUNCTIONS) we consider the case where a and b are Hilbert
spaces, and we recall some facts from [DLS1].
For this case we show in Section 2 (POSITIVE SQUARES AND INJECTIVITY) that if the kernels have a finite number of positive squares, then the operator function Q9(z) 10_ is invertible. As an aside we prove a simple result about the
positivity of the Schur product of nonnegative matrices.
The main theorems of this paper,
Theorem 0.1 and Theorem 0.2, are proved in Section 3 (APPLICATION OF THE POTAPOV-GINZBURG
TRANSFORM) by invoking the results of the previous sections.
There we also recall the
definition of the Potapov-Ginzburg transform.
In a sequel to this paper we give an interpretation of our results from the point of view of reproducing kernel Pontryagin spaces and operator ranges. We show how this leads to models for contractions in Pontryagin spaces in which certain maximal negative invariant subspaces can be displayed explicitly. 1.
REALIZATIONS OF A CLASS OF SCHUR FUNCTIONS
A colligation (system or node) is a quadruple A=(St,a,1b;U) consisting of three Krein spaces Sz (the state space), Ll (the incoming space) and 6 (the outgoing space) and an operator UeL(S i ,AE)@) (the connecting operator). We often write a=(St,J,0;T,F,G,H) to indicate that U has the operator matrix decomposition
U=1G where TaL(t) (the basic operator), FaL(a,SL), GaL(S2,(t3) and HeL(jy,(53). The colligation is called isometric, coisometric or unitary if the connecting operator has this property. A colligation A_(.t,R,(t3;T,F,G,H) is called closely innerconnected (controllable) if
SZ==VN9t((I-zT)-nF)=V{T"Ff IneNu{0}, fea}, closely outerconnected (observable) if R
3?((I-zT')-nG")=V{T'"G'glneNu{0}, .CA(
ge($},
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and it is called closely connected if
ZeN(R((1-zT)-'F)u9t((I-zT')-'G'")).
A
In these cases N stands for a small neighborhood around 0 and V signifies the closed linear ZEN span of the sets with index zaN. We call an isometric (coisometric, unitary) colligation minimal
if
it
is
closely
innerconnected
(closely
outerconnected,
closely
connected,
respectively).
A weak isomorphism V from a Krein space Sz to a Krein space SZ' is a mapping with dense domain
D(V) in St and dense range N(V) in St' such that [Vx,Vy]q,=[x,y]ff, x,yaD(V). If S2 and St' are isomorphic, then ind+(,S2)=ind+(S'). Two colligations 4=(St,,(S3;T,F,G,H) and with the same incoming and outgoing spaces a and 0 are called weakly isomorphic if H=H' and there exists a weak isomorphism V from St to St' such that weakly
0
(0 1)- (0
IG' H'1
11
(G H) on ( ZaV)
If V can be extended to an isomorphism (unitary operator) from St' onto St, then of course the
colligations 0 and
A' are called isomorphic or unitarily equivalent. colligation A= (R,a,63;T,F,G,II) is the socalled characteristic function
Associated
with a
e(z) =O,a(z) =H+zG(I -zT) 'F,
defined for all z in some neighborhood of 0. Clearly, 0ES°(R,t ). If for 0ES°(a,($) there is a colligation A such that O(z)=OA(z) in a neighborhood of 0, then the colligation is called a
realization of 0.
Realizations
in a general Krein space case are studied
in, e.g., [Azl,2],
[Bl,2,3], [CDLS], [DLS3], [M], [Y] and for a special class in [DLSI].
If a and t3 are Krein spaces, 0ES°(af,0) and 0=Oa on 1((9)n1(O4) for some isometric (coisometric, unitary) colligation 0=(St,a,0;U), then ind+.S2>sq+(a,,) (ind..S2>sq.(oe), ind+k sq±(S(4), respectively). If the colligation is minimal then in the corresponding REMARK 1.1.
inequality the equality sign prevails. These (in-)equalities can easily be deduced from the following relations between the kernels and the inner product of the state space St of A. For and
(i)
we have
[oH(z,w)fi,f2]g=[(I-zT)-'Ffl,(I-wT)-'Ff2]c,
(ii)
(iii)
if A is isometric; if A is coisometric;
[S0(z,w)
(f'.), (f) ]30= [(I -zT) 'Ffl+(I -zT') 'G 9i,(1-wT) 'Ff2+(I -wT`) 'G*92]R, 9
92
if A is unitary. In this section we summarize some of the results from [DLS1].
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17
Assume that R and 0 are Hilbert spaces and let OES°(jY,@). Then
THEOREM 1.2. (1.1)
sq-(ae) = sq-(ae) = sq-(Se)
That is, either these three numbers are infinite or one of them is finite and then they all are finite and equal. If one of them is finite and equal to #ENu{O}, say, then e admits the following realizations.
(a) O=ea,on D(O) for some isometric colligation Zj=(t1,j,t3;U,), and then ind+Sm>sq+(ae); the isometric colligation dl can be chosen closely innerconnected, in which case it is uniquely determined up to isomorphisms and ind+S21 = sq+(ae). (b) 0=0'd2 on D(O) for some coisometric colligation z12=(52,f3,(g3;U2), and then ind+l2>sq+(ae); the colligation d2 can be chosen closely outerconnected, in which case it is uniquely determined up to isomorphisms and ind+S2=sq+(ae). (c) O=Oe on D(O) for some unitary colligation %=(52,J,(53;U), and then ind+S2>sq+(Se); the unitary coisometric
colligation z can be chosen closely connected, in which case it is uniquely determined up to isomorphisms and ind+S2 = sq+(Se ).
One of the key tools in the proof of Theorem 1.2 given in
[DLS1] is formulated in the A Hilbert space version of it is given by de Branges and Shulman in [BS]. It relates isometric colligations to unitary colligations.
following lemma.
LEMMA 1.3.
Let D1=(521, ,(53;01) be a closely innerconnected isometric colligation, in which jV
and 6 are Hilbert spaces and Sl is a Pontryagin space.
Then there exists a closely connected unitary colligation z1=(52,5,b;U) such that ind-k=ind-.11i Ulg=V and Oa=0olon D(ea)nD(Ool).
Sketch of the proof of Theorem 1.2. (1) We first consider the case where sq_(ae) is finite, and we briefly describe the construction of the realization described in (a). Consider the linear space 13 of finite sums ze f2f where fZe{ and ez is a symbol associated with each
and provide £ with the (possibly degenerate, indefinite) inner product [EZe
=E2 W[ae(z,w)fz,gw)]g.
Define the linear operators To, F0, Go and H via the formulas T 06.f = z (Ezf - e0f ),
F0f = e0f ,
GOezf = z (®(2 )f - O(O )f
Hf =O(0)f,
where z # 0 and f r= tY. Then To and Go are densely defined operators on P- with values in 13 and 18,
respectively, Fo : a -s 2, H: a -* (9,
Uo=IG0 H
al>(0
is an isometric operator on a dense set and H+zG0(I-zTo)-1F0=O(z) on a. Now we consider the
Alpay et al.
18
quotient space of i over its isotropic part and redefine the operators on this space in the usual way. Then completing the quotient space to a Pontryagin space and extending the operators by continuity to this completion we obtain a closely innerconnected isometric colligation with the desired properties.
is given that sq_(ae) is finite we apply the above construction to 0 to obtain a realization in terms of a closely innerconnected isometric colligation _ (1t1,($, ;U1 ). Then (2) If it
,A2=(S2,j,t3;U2) with %=S1 and U2=U1 is the closely outerconnected coisometric colligation with the properties mentioned in (b). If sq_(Se) is finite, then also sq_(ae) is finite and a
realization of t9 described in (c) can be obtained by constructing a realization of type (a) as done in part (1) and by invoking Lemma 1.3.
(3) We omit the details of the proof that the minimality condition implies the essential uniqueness of the colligation in the realization of O.
This uniqueness property together with the constructions indicated in steps (1) and (2) imply that ind_S1= ind_S2 = ind A. The equalities This completes the sketch of in (1.1) now follow directly from the equalities in Remark 1.1. the proof. The following factorization theorem is an application of Theorem 1.2. A proof can he found in Below we sketch the proof given in [DLSl]. To formulate it we recall the notion of a Blaschke-Potapov product. For a E D and B e (0, 27r) we define the holomorphic bijection bn,e : D - D by bae(z)=(z-a)ese/(1-za). An operator B:D-*L() ,a) is called a (finite) Blaschke-Potapov product on ) of degree KEN if B can be written as a (finite) product of factors of the form (I-P)+ba,g(z)P for some aeD (a is called a zero of the product), Oe[0,2r) and some projection P [KL].
on ) (of finite rank), and ErankP=K, where the sum is taken over the projections P appearing in the factors forming B. Note that such a product assumes unitary values on the boundary T of the open unit disc D. THEOREM 1.4.
Assume that a and (t5 are Hilbert spaces and let O e S°(jY, (53 ). If one of the numbers
sq_(ae), sq_(ag) and sq_(SA) is finite and equal to KENu{0}, say, then e admits the strongly regular factorizations e(z) =BL(z)-'OL(z) =OR(z)BR(z)-', zE1(O), where
BL:D*L((53,(t3) are Blaschke-Potapov products of degree K with zeros in D\{0} and OR, OL:D-.L(a,(t3) are holomorphic contraction operators. In particular, e can be extended to a meromorhpic operator function on D with K poles (counting multiplicities) in D\{0}.
In the theorem strongly regular essentially means that the zero's of OL (OR) and the projections appearing in the Blaschke-Potapov product BL (BR) do not cancel any of the poles of Bi' (BR', respectively).
For the precise definition we refer to [DLS1], Section 7, but in the sequel the
above indication of what it means should be sufficient.
Alpay et al.
19
Sketch of the proof of Theorem 1.4. Apply Theorem 1.2 and let A=(.t,jr,t ;T,F,G,H) be a closely connected unitary colligation whose characteristic function coincides with e. Then k is a Pontryagin space with ind_SZ = ic. From the equality T'T +G'G = I and the fact that G maps the Pontryagin space .. into the Hilbert space lb it follows that T is a contraction. Hence there exists a rc-dimensional nonpositive subspace H of S which is invariant under T and such that the spectrum of the restriction of T to H lies outside D; see [Ilk.] or [DLS1]. This leads to a decomposition of 0 as the product of two unitary colligations, one of them on the left- or right-hand side corresponding to the rc-dimensional invariant subspace Ho, whose characteristic function is a Blaschke-Potapov product. This completes the sketch of the proof.
To remove the restriction "OEZ(0)" in Theorem 1.4 we use the following lemma. amounts to straightforward substitution and is therefore omitted. LEMMA 1.5.
Its proof
0 are Krein spaces. Assume that zoeD(0), denote the
Let
mapping b_,0:D-D by b so that b(z)=(z+z0)/(l+zzo), zED, and define the operator function OoES(ta,O by Oo(z)=O(b(z)), zelr(0o)=b,,,o(Z(0)). Then: (i)
Oo-so J,6)),
(ii)
(1-IzoI2)ae(b(z),b(w))=(l+wzo)aeo(z,w)(1+zzo) and sq+(ae)=sq±aeo),
(iii) (iv)
(1-Iz0I2)ae(b(z),b(w))=(l+wzo)a6o(z,w)(l+zzo) and sq+(ae)=sq±(aeo),
(1- I2012)0(b(0))-0(b(z))=(l+wzo) 00(x) -eo(z)(1+zzo).
w-z
b(w)-b(z)
If zoERn1(O), so that b(x)=b(z), then (v)
(1- IzoI2)Se((b(z),b(w)) =diag((l+wfo),(l+wzo))tSeo(z,w)diag((l+z2o),(1+zzo))
and sq±(Se)=sq±(Seo)
Applying Lemma 1.5 to Theorems 1.2 and 1.4, we obtain the following result. COROLLARY 1.6.
Assume that a and 0 are Hilbert spaces and let OES(J,t3). Then
sq-(oe) = sq-(a&) = sq-(Se )
(with the same interpretation as in Theorem 1.2).
If these numbers are finite and equal to
9 E Nu { 0 }, say, then a admits the strongly regular factorizations 0(z) =BL(z)-'OL(z) =eR(z)BR(z)-',
zEZ)(e),
where BR:D-o-L(ar,f ), BL:D-.L((s3,(3) are Blaschke-Potapov products of degree rc and 0R, OL:D-'L(a,( )
are holomorphic contraction operators. In particular, a can be extended to a meromorphic operator function on D with is poles (counting multiplicities) in D.
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20
2.
POSITIVE SQUARES AND INJECTIVITY
For the main theorem in this section we use the following result. A more general version involving Schur products will be given at the end of this section.
Let z1iz2,.... z be n different points in D and let Q=(Q;j) be a nonnegative nxn Then the nxn matrix (Q;j/(1-z;zj)) is matrix with diagonal elements Q,,>0, i=1,2,...,n. LEMMA 2.1.
positive.
Proof.
Let P be the matrix (Q,,/(1-z;zj)). Then P= Ek=ODkQD"k, where D=diag(z1,z2,...,zn).
Hence P>0, and to show that P>0, it suffices to prove that 9t(P)={0}, where R(P)cCn is the null space of P. Let u = (u;) a 9t(P). Then 0 = u'Pu = Ek=0 (D*ku )`QD"`ku, and, since each summand in the series is nonnegative, the vector QD'ku = 0, k = 0,1, .... It follows that for every polynomial P with complex coefficients the vector QP(D")u=0. Let ie{1,2,...,n} and let P; be a polynomial with the property that P;(13)=6i,, the Kronecker delta. Then 0=(QP;(D`)u),=Q;;u;, which implies that u; = 0, since, by assumption, Q i; > 0. It follows that u = (u;) = 0, i.e., 92(P) = { 0 }.
Note that in the proof of Lemma 2.1 we have used the Taylor expansion 1/(1-zw)=En=1(ti)z)n. A
similar proof can be given based on the integral representation of 1/(1-zw) given in the Introduction just above Theorem 0.2.
Let a and ( be Hilbert spaces and let 9ES(J,6). If sq+(oe)
e(z)I. (ii) If sq+(ae)6(z)I. THEOREM 2.2.
(i)
To prove (i) we let K=sq+(ae), and we assume that there are ic+1 different points zieb(9), i=1,2,...,rc+1, for which there is no positive constant e, such that 9(z;)s9(z;)>eI. Proof.
Then, for each n we can find elements x; e 15 such that II x; II,, = 1 and IIe(z; )x; I1jy< 1/n. Define the Gram matrix Then Q. is nonnegative and of size (K+1)x(ic+1) by (Q,);i=1. If necessary by going over to subsequences, we see that the limit exists and is a nonnegative (/c+l)x(K+1) matrix with diagonal entries Q;; = 1. Now consider the (ic +1)x(/c+1) matrix Pn=((Pn)ij) with
(Pn)if=[ae(zi,z?)xi Xnjia= ((Qn)q/(1-ziz,i))-((e(zi)xi,e(zj)xil@l(1-zizl))
Then the limit lim,,.,,oPn exists and equals the (16+1)x(K+1) matrix P=((Q;i/(1-ziz3)). On account of Lemma 2.1 the matrix P has precisely K+1 positive eigenvalues. However, this is impossible, since, by assumption, for each n the approximating (IC+1)x(K+l) matrix P has at most
i positive eigenvalues. This contradiction proves (i).
e by 9. This completes the proof.
Part (ii) follows from (i) by replacing
Alpay et al.
21
Let a and 0 be Krein spaces and 0F=S(),@). Assume that 0(z) is invertible for all
LEMMA 2.3.
but finitely many ze(e) and put P(z)=0(z)-1. Then IPES((t3,)) and (ii)
ae(z,w) = -0(w)*nw(z,w)0(z), oc;(z,w) = -0(w)aw(z,w)0(z)*, Se(z,w)= -diag(e(w),e(w)*)*Sy(z,w)diag(e(z),e(z)*),
(iii)
sq±(o(4)=sq+(av),
(i)
sq±(Se)=sq+(SW)
The proof of this lemma is straightforward and therefore omitted. COROLLARY 2.4.
Let a and 0 be Hilbert spaces and let 0ES(jy,(t3). Then
(i) sq+(Se)
If (i) and (ii) are valid, then sq+(oe)=sq+(7)=sq+(Se) and for each zED(0)nZ(0)*, with the exception of at most 2sq+(Sg) points, 0(z) is invertible. Proof.
Clearly, (i) implies (ii).
Now assume (ii). Combining parts (i) and (ii) of Theorem
2.2, we obtain that 0(z) is invertible for all, but at most sq+(ae)+sq+(oe), points z in D(0); see, e.g., Lemma 2.5 in [An]. Hence we may apply Lemma 2.3. Using this lemma and the first statement in Corollary 1.6 with P(z)=0(z)-' instead of 0(z), we obtain the equalities
They imply (i) and show that in the last statement of the corollary the number of exceptional points is at most sq+(oe)+sq+(oe)=2sq+(ae). sq+(oe) = sq+(oA) = sq+(Se ).
Using the same kind of arguments as in the proof of Corollary 2.4 we can prove the following Recall that an operator TEL(a,t) is called expansive if T*T>I0; it is called biexpansive if both T and T* are expansive. result.
COROLLARY 2.5. Let jy and (t3 be Hilbert spaces and let 0:D-*L(j,(t3) be a holomorphic mapping. Then the following assertions are equivalent. (1)
0(z) is a biexpansion for all zED.
(ii)
Both kernels oe(z,w) and o,4(z,w),
are nonpositive.
The remaining part of this section is devoted to a generalization of Lemma 2.1 to the Schur product of two nonnegative matrices. It is not needed in the sequel. Recall that the Schur product of two nxn matrices P = (P;3) and Q = (Q,) is the nxn matrix P*Q = ((P*Q );j) with (P*Q);j=P;jQ;j, i,j=1,2,...,n. That is, P*Q is defined as the entry-wise product of P and Q. For example: (1)
The matrix (Q,,/(1- z12j)) in Lemma 2.1 is the Schur product of the nxn matrix (1/(1- z2 ,) ) and Q.
(2)
If a = (a;), b = (b,) E c" (in the sequel considered as the space of nxl vectors) P = aa* and Q = bb*, then P*Q = cc*, where c = (c;) E C" with c; = a;b;, i =1, 2, ... n.
(3)
10\
0
1111
0
If a= (0), b = (1) , C= (1), P = aa* + bb' and Q = aa* + cc*, then P and Q are nonnegative 3x3 1
Alpay et al.
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101
matrices with rank P = rank Q = 2 and P.Q = ( 0 1 0) is a positive matrix.
`1021
The following result concerns the decomposition of nonnegative matrices. We give a proof without recourse to the spectral decomposition theorem for Hermitian matrices. The method makes use of the nonnegativity of the matrix and is very much like the methods of Lagrange and Jacobi which apply to general Hermitian forms; see, e.g., Gantmacher [G], p. 339. PROPOSITION 2.6.
Let P be a nonnegative nxn matrix with rank P = r.
Then there exist r
linearly independent vectors piEC" such that p = i=1 pip;.
Consider a block decomposition of P of the form P = (j ), where A is a nonnegative D matrix of size kxk, say. (We actually only need the case where k = 1.) If x r= 92(A ), the null space of A, and y e C" k is arbitrary, then the nonnegativity of P and the Cauchy-Schwarz inequality Proof.
imply that
Iy"B*xI2= I(Oy*)P( 0) I25(x*0)P(0).(0y*)P(0) =(x*Ax)(y*Dy)=0. It follows that B*x=0 and this shows that 9t(A)c l(B`). Taking orthogonal complements we see that R(B)c92(A), where 9t(A) denotes the range of A. Hence there exists an kx(n-k) matrix G such that B = AG.
It is straightforward to verify that P can be written as 1
((
P= (B* D)
((
G'* I).0 D-G*AG)l0 I)=(G*A2)(G*A2)*+10
(For more details, see for instance [Dy], Lemma A.1.)
D-G*AG
To prove the proposition we start with
the block decomposition for P = (Pij) in which A = P1i. Then we obtain 1.
s
P=P1Pi+( 0 D-G*AG)' P1=
(( (GAA2)EC".
Note that p, is the zero vector if P11= 0. We repeat the same procedure to the nonnegative (n-1)x(n-1) matrix D-G*AG. After n steps we have constructed vectors piEC", i=1,2,...,n, such that P = E;-1 pip;. Since the first i -1 entries of each piEC" are zero, 1 < i!5 n, it is clear from the construction that the nontrivial vectors among PI,p2,...,p" are linearly independent. By deleting the trivial vectors and reordering we obtain the decomposition P=Ei= pip; for some r5n, in which are linearly independent. As P>0, we have that xE92(P) if and only if x'Px = 0. Since this last relation holds if and only if x*pi = O, i= 1, 2, ... , r, it follows that dim R(P) = n - r, so necessarily r = rank P. This completes the proof.
Proposition 2.6 implies Schur's lemma; see [D], p.9. The proof of this lemma can be used to prove the desired generalization. For the sake of completeness, we give the proofs of both results.
Alpay et al.
23
(i)
Let P and Q be two nonnegative nxn matrices. Then: (Schur's lemma) The Schur product P.Q is nonnegative.
(ii)
If P is positive, then P.Q is positive if and only if Qii # 0 for all i = 1....,n.
THEOREM 2.7.
Proof We first prove (i). Put rank P = r and rank Q = s. Then by Proposition 2.6 r
*
s
Q=Ej=i4jgj,
p,.... p, are linearly independent vectors in C" and q,,...,q, are also linearly independent vectors in C". We denote the k-th entry of the vector pi by (Pz)k, so that Pi=((P.)k)k=1+ i=1,2.... r, and we use a similar notation for each vector qj. It follows that where
P.Q = j= j=i
Ei=I
vijv; j>
where vij is vector in C" and the k-th entry of vij is given by (vij)k=(Pi)k(gj)k. This shows that P.Q is nonnegative. We now prove (ii). We use the same decompositions of the matrices P and Q as in the proof of
part (i). Since P is positive, we have that r=rankP=n. Note that, on account of (i), P.Q is nonnegative. The "only if" statement in (ii) follows easily from the fact that all diagonal elements of a positive (nonnegative) matrix are positive (nonnegative, respectively). To prove the "if" statement, we suppose that Qii # 0 and hence Qii > 0 for all i = 1,... , n. Since P.Q > 0, it
suffices to show that P.Q has a trivial null space 9l(P*Q).
Let a=(ak)k=1e92(P*Q).
Then
a*P.Qa = 0, and it follows from P.Q = En=j E;=, vijv; j that 0 = a*vij = k=1ak(vij)k = k=1(ak(gj)k)(Pi)k,
i = 1,2,...,n, j= 1,2,...,s.
that dk(gj)k=0, j=1,2,...,s, k=1,2,...,n, and from Esj=i I(gj)kI2=Qkk>0 it follows that to each k, 1
linear independence of the vectors
vector and 91(P*Q) = { 0 }.
P" implies
This completes the proof.
Lemma 2.1 is a special case of Theorem 2.7 (ii): take P=(1/(1-zizj)). Example (3) given above shows that if the Schur product of two nonnegative matrices is positive then it is not true that at least one of these matrices is positive also. 3.
APPLICATION OF THE POTAPOV-GINZBURG TRANSFORM
In this section we prove Theorems 0.1 and 0.2. The basic idea behind the proofs is a reduction of the Kreln space situation to the Hilbert space situation. This reduction is obtained by applying the Potapov-Ginzburg transform. Under various different names this transform has been used in, for example, [AG1,2], [DR), [Dy], [IKL]. We first briefly describe this transform and introduce the convenient notation used in [DR], Section 1.
Alpay et al.
24
Let a be a Krein space and let a = a+ ®a- be a fixed fundamental decomposition of a. The operator Jg = P+ - P_, where P± is the orthogonal projection on a onto a±, is called the fundamental symmetry associated with the fundamental decomposition. Note that J =10. The linear space a provided with the inner product [f,g]lal = [Jaf,g]a, f,ga , is a Hilbert space and will be denoted by I a I . The definition of J a i depends on the fundamental decomposition, I a I = a+ ® I a- I and I a-I
is the anti-space of a-.
For the Krein space tB we also fix a fundamental decomposition
($ = ($+ ®6_ and we denote the orthogonal projection on (B onto (B± by Q±. Then J&=Q,-Q- is the fundamental symmetry corresponding to the fundamental decomposition by means of which the Hilbert space 101 is defined. By definition T E L(a, B) if and only if T E L(I a 1, I 0 I ). If T E L(?r, 0), then by 1 we denote the adjoint of T with respect to the Kreln space inner products on a and 0, and by T" we denote the adjoint of T with respect to the Hilbert space inner products on IaI and 101. It is easy to see that T`, T" E L((, 1) and that
r = JOT"Jr,
T" = JOT'"Jg.
If TEL(a,B), then with respect to the fundamental decompositions of a and B we often write T in following operator matrix form
T=(Ti1 Tiz Tz1 T. 11
//,
1
11
If T22 is invertible then the operator
where, for example((, T22=Q_TIO EL(5_,6_).
P++Q_T=(T21 T22/'1 also invertible and S=(Q+T+P-)(P++Q-T)-1. is
S-(
the Potapov -Ginzburg transformation S of T is the operator Clearly, SEL(a+E)(t3_,B+®ta-) and S has the operator matrix form
T11-T1 z Tz2T21 T12T22 -T-22 T21
Tz2
1
I
1
II
(-
l
1
+
R_ l
It is straightforward to verify that if we apply the Potapov-Ginzburg transform to S we get the operator T back. For proofs of these facts we refer to [DR], where a complete survey of the Potapov-Ginzburg transform in connection with contractions is presented.
Below we consider a function 9ES(a,B) and we use the same notation as above. For example, 921 stands for the operator function e21(z)=Q_9(z)I0+ defined for z in a deleted neighborhood of 0. The following theorem is an analog and a small extension of [DR], Theorem 1.3.4. We recall that a and (9 are Kreln spaces with fundamental decompositions a = ta+ ®a_ and (B = lfi+ ®l9_ and corresponding fundamental symmetries Jky and J.
Alpay et al.
25
THEOREM 3.1.
Let 0ES(jY,c3) and assume that 022(z)=Q_0(z)1,
is invertible for z in an open set
b=b*c1(0). Then E defined by (Q+e(z)+P_)(P++Q-e(z))-'
(i)
E(z) _
is a well-defined and holomorphic function on b with values in the space L(
and
satisfies the following identities (ii) E(z)x=(P+0(z)*+Q-)(Q++P_0(z)*) I,
(iii)
0(z) = (Q+E(z)+Q_)(P++P-E(z))-',
(iv)
0(z)*=(P+E(z)"+P-)(Q++Q-E(z)x)-1.
Moreover, we have
(a)
I -0(w)*0(z)
(b)
I -0(w)0(z)
(c)
0(z)*-0(w)*=Ja(P++Q-0(w))%(E(z)x-E(w)x)(Q++P_0(z)*), 0(w)-0(z)=JS((Q++P_0(w)*)x(E(w)-E(z))(P++Q_e(z))-
(d)
-E(w)"E(z))(P++Q-0(z)),
It is clear that the function f is well-defined and holomorphic on Z. The relations To prove (a) we consider its right-hand side and substitute for E(z) and E(w) the expression in (i). We Proof.
(ii)-(iv) can be verified in the same manner as the corresponding results in [DR].
obtain (P++Q-0(w))x(I E(w)xE(z))(P++Q-(9(z)) = = (P++Q-0(w))x(P++Q-0(z)) - (Q+0(w)+P_)x(Q+0(z)+P-)
_
=P++Ja0(w)*JBQ_0(z)-P_-Ja0(w) JtQ+0(z)=Ja(I-0(w) (9(z)),
which is equivalent to (a). Similarly, one can obtain (b) by substituting for E(z) and E(w) in its right-hand side the expression in (ii). To show (c) we consider its right-hand side and substitute for E(w) the expression in (i) with z replaced by w and for E(z) the expression in (ii) with z replaced by z. We obtain (P+ +Q_0(w))x(E(z)x E(w)x)(Q++P_0(z )*) _
-
= (P++Q-e(w))x(P+e(z)*+Q-) - (Q+0(w)+P-)x(Q++P-e(z)*) =
=P+0(z)*+Jg0(w)*JgQ_-J 0(w)*JgQ+-P_0(z)*=Ja(0(z)*-0(w)*),
which is equivalent to (c). Finally, (d) can be shown by substituting in its right-hand side for E(z) the expression in (i) and for E(w) the expression in (ii) with z replaced by w. This completes the proof of the theorem. Combining the identities (a)-(d) of Theorem 3.1 we get a useful relation between the kernels
26
Alpay et al.
associated with the operator functions 9 and E.
COROLLARY 3.2.
Under the conditions of Theorem 3.1 the following kernel identities are valid
for all z,we2). (i)
ae(z,w) =Jg(P++Q_e(w))"aE(z,w) (P++Q_e(z))
(ii)
ae(z,w) = Je(Q++P_e(i)*)xaE(z,w) (Q++P_e(z)'). Se(z,w)=diag(Jg,Jo)(diag(P++Q-e(w),Q++P_e(fb)*)xSE(z,w)(diag(P++Q_e(z),Q++P_e(z)*).
(iii)
In order to use Theorem 3.1 and Corollary 3.2, we must make sure that 0922 is invertible on some open set 2)c2)(e). We first make the following observation. LEMMA 3.3. If A and B are Hermitian matrices and A
of B, where the eigenvalues are counted according to their multiplicities.
Lemma 3.3 is an immediate consequence of the minimax characterization of the eigenvalues of a Hermitian matrix, which implies that if A:5 B, then A,(A)
details see [G], Anhang, and, in particular, p. 621, Satz 6. A simple and direct proof of the more global statements in the lemma here can be given by showing that if E_(A) (E_(B)) stands for the linear span of the eigenspaces of A (B) corresponding to all negative eigenvalues of A (B, respectively) and PA denotes the orthogonal projection onto E_(A), then the restriction This implies the PAIE_(B) to E_(B) is injective on E_(B) and hence dimE_(B)
Let 0 S(jY, 0) and set e22(z) = Q-e(z) 1,
.
If sq_(oe)e(z)I. (ii) If sq_(ae)6(z)I. (i)
Proof.
The following chain of (in-)equalities is valid. All kernels here are defined on the domain 2)(9) and in brackets we indicate on what inner product space they act. sq_(ae) >_ sq-(P_ae 1,3_)
= sq_(- P_oe I g_) = sq+(P-ae I g_) > sq+(a )
(on the Krein space a) (on the anti-Hilbert space (_ ) (on the Hilbert space I:_ I ) (on the Hilbert space I a- I ) (on the Hilbert space _ I ). I
Alpay et al.
27
The last inequality follows from the identity
_=I-022()%022(x)+e12()%012(x)=aen(zw)+e12(w)xe12(z) 1-zw 1-zw 1-zw
P-ae(z,w)I
z,wea(e),
the nonnegativity of the kernel
012(_)%012(x)
(on the Hilbert space
Ita-I )
1-zvw
Hence the kernel ae.(z,w) has at most sq_(oe) positive squares on I)LI. Part (i) now follows from Theorem 2.2(i). If we replace in the above argument e by by 0 and P by Q, we obtain that the kernel and Lemma 3.3.
oe22 (z
w)=1-022(f)022(2)X
z,wez(e),
1-zw
has at most sq_(ae) positive squares on the Hilbert space Theorem 2.2(ii). COROLLARY 3.5.
e22(x)=Q_19(z)I3
Part (ii) now follows from
This completes the proof.
Let 0ES(),(S3) and assume that sq_(ae)
sq-(ae)+sq-(ae) points. We now come to the proofs of the main theorems of the paper. Proof of Theorem 0.1. Clearly, the inequality in (i) implies the inequalities in (ii). Assume (ii). Then by Corollary 3.5 the operator 022(x) is invertible for each zeb(0)na(®)
except for at most sq_(ae)+sq_(oe) points. Hence the conditions of Theorem 3.1 are satisfied, and the operator function E(z) in (i) of Theorem 3.1 is well defined. Corollary 3.2 implies the equalities sq-(aE)=sq-(oe),
sq-(o
)=sq_(ae),
sq-(SE) =sq-(Se)
Here the kernels on the left-hand side are considered as operators on Hilbert spaces. According to Corollary 1.6 their numbers of negative squares are equal. It now follows that sq_(ae)=sq_(oe)=sq_(S9). Hence, in particular, (i) is valid and the last statement in Theorem 0.1 holds true. This completes the proof. Proof of Theorem 0.2. The implications (i)=o.(ii) and (i)=*(iii) follow immediately from Theorem If (ii) (or (iii)) is valid then, on account of Theorem 3.4(i) (or (ii), respectively), 022(z) is invertible for all but at most lc points in b(0). Again we may apply the 0.1.
Potapov-Ginzburg transform to conclude, as in the proof of Theorem 0.1, that (i) is valid. If E(z) is the transformation of 0(z) then, by Corollary 1.6, E(z) can be extended to a meromorphic operator function on D with at most k poles. The lower right corner E22(z) of E(z) is a square
Alpay et al.
28
matrix function of size ind_1=ind_ and, since detE22(z)O0 on D, it is invertible for all z outside a discrete subset of D. Applying the inverse Potapov-Ginzburg transform to E we find that 0 has the property mentioned in the last statement of the theorem. This completes the proof. REFERENCES Alpay, "Some Krein spaces of analytic functions problem", Michigan Journal of Math., 34 (1987), 349-359.
and
an
inverse
scattering
[Al
D.
[AD1]
D. Alpay, H. Dym, "Hilbert spaces of analytic functions, inverse scattering and operator models I", Integral Equations Operator Theory, 7 (1984), 589-641. D. Alpay, H. Dym, "Hilbert spaces of analytic functions, inverse scattering and operator models II", Integral Equations Operator Theory, 8 (1985), 145-180. D. Alpay, H. Dym, "On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization", Operator Theory: Adv. Appl., 18 (1986), 89-159. D.Z. Arov, L.Z. Grossman, "Scattering matrices in the theory of dilations of isometric operators", Dokl. Akad. Nauk SSSR, 270 (1983), 17-20, (Russian) (English translation:
[AD2] [AD3] [AG1]
Sov. Math. Dokl., 27 (1983), 518-522). [AG2]
D.Z. Arov, L.Z. Grossman, "Scattering matrices in the theory of unitary extensions of
[An]
T. Ando, De Branges spaces and analytic operator functions, Lecture Notes, Hokkaido
[Azl]
T. Ya. Azizov, "On the theory of extensions of isometric and symmetric operators in spaces with an indefinite metric", Preprint Voronesh University, 1982; deposited paper
isometric operators", manuscript. University, Sapporo, 1990.
no. 3420-82 (Russian). [Az2j
T.Ya. Azizov, "Extensions of J-isometric and J-symmetric operators", Funktsional. Anal. i Prilozhen, 18 (1984), 57-58 (Russian) (English translation: Functional Anal. Appl., 18
[Bl]
L. de Branges, "Krein spaces of analytic functions", J. Functional Analysis, 81 (1988),
[B2]
L. de Branges, "Complementation in Krein spaces", Trans. Amer. Math. Soc., 305 (1988),
[B3]
277-291. L. de Branges,
[BC]
J.A. Ball, N. Cohen, "De Branges-Rovnyak operator models and systems theory: a survey",
[BR]
Operator Theory: Adv. Appl., 50 (1991), 93-136. L. de Branges, J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New
(1984), 46-48). 219-259.
Square summable
power
series,
manuscript.
York, 1966. [BS]
L. de Branges, L.A. Shulman, "Perturbations of unitary transformations", J. Math. Anal. Appl., 23 (1968), 294-326.
[CDIS]
B. (urgus, A. Dijksma, H. Langer, H.S.V. de Snoo, "Characteristic functions of unitary colligations and of bounded operators in Krein spaces", Operator Theory: Adv. Appl., 41
[D]
W.F. Donoghue, Monotone matrix functions and analytic continuation, Springer-Verlag,
[DLS1j
A. Dijksma,
(1989), 125-152.
Berlin-Heidelberg-New York, 1974. H. Langer, H.S.V. de Snoo, "Characteristic functions of unitary operator
colligations in H,,-spaces", Operator Theory: Adv. Appl., 19 (1986), 125-194. [DLS2]
A.
[DLS3]
A. Dijksma, H. Langer, H.S.V. de Snoo, "Unitary colligations in Kreln spaces and their
[DR]
Dijksma, H. Langer, H.S.V. de Snoo, "Unitary colligations HK spaces, in characteristic functions and Straus extensions", Pacific J. Math., 125 (1986), 347-362.
role in the extension theory of isometries and symmetric linear relations in Hilbert spaces", Functional Analysis II, Proceedings Dubrovnik 1985, Lecture Notes in Mathematics, 1242 (1987), 1-42. M.A. Dritschel, J. Rovnyak, Extension theorems for contractions on Kreln spaces, Operator Theory: Adv. Appl., 47 (1990), 221-305.
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[Dy]
H.
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Dym,
J
interpolation,
contractive
Regional
matrix functions, conference series
in
reproducing kernel mathematics, 71,
Hilbert
Amer.
spaces and Math. Soc.,
[G]
Providence, R.I., 1989. F.R. Gantmacher, Matrizentheorie, 2nd ed., Nauka, Moscow, 1966 (Russian) (German translation: VEB Deutscher Verlag der Wissenschaften, Berlin, 1986).
[IKL]
I.S.
Iohvidov, M.G. Kreln, H. Langer, Introduction to the spectral theory of operators
in spaces with an indefinite metric, Reihe: Mathematical Research 9, Akademie-Verlag, Berlin, 1982.
[KL]
M.G. Krein, H. Langer, "Uber die verallgemeinerten Resolventen and die charakteristische Funktion eines isometrischen Operators im Raume H,.", Hilbert Space Operators and Operator Algebras (Proc. Int. Conf., Tihany, 1970) Colloquia Math. Soc. JAnos Bolyai, no. 5, North-Holland, Amsterdam (1972), 353-399.
[M]
S.
[Mc]
B.W. McEnnis, "Purely contractive analytic functions and characteristic functions of
Marcantognini, "Unitary colligations of Equations Operator Theory, 13 (1990), 701-727.
operators
in
Kreln
spaces",
Integral
non-contractions", Acta. Sci. Math. (Szeged), 41 (1979), 161-172. [SF]
B. Sz.-Nagy, C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Company, Amsterdam-London, 1970.
[Y]
A. Yang, A construction of Kretn spaces of analytic functions, Dissertation, Purdue
University, 1990. D. ALPAY DEPARTMENT OF MATHEMATICS BEN-GURION UNIVERSITY OF THE NEGEV POSTBOX 653 84105 BEER-SHEVA ISRAEL
MSC: Primary 47B50, Secondary 47A48
A. DIJKSMA, J. VAN DER PLOEG, H.S.V. DE SNOO DEPARTMENT OF MATHEMATICS UNIVERSITY OF GRONINGEN
POSTBOX 800 9700 AV GRONINGEN THE NETHERLANDS
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
30
ON REPRODUCING KERNEL SPACES, THE SCHUR ALGORITHM, AND INTERPOLATION IN A GENERAL CLASS OF DOMAINS Daniel Alpay and Harry Dym* This paper develops the theory of reproducing kernel Pontryagin spaces
with reproducing kernels Au,(\) = X(A)JX(w)*/p,(A) based on a k x m matrix valued function X(\), a signature matrix J and a denominator of the general form a(A)a(w)* - b(A)b(w)*. This both unifies and generalizes earlier studies of such kernels wherein the denominator was taken to be either 1 - Aw* or -27ri(.\ - w*). A Schur-like algorithm is then interpreted in terms of a recursive orthogonal direct sum decomposition of such spaces. Finally, these spaces, in conjunction with a corresponding class of 1C(0) spaces which were introduced earlier (in [AD4]), are used to solve a general one-sided interpolation problem in a fairly general class of domains.
CONTENTS
4.
INTRODUCTION PRELIMINARIES B(X) SPACES RECURSIVE EXTRACTIONS AND THE SCHUR ALGORITHM
5.
fp(S) SPACES
6. 7.
LINEAR FRACTIONAL TRANSFORMATIONS ONE SIDED INTERPOLATION
8.
REFERENCES
1.
2. 3.
*H. Dym would like to thank Renee and Jay Weiss for endowing the chair which supports his research.
Alpay and Dym
31
1. INTRODUCTION In this paper we shall study reproducing kernel Pontryagin spaces of m x 1 vector valued meromorphic functions with reproducing kernels of the form
A,(A) - X(A)JX(w)*
(1.1)
p.(A) where X is a k x m matrix valued function, J is an m x m signature matrix (i.e., J = J* and JJ* = I,,,), the denominator p,,,(A) is of the form p,,,(A) = a(A)a(w)* - b(A)b(w)*
,
(1.2)
and it is further assumed that: 1. a(A) and b(A) are analytic in some open nonempty connected subset Sl of U. II. The sets St+ _ {w E S2 : p,,,(w) > 0}
and
St_ _ {w E St : pw(w) < 0}
are both nonempty. Because of the presumed connectedness of Sl, it follows further (see [AD4] for
details) that: The set III.
1lo={wESZ: pw(w)=0} contains at least one point p such that p,, (A) 0 0. Any function p,,(A) of the form (1.2) which satisfies I and II (and hence III) will be said to belong to Dn. Reproducing kernel Pontryagin spaces with reproducing kernels of the form (1.1) were extensively studied in [AD3] for the special choices p,,,(A) = 1 - Aw* and p,,,(A) = -2iri(A - w*). Both of these belong to Do with S1 = C: 1- Aw* is of the form (1.2) with a(A) = 1, b(A) = A, Sl+ = ID and SZo = a.
-2ai(A -w*) is of the form (1.2) with a(A) = f(1- iA), b(A) = f(1 +iA),
A+_ C+and110=1R. In this paper we shall extend some of the results reported in [AD3] to this new more general framework of kernels with denominators in Dn and shall also solve a general one-sided interpolation problem in this setting. Many kernels can be expressed in the general form (1.1); examples and references will be furnished throughout the text. In addition to these, which focus on A,,,(A) as a reproducing kernel, the form (1.1) shows up as a bivariate generating function in the study of structured Hermitian matrices; see the recent survey by Lev-Ari [LA] and the references cited therein. In particular, Lev-Ari and Kailath [LAK] seem to have been the first to study "denominators" p,,,(A) of the special form (1.2). They showed that Hermitian matrices with bivariate generating functions of the form (1.1) can be factored efficiently whenever p,,(A) is of the special form (1.2). The present analysis gives a geometric interpretation of the algorithm presented in [LAK] in terms of a direct sum orthogonal decomposition of the underlying reproducing Pontryagin spaces.
Alpay and Dym
32
As we already noted in [AD1], the important kernel K. (A) =
J - O(A)JO(w)*
can also be expressed in the form (1.1), but with respect to the signature matrix J
by choosing
J
0
0
-J
'
X=[I..O].
For the most part, however, we shall take J equal to
_ I JP9
0
0
-Iq
and shall accordingly write
X(\) _ [A(,\) B(A)] with components A E (Ckxp and B E Ckxq, both of which are presumed to be meromorphic in SZ+. Then (1.1) can be reexpressed as A,(A)
= A(.\)A(w)* pw(A)
-
B(.)B(w)* pw(.\)
which serves to exhibit A,,,(.) as the difference of two positive kernels on S2+ (since l/p,,,(.\) is a positive kernel on Q+, as is shown in the next section). Therefore, by a result of L. Schwartz [Sch], there exists at least one (and possibly many) reproducing kernel Krein space with A,,,(.\) as its reproducing kernel. However, if the kernel is restricted to have only finitely many negative squares (the definition of this and a number of related notions will be provided in Section 2), then there exists a unique reproducing kernel Pontryagin space with A,( A) as its reproducing kernel. This too was established first by Schwartz, and independently, but later, by Sorojonen [So] (and still independently, but even later, by the present authors in [AD3]). If A,,,(.\) has zero negative squares, i.e., if A,,,(A) is a positive kernel, then the (unique) associated reproducing kernel Pontryagin space is a Hilbert space, and the existence and uniqueness of a reproducing kernel Hilbert space with A,,,(A) as its reproducing kernel also follows from the earlier work of Aronszajn [Ar]. Throughout this paper we shall let $(X) [resp. IC(&)] denote the unique reproducing kernel Pontryagin (or Hilbert) space associated with a kernel of the form (1.1) [resp. (1.3)]. The reproducing kernel Hilbert spaces 5(X) and IC(E)), but with p,,,(.1) restricted to be equal to either 1- )w* or -27ri(.1 - w* ), originate in the work of de Branges, partially in collaboration with Rovnyak; see [dBl], [dBR], [dB3], the references cited therein, and also Ball [Bal]. Such reproducing kernel Hilbert spaces were applied to inverse scattering and operator models in [AD1] and [AD2], to interpolation in [Dl] and
Alpay and Dym
33
[D2], to the study of certain families of matrix orthogonal polynomials in [D3] and [D4], and to the Schur algorithm and factorization in [AD3]; the latter also extends a number of basic structural theorems from the setting of Hilbert spaces to Pontryagin spaces. In [AD4] and [AD5], the theory of K (O) spaces was extended beyond the two special choices of p mentioned above, to the case of general p E Dn. The parts of that extension which come into play in the present analysis (as well as some other prerequisites) are reviewed in Section 2. In this paper we shall carry out an analogous extension for the spaces
5(X). This begins in Section 3. Recursive reductions and a Schur type algorithm are presented in Section 4. Section 5 treats the special case in which A,,,(A) is positive and of the special form A,(A) = {Ip - S(A)S(w)*}/pu,(A). Section 6 deals with linear fractional transformations, and finally, in Section 7, we apply the theory developed to that point to solve a general one-sided interpolation problem in Q+.
The basic strategy for solving the interpolation problem in H+ is much the same as for the classical choices of the disc or the halfplane except that now we seek interpolants S for which the operator MS of multiplication by S on an appropriately defined analogue of the vector Hardy space of class 2 is contractive: IIMSII < 1. Although this implies that S is contractive, the converse is not generally true; see the examples in Section 5. Moreover, 52+ need not be connected. Interpolation problems in nonconnected domains have also been considered by Abrahamse [Ab], but both the methods and results seem to be quite different.
Finally, we wish to mention that there appear to be a number of points of contact between the interpolation problem studied in this paper and the interpolation problem described by Nudelman [N] in his lecture at the Sapporo Workshop. However, we cannot make precise comparisons because we have not yet seen a written version. The notation is fairly standard: The symbols IR and C denote the real and complex numbers, respectively; ID = {A E C : JAI < 11, 'II' _ {A E C : JAI = 11, E= {A E C : JAI > 11 and C+ [resp. C_] stands for the open upper [resp. lower] half plane. Cpx9 denotes the set of p x q matrices with complex entries and C' is short for CPX 1. A` will denote the adjoint of a matrix with respect to the standard inner product, and the usual complex conjugate if A is just a number.
2. PRELIMINARIES To begin with, it is perhaps well to recall that a vector space V over the complex numbers which is endowed with an indefinite inner product [ , ] is said to be a Krein space if there exist a pair of subspaces V+ and V_ of V such that (1)
V+ endowed with [
(2)
V+ n V_ _ {0}.
,
] and V_ endowed with -[
,
] are Hilbert spaces.
V+ and V_ are orthogonal with respect to [ , ] and their sum is equal to V. V is said to be a Pontryagin space if at least one of the spaces V+, V_ is finite dimensional. In this paper, we shall always presume that V_ is finite dimensional. In this instance, the dimension of V_ is referred to as the index of V. (3)
Alpay and Dym
34
A Pontryagin space P of m x 1 vector valued meromorphic functions defined on an open nonempty subset A of C, with common domain of analyticity A', is said to be a reproducing kernel Pontryagin space if there exists an m x m matrix valued function L,,,(A) on A' x A' such that for every choice of w E A', v E C'° and f E P: (1) L,,,v E P, and (2)
[f, Lwv]P = v*f(w)
The matrix function L,,,(.\) is referred to as the reproducing kernel; there is only one such. Moreover, La(13) = LA(a)* , for every choice of a and /3 in A'.
The kernel L,,,(A) (or for that matter any Hermitian kernel) is said to have v negative squares in A' if (1) for any choice of points in A' and vectors
vl,... , v in i' the n x n matrix with ij entry equal to v, Lwj (w; )vj has at most v negative eigenvalues and, (2) there is a choice of points w l , ... , wk and vectors v1, ... , vk for which the indicated matrix has exactly v negative eigenvalues; it should perhaps be emphasized here that is is also allowed to vary. In a reproducing kernel Pontryagin space,
the number of negative squares of the reproducing kernel is equal to the index of the space.
For additional information on Krein spaces and Pontryagin spaces, the monographs of Bognar [Bo], Iohvidov, Krein and Langer [IKL], and Azizov and Iohvidov [AI] are suggested.
Next, it is convenient to summarize some facts from [AD4] about the class Dp which was introduced in Section 1 and on some associated reproducing kernel spaces.
First, it is important to note that the definition of the class DU depends only upon p and not upon the particular choice of functions a and b in the decomposition (1.2). In particular, if p,,,(.\) can also be expressed in terms of a second pair of functions c(A) and
d(.): if pw(\) = c(7)c(w)* - d(\)d(w)* then there exists a J11 unitary matrix M such that [c(.\)
d(.\)] = fa(.\) b(.)]M
for every A E 11; see Lemma 5.1 of [AD4].
We have already remarked that the functions p,,,(.\) = 1 - ,w* and pw(A) _ -27ri(A - w*) belong to Dc. So does the less familiar choice pw(A) = -27ri(A - w*)(1 - \w*) .
(2.1)
The latter is of the form (1.2) with
a(.\) = /{a + i(A2 + 1)} and b(A) = f{A - i(A2 + 1)} Moreover, in this case,
1l+=(IDn (C+)U(IEn U-)
.
(2.2)
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35
is not connected.
Now if p E D1 with decomposition (1.2), then a(A) # 0 for A E 52+ and s(A) = b(A)/a(A) is strictly contractive in Sl+. Therefore, the kernel 1
kw(A)
=
p,,,
(A) =
a(1A)
`
1
s(A)ts(w):t
a(w)
is positive on Sl+: for every positive integer n, and every choice of points wl, ... , w in S2+ and constants cl, ... , cn in IV, n
E c k,,,. (wj)ci > 0 . i,7=1
Thus, by one of the theorems alluded to in the introduction, there exists a unique reproducing kernel Hilbert space, with reproducing kernel k,,,(A) = 1/pu,(A). We shall refer to this space as Hp and shall designate its inner product by ( , ) Up. Recall that this means that, for every choice of w E S2+ and f E Hp, (1) 1/p,,, belongs to Hp,
and (2) (f,1/Pw)H, = f(w) The space Hp plays the same role in the present setting as the classical Hardy
spaces H2(ID) for the disc and H2( (C+) for the open upper half plane. Indeed, it is identical to the former [resp. the latter] when p,,,(A) = 1 - Aw* [resp. p,,,(A) = -2iri(A w*)]
More generally, HP will denote the space of m x 1 vector valued functions
9 =
with coordinates fi and gi, i = 1, ... , m, in Hp and inner product m
(fi,9i)H, i=1
From now on, we shall indicate the inner product on the left by (f, g) H, (i.e., we drop the superscript m), in order to keep the notation simple. For any m x m signature matrix J, the symbol Hp j will denote the space Hp' endowed with the indefinite inner product [f, 91 HP,,, = (Jf, 9)H,
The space H, is a reproducing kernel Hilbert space with reproducing kernel K,,,(A) = whereas HPj is a reproducing kernel Krein space with reproducing kernel K,,,(.1) = J/p,(A)
Alpay and Dym
36
Because 1/p,,,(A) is jointly analytic for A and w* in Q+, it follows (as is spelled out in more detail in [AD4]) that ak
1
4Ow,k =
k
1
aw*k
Pw
belongs to Hp for every integer k > 0 and that
(.fk)H, =
f(k)(w)
for every f E Hp and every w E Q+. We shall refer to the sequence 'Pw,O,
, 4Pw,n-1
as an elementary chain of length n based on the point w E Q+-
More generally, by a chain of length n in Hp we shall mean the columns fl, ... , fn of the m x n matrix valued function
F(A)=V-P,,n(A), wherein V is a constant in x n matrix with nonzero first column and
0
"Dw,n(A) =
0
V,,,,o(.1)
0
J
... 'P ,,n_1(A) as indicated just
is the n x n upper triangular Toeplitz based on co,,,,o(A),
above. It is important to note that
= {a(A)A, - b(A)B,}-1 ,
(2.5)
where A,,, and B,,, are the n x n upper triangular Toeplitz operators given by the formulas «o
0
...
0
QO
0
...
0
and B,,, =
Aw --
,
0
where
aj =
(2.6)
0
a0
.an-1
*
Nn -1
a(.i)(w) 3.
and Qj =
QO
W) (w) J
These chains are the proper analogue in the present setting of the chains of rational functions considered in [AD3] and [D2]. They reduce to the latter in the classical cases, i.e., when p,.,(A) = 1 - Aw* or p,,,(A) = -2iri(A - w*).
Alpay and Dym
37
In the classical cases, every finite dimensional space of vector valued meromorphic functions which is invariant under the resolvent operators (Ra.f)(A) = .f (A) -.f (a) A - a
(for every a in the common domain of analyticity) is made up of such chains. An analogous
fact holds for general p E D11, but now the invariance is with respect to the pair of operators
a(t)f(A) - a(a).f(a)
{r(a, b; a) f
a(a)b(A) - b(a)a(A)
and
{r(b, a; a) f }(A) =
b(a)a(A) - a(a)b(A)
'
see [AD5] for details.
Just as in the classical cases, a nondegenerate finite dimensional subspace of Hp j with a basis made up of chains is a reproducing kernel Pontryagin space with a reproducing kernel of the form (1.3). More precisely, we have:
THEOREM 2.1. Let p E Do and let A E Vxn B E (r'ixn and V E Tmxn be a given set of constant matrices such that (1)
(2)
det{a(p)A - b(s)B} # 0 for some point p E Sio, and the columns of F(A) = V{a(A)A- b(\)B}-1
(2.10)
are linearly independent (as analytic vector valued functions of A) in Q1, the domain of analyticity of F in ci+.
Then, for any invertible n x n Hermitian matrix P, the space
.F = span{columns of F(.\)} endowed with the indefinite inner product [FE ,
Frl]z' = ?*P
(2.11)
is an n dimensional reproducing kernel Pontryagin space with reproducing kernel Ii,,,(A) =
F(A)P-1 F(w)*
.
The reproducing kernel can be expressed in the form
K.(.\) - J - O(A)JO(w)* p.(A)
(2.12)
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38
for some choice of m x m signature matrix J and m x m matrix valued function O(A) which is analytic in S4 if and only if P is a solution of the equation
A*PA - B*PB = V*JV.
(2.13)
Moreover, in this instance, 0 is uniquely specified by the formula
O(A) = Im - pµ(.\)F(\)P-1F(µ)*J
,
(2.14)
with µ as in (1), up to a J unitary constant factor on the right. PROOF. This is Theorem 4.1 of [AD4]. An infinite dimensional version of Theorem 2.1 is established in [AD5]. Therein, the matrix identity (2.13) is replaced by an operator identity in terms of the operators r(a, b, a) and r(b, a, a).
In the sequel we shall need two other versions of Theorem 2.1: Theorems 5.2 and 5.3 of [AD4], respectively. They focus on the special choice of A = A,,, and B = B. THEOREM 2.2. Let it and F be as in Theorem 2.1, but with A = A,,, and B = B,,, for some point w E 52+. Then the columns f 1, ... , fn of F belong to HP and the n x n matrix P with ij entry
Pij = (Jfj, fi) H, is the one and only solution of the matrix equation
A* PA,,, - B ,PB,,, = V*JV
.
(2.15)
THEOREM 2.3. Let P E Dc, µ E S20 and w E S2+ be such that pµ(w) # 0 and suppose that the n x n Hermitian matrix P is an invertible solution of (2.15) for some m x m signature matrix J and some m x n matrix V of rank m with nonzero first column. Then the columns fl,..., fn of
F(A) = V{a(A)A,,, - b(A)B}-1 are linearly independent (as vector valued functions on S2+) and the space F based on the span of fl,..., fn equipped with the indefinite inner product [Fe, F,7].17= 7*PC
(for every choice of l: and 71 in In) is a K(O) space. Moreover, O is analytic in S2+ and is uniquely specified by formula (2.14), up to a constant J unitary factor on the right. From now on we shall say that the m x m matrix valued function O belongs to the class PP(St+) if it is meromorphic in SZ+ and the kernel (1.3) has v negative squares in SZ+.
Alpay and Dym
39
3. 8(X) SPACES Throughout this section we shall continue to assume that p E DO and that J is an m x m signature matrix. A k x m matrix valued function X will be termed (Sl+, J, p)v admissible if it is meromorphic in Sl+ and the kernel X(A)JX(w)*
A, (A) =
(3.1)
pw(A)
has v negative squares for A and w in Sl+, the domain of analyticity of X in Sl+. Every such (Sl+, J, p)v admissible X generates a unique reproducing kernel Pontryagin space of index v with reproducing kernel A,,,(A) given by (3.1). When v = 0, X will be termed (fl+, J, p) admissible. In this case, the corresponding reproducing kernel Pontryagin space is a Hilbert space. We shall refer to this space as 13(X) and shall discuss some of its important properties on general grounds, in the first subsection, which is devoted to preliminaries. A second description of 13(X) spaces in terms of operator ranges is presented in the second and final subsection. The spaces 13(X) will play an important role in the study of the reproducing kernel space structure underlying the Schur algorithm which is carried out in the next section. It is interesting to note that the kernel Ip - S(A)S(w)* 11-A
KS(A
S(A) - S w* A-w
S(A*)* - S(w)*
Ig -
S(A*)*S(w*)
A - w1 -Aut based on the p x q matrix valued Schur function S can be expressed in the form (3.1) by choosing
X(A)
=
Ip
AS(A)
AS(A*)*
Iq
AIp
2n I s(A*)*
J = (-i)
0
Ip
0
0
-Ip
0
0
0
0 0
-Iq
0
0
Iq
0
S(A)
l
AIq J
0
and p,,,(A) as in (2.1). This kernel occurs extensively in the theory of operator models; see e.g., [Ba2], [DLS] and [dBS].
3.1. Preliminaries. THEOREM 3.1. If the k x k matrix valued function A,,,(A) defined by (3.1) has v negative squares in Sl+, the domain of analyticity of X in Sl+, then there exists a unique reproducing kernel Pontryagin space P with index v of k x 1 vector valued functions which are analytic in Sl+. Moreover, A,,,(A) is the reproducing kernel of P and {A,,,v : w E Sl+
and v E Ck}
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40
is dense in P. PROOF. See e.g., Theorem 6.4 of [AD3].
I
Schwartz [Sch] and independently (though later) Sorojonen [So] were the first to establish a 1:1 correspondence between reproducing kernel Pontryagin spaces of index v and kernels with v negative squares. We shall, as we have already noted, refer to the reproducing kernel Pontryagin space P with reproducing kernel given by (3.1), whose existence and uniqueness is established in Theorem 3.1, as B(X ).
THEOREM 3.2. If X is a k x m matrix valued function in S2+ which is (52+, J, p)v admissible and if f belongs to the corresponding reproducing kernel Pontryagin
space B(X) and w E Sl+, then, for j = 0, 1.... and every choice of v E Gt`, A(j)(A)v : = a&
Ap(A)vLW
belongs to B(X) and [f,
A(j)v]B(x)
= v` f(j)(w)
(3.2)
.
in Q+ and PROOF. By definition there exists a set of points vectors v1,. .. , v in '1 such that the n x n Hermitian matrix P with ij entry pij = [Awj vj, Aw; vi]B(x) = vi Awj (wi )vj
has v negative eigenvalues \ 1 , ... , A. Let u 1 , ... , u, be an orthonormal set of eigenvectors
corresponding to these eigenvalues and let
fj =
[A,,vi...Aw.vn]uj
j = 1,...,v .
,
Then, for any choice of constants cl,... , c, not all of which are zero, v
v
1: cjfj,
j=1
E i=1
v
ciu,Pu;cj
=
cifi B(x)
i,j=1 AiIcil2
<
0.
i=1
Thus the corresponding Gram matrix Q with ij entry
qij = [fj, fi]B(x) ,
i,j = 1,...,v ,
is negative definite and the span N of the columns fl,..., f,, of F = [fl dimensional strictly negative subspace of B(X) with reproducing kernel
Nw(A) = F(A)Q-'F(w)'
,
A,w E S2+ .
.
fv] is a v
Alpay and Dym
41
Consequently,
H=B(X)oN,
the orthogonal complement of N in B(X ), is a reproducing kernel Hilbert space with reproducing kernel
),wESZ+.
H,,,(A)=A,(A)-N,,,(A),
Clearly H,,,(\) is jointly analytic in H+ x S2+ since A,(A) is (by its very definition) and N,,,(J) is (since it involves only finite linear combinations of vector valued functions which are analytic in n'). Therefore, since H is a Hilbert space,
HPv E 7.1
[g,H(i)v]B(X) = v*g(.i)(w)
and
for every choice of w E 9', v E & and g E 7{; see e.g., [AD4] for more information, if need be. Similar considerations apply to N since it is a Hilbert space with respect to
-[
]B(x), or, even more directly by explicit computation:
N(i)()) =
F(A)Q-1F(i)(w)*
and hence, since every h E N can be expressed as h = Fu for some u E [h, NW( )v]B(x)
= [Fu, FQ-1F(i)(w)*v]B(X) = v*F(i)(w)Q-1Qu = v*h(i)(w)
Thus
.
A(j)(A)v = H(j)(A)v + NP)(A)v
clearly belongs to B(X) for every choice of w E H+ and v E Ck. Moreover, as every f E B(X) admits a decomposition of the form
f=g+h with g c H and h E N it follows readily that [f, A(i)v]B(x) = [g, H(i)v]B(x) + [h, N(i)v]B(x)
= v*g(i)(w) + v*h(i)(w) = v* f(i)(w)
,
as claimed.
In order to minimize the introduction of extra notation, the last theorem has been formulated in the specific Pontryagin space B(X) which is of interest in this paper. A glance at the proof, however, reveals that it holds for arbitrary Pontryagin spaces with
Alpay and Dym
42
kernels Aw(\) which are jointly analytic in A and w* for (A,w) E A x A. In particular the decomposition A. (A) = H,,,,(\) - {-N,,J(\)} , which continues to hold in this more general setting, exhibits A,,,(A) as the difference of two positive kernels both of which are jointly analytic in .A and w* for (A,w) E A X A. The conclusions of Theorem 3.2 remain valid for those points a E S20 at which AI(A) is jointly analytic in A and w* for .A and w in a neighborhood of a. This is because, if wl, w2i ... is a sequence of points in 12+ which tends to a, then
HIn v, HIn (k)
(k)
v J
B(X)
=
AWn
(k)
L
A,in In v (k)
J
B(X)
-
v NIn (k'v, NIn (k) J B(X)
`
stays bounded as n T oo. Thus at least a subsequence of the elements H(k)v tends weakly to a limit which can be identified as since weak convergence implies pointwise convergence in a reproducing kernel Hilbert space. Thus v belongs to B(X ), as does N.k)(.\)v = F(A)Q-1F(k)(a)*v and hence also Ack)v
=
N(k)v
.
It remains to verify (3.2) for w = a, but that is a straightforward evaluation of limits.
3.2. B(X) Spaces In this subsection we give an alternative description of B(X) under the supplementary hypothesis that the multiplication operator MX : f ---* X f
is a bounded operator from Hp into HP . Then X is automatically analytic in S2+ and
r = MXJMX.
(3.3)
is a bounded selfadjoint operator from Hp into itself. The construction, which is adapted from [Al], remains valid even if the kernel A,,,(,\) defined in (3.1) is not constrained to have a finite number of negative squares. However, in this instance the space will be a reproducing kernel Krein space, i.e., it will admit an orthogonal direct sum decomposition
B(X) = B+[+]B-
where B+ is a Hilbert space with respect to the underlying indefinite inner product , ]B(X) and B_ is a Hilbert space with respect to -[ , ]B(X) and both B+ and B_ are presumed to be infinite dimensional. Moreover, in contrast to kernels with a finite number of negative squares, there may now be many different reproducing kernel Krein spaces [
with the same reproducing kernel. Examples of this sort were first given by Schwartz
Alpay and Dym
43
[Sch]. For another example see [A2], and for other constructions of reproducing kernel Krein spaces, see [A3], [dB4], [dB5] and [Y]. Let
Rr={rg:9EHP}
and let Kr denote the closure of Rr with respect to the metric induced by the inner product (3.4)
(rf, r9)r = ((r*r) Z f, g)H, . It is readily checked that Rr is a pre-Hilbert space:
(rf,rf)r = 0
if and only if I'f = 0
,
and hence that Kr is a Hilbert space. Next let C(X) = Rr endowed with the indefinite inner product
[rf,r9]r = (rf,9)H, which is first defined on Rr and then extended to Rr by limits. LEMMA 3.1. If w E S2+ and v E Irk, then *
Mx
v Pu
v
X (w)
Pw
PROOF. Let k,(A) = 1/p,(A). Then, for every choice of a E SZ+ and u E Vn, (MXkuv, kau)H, _ (k,,,v, Mxk.u)H, _ {v*X(w)ka(w)u}*
= u*ku(a)X(w)*v .
On the other hand, by direct calculation, the left hand side of the last equality is equal to
u*(MXkw)(a)v
.
This does the trick since both u and a are arbitrary. I THEOREM 3.3. C(X) is a reproducing kernel Krein space (of k x 1 vector valued analytic functions in 1 +) with reproducing kernel Au,(.\) =
X(A)JX(w)*
(3.7)
Pw(.A)
for every choice of w and A in Q+.
PROOF. Since r is a bounded selfadjoint operator on the Hilbert space HP it admits a spectral decomposition: r = f tdEt with finite upper and lower limits. Let !0
r_ = /
tdEt 00
and
r+
roo
tdEt
.
Alpay and Dym
44
Then r_ and r+ are bounded selfadjoint operators on HP,
r=r_+r+
and
r_r+=r+r_ =o.
It now follows readily that
Rr = Rr+ [+]Rris a Krein space since the indicated sum decomposition is both direct and orthogonal with respect to the indefinite inner product [ , ]r given in (3.5), and Rrf is a Hilbert space
with respect to ±[
,
Jr. It remains to show that
(1) Av E C(X) and (2)
[f,A,,v]r = v*f(w)
for every choice of v E irk, w E H+ and f E C(X). The identification
r
v
v
= MXJMjr
Pw
Pw
= MXJX(w)*
v ,
Pw
which is immediate from Lemma 3.1, serves to establish (1).
Suppose next that f = rg for some g E Hp . Then [f, Awv]r = [r9, r
_ (r9,
Pte,
]r Hp
Pw
= v* f (w) .
This establishes (2) for f E Rr. The same conclusions may be obtained for f E Rr by a limiting argument. I THEOREM 3.4. If X is a k x m matrix valued function which is (52+, J, p) admissible and if also the multiplication operator MX is bounded on HP', then 13(X) = C(X)
.
PROOF. Under the given assumptions, both 8(X) and C(X) are reproducing kernel Pontryagin spaces with the same reproducing kernel. Therefore, by the results of Schwartz [Sch] and Sorojonen [So] cited earlier, or Theorem 3.1, 13(X) = C(X). LEMMA 3.2. Let X be a k x m matrix valued function which is analytic in S2+ such that MX is a bounded linear operator from Hp to Hp H. Then
a, MX
v
= MX
i
v
(3.8)
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45
for j = 0,1,... and every choice of v E Tk and w E 52+. PROOF. It is convenient to let
Di =
03
v
and f
aw*j
Pw
Then, for every choice of u E C' and a E Q+,
u*(MMDJf)(a) = (Mj D'f, uPcv)H0 _ (D)f,MXPa)HP = DJ V, MX Pte) HP
Dj
v*X(w)u Pa(w)
= u*DjX(w)*
v P"'(a)
= u*D'(Mxf)(a) , where the passage from line 2 to line 3 is just a special case of (3.2) applied to the Hilbert space H, . Therefore,
MMD3f = D'MXf as claimed. COROLLARY 1. If X,w and v are as in the preceding lemma and if A,(.\) and r are as in (3.1) and (3.3), respectively, then
r as j v= *
Pw
a
*
j r PW v= aj
8w * j
COROLLARY 2. If X, w and v are as in the preceding lemma and if fj = JMX4Ow,jv
,
j = 0,1,...
,
then J
fj(') = JE X(i-s)(,,,)* pw,s(A)v (j
(3.10)
s-o
and 1
(n-1)
[f0 ... fn-11 = [Jx(W)sV ... JX(n
- ijw) yl (Dw n ,
(3.11)
Alpay and Dym
46
PROOF. Let Di = c3/8w*j. Then, by Lemmas 3.2 and 3.1,
fj = =
i
Dj JMXV,,ov
DjJX(w)* Pw
which, by Leibnitz's rule, is readily seen to be equal to the right hand side of Formula (3.11) is immediate from (3.10) and the definition (2.4) of
(3.10).
1
Suitably specialized versions of formulas (3.8)-(3.10) play a useful role in the study of certain classes of matrix polynomials; see Section 11 of [D1] and Section 6 of [D4].
LEMMA 3.3.
If X is as in Lemma 3.2 and if f = JMXcpa ju and g =
JMj o19 iv for some choice of a,# in H+ and u, v in irk, then
[Mxf, Mx9]B(x) = (Jf, 9)H, .
(3.12)
PROOF. This is a straightforward calculation based on the definitions: [M x f, Mx9]B(x) _ [r o ,ju, rVp,iv]B(x)
_ (r'Pa,ju,Vp,iv)Hp _ (MXJMXPP, ,ju,V,3,iv)HP,
_ (Jf,9)H, . 0
Formula (3.12) exhibits Mx as an isometry from the span M of fj =
JM!Wa ju, j = 0, ... , n in HP into 8(X). This is an important ingredient in the verification of the orthogonal direct sum decomposition
8(X) = 8(X6)[+]XlC(O)
(3.13)
which holds when M = IC(E)). In fact M =)C(O) when M is a nondegenerate subspace of HP j, as follows from Theorems 2.2 and 2.1. THEOREM 3.5. Let X = [F G] be a k x m matrix valued meromorphic function on f2+ with components F()) E crk"P and G(.\) E Xkx9 such that Mx is a bounded operator from HP into Hp . Then X is analytic on H+ and the following are equivalent:
(1) X is (52+, J, p) admissible. (2) r = MFMF - MGMM is positive semidefinite on H. (3) There is a p x q matrix valued analytic function Son S2+ such that IIMSII < 1 and MG = MFMS.
Alpay and Dym
47
PROOF. X is analytic on S2+ because MX a E HP
for every choice of v E 4r'". The equivalence of (1) and (2) is an easy consequence of the evaluation
v; Aw; (wi)v; = r
v7
vi
1 pw;
8(X)
Pwi
'j =(r, Pw;
vt)HP Pw;
and the fact that finite sums of the form EA,1 vj are dense in 5(X). Suppose next that (2) holds. Then, by a (slight adaptation - to cover the case p # q of a) theorem of Rosenblum [Ro], there exists an operator Q from Hp to HP such that: MG = MFQ, IIQII < 1 and QM, = MsQ ,
where, in the last item M9 denotes the (isometric) operator of multiplication by s = b/a on Hp, regardless of the size of the positive integer r. Thus, by Theorem 3.3 of [AD4], Q = MS for some p x q matrix valued function S which is analytic on Q+, as needed to complete the proof that (2) = (3). The converse is selfevident. I A more leisurely exposition of the proof of this theorem for p,,,(A) = 1 - Aw* may be found e.g., in [ADD].
4. RECURSIVE EXTRACTIONS AND THE SCHUR ALGORITHM In this section we study decompositions of the form (3.13) of the reproducing kernel Pontryagin space 13(X) based on a k x m (52+, J, p), admissible matrix valued function X; the existence and uniqueness of these spaces is established in Theorem 3.1. Such decompositions originate in the work of de Branges [dB2, Theorem 34], [dB3], and de Branges and Rovnyak [dBR] for the case v = 0 (which means that 13(X) is a Hilbert space) and p,,,(A) = -21ri(A - w*) for a number of different classes of X and O. Decompositions of the form (3.13) for finite v (i.e., when 13(X) is a reproducing kernel Pontryagin space) and the two cases p,,,(,\) = 1 - \w* and p,,,(A) = -2-7ri(A - w*) were considered in [AD3]; such decompositions in the Krein space setting were studied in [Al] and, for Hilbert spaces of pairs, in [A4]. If 13(X) is a nonzero Hilbert space, then it is always possible to find a one dimensional Hilbert space 1C(O1) such that XJC(O1) is isometrically included inside B(X ).
This leads to the decomposition B(X) = B(xe1) ED XIC(e1)
.
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48
Then, if 5(X01) is nonzero, there is a one-dimensional Hilbert space K(02) such that X01K(02) sits isometrically inside 8(X01), and so forth. This leads to the supplementary sequence of decompositions
8(X01) = 5(x0102) ® XO1K(02)
5(X0102) = 5(X010203) ® x0102K(03)
which can be continued as long as the current space 5(X01 0n) is nonzero. In this decomposition, the 0j are "elementary sections" with poles (and directions) which are allowed to vary with j. The classical Schur algorithm corresponds such a sequence of decompositions for the special case in which p,,,(P) = 1-.\w*, X = [1 S] with S a scalar analytic contractive function in ID and all the Oj have their poles at infinity. For additional discussion of the Schur algorithm from this point of view and of decompositions of the sort considered above when 5(X) is a reproducing kernel Pontryagin space, see [AD3]. In particular, in this setting it is not always possible to choose the K(Oj) to be one-dimensional, but, as shown in Theorem 7.2 of [AD3] for the two special choices of p considered there, it is possible to choose decompositions in which the K(Oj) are Pontryagin spaces of dimension less than or equal to two. The same conclusions hold for p E Do also as will be shown later in the section. THEOREM 4.1. Let X be a k x m matrix valued function which is (St+, J, p),, admissible and let 5(X) be the (unique) associated reproducing kernel Pontryagin space with reproducing kernel given by (3.1). Let a E 52+, the domain of analyticity of X in fl+, let M denote the span of the functions
fj = JMXcpa j-1v =
1-1
(j - 1)!
aw=j-1
JX(w)
v Pw
w=a
j = 1, ... , n, endowed with the indefinite inner product [ffj,fi]m = (Jfj,.fi)H, = Pij and suppose that the n x n matrix P = [pij] is invertible. Then:
(1) M is a K(0) space. (2)
The operator MX of multiplication by X is an isometry from K(O) into 5(X). (3) X0 is (1 +, J, p)µ admissible, where p = v - the number of negative eigenvalues of P.
(4) 5(X) admits the orthogonal direct sum decomposition
5(X) = 5(XO)[+]XK(0) PROOF. Let V = [vl,...,vn]
.
(4.2)
Alpay and Dym
49
denote the m x n matrix with columns JX(3-1)(a)*v (j - 1)! Then, by Leibnitz's rule, it is readily checked that F = [f1,...,fn]
= Van By (2.4), this can be reexpressed as
F(A) = V{a(A)Aa - b(A)Ba}-1
where Aa and Ba are as in (2.6). Therefore, by Theorems 2.2 and 2.3, M is a !C(O) space.
Next, since
Xfj=(j-1)! 1 A(j-1)v a in terms of the notation introduced in Section 3, it follows from Theorem 3.2 that X f j E B(X) and 1
1
aj-1
ai-1
[Xfj, Xfi]B(X) _ (i - 1)! (j - 1)! aAi-1
aw*j-1
v*X(A)JX(w)*v Pw(A)
A=w=a
But the last evaluation is equal to
(Jfj,fi)H, , as follows by writing
fj =
j-1 k(t) t=0
t!
vj-1-t
with k,,,(A) = 1/p,,,(A) and applying Theorem 3.2 to H,,. This completes the proof of (2).
The proofs of (3) and (4) are much the same as the proofs of the corresponding assertions in Theorem 6.13 of [AD3] and are therefore omitted.
If n = 1, then the matrix P in the last theorem is invertible if and only if X(a)*v is not J neutral. In this case, the 0 which intervenes in the extraction can be expressed in the form
O(A) = Im + {ba(A) - 1}u(u*Ju)-1u*J
where u = JX(a)*v and ba(J1) =
1 - (A)s(a)
,
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50
with s(A) = b(A)/a(,\); see (2.24) of [AD4], and hence
v*X(a)O(a) = 0
.
Thus it is possible to extract the elementary Blaschke-Potapov factor
G(A) = Ik + {ba(.\) - 1}v(v*v)-lv* from the left of XO to obtain
X1(\) = G(\)-'X(A)O(A) , which is analytic at a and is (52+, J, p)µ admissible, where y = v if u* Ju > 0 and tt = v-1
ifu*Ju<0. THEOREM 4.2. Let X be a k x m matrix valued function which is (52+, J, p)v
admissible and suppose that B(X) # {0} and H+ is connected. Then X(A)JX(A)* 0 0 in 52+, the domain of analyticity of X in Q+.
PROOF. Suppose to the contrary that X(A)JX(.)* - 0 in Q+. Then, for any w E 52+, there exists a 6 > 0 such that the power series
Xs(A - ,)s
X(A) S=0
with k x m matrix coefficients converges, and 00
E Xs(A - w)SJ(,1* - w*)iXt = 0 s,t=0
for J\ - wI < S. Therefore 00
j
s,t=0
es+te'(s-t)0XsJX*t = 0
for 0 < e < 6 and 0 < 0 < 2a. But this in turn implies that XSJXt = 0 for s,t = 0,1,..., and hence that X(a)JX(/1)* = 0 for Ja - wl < 6 and I# - wl < 6. Since Si+ is connected, this propagates to all of f2+ and forces B(X) = {0}, which contradicts the given hypotheses. Thus X(A)JX(A)* 0 0 in 52+, as claimed. I COROLLARY 1. If, in the setting of Theorem 4.1, S2+ is connected, then there exists a point a E f2+ and a vector v E Ck such that v*X(a)JX(a)*v # 0. COROLLARY 2. If, in the setting of Theorem 4.1, S2+ is connected, then the set of points in f2+ at which the extraction procedure of Theorem 4.1 can be iterated arbitrarily often (up to the dimension of B(X)) with one dimensional IC(E)) spaces is dense in 52+.
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If S2+ is not connected, then it is easy to exhibit nonzero 13(X) spaces for which X(A)JX(A)* = 0 for every point A E 52+. For example, if S2+ has two connected components: 521 and 522i let
Kl = 1 [IP
IP]
X(A) = [a(A)Kj
K2 =
,
b(A)Kj]
1f
- IP] ,
[IP
A E 1j ,
for
,
and [jopp
j=
O - JPP 1
Then, forAE52j andwESl;, X(A)JX(w)* = p,,(A)KjJPPK: 0
P,(A)IP
i=j i#j.
if if
Thus X(A)JX(A)* = 0 for every point A E 5l+, while 8(X) is a 2p dimensional Pontryagin space of index p.
THEOREM 4.3. Let X be a k x m matrix valued function which is (52+, J, p), admissible such that 13(X) # {0} and yet X(a)*v is J neutral for every choice of a E S2+
and V E Ck, then there exist a pair of points a,/3 in S2+ and vectors v,w in
&
such
that
(1) w*X(/3)JX(a)*v # 0. (2) The two dimensional space
M = span
JX(a)*v Pa
JX(Q)*w 1
J
Pp
endowed with the J inner product in Hp will be a IC(O) space. (3) The space 13(X) admits the direct orthogonal sum decomposition 13(X) = B(XO)[+]XKC(O)
.
(4) XO is (52+, J, p)i_1 admissible. PROOF. The proof is easily adapted from the proof of Theorem 7.2 in [AD3].
I There is also a nice formula for the O which intervenes in the statement of the last theorem: upon setting ul = JX(a)*v, u2 = JX(/3)*w and W;j =
for
i#j,
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52
we can write
O(A) -
+ (Im
s(A) - s(A) _ 1} W12 f 1 - s(A)s(a)* f /
{
(I.+
s(-') - s(a) S
l 1 - s(A)s(i)*
- 1 } W21)
.
(4.4)
J11
This the counterpart of formula (7.5) of [AD3] in the present more general setting; for more information on elementary factors in this setting, see [AD4]. The reader is invited to check for himself that
J - O(A)JO(w)* _
1
0 F(A) 17*
0J
1
F(w)*
,
where F(A) = [u1 u21(a())A - b(A)B)-1
with 0 a(#)
I
'
B =
[
b(a)
0
0
b(Q),
*
and -y = u1* Ju2/pp(a). The formula
____a(,8) _- 1 -
j s(A) - s(Q) l 1 s(w) - s(a) l
pa(A)p,,(#)- s(A)s(a)* J l 1 - s(w)s(f )* J which is the counterpart of (7.3) of [AD3], plays an important role in this verification.
The matrix valued function
L1
(2)2
c2A2
... c2vA2v
c1A ... c2v-1A 2v-1
1
with
cj =
is (+, J, p)admissible with respect to p(A) = 1 - Aw* and
J=
Thecorresponding space 8(X) is a 2v dimensional Pontryagin space (of polynomials) of index v with reproducing kernel A,(A) = (1 Aw*)2i-1. It does not isometrically include a one dimensional 1C(O) space such that XO is (52+, p, J)v_1 admissible, since X(A)JX(A)* > 0 for every point A E SZ+ = ID. This serves to illustrate
-
the need for the two dimensional 1C(O) sections of Theorem 4.3.
We remark that Theorem 4.1 can be extended to include spaces M in which the chain f j, j = 1, ... , n, is based on a point a E no such that Au, ()) is jointly analytic in ) and w* for (A, w) in a neighborhood of the point (a, a) E f o x 520. In this neighborhood A,,,(A) admits a power series expansion
i,j=O
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53
with the k x k matrix coefficients A23. Now if v*Aoov
...
v*Ao n-lv
P= v*An_1 Ov
v*An_1 n_lv
and if Aa and Ba are defined as in (2.6), and
V = {JX(a) * v... JX(n-1)(a)*v J (n - 1)!
'
then
AQ.PA0 - B*PBa = V*JV.
(4.6)
Formula (4.6) may be verified by differentiating {a(A)a(w)* - b(A)b(w)*}A.(A) = X(A)JX(w)*
i times with respect to A, j times with respect to w* for i, j = 0, ... , n - 1 and then evaluating both sides with A = w = a. Therefore, by Theorem 2.3, the span M of the columns of
F(A) = V{a(A)Aa - b(A)Bo,}-1 endowed with the indefinite inner product
[Fe, F77]M = rl*P is a K(O) space, whenever the hypothesis of Theorem 2.3 are satisfied. It is important to bear in mind that the indefinite inner product M is now defined in terms of derivatives of A,(A) and not in terms of evaluations inside Hp which are no longer meaningful since
the columns of F (which are just the fj of (4.1)) do not belong to Hp when a E Slo. Nevertheless formula (4.3) is still valid (as follows from the remarks which follow the proof of Theorem 3.2) and serves to justify the assertion that MX maps K(O) isometrically into
5(X) in this case also.
Recall that a subspace M of an indefinite inner product space is said to be nondegenerate if zero is the only element therein which is orthogonal to all of M. It is readily checked that if M is finite dimensional with basis fl, ... , fn, then M is nondegenerate if and only if the corresponding Gram matrix is invertible. THEOREM 4.4. Let P be a reproducing kernel Pontryagin space of k x 1 vector valued functions defined on a set A with reproducing kernel L,,,(A). Suppose that
M = span{Laul, ... , Laun } is a nondegenerate subspace o f P f o r some choice of a E A and u1, ... , un in Uk, let A/ = ME'] and let U = [ul, ... , un] denote the k x n matrix with columns ul,... , un. Then:
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54
(1)
The matrix U*La(a)U is invertible.
(2)
The sum decomposition
P = M [+]JV
is direct as well as orthogonal. (3)
Both the spaces M and JV are reproducing kernel Pontryagin spaces with reproducing kernels
LM(\) = La(\)U{U*La(a)U}-l U*Lw(a)
(4.7)
and
LN(A) = L,, (A) - LM (A)
,
(4.8)
respectively.
PROOF. The matrix U*La(a)U is the Gram matrix of the indicated basis for M. It is invertible because M is nondegenerate. The fact that M is nondegenerate further guarantees that M fl H = {0} and hence that assertion (2) holds. Next, it is easily checked that LM (A), as specified in formula (4.7), is a reproducing kernel for M and hence that M is a reproducing kernel Pontryagin space. Finally, since L,,,u - LMu belongs to H for every choice of w E A and u E Irk and [.f, Lwu - LMu]P = [.f, L,,,u]v = u*.f (w)
for every f E H, it follows that JV is also a reproducing kernel Pontryagin space with reproducing kernel LW (A).
I
THEOREM 4.5. If, in the setting of Theorem 4.4, P = B(X) and Aw(A) _ X(A)JX(w)* pw(A)
for some choice of p E Dc and k x m matrix valued function X which is (5l+, J, p)v admissible, and if a E S2+ (the domain of analyticity of X in Sl+), then there exists an m x m matrix valued function O E Pj(1l+) for some finite v such that M = XKC(O).
H = B(XO). LM (A) = X(A) { J-enW Je w ' } X(w)*. AN(A) = X(A)O(A)JO(w)*X(w)*lpw(A).
PROOF. To begin with, let
V = JX(a)*U and
F(A) =
Vpa(A)-l
.
Then, upon setting A = a(ce)*In and B = b(ax)*In
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55
F can be expressed in the form
F(A) = V{a(A)A-
b(A)B}-1
,
which is amenable to the analysis in Section 4 of [AD4]. The latter is partially reviewed in Section 2. In particular, hypotheses (1) and (2) of Theorem 2.1 (above) are clearly met: (1) holds for any p E Slp for which Ia(p)J = jb(p)I # 0 since a V c10i whereas (2) holds because M (which equals the span of the columns of X F) is nondegenerate by assumption. Thus since V*JV
P
Pa(a)
is a Hermitian invertible solution of the matrix equation
A*PA - B*PB = V*JV, it follows from Theorem 2.1 that the span .F of the columns of F endowed with the indefinite inner product
[Fu, Fv]z- = v*Pu is a 1C(O) space. This proves (1) and further implies that 1
F(A)P- F(w) =
J - O(A)JO(w)* Pw(\)
Thus Aw
X(A)V
V*JV -1 V*X(w)*
Pa(A) { Pa(-) }
Pa(w)
= X(A)F(A)P-1F(w)*X(w)* , which proves (3).
The remaining two assertions follow easily from the first two.
I
We remark that if the matrix U which appears in the statement of Theorem 4.4 is invertible, then A'"(A) = Aa(A)Aa(n)-1Aw(a) (4.9) and
AN(A) = A, (A) - Aa(A)A,x(a)-1Aw(a)
(4.10)
Conclusion (4) of the last theorem exhibits the fact that (whether U is invertible or not) AN(A) has the same form as A,(\). Fast algorithms for matrix inversion are based upon this important property. Lev-Ari and Kailath [LAK] showed that if a kernel A,(\) is of the form X(A)JX(w)* Aw(A) P,,(\) for some p,,,(A) with p,,,(A)* = pa(w), then the right hand side of (4.10) will be of the same form if and only if p,,,(A) admits a representation of the form (1.2). The present analysis
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56
gives the geometric picture in terms of the reproducing kernel spaces which underlie the purely algebraic methods used in [LAK]. We complete this section with a generalization of Theorem 3.5.
THEOREM 4.6. Let X = [C D] be a k x m matrix valued function which is (ft+, J, p) admissible and for which Mg is a bounded operator from HP into HP . Then there exists a p x q matrix valued meromorphic function S on fZ+ such that
(1) [Ip - S] is (H+, J, p)., admissible, and
(2) D = -CS. PROOF. If v = 0, then the assertion is immediate from Theorem 3.5. If v > 0, then, by repeated applications of either Theorem 4.1 or 4.3, whichever is applicable, there exists an m x m matrix valued function e E P j(fl+) which is analytic
in fZ+ such that X8 is (fl+, J, p) admissible and the multiplication operator Me is bounded on HP. The last assertion follows from Theorem 6.1 and the formulas for 8 which are provided in and just after the proofs of Theorems 4.1 and 4.3, respectively. Thus, the multiplication operator M%e is also bounded on HP and so, by Theorem 3.5, there exists a p x q matrix valued analytic function S, on fl+ with II M3, II : 1 such that Ce12 + D822 = -(C811 + De21)So Therefore,
D(e21So + 022) = -C(e11So + 812) which in turn implies that
D = -CS with S = (01150 + 012)(021 S0 + 822)-1
.
The indicated inverse exists in 11+ except for at most a countable set of isolated points because det(821So + e22) is analytic and not identically equal to zero in fZ+. Indeed, since 0 is both analytic and J unitary at any point it E 1o at which Ia(, )I = Ib(µ)I # 0, it follows by standard arguments that 02-21821 is strictly contractive at p and so too in a little disc centered at p. This does the trick, since every such disc has a nonempty intersection with f1+ (otherwise Ia(A)/b(A)I < 1 in some such disc with equality at the center; this forces b(.1) = ca(A), first throughout the disc by the maximum modulus principle, and then throughout all of ft since it is connected) and So is contractive in H+. Now, let F = [fl be an m x n matrix valued function whose columns form a basis for X (O), let Q denote the invertible n x n Hermitian matrix with ij entry
qij = (fI, f:)ic(e) , and finally, let
Y=[Ip -S] and G=811-5821.
57
Alpay and Dym
Then it follows readily from the decomposition Y(A)JY(w)* - 1,(A) J - O(A)JO(w)* Y(w)* O(A)JO(w)* X(w)* + Y()) P"(A) Pw(A) Pw(A)
= Y(A)F(a)Q-1F(w)*Y(w)* + G(a)
IP -
G(w)*
that the difference between the kernel on the left and the first kernel on the right is a positive kernel. Therefore, for any set of points a1, ... , at in the domain of analyticity of Sin S2+ and any set of vectors 1, ... , 6t in k, the t x t matrices D
P,1 - [CiY(ai)JY(aj)%j 1 and P2 - I C; Y(ai)F(ai)Q Pad ai) Pad ai) J i,j = 1, ... ,t, are ordered: P1 > P2.
1
J
Thus, by the minimax characterization of the
eigenvalues of a Hermitian matrix, Aj (P1) ? Aj (P2) ,
j = 1, ... , t
,
in which Aj denotes the j'th eigenvalue of the indicated matrix, indexed in increasing 0 and hence the kernel based on S has at most v negative size. In particular, squares.
On the other hand, since X is (52+, J, p) admissible, there exists a set of points /31 i
. . .
, /ir in S2+ and vectors 771,..., 77,. in Ck such that the r x r matrix with Z 'J'
entry equal to
_2 X(ai)JX(aj)*nj - 71 C(Qi)IP - S(Qi)S(aj)* C(aj)*77j P/3; (Qi)
-
Ppj (Qi)
has exactly v negative eigenvalues. This shows that the kernel based on S has at least v negative eigenvalues, providing that the exhibited equality is meaningful, i.e., providing that the points 31, ... , /j,. lie in the domain of analyticity of S. But if this is not already the case, it can be achieved by arbitrarily small perturbations of the points ,Q1.... , Or because S has at most count ably many isolated poles in 52+. This can be accomplished without decreasing the number of negative eigenvalues of the matrix on the left of the last equality because the matrix will only change a little since X is analytic in 52+, and therefore its eigenvalues will also only change a little. In particular, negative eigenvalues will stay negative, positive eigenvalues will stay positive, but zero eigenvalues could go either way. This can be verified by Rouche's theorem, or by easy estimates; see e.g., Corollary 12.2 of Bhatia [Bh] for the latter.
5. fp(S) SPACES In this section we shall first obtain another characterization of the space Rr endowed with the indefinite inner product (3.5) in the special case that r = MxJMX is positive semidefinite. We shall then specialize these results to X = lip
-S]
(5.1)
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58
and
J
Jpq
Op
0e J
(5.2)
where S is a p x q matrix valued function which is analytic in St+ such that the multiplication operator MS from Hp to Hp is contractive. The resulting space C([Ip - S]) will then be designated by the symbol 74 (S). THEOREM 5.1. If X is a k x m matrix valued function which is analytic in ci+ such that the multiplication operator MX from HP to Hp is bounded and if
r=MXJMX>0, then
(1) C(X) = ran r2 with norm
IIr29IIr = II(I - P)9IIHp , where P denotes the orthogonal projection of Hp onto the kernel of F. (2)
ran r is dense in ran r2 and (
r9, rh )r = ( r 9, h)Hp
for every choice of g and h in HP . (3) C(X) is the reproducing kernel Hilbert space with reproducing kernel given by (3.1). (4) X is (52+, p, J) admissible. (5) C(X) = B(X)-
PROOF. Since ker r = ker r2 , it is readily checked thatII IIr, as defined in (1), is indeed a norm on ran r 2 . Moreover, if r s fn, n = 1, 2, ..., is a Cauchy sequence in ran r2, then (I - P)fn is a Cauchy sequence in the Hilbert space Hp, and hence tends to a limit g in HP as n T oo. Therefore, since I - P is an orthogonal projector, it follows by standard arguments that
9=rim(I-P)fn= lim(I-P)2fn=(I-P)9 and hence that
IIr2fn - r29IIr = II (I - P)(fn - 9)II Hp
=II(I-P)fn-9IIHp Thus ran r2 is closed with respect to the indicated norm; it is in fact a Hilbert space with respect to the inner product
(r2f,r2g)r = ((I -P)f,9)Hp .
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59
For the particular choice g = v/p,., with v E (k and w E 52+, the identity
(r2 f, r29)r = ((I - P)f, r29)H,
_ (r
i2f,9)H,
= v*(rzf)(w) serves to exhibit
rg= XJX(w)*v=Awv Pa
as the reproducing kernel for ran r . This completes the proof of (1), since there is only one such space. (2) is immediate from (1) and the fact that ker 1''2 = ker r; (3), (4) and (5) are covered by Theorems 3.3, 3.5 and 3.4, respectively. For ease of future reference we summarize the main implications of the preceding theorem directly in the language of the p x q matrix valued function S introduced at the beginning of this section. THEOREM 5.2. If S is a p x q matrix valued function which is analytic in S2+ such that the multiplication operator Ms from Hp to HP is contractive and if X and J are given by (5.1) and (5.2), respectively, then:
(1) r=MXJMX=I - MsMM (2) 9-[p(S) = ran r2 with IIri2fIIW,(s)
= II (I - P)fII H, where P designates the orthogonal projection of HP onto ker I72. (9)
ran r is dense in ran r2 and (r9, rh)rt,(s) = (r9, h) H, for every choice of g and h in H.
(4)
1-lp(S) is a reproducing kernel Hilbert space with reproducing kernel AW(A) = Ip - S(A)S(w)* P,,,('\)
The next theorem is the analogue in the present setting of general p E DU of a theorem which originates with de Branges and Rovnyak [dBR1] for p,,,()) = 1 - A. THEOREM 5.3. Let S be a p x q matrix valued function which is analytic in Q+ such that the multiplication operator Ms from Hp to Hp is contractive and for
f E Hp let
r.(f)=sup{][f+MsgllH,-119[1X,: 9 E Hp} .
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60
Then
Ilp(S) _ {f E HP: r(f) < oo} and
IIfIIfP(s) _ ic(f) PROOF. Let X and J be given by (5.1) and (5.2), respectively. Then clearly Theorem 5.1 is applicable since
r=MXJMX=I-MSMM>0. Moreover, since r < I, it follows that 172 is a contraction and hence, by Theorem 4.1 of Fillmore and Williams [FWI, that f E ran 1'2 if and only if
sup{IIf+(I-rZrz)z9IIHp-II9IIHp: 9EHP}<00, or equivalently, if and only if
sup{IIf+(MsMs)29IIHII9IIHp
9EHP}
Therefore, since
HP = ran(MSMS) z + ker(MSMS) 2
any g E HP can be approximated arbitrarily well by the elements of the form
(MsMs)2U+v with u E HP and v E ker(MSMS)2. Thus the sup in (5.5) can be reexpressed as sup{IIf +( MS MS)2(MSMS)2UIIHp
- II(MsMM)ZU+vIIHp
uEHPandvEker(MSMS)4} =sup{IIf +Ms11IsuIIHp - II(MsM.)ZUIIHp : u E HP} = sup{IIf + MSMsuII HP - IIM.UIIH0 : u E Hp )
=sup{IIf+MS(Mg*u+w)IIHp-IIM.u+wIIHP: uEHP and wEkerMS} =sup{IIf+MshjI1- IIhIIHp: h EHP} This proves that f E ran r =1-lp(S) if and only if ac(f) < oo.
(5.5)
Alpay and Dym
61
Next, if f = I' 2 h for some h E Hp, then (as is also shown in [FW]) is(f) = inf{II hIIH, : (I - MsMs)2 h = f } =II(I-P)hIIH,
where P denotes the orthogonal projection of Hp onto the kernel of r. This serves to establish (5.4), thanks to item 2 of Theorem 5.2. 1 THEOREM 5.4. Let S be a p x q matrix valued function which is analytic in S2+ such that IIMsII < 1 and let J be given by (5.2). Then (1) [I - Ms] f E l p(S), and (2) 11 [1 -Ms]fllrt,(s) for every choice of
< (Jf,f)H,
f = [g] E Hp with components g E Hp and h E Hp, if and only if
h=MSg
.
PROOF. Suppose first that (1) and (2) hold. Then, by Theorem 5.3, 11 [1
- Ms]fll;c,(s) = IIg - Mshllv(s)
=sup{IIg-Msh+MsuIIH,-IIuII23r,: uEHP} =sup{II9+MsvIIHp -IIh+vIIHp : veHp} Therefore, by the prevailing assumptions, IIgII2
2 + 2Re(g, Msv)H, + IIMsvIIH ,
2 2 - 2Re(h, v)H, - IIvIIH, - IIhiIH,
=
for every v E H. But this in turn implies that IIMsvII2Hp + 2Re(Msg - h,V)H, - IIVI12p < 0
for every v E Hp and hence in particular, upon choosing
v=e(Msg-h) with a>0, that
e2IIMs(Msg-h)Ilii,+2eJJMsg-hljH,-e2IIMsg-hII Hp <0 -
Alpay and Dym
62
for every e > 0. The desired conclusion (5.6) now follows easily upon first dividing through by a and then letting e 10.
Next, to obtain the converse, suppose that (5.6) holds. Then, by Theorem 5.2,
[I -Ms]f =(I -MsMs)9=r9 belongs to 'Hp(S) and II [I - Ms]f II,H,(s) = (r9, r9)7-(,(s) = (r9, 9) H,
=IIIIH,-IIMs9IIH, _ (Jf, f) H, . Thus (1) and (2) hold and the proof is complete.
We remark that the inequality in (2) of the last theorem can be replaced by equality.
Finally, to complete this section we observe that if S is a p x q matrix valued function which is analytic in Q+, then the multiplication operator MS is contractive if and only if the kernel (5.3) is positive, i.e., if and only if it has zero negative squares. THEOREM 5.5. Let S be a p x q matrix valued function which is analytic on 52+. Then the kernel A(l) = Ip - S(A)S(w)' W
Pw(A)
is positive on 92+ if and only if IIMSII < 1.
PROOF. Suppose first that II MS 115 1 and let f = E' 1 Cj l p,, for any choice of points w1,. .. , wn E 52+ and vectors E V. Then it is readily checked that n
C;Awi(w;)Ci = IIfIIH, - IIMsfIIH, > 0 i,j=1
which establishes the positivity of the kernel. Next, to go the other way, we define a linear operator T on finite sums of the form f given above by the rule T -L = S(w)' Pw
6
.
Pw
By the presumed positivity of the kernel A,,,(A), T is well defined and contractive on finite
sums of this form and hence, since such sums are dense is AfV. a,An be extended by
Alpay and Dym
63
limits in the usual way to a contractive operator (which we continue to call T) on all of Hp. Finally the evaluation C*(T*g)(-)
_ (T*g, _ (9, T
)HP
P ) HP
P _ (9, S(-')*
) HP PLO
= E*S(w)9(w) ,
which is valid for every choice of C E V, w E S2+ and g E Hp serves to identify T* with the multiplication operator Ms. Therefore
IiMsil=IIT*II<1, as claimed.
I COROLLARY. If S is a p x q matrix valued function which is analytic on
S2+ such that JIMSII < 1, then
Ip - S(w)S(w)* > 0
(5.7)
for every choice of w E l+. It is important to bear in mind that even through (5.7) implies that 1IMSII
1
for p,,,(.\) = 1 - )w* and p,(.\) = -2iri(A - w*), the last corollary does not have a valid converse for every choice of p E Dc, as we now illustrate by a pair of examples. EXAMPLE 1. Let a(A) = 1 and b(\) = A2 so that S2+ = ID and let S be any scalar contractive analytic function from ID into ID such that S(") = -S(-") # 0. Then MS is not a contraction.
DISCUSSION. It follows from the standard Nevanlinna-Pick theory (see e.g.
[D2]) that there exists an S of the desired type with S(1) = c if and only if the 2 x 2 matrix
11 - S(wi)S(wj)*1 IL
Pwj (wt)
i,7 = 1,2
(5.8)
J
with wl = -w2 = ', S(w1) = -S(w2) = c, and p,(A) = 1 -Aw* is positive semidefinite. The matrix of interest: 1-c2 1+c2 1-14 1+14 1+c2 1-c2 1+1/4
1-1 4
is readily seen to be positive semidefinite if and only if Icl <
.
Alpay and Dym
64
On the other hand, if IIMsil < 1, then the matrix (5.8) must be positive semidefinite for the same choice of points and assigned values as before but with p,,,(\) _ 1 - \2w" 2. But this matrix: 1-c2 1+c2 1-1 16
1-1 16
1+c2 1-1 16
T--17-16
1-c2
is not positive semidefinite for any c # 0 as is readily seen by computing its determinant. EXAMPLE 2. Let p,,,(.\) = -27ri(\ - w`)(1 - Aw*) with a(\) and b(\) as in (2.2), and let S(w) = c for w E ID fl and S(w) = -c for w E En C_, where Ici < 1. Then S is a contractive analytic function in SZ+ but Ms is not a contractive mapping of Hp into itself for c # 0.
DISCUSSION. If liMsil < 1, then the matrix (5.8) will be positive semidefinite for any pair of points w1,w2 in SZ+. For the particular choice w1 = i/a, w2 = -ia with a > 1, the matrix of interest is equal to 1-c2 1+c2 47r a -1 a 41r 1-a a 1+c2 1-c2 47r1-a a 4aa _1a which is not positive semidefinite if c # 0.
6. LINEAR FRACTIONAL TRANSFORMATIONS In this section we shall recall a number of well-known properties of the linear fractional transformation (6.1)
TO[So] _ (0115. + 012)(021So + 022)-1 for
0-
0121
1011
022
021
which is analytic and J = Jpq contractive in Q+. In particular it is well known that the indicated inverse exists if (the p x q matrix valued function) S, is analytic and contractive in SZ+ and that in this case Te[So] is also analytic and contractive in SZ+. THEOREM 6.1. If 0 is given by (2.14) and the matrices A and B in (2.10) are such that A is invertible and the spectral radius of BA-1 is less than one, then 0 is analytic in SZ+ and the multiplication operator Me is bounded on HP . PROOF. By (2.14) and (2.10),
0(\) = I. - a(µ)`{1 - s(\)s(µ)'}VA-1{Im where, as before, s(\) = b(\)/a(\). In particular ao
{Im - s(\)BA-1 }-1 =
(A)n(BA-1)n n=0
s(\)BA-1}-1P_1F(1,)"J
65
Alpay and Dym
and since by assumption 00
E II(BA-1)"II < 00 n=0
it is easily seen that O(A) admits a representation of the form 00
O(A) = E s(A)nOn n=0
with m x m matrix coefficients On which are summable: 00 EIIOnll
n=0
It now follows readily by standard estimates that if
1E 00
f(') =
s(A)iuj
j=o
belongs to HP', i.e., if the m x 1 vector coefficients uj are norm square summable, then 2
n
IIMef IIH, _
0"
On-j uj
n=0
j=0
0o
n
n
< E > IIOn-jll
3
j=o
k:
n=o
2
00
c1: Iluk112IlOnll} k=0
n-0
But this proves that 00
11M011:5 1:
IIOniI<00;
n=0
as claimed.
I
THEOREM 6.2. Let So be a p x q matrix valued function which is analytic in S2+ such that the multiplication operator MSo is contractive, let 0 E P3(Sl+) be analytic in 52+, and let S = Te1So]. Then IIMsII < 1. PROOF. If S = Te[So], then it is readily checked that [Ip
- S]e = E[Ip - S.] ,
Alpay and Dym
66
where
E=011-5021.
Now let
X = [Ip - S] . Then
Ip - S(a)S(,Q)* Pp(a)
_ X(a)JX(,Q)* P1(a)
= X(a)
J - O(a)JO(,0)** Pa(#)
= X(a) J -
X(#)+
X(a)0(a)JO(/3)*X(/3)*
p (a)
)J o(#)** X(#)* + E(a) j Ip O(a
of
)j o(13)* 1 E(Q)*
.
By Theorem 5.5, J I Ms I I < 1 if and only if n
- S(a;)S(a.i)* Cj > 0
e_
{Ip
f
Paj (ai)
(6.3)
for every choice of points al, ... , an in H+ and vectors fl,.. . , Cn in V. Since IIMso II < 1
by assumption, the inequality (6.3) holds with S. in place of S. Therefore, by (6.2) it holds for S also, since 0 E P5(S2+). Thus JIMsII < 1. 1 Now as
O = [011 021
012 1
022
is J = Jpq contractive in Q+, it can be expressed in the form O
01FII
1Ip
0
011[51
0
Iq
E2
0.2
01 Iq
where
62 = 022 is invertible in SZ+,
01=012022, a2=022021 C1 = 011 - 012022021 and e1i o1, 02 and e21 are all contractive in 5l+; see e.g., p.15 of [D2]. THEOREM 6.3. If O is as in Theorem 6.2, then the multiplication operators Moa, Mol and M _1 are all contractive. Moreover, Mgt is a strict contraction. a
PROOF. Since
K.(A) -
J - 19(.1)JO(ca)*
P-(A)
Alpay and Dym
67
has zero negative squares in Sl+, the same holds true for its 22 block:
Iq - 021(ai)021(aj)* + o22(ai)022(aj)*
qi k
qj > 0
Pa; (ai)
i,j=1
(s.5)
f o r every choice of al, ... , ak in Sl+ and 771, ... 01k in V. Thus clearly k
71i O22(ai) 1
ll
i,j=1
Iq - p(aj .' (ai)
e22(aj)*'77j > 0
and k
*
i 022(ai)
1 Iq - E2(ai)-lE2(aj)-* l ()22(aj)*'J)j Paj(ai) Sl 1
0
for every such choice of a1 , ... , ak and 771, ... , Ilk. Thus, by Theorem 5.5, the multiplica-
tion operators M _1 and Mat are both contractive. 2
The proof that M1 is contractive is immediate from Theorem 6.2 and the observation that a1=TO[0]To check that Mot is strictly contractive, it suffices to note that (6.5) implies that I - Mot Mot > ME21 ME2* .
U
7. ONE-SIDED INTERPOLATION In this section we study the following general one-sided interpolation problem for a fixed p E D. The data for this problem consists of a set of (not necessarily distinct) points W1i...,Wn in Sl+
and two sets of associated vectors in
C' in
Q:9
such that
.;SnWwn 07 ...,Cncw,,rn-1
are linearly independent. The problem is to find necessary and sufficient conditions on the data for the existence of a p x q matrix valued function S such that: (1) S is analytic in fl+ (which is actually automatic from (3)), (2)
* .S(t-1)(x+
ej
t_1
= *Ijt
,
t = 1,...,rj ;
7 = 1,...,n ,
Alpay and Dym
68
The operator MS of multiplication by S is a contractive map of Hp into H. It is also desirable to give a description of the set of all solutions to this problem when these conditions are met. We shall solve this interpolation problem by adapting the strategy of Section 7 of [D3] to the present setting. We begin by introducing the auxiliary notation (3)
vjl =
[
rlj1 J
and vjt = I
()
Tljt
t = 2,...,rj
,
J
7 = 1,...,n
Vj = [vj l ... vjrj ]
,
7 = 1, ... , n ,
Fj = Vj ,rj [fl ... f ] = [Fl ... Fn]
,
v = r1 + ... + rn ,
and
ft= [htI
t=1,...,v
,
where gt E Hp and ht E Hp H.
LEMMA 7.1. Let S be a p x q matrix valued function which is analytic in 11+ such that JIMSII < 1. Then S is a solution of the stated interpolation problem if and only
if Msgt = ht
(7.1)
for t = 1, ... , v.
PROOF. In order to keep the notation simple we shall deal with the first block, i.e., the columns in F1, only. The rest goes through in just the same way. By definition, t
ft = > Pwi,t-j vl j
t = 1.... ,rl
.
j=1
with components 9t = IPW,,t-1f1 and
t
ht = 1: Wwi,t-j?11j . j=1
Thus, by the evaluation in Corollary 2 to Lemma 3.2, Msgt
t - ht = F j=1
I
(j - 1)!
- ]lj
The rest is selfevident since the functions V,,,, ,o, . . . ,'Pwl independent in the formulation of the problem.
are assumed to be linearly
Alpay and Dym
69
THEOREM 7.1. The one-sided interpolation problem which was formulated at the beginning of this section is solvable if and only if the v x v matrix P with ij entry (7.2)
Pij = (Jfj, fi)Hp is positive semidefinite.
PROOF. Suppose first that the interpolation problem is solvable, and let S be a solution. Then, by the preceding lemma, the components gt and ht of ft are related by formula (7.1). Thus if v
v
f = > cj fj
and
cjgj
g=
j=1
j=1
for some choice of constants c1, ... , c,,, then
Therefore
iPijcj = (Jf,f)H, i,j=1
=
lIgI12
- IIMsgIIH,
>0, since IIMs1I=IIMsil<1. Now suppose conversely that P > 0 and let M = span{ f l , ... , f.)
endowed with the J inner product
(fj, fi)M = (Jfj, fi)H, = Pij Then, with the help of Theorem 2.3, it is readily checked that conditions (1) and (2) of Theorem 2.1 are met. Now let A = diag{A,,,1, ... , A,,,, }
and B = diag{B,,,1, ...
, B,,,,, }
be the block diagonal matrices with entries A,,,, and B,,,, of size rj x rj given by (2.6). Then F(,X) = V {a(.\)A - b(A)B}-1 (7.3) and, by Theorem 2.2,
A*PA - B*PB = V`JV.
(7.4)
Alpay and Dym
70
Therefore, by Theorem 2.1, M is a finite dimensional reproducing kernel Hilbert space with reproducing kernel J - O(A)JO(w)*
K,,(A) -
p,,(.A)
where 0 is uniquely specified up to a right J unitary factor by (2.14). Moreover, since (A)rn
det{a(A)A - b(.\)B} = Pwi (A)r' ... Pwn
,
it follows easily from (2.14) that O is analytic in all of Si except at the union of the zeros of the functions p,,, .... , Pwn . In particular, O is analytic in Si+ and is both analytic and
invertible at the special point µ E Sip which is used in the definition of 0. Thus 0 is invertible in all of c except for an at most countable set of isolated points. Moreover, it is readily checked that Theorem 6.1 is applicable for the current choices of A and B, and hence that the multiplication operator ME) is a bounded mapping of HP into itself. Next, since
ka(A) = IM
,
A E 52+
Pa(')
is a reproducing kernel for HP , it is easily seen that
v*ft(a) = (J.ft,
_ (ft,
J - OJO(a)*v)H, Pa
Pa
)H, - (Jft,
= v*ft(a) - (Jft,
OJO(a)*y
OJO a
)H,
v)Hp
Pa
and hence that
v*J0(a)J(MMJft)(a) = (MMJft,
_ (Jft,
JO(a)*Jv)H, Pa
OJO(a)*v)H, Pa
=0 for every choice of a E c+ and v E C"`. Therefore, since 0 is invertible except for at most a set of isolated points in c+, it follows that (the analytic vector valued function)
MMJft =0. Now, by (6.4), 0 = `I'1%F2 ,
where
$+l= [Cl 0 ole2 62
and 2=
Ip a2
01 IqI
Alpay and Dym
71
By Theorems 6.1 and 6.3, M,yl and M*z are both bounded multiplication operators on HP . Thus it is meaningful to write MM = M,*gz My1
and therefore, since M;z is clearly invertible, it follows from (7.5) that
M,,1 Jft = 0
.
But this in turn reduces to the pair of constraints Mfg 9t = 0 and
MEZ {MQI gt - ht} = 0
But now as ME2 is invertible (with inverse (M _,)*) the latter constraint implies that z
Molgt=ht This exhibits of as a solution to the given interpolation problem, thanks to Lemma 7.1 and Theorem 6.3. This completes the proof of the existence of a solution to the stated interpolation problem when P > 0. It remains to show that the interpolation problem is also solvable when P > 0, by passage to limit. To this end, let us first write
P=G-H with entries corresponding to the decomposition
Pij = (gj,gi)H, - (hj,hi)H, By assumption G > 0. Now, just as in [D2], let us replace the t]ij in the formulation of the problem by gij(1 +E)-z with E > 0. Then the new Pick matrix, corresponding to the perturbed data, PE
G
1 1
H
E
1
1+e G+ 1+e P >
+ EG>0
for every choice of e > 0. Therefore, by the preceding analysis, there exists a p x q matrix valued analytic function Se with 11 Ms, 11 < 1 such that
MS.g1 = (1 +e)-4hj
Alpay and Dym
72
for j = 1, ... , v. The proof is completed by letting e 1 0 and invoking the fact that the unit ball in the set of bounded operators from Hp to Hp is compact in the weak operator topology (see e.g., Lemma 2.3 on p.102 of [Be]). This means that there exists a contractive operator Q from Hp to Hp such that Elm (Mss f, g) Hp = (Qf, g) Hp
for every choice of f E Hp and g E Hp as e 1 0 through an appropriately chosen subsequence (which we shall not indicate in the notation). In particular, (Qgt, u)Hp = lim (Msc gt, u)Hp CIO = lim(l + F) -7 (ht, U) Hp CIO
= (ht,u)HP .
Now let T = Ms denote the operator of multiplication by s = b/a on HP, regardless of the size of r. Then since MS.T = TMS, for every e > 0, it is readily checked that Q*T = TQ* and hence, by Theorem 3.3 of [AD4], that there exists a p x q matrix valued function S which is analytic in Sl+ such that Q* = Ms. This completes the proof, since S is a solution to the one-sided interpolation problem. THEOREM 7.2. If the v x v matrix P with entries given by (7.2) is positive definite and if O is specified by (2.14) for the space M considered in the proof of the last theorem, then {Te[So] : S,, is analytic in S2+ and IIMs0II < 1}
is a complete list of the set of solutions to the given interpolation problem.
PROOF. The proof is divided into steps. STEP 1. If S is a solution of the interpolation problem, then Y = [Ir,
- S]O
is (SZ+, J, p) admissible.
PROOF OF STEP 1. The argument is adapted from the proof of Theorem 6.2 in [D2]. Let X = [Ip - S] and let A,.,(A) _ X(,\)JX(w)* Pw(A)
denote the reproducing kernel for lp(S). Then it is readily seen that Y(a)JY(Q)* Pp(a)
X(a)JX(,6)* _ X(a){J - o(a)JO(p)*}X(Q)* PO(a)
pp(a)
= Af (a) - X (a)F(a)P-1F(13)*X (,6)*
Alpay and Dym
73
for a and Q in SZ+. Thus if
n
f = E Aa, It t=1
for any choice of a 1 i ... , an in S2+ and a , ... , Cn in IV' and if ryij denotes the ij entry
of P-1, then
= ` sAat(ae)ft n
c*Y(aa)JY(at)"et B
tL=1
n
-
v
EE s,t=1 i,j=1
Moreover, since S is an interpolant, it follows from Lemma 7.1 and Theorem 5.4 that X fi E 7-lp(S) and hence that the right hand side of the last equality is equal to n
Ill I7 (S) -
v
s,t=1 i,j=1 v
= IIfll,cp(s) -
(Xf1,f)fp(S)7i,(f,Xfj)fp(s)
i,j=1
But this in turn is equal to Ilfll,tp(S) -
IIJIfI12
in which II denotes the orthogonal projection of f onto the span of the X fi in 13(X). This last evaluation uses the fact that MX is an isometry from JC(O) into B(X), as noted in the remark following Theorem 5.4. The rest is selfevident since IIfII2
rtp(S) - IIIIf Il,cp(S) >- 0 .
STEP 2. If S is an interpolant, then lip
- S]O = Clip - S-1
where C and So are analytic in ci+, and MC is bounded and IIMso II C 1.
PROOF OF STEP 2. This is immediate from Theorem 3.5 since My is bounded (because both MS and Me are; the former by assumption and the latter by Theorem 6.1), and Y is (SZ+, J, p) admissible, by Step 1.
STEP 3. If S is a solution of the given interpolation problem, then S = Te[So] for some p x q matrix valued function So with II Mso II 1.
PROOF OF STEP 3. By Step 2, 611 - S®21 = C
Alpay and Dym
74
and
012 - 5022 = -CSo Therefore
5(021So+022) = 01150+012
,
which is the same as the asserted formula. The term 021 So + 022 is invertible in Sl+ by standard arguments which utilize the fact that O is J contractive and So is contractive, both in fl+. STEP 4. If S = Te[So] for some pxq matrix valued function So with II MSo II 1, then S is a solution of the given interpolation problem.
PROOF OF STEP 4. By Theorem 6.2, IIMSII < 1. In the proof of Theorem 7.1 it is shown that
Ul = NO] is a solution of the interpolation problem. This means that MM, gt = ht
for t = 1, ... , v, and hence that S is a solution of the given interpolation problem if and only if
(MM-Mo,)gt=0 for t = 1, ... , v. But now as S
al =-'I So(Iq + a25o)-lE2
(1
M* iM*(I9+o2So)-i M* of )gt = M*S - M* S.M*El gt EZ
=0, by (7.6). The calculation is meaningful because all of the indicated multiplication operators are bounded thanks to Theorem 6.3. This completes the proof of the step and the theorem.
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[A2]
-, Some remarks on reproducing kernel Krein spaces, Rocky Mountain Journal of Mathematics, in press. -, Some reproducing kernel spaces of continuous functions, J. Math. Anal.
[A3]
Appl. 160 { 1991), 424-433.
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[A4]
-, On linear combinations of positive functions, associated reproducing kernel spaces and a non-Hermitian Schur algorithm, Archiv der Mathematik (Basel), in press.
[AD1]
D. Alpay and H. Dym, Hilbert spaces of analytic functions, inverse scattering, and operator models I, Integral Equations and Operator Theory 7 (1984), 589641.
[AD2]
-, Hilbert spaces of analytic functions, inverse scattering, and operator models II, Integral Equations and Operator Theory 8 (1985), 145-180.
[AD3]
-, On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization, in: I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications OT18, Birkhauser Verlag, Basel, 1986, pp. 89-159.
[AD4]
-, On a new class of reproducing kernel spaces and a new generalization of the Iohvidov laws, Linear Algebra Appl., in press.
[AD5]
-, On a new class of structured reproducing kernel spaces, J. Functional Anal., in press.
[ADD]
D. Alpay, P. Dewilde and H. Dym, On the existence and convergence of solutions to the partial lossless inverse scattering problem with applications to estimation theory, IEEE Trans. Inf. Theory, 35 (1981), 1184-1205.
[Ar]
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950),337-404.
[Al]
T.Ya. Azizov and I.S. Iohvidov, Linear Operators in Spaces with an Indefinite Metric, Wiley, New York, 1989.
[Ball
J.A. Ball, Models for non contractions, J. Math. Anal. Appl. 52 (1975), 235-254.
[Ba2]
, Factorization and model theory for contraction operators with unitary part, Memoirs Amer. Math. Soc. 198, 1978.
[Be]
B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North Holland, 1988.
[Bh]
R. Bhatia, Perturbation bounds for matrix eigenvalues, Pitman Research Notes in Math., 162, Longman, Harlow, UK, 1987.
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J. Bognar, Indefinite Inner Product Spaces, Springer-Verlag, Berlin, 1974.
[dBl]
L. de Branges, Hilbert spaces of analytic functions, I, Trans. Amer. Math. Soc. 106 (1963), 445-468.
[dB2]
-, Hilbert Spaces of Entire Functions, Prentice Hall, Englewood Cliffs, N.J. 1968.
[dB3]
-, The expansion theorem for Hilbert spaces of entire functions, in: Entire Functions and Related Topics of Analysis, Proc. Symp. Pure Math., Vol. 11, Amer. Math. Soc., Providence, R.I., 1968.
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Complementation in Krein spaces, Trans. Amer. Math. Soc. 305 (1988),
277-291. [dB5]
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Krein spaces of analytic functions, J. Functional Anal. 81 (1988), 219-259.
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L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, in: Perturbation Theory and its Applications in Quantum Mechanics (C. Wilcox, ed.), Wiley, New York, 1966, pp. 295-392.
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L. de Branges and L. Shulman, Perturbations of unitary transformations, J. Math. Anal. Appl. 23 (1968), 294-326. A. Dijksma, H. Langer and H. de Snoo, Characteristic functions of unitary operator colligations in irk spaces, in: Operator Theory and Systems (H. Bart, I. Gohberg and M.A. Kaashoek, eds.), Operator Theory: Advances and Applications OT19, Birkhauser Verlag, Basel, 1986, pp. 125-194.
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H. Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, in: Orthogonal MatrixValued Polynomials and Applications (I. Gohberg, ed.), Operator Theory: Advances and Applications OT34, Birkhauser Verlag, Basel, 1988, pp. 79-135.
[D2]
, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Regional Conference Series in Mathematics, No. 71, Amer. Math. Soc., Providence, 1989.
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-, On reproducing kernel spaces, J unitary matrix functions, interpolation and displacement rank, in: The Gohberg Anniversary Collection (H. Dym, S. Goldberg, M.A. Kaashoek and P. Lancaster, eds.), Operator Theory: Advances and Applications OT41, Birkhauser Verlag, Basel, 1989, pp. 173-239.
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-, On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy, Integral Equations and Operator Theory 12 (1989), 757-812. P.A. Fillmore and J.P. Williams, On operator ranges, Adv. Math. 7 (1971), 254-281.
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I.S. Iohvidov, M.G. Krein and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Mathematische Forschung, Vol. 9, Akademie-Verlag, Berlin, 1982.
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H. Lev-Ari and T. Kailath, Triangular factorization of structured Hermitian matrices, in: I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications OT18 (1986), 301-324.
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A.A. Nudelman, Lecture at Workshop on Operator Theory and Complex Analysis, Sapporo, Japan, June 1991. M. Rosenblum, A corona theorem for countably many functions, Integral Equations and Operator Theory 3 (1980), 125-137. L. Schwartz, Sous espaces hilbertiens d'espaces vectoriels topologiques et noyaux associ s (noyaux reproduisants), J. Analyse Math. 13 (1964), 115-256.
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D. Alpay Department of Mathematics Ben Gurion University of the Negev Beer Sheva 84105, Israel
MSC: Primary 47A57, Secondary 47A56
H. Dym Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot 76100, Israel
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
78
THE CENTRAL METHOD FOR POSITIVE SEMI-DEFINITE, CONTRACTIVE AND STRONG PARROTT TYPE COMPLETION PROBLEMS Mihaly Bakonyi and Hugo J. Woerdeman In this paper we obtain a new linear fractional parametrization for the set of all positive semi-definite completions of a generalized banded partial operator matrix. As applications we obtain a cascade transform parametrization for the set of all contractive completions of a triangular partial operator matrix satisfying possibly an extra linear constraint (thus extending the results on the Strong Parrott problem). In each of the problems also a maximum entropy principle appears.
1. Introduction. In this paper we establish a new parametrization for the set of all positive semi-definite completions of a given "generalized banded" operator matrix. Before we elaborate on this problem we start with an important application, namely a generalization of the Strong Parrott problem introduced by C. Foia§ and A. Tannenbaum. It concerns the following. For 1 < i < j < n let Bi3 : fl3 --+ Ki be given bounded linear operators acting between Hilbert spaces. Further, let also be given the operators T1
S1
(1.1)
S=
7-l -+ ® 17-11.
: 7-l -> (Dn 11Ci, T Sn
T.
We want to find contractive completions of the following problem: B11
B12
"'
Bln
B22
...
B2n
7
..
Bnn
Sl
Sn
i.e., we want to find Bi 1 < j < i < n, so that B = (Bi;); =1 is a contraction satisfying the linear constraint BS = T. The introduction of the Strong Parrott problem was a consequence of questions arising in the theory of contractive intertwining dilations (see, e.g., the recent book by C. FoiaM and A.E. Frazho [10]).
Bakonyi and Woerdeman
79
For the problem (1.2) we derive neccessary and sufficient conditions for the existence of a contractive solution. In case these conditions are met we build a so-called "central completion", a solution with several distingueshed properties. From the central completion we construct a cascade transform parametrization for the set of all solutions. As we mentioned before the above results appear as application of our results on positive semi-definite completions. The (strictly) positive definite completion problem is a well studied subject. The first results in this domain were by H. Dym and I. Gohberg in [8]. These results were in many ways generalized by H. Dym and I. Gohberg (see the references in [12]) and I. Gohberg, M.A. Kaashoek and H.J. Woerdeman in [12], [13], [14]. A complete Schur analysis of positive semi-definite operator matrices was given by T. Constantinescu in [7], and these results were later used by Gr. Arsene, Z. Ceau§escu and T. Constantinescu in [1] in positive semi-definite completion problems. An analysis of positive semidefinite completions in the classes of so-called U'DU- and L*DL-factorable positive semi-definite operator matrices was recently given by M. Bakonyi and H.J. Woerdeman in [3]. The methods in [3] cover the finite dimensional case, but do not extend to the general case described in this paper. In the study of positive semi-definite "generalized banded" completions, we first develop some distingueshed properties which uniquely characterize a so called central
completion, a notion that appeared in different settings and with different names in [8], [1], [12] and [3]. Next we present a result on which the rest of the paper is based, namely a linear fractional transform parametrization for the set of all solutions. The coefficients of the transformation are obtained from the Cholesky factorizations of the central completion. This is a generalization of results in [12], where the positive definite case was considered, and of results in [3]. Our paper is organized as follows. In Section 2 we treat positive semi-definite
completions and in Section 3 contractive completions. Section 4 is dedicated to the study of a generalized Strong Parrott problem.
2. Positive Semi-Definite Completions. Consider the following 3 x 3 problem: All
A12
A21
A22 A23 A32
?
0,
A33
where
C All A121>0 (A22 A21
A22 J
A32
A33
By this we mean that we want to find the (1,3) entry A13 of the operator matrix in (2.1) such that with this choice (and with A31 = A13) we obtain a positive semi-definite 3 x 3 operator matrix. Note that the positivity of the 2 x 2 operator matrices in (2.2) implies that A12 = A1i2G1A22 A23 =
G2A313 2
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80
where G1 : R(A22) -* 7Z(A11) and G2 : R(A33) -+ R(A22) are contractions. For a linear
operator A we denote by R(A) its range and by R(A) the closure of its range, and if K A > 0 then A' /2 is its square root with Al/2 > 0. Let also for a contraction G : G
denote DG = (Ic - G`G)1/2 : G -* G and DG = R(Dc). It was proved in [1] that there exists a one-to-one correspondence between the set of all positive semi-definite completions of (2.1) and the set of all contractions G : Dc, -> Dc, via (2.3)
A,3 = A'11/2(G,G2 + Dc,GDcz)A1/2. 33
With the choice G = 0 we obtain the particular completion A13 = Al11/2G1G2A332.
(2.4)
We shall call this the central completion of (2.1), referring to the fact that in the operator ball in which A13 lies (namely the one described by (2.3)) we choose the centre. If F is a positive semi-definite operator matrix it is known that there exist an upper
triangular operator matrix V and a lower triangular matrix W such that F = V*V = W*W.
(2.5)
The factorizations (2.5) are called lower-upper and upper-lower Cholesky factorizations,
respectively. Moreover if V and W are upper (lower) triangulars with F = V*V = W*W, then there exists block diagonal unitaries U : R(V) R(V) and U : R(W) -> R(W) with UV = V and UW = W. This implies that if F is a positive semi-definite n x n operator matrix, then the operators
Au(F) diag(V;V;)"
(2.6)
1
and (2.7)
AL(F) := diag(W;;Wjj),"_1
do not depend upon the particular choice of V and W in (2.5). Returning to our problem (2.1), if F is an arbitrary completion corresponding to the parameter G in (2.3) then F admits the factorization (2.5) with (2.8)
V=
Aii2
G1 A222
0
Dc1 A222
0
0
(G1G2 + DGt GDc, )A332 (Dc, G2 - G-1 GDG, )A332 Dc Dc, A33 2
and
W=
Dc.DciA;i2 0 (DGIGi - G2G'DG )Aii2 Dc;Azz2 (GzGi + Dc,G*Dc; )Aii2
GZA222
0 0 A132
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81
Further, using relations like G;(DG,.) C DG one easily obtains that 1 (V,) C A (Vii) and R(W,) C 1(Wii), for all i and j. The triangularity of V and W now yields R(V) = '&(A11 '/') E Dc, E DG, A(W) = DG. E DG; E 1.(A332)
(2.10)
One immediately sees from these equalities that when G = 0 the closures of the ranges of the Cholesky factors of the completion are as large as possible. R(V) with UW = V. Relation (2.5) implies the existence of a unitary U :7Z(W) A straightforward computation gives us the explicit expression of U, namely Dc; DG.
G1 Dc2 - Dc. GG2 G1 G2 + DG. GDG2
-G; Dc. Dc, DG2 - GiGG2 Dc, G2 - G;GDG2
(2.11)
-G'
-DGG2
DGDG
Note that the (3, 1) entry in U is zero if and only if G = 0. As it will turn out, this will be a characterization for the central completion, thus providing a generalization of the banded inverse characterization in the invertible case, discovered in [8]. We will state the result precisely in the n x n case. Before we can do this we have to recall the following.
We remind the reader of the Schur type structure of positive semi-definite matrices obatined in [7]: There exists an one-to-one correspondence between the set of positive semi-definite matrices (Aij); j=1 with fixed block diagonal entries and the set of all upper
i = 1, ..., n, triangular families of contractions G = {r, }1<1<; 0 and G = {f1 }1
(2.12)
V
1IZ(Aii) - R(A11) E (Ek=2V'lk)
upper triangluar and (2.13)
W : E 1R(Aii)
Ek=1Dr;,,
R(Ann)
lower triangular. The operators V and W have dense range, and their block diagonal entries are given by (2.14)
Vii = Dr ... Dr,_,,.A1ii 2
and (2.15)
Wii = Dr D. ... Dr,_l , A;12.
Let us now consider the n x n generalized banded positive semi-definite completion problem. Recall that S C n x n (a = {1, ..., n}) is called a generalized banded pattern
if (1) (i,i) E S, i = 1,...,n; (2) if (i, j) E S then (j,i) E S; and (3) (i, j) E S and
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82
i < p, q < j imply (p, q) E S. The problem is the following. Given are Aii : 7{i --+ Ri for (i, j) in a prescribed generalized banded pattern S. We want to find Ai (i, j) E
(n x n)\S such that A = (Aii);,=1 > 0. Such an operator matrix A will be called a positive semi-definite completion of the band {Aii, (i, j) E S}. It is known (see [8]) that a positive definite completion of {Aii, (i, j) E S} exists if and only if (2.16)
(Aii)i,jEJ > 0
for all J C n with J x J C S. When {Aii, (i, j) E S} verifies condition (2.16) we shall call this band positive semi-definite. In [1] a parametrization was given for the set of all positive semi-definite completions of {Aii, (i, j) E S} as follows. This parametrization is based on the result in [7] quoted above and the fact that making a completion of {Aii, (i, j) E S} precisely corresponds
to choosing the parameters {f,,,1 < i < j < n, (i, j) V S}. Thus there exists an one-to-one correspondence between the set of all positive semi-definite completions of
{Aii, (i, j) E S} and the completions of {I'ii,1 < i < j < n, (i, j) E S} to a (Aii); 1 choice triangle. This parametrization is recursive in nature, because of the way the choice triangles are constructed. The completion corresponding to the choice Iii = 0 whenever 1 < i < j < n with (i, j) V S is called the central completion of {Aii, (i, j) E S}. It shall be denoted by FF, where the subscript "c" stands for central. An alternative way to obtain the central completion is described below. For a given n x n positive generalized band {Ai (i, j) E S} one can proceed as follows: choose a position (io, jo) V S, io < jo, such that S U {(io, jo), (jo, io) } is also generalized banded. Choose Aio,i0 such that (Aii)°ii is the central completion of {Ai (i, j) E S and io < i, j < jo}. This is a 3 x 3 problem and Ai,,,i0 can be found via a formula as in (2.4). Proceed in the same way with the thus obtained partial matrix until all positions are filled. It turns out (see [1]) that the resulting positive semi-definite completion is the central completion F. Note that for (io, jo) V S, io < jo, the entry Ai,,,i0 only depends
upon {Aii, (i, j) E S and io < i, j < jo}. This implies that the submatrix of FF located in the rows and columns {k, k + 1, ..., 11 is precisely the central completion of { Aii, (i, j) E S fl { k, k + 1, 1} x { k, k + 1, ..., 111. This principle is referred to as the "inheritance principle". Our first result gives four equivalent conditions which characterize the central completion. This is a positive semi-definite operator analogue of Theorem 6.2 in [8]. THEOREM 2.1. Let S be generalized banded pattern and F a positive semi-definite completion of {Aii, (i, j) E S}. Let F = V'V = W'W be the lower-upper and upperlower Cholesky factorizations of F. Then the following are equivalent: (i) F is the central completion of {Aii, (i, j) E S}. (ii) Au(F) > Au(F) for all positive semi-definite completions P of {Aii, (i, j) E S};
(iii) AL(F) AL(P) for all positive semi-definite completions P of {Ai (i, j) E S}; (iv)
The unitary U : 7Z(W) -* 7Z(V) with UW = V verifies Uii = 0 for i > j, (i,7) V S.
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Note that the uniqueness of the central completion implies that Du(F) = Au(F) (or OL(F) = OL(F)) yields F = F. The maximality of 0u(F) (OL(F)) can be viewed as a maximum entropy principle (see, e.g., [6]). Proof. The equivalence of (i) and (ii) can be read off immediately from (2.14), and similarly the equivalence of (i) and (iii) can be read off immediately from (2.15).
We prove the equivalence if (i) and (iv) by induction on the number of missing entries in the pattern S. For the 3 x 3 problem (2.1), discussed at the beginning of this section, formula (2.11) proves immediately the equivalence. Let S C n x n be a generalized banded pattern and {Ai;, (i, j) E S} positive semidefinite. Let F, denote the central completion of {Ai (i, j) E S}, and let V. and WW be upper and lower triangular operator matrices such that F, = V,"V, = W: W,.
(2.17)
Consider the unitary operator matrix U : 1Z(W,) -> 1Z(V,) so that UW. = V. Let S n-1 (F;.,)obtained from F, by denote the pattern S = S fl (n - 1 x n - 1), and P = compressing its last two rows and columns. So, F;; = (F,)ij for i,j < n - 1, Pi,n-1
= P.-l,i = ((F,)i,n-1 (F,);n), i < n - 1,
and
Fn_1 n_1
(F,)n_ (F'c)n,n-1
(F,)nn
Consider the data {Fib, (i, j) E S}. From the way the central completion is defined one sees that F(= F,) is the central completion of {F13, (i, j) E S}. Now, in the same way, consider the operator matrices U = (U13)i)..1, V = (V )i 7-1 and W = j-1 obtained by the compression of the last two rows and columns of U, VV and W,, respectively. We obtain by the induction hypothesis that (Jr, = 0 for (i, j) V S with
i > j. Thus it remains to show that U, = 0 for j with (n, j) V S and (n - 1, j) E S. For this purpose let -y = min{j, (n, j) E S} and consider the decomposition
E11E12 E13
U=
E21 E31
E22 E32
E23 E33
with E11 = (Ui;)i i1,, E22 = (U1)°-try and E33 = Unn. Consider also the corresponding decomposition of F, _ Again we have that F, is also the central completion (0ii)3.
of O11
-01z
021
022
? 023
?
032
033
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84
But then from the 3 x 3 case we obtain that E13 = 0 and, consequently, U; = 0 for j < y - 1, proving (iii). Implication (iii) --+ (i) can be proved by the same type of induction process. One needs to use the observation that if S1 and S2 are two generalized banded patterns and F is the central completion of both {A,3, (i, j) E S1} and {Ai (i, j) E S2}, then F is the central completion of {A;, (i, j) E S1 fl S2}. We omit the details. THEOREM 2.2. Let S C n x n be a generalized banded pattern and {Ai (i, j) E S} be positive semi-definite. Let FF denote the central completion of {Ai (i, j) E S}, and VV and We be upper and lower triangular operator matrices such that
Fr =VcVC=W:W,,.
(2.18)
Further, let U : R(W,,) -' 1.(V,,) be the unitary operator matrix so that (2.19)
UWW = Vt,.
Then each positive semi-definite completion of {Ai (i, j) E S} is of the form (2.20)
T(G) = VV (I + UG)'-1(I - G'G)(I + UG)-1V,, = W,, (I + GU)-'(I - GG')(I + GU)*-1Wc,
where G = (Gi;); i=1 : R(Vc) -4 R(W,,) is a contraction with Gi, = 0 whenever i > j or (i, j) E S. Moreover, the correspondence between the set of all positive semi-definite completions and all such contractions G is one-to-one. The decompositions of R(V) and R(W) are given by (2.21)
R(V) = 7Z(Ail) e
(Ek=2Dr,k)
and (2.22)
R(W) = Ek=i Drkn E R(A,,,,).
Before starting the proof, we need additional results. PROPOSITION 2.3. Let S C n x n be a generalized banded pattern and {Ai;, (i, j) E S} positive semi-definite. Let F. denote the central completion and F an arbitrary positive semi-definite completion of {Ai;, (i, j) E S}. Then (2.23)
R(F1/2) C R(FF /2).
We remark first that if 0 is an operator on Tl and A = 0'0 then there exists a unitary U on h such that A'12 = U4, and thus R(4') = R(A'/2). Thus (2.23) is equivalent with the fact that if F,, = E,,EC and F = E'E with E,, and E upper (lower)
triangular then R(E') 9 R(ED).
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85
Proof. We start the proof with the 3 x 3 problem (2.1). Let F be the positive semi-definite completion of (2.1) corresponding to the parameter G in (2.3). Then, as we have already seen, F = V*V where V is given by (2.8). Thus 0
DGi G
0
I -GIG
0
0
DG
1
(2.24)
V=
V
which yields 7Z(V*) C 7Z(V, ). The result now follows from the remark preceeding the proof.
Consider now a given generalized banded pattern S C n x n. We prove our result by induction assuming the statement is correct for all generalized banded patterns S
which have S as a proper subset. The case S = n x n\{(1,n),(n,1)} reduces to the 3 x 3 problem. Let (io, jo) S, io < jo, be such that S = S U {(io,jo),(jo,io)} is also generalized banded. Let {Aii,(i,j) E S}, F. and F be as in the statement of the proposition. Consider the partial matrix {Bi (i, j) E S}, where Bi, = Fi, for (i, j) E S. Let FF denote the central completion of this latter partial matrix. By the induction hypothesis, since clearly F is a completion of {Bii, (i, j) E S} we have that (2.25)
7Z(F'/2) C 7Z(F,/2).
Observe that the matrices P. and FF differ only on the positions (i, j) and (j, i), where 1 < i < io and jo < j < n. Moreover, defining _S = (n x n)\{(i, j), (j,i), 1 < i <_ io, jo <
j < n} and the partial matrix {Cii, (i, j) E S}, where Ci, = (F),, for (i, j) E S, we have that FF is also the central completion of this latter partial matrix. We can now use the 3 x 3 case to conclude that (2.26)
7Z(F'1/2) c R(FF/2),
since F. can be viewed as a completion of {Ci (i, j) E S}. Now (2.23) is a consequence of (2.25) and (2.26), and the remark preceeding the proof.
PROPOSITION 2.4. Let S C n x n be a generalized banded pattern and {Ai (i, j) E S} positive semi-definite. Let FF denote the central completion and F an arbitrary positive semi-definite completion of (Aii, (i, j) E S), and let (2.27)
Fc=V,*Vc=W,,WC
be lower-upper and upper-lower Cholesky factorizations of F,,. Then, if F-Fc = Sl*+Sl, where Sl = (Slii) j-i with Slii = 0 whenever i > j or (i, j) E S, there exists an operator
matrix Q = (Qii);i_i : 7Z(Vc) -. 1Z(WW), with (2.28)
S2=W'-QV,
and Qii = 0 whenever i > j or (i, j) E S.
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86
Proof. We prove the proposition by induction in a similar way as Proposition 2.3. The 3 x 3 case is straightforward to check. (Using (2.3), (2.8) and (2.9), we obtain that only the (1, 3) entry of Q is nonzero, and equals G.) Consider an arbitrary generalized banded pattern S C n x n and assume that the
proposition is true for all generalized banded patterns S which have S as a proper subset. The case S = (n x n)\{(1, n), (n, 1)} reduces to the 3 x 3 problem. Let (io, jo) 0 S, io < jo, be such that S = S U {(io, jo), (jo, io)} is also generalized banded.
Let {Ai (i, j) E S}, FF, F, WW and VV be as in the statement of the proposition. Consider the partial matrix {Bi (i, j) E S}, where Bid = Fi; for (i, j) E S. Let F denote the central completion of this latter partial matrix. By the induction hypothesis, (2.29)
Q = WNV,
where S2 and Q are upper triangular with support outside the band S, 12* + S2 = WW We are lower-upper and upper-lower R(k), and F. = F -,P,, Q : Cholesky factorizations of F. By Proposition 2.3 and the remark preceeding proof of Proposition 2.3 we have that R(Vc*) C R(VV) and R(W') C_ R(W' ). But this yields that there exists an upper triangular a and a lower triangular /3 such that VV = aVV and We = /9Wc. Now, taking Q1 = Q*Qa we obtain from (2.29) that (2.30)
=W:Q1VC
and clearly Q1 is upper triangular with support outside S. As in the proof of Proposition 2.3, F. is also the central completion of the partial matrix {Ci (i, j) E S}, where S = (n x n) \{(i, j), (j, i), 1 < i < io, jo < j < n} and Ci2 = (F)ib for (i, j) E S. By the 3 x 3 case we may conclude that (2.31)
fl = W, Q2 V,
where fl and Q2 are upper triangular with support outside the band S,
+ 1 = F-F,,
Q2 : R(K) - R(Wc) Since F - Fc = (F - F) + (Fc - Fc), we have that fZ = S2 + Il, and thus (2.30) and (2.31) imply the desired relation (2.28) with Q = Q1 + Q2, which clearly is of the desired form. 0 We are now ready to prove the parametrization result. Proof of Theorem 2.2. Write FF = C + C*, where C is upper triangular with Cii = 1/2Fii, i = 1, ..., n, and define for a contraction G = (Gij)'i,j=1 : R(WO with Gi,, = 0 whenever i > j or (i, j) E S, (2.32)
,C(G) = C - WW (I + GU)-1GV,.
Since Ui,, = 0 for (i, j) 0 S with i > j, one easily sees that GU is strictly upper triangular and so (I + GU)-1 exists and is upper triangular. Since WW and Vc are both also upper triangular one readily obtains that (2.33)
(,C(G))i, = Ci (i, j) E S.
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Further, using (2.32) and the unitarity of U it is straightforward to check that £(G) + ,C(G)* = T(G). This together with (2.33) yields that T (G) is a completion of {A1,, (i, j) E S} and since IIGII < 1 the operator matrix T(G) is positive semi-definite. Assume that for two contractions G1 and G2 (of the required form) we have that T(G1) = T(G2). Then also,C(G1) = .C(G2) and since WW and V,* are injectiveon R(WW) and R(VV), respectively, equation (2.32) implies that (I +G1U)-'G1 = (I +G2U)-1G2. Thus G1(I + UG2) = (I + G1U)G2 which yields G1 = G2. Conversely, let F be an arbitrary positive semi-definite completion of {Ai (i, j) E S}. Consider 11 = (11i;)i j=1 such that 1l = 0 whenever i < j or (i, j) E S, and Fe-F = S1+f2*. Then by Proposition 2.4 there exists an operator Q = (Qi,) ' : 7Z(WW) -' R(V,) with Qij = 0 whenever i > j or (i,j) S and Q = WWQVC. Since UQ is strictly upper triangular, we can define
G = Q(I - UQ)-1, which will give that S2 = WW (I + GU)-1GVV. Since F = FF - 11 - 52*, and taking into account (2.32) we obtain that F = T(G). Since F = T(G) is positive semi-definite, the relation (2.20) implies that G is a contraction. This finishes our proof. fl
3. Contractive Completions. Consider the following 2 x 2 problem: B22/)
II
<_ 1
where
Bll )II:1,II( B21 B22)II<1.
II1 B21
Note that the contractivity of the latter operator matrices implies that B11 = G1 DB21, B22 = DB2, G2
where G1 and G2 are contractions. It was proved in [2] and [9] that there exists a one-to-one correspondence between the set of all contractive completions of (3.1) and the set of all contractions G : DG2 --' DGi given by (3.2)
B12 = -G1B21G2 + DG GDG2.
With the choice G = 0 we obtain the particular completion B12 = -G1 B21 G2. We shall call this the central completion of (3.1).
Let {Bi 1 < j < i < n} be a n x n contractive triangle, i.e., let Bi3 1C; -+ 'Hi, 1 < j < i < n, be operators acting between Hilbert spaces with the property that :
II(Bii) n=1II <_ 1,p = 1,...n.
In order to make a contractive completion one can proceed as follows: choose a position (io, jo) with io = jo-1, and choose Bio,h such that (Bij)' 101 1 is the central completion
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88
of {B;;, i > io, j < jo} as in the 2 x 2 case. Proceed in the same way with the thus obtained partial matrix (some compressing of columns and rows is needed) until all positions are filled. We shall refer to FF as the central completion of {B; (i, j) E T}. THEOREM 3.1. Let {B;;,1 < j < i < n} be a contractive triangle. Let Fc denote the central completion of {B23, 1 < j < i < n} and let 40c and '1c be upper and lower triangular operator matrices such that (3.3)
ID c=I-FFF:.
0c-
Further, let w1 : DF -4 7Z(4),) and W2 : DF. -+ 1 (Wc) be unitary operator matrices so that (Dc=w1 DF,1kc= W2DF.,
(3.4)
and put TT = -w1 FF w2.
(3.5)
Then each contractive completion of {B;;,1 < j < i < n} is of the form S(G) = Fc - 'TG(I + TcG)-14)c.
(3.6)
where G = (G13)n.,-1 : R(Oc) --* R(Tc) is a contraction with G;; = 0 whenever (i, j)
T. Moreover, the correspondence between the set of all positive semi-definite completions
and all such contractions G is one-to-one. Furthermore, S(G) is isometric (co-isometric, unitary) if and only if G is. The decompositions of R(40c) and 1 (Wc) are simply given by R(oc) = ® 1R((41c);,),R(Tc) = ® 1R((`Pc)i=).
Proof. We apply Theorem 2.2 using the correspondence (3.7)
I
) > 0 if and only if IIBII :5 1.
B
1
Consider the (n+n) x (n+n) positive semi-definite band which one obtains by embedding
the contractive triangle {B;;,1 < j < i < n} in a large matrix via (3.7). It is easy to check that when applying Theorem 2.2 on this (n + n) x (n + n) positive semi-definite band one obtains V` =
(0
W` -
-t'
F.
U
r.
't
I
(use FF DF. = DF. FF). It follows now from Theorem 2.1 that (rr);j = 0 for i > j. Further, it is easy to compute that (3.8)
((0 G
T1
I
0
0
))
=
(
I
S(G)`
S(G)
Q(G) S(G)
Q(G)) _- ( S(G)-
I
Bakonyi and Woerdeman
89
where we have
I = Q(G) = S(G)*S(G) + I (I + TG)'-' (I - G'G)(I + -rG)-14,
(3.9)
and (3.10)
I = Q(G) = S(G)S(G)* + WY,(I + GTR)-'(I - GG`)(I + GTR)'-'',
We obtain the first part of the theorem from (3.8) and Theorem 2.2. From relation (3.9) one immediately sees that G is an isometry if and only if S(G) is. Similarly, one obtains from (3.10) that G is a co-isometry if and only if S(G) is. This proves the last statement in the theorem. 0 The existence of an isometric (co-isometric, unitary) completion is reduced to the existence of a strictly upper triangular isometry (co-isometry, unitary) acting between the closures of the ranges of (D, and IF,. Taking into account the specific structures of ,D, and ID, one recovers the characterizations of existence of such completions given in [5] and [1] (see also [3]).
REMARK 3.2. We can apply Theorem 2.1 to characterize the central completion.
We first mention that for an arbitrary completion F of {Bi 1 < j _< i < n} one can define 4D, 'Y and r analogously as in (3.3), (3.4), and (3.5). The equivalence of (i), (ii)
and (iii) in Theorem 2.1 implies that the central completion is characterized by the maximality of 1 or diag(W;i'Yii),"_1. This is a so-called "maximum entropy principle". From the equivalence of (i) and (iv) in Theorem 2.1 one also easily obtains that the uppertriangularity of r characterizes the central completion.
4. Linearly Constrained Contractive Completions. We return to the problem (1.2). The next lemma will reduce this linearly constrained contractive completion problem to a positive semi-definite completion problem. The lemma is a slight variation of an observation by D. Timotin [15]. 1C be linear operators acting LEMMA 4.1. Let B : f -p AC, S : Gj -* 7-l and T between Hilbert spaces. Then JIBIM < 1 and BS = T if and only if I
S
B'
S' S'S T* B T I
>0.
Proof. The operator matrix (4.1) is positive semi-definite if and only if (4.2)
(T
I*)-(B)(S B')=I T OBST I-BB'»>0
and this latter inequality is satisfied if and only if JIBII < 1 and BS = T.
THEOREM 4.2. Let Bi, : 7i -' 1Ci, 1 < i < j < n, Si : ? -+ 7{i, i = 1,...,n and T; : 7-l -+ 1C; be given linear operators acting between Hilbert spaces and S and T be as
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90
in (1.1). Then there exist contractive completions B of {Bid, 1 < i < j < n} satisfying the linear constraint BS = T if and only if (4.3)
\
S'S - S(i)`S(i) T(i)' - S(i)`B(t)* T(i) - B(i)S(i) I - B(i)B(i)`
/
?0
for i = 1,.., n,where B1,
(4.4)
B(1) = Bii
.
.
.
.
.
.
.
Bln
Tl
Si
T(,)
S(i) .
Bin
Ti
Sn
for i = 1, ..., n.
Proof. By Lemma 4.1 there exists a contractive completion B of {Bi 1 < i < j < n} satisfying the linear constrained BS = T if and only if there exists a positive semi-definite completion of the partial matrix
(4.5)
I
0
0
1
...
0
S1
B11
0
S2
Biz
Bin B2n
..
B,*,
?
?
0
0
I
Sn
Si
S2
Sn
S`S
T1
TZ
...
T,
B11
B12
T1
I
0
...
0
?
B21
B1n Bzn
T2
0
I
...
0
Bnn
Tn
0
0
...
I
?
?
...
...
Bnn
As it is known, the existence of a positive semi-definite completion of (4.5) is equivalent to the positive semi-definiteness of the principal submatrices of (4.5) formed with known entries. This latter condition is equivalent with (4.3). Let us examine the 2 x 2 case a little further, i.e.,
(
B11
B1z
S1
?
Bzz
Sz
TT1z
The necessary and sufficient conditions (4.3) for this case reduce to (4.7)
B11S1 + B12Sz = T1, I I ( B11
B12 ) 11 < 1
and
(4.8)
(
1 - B12B12 - B22B22
Sz - Bi2T1 - B;2T2
S2 - T; B1z - T2 *B22
Si S1 + S2 *S2 - T; T1 - T2 .T2
> 0.
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91
Assume that (4.7) and (4.8) are satisfied. Similar to Section 3, let G1 : 9-11 -> DB12 and G2 : DB12 --i )C2 be contractions such that (4.9)
B11 = DB12G1, B22 = G2DB12
Any solution of the constrained problem (4.6) is in particular a solution of the unconstrained problem (the lower triangular analogue (3.1)), and therefore we must have that (use the analogue of (3.2)) (4.10)
B21 = -G2B12G1 + Dc,rDc1,
where IF : DG, -9 DGz is some contraction. The equation B21 S1 + B22 S2 = T2 implies
that r is uniquely defined on 1Z(DG,S1) by (4.11)
DG,rDG, S1 := T2 - B22S2 + G2B12G1S1.
We define 1'0 : DG, -- DGz to be the contraction defined on 1Z(DG, S1) as above, and 0 on the orthogonal complement, i.e., (4.12)
ro 1 Dc, e R(DG,S1) = 0
We let BZ°I denote the corresponding choice for B21, that is, (4.13)
Biil = -G2Bi2G1 + DG, roDG, .
We shall refer to B11
(4.14)
B12
B210) 0 B22
as the central completion of problem (4.6). In then x n problem (1.2) (assuming conditions (4.3) are met) we construct step by step the central completion of (1.2) as follows. Start by making the central completion of the 2 x 2 problem Si B11
(4.15)
(
B12
B22
...
...
S2
B1n
B2n
)
_ = (
T1
T2
)
Sn
and obtain in this way B2°). Continue by induction and obtain at step p, 1 < p < n - 1, BP°l
BP(OP)_1 by taking the central completion of the 2 x 2 problem Si
,P
BI,P-1
(4.16) p
-
P
... B ...
1
B1P Sp_1
(O)1'
?
P -1
BP_1,P Bpp
Sp
Tp_1
Tp
Sn I
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92
The final-result Bo of this process is the central completion of the problem (1.2). LEMMA 4.3. Let Bo be a contractive completion of (1.2). Then Bo is the central completion of (1.2) if and only if S
I
Bo
S` S'S T'
(4.17)
T
Bo
I
is the central completion of the positive semi-definite completion problem (4.5). Proof. By the inheritance principle and the way the central completion is defined it suffices to prove the lemma in the 2 x 2 case. Take an arbitrary contractive completion B of (4.6), corresponding to the parameter 1, in (4.10), say. The lower-upper Cholesky factorization of the corresponding positive semi-definite completion problem is given by S
I
(4.18)
V`V =
B*
S' S'S T'
B
T
I
where (4.19)
I S B'
V=
0
0
0
0 V
0
and 4D is lower triangular such that I - BB' = 'I (4.20)
.
It is straightforward to check that
0 DB12DG, = ( DG.rG; DG2Dr. \ -G2B12DG,' -
Since for F = Fo the operator D. is maximal among all t satisfying (4.11), the lemma follows from the equivalence of (i) and (ii) in Theorem 2.1. 0 THEOREM 4.4. Let Bo be the central completion of the linearly constrained contractive completion problem (1.2) (for which the conditions (4.3) are satisfied). Let
p : f1 ®rl2 , 1 ((S-S - T`T)1/2) be such that (4.21)
(S"S - T"T)1/2p = S'DBo,
and T and 4) lower triangulars such that (4.22)
VIF=I - p'p - BoBo
and (4.23)
W =I - BoBo.
Consider the contraction w1 : DBo -+ 1.(') and the unitary w2 : 1Z(V) -' DBo with the properties (4.24)
%F = w1DB.
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93
and (4.25)
4) = DBOWZ
Finally, define
r=
(4.26)
Then there exists an one-to-one correspondence between the set of all contractive solutions of the problem (1.2) and the set of all strictly lower triangular contractions 7Z((D') given by
G : 1Z(W)
V(G) = Bo - I(I + Gr)-1G%k
(4.27)
Moreover, V(G) is a co-isometry if and only if G is a co-isometry and V(G) is an isometry if and only if S'S = T'T and G is an isometry. and 1 ('Y) are simply given by
The decompositions of 7
IZ(41') = ®t iTZ(D«),IZ(IF) = ®
Proof. We shall obtain our results by applying Theorem 2.2 for the positive semidefinite completion problem (4.5). Straightforward computation yield that I S Bo
V=
(4.28)
0
0
0
0
0
4D'
and 0
0
W. =
(4.29)
(S'S - T'T)1/2 0 I T
%pp
Bo
We remark here that the relation
S'S - T'T = S'DBOS> S'DB0S
(4.30)
gives the existence of the contraction p with (4.21). Now we have to determine the unitary U = (U;, ); j=1 so that UW, = V. Note that the existence of w1 and w2 is assured by the relations (4.22) and (4.23). An immediate computation shows that
U=
I
qj '
p'
Bo
0
0
0
-w2 Bowi -w2 Bowi V where
Wl
) is unitary with
()=
(').w_' )
DBE
l
-
Bakonyi and Woerdeman
94
Substituting these data in the first equality of (2.20) gives
(4.31)
T(
0
0 G*
0
0
0
0
0
0
) =
I
S
S*
S*S
T*
V(G)
T
Q(G)
V(G)'
where V(G) is given by (4.27) and (4.32)
I = Q(G) = V(G)V(G)* + 4i(I + Gr)-' (I - GG*)(I + Gr)*-'4?*
The first part of the theorem now follows from (4.31) and Lemma 4.1. Further, (4.32) implies that V(G) is a co-isometry if and only if G is. If the contractive solution V(G) to the constrained problem (1.2) is isometric, then clearly we must have that S*S = T*T and thus p = 0. In this case,
t 'D
0
0
0
0
0
WW =
(4.33)
Bo T I Using the second inequality in (2.20) in this special case, we obtain that (4.34)
T(
0
0
G`
0
0
0
0
0
0
)=
Q(G)
S
S* V(G)
S*S
V(G)* T*
T
I
where (4.35)
I = Q(G) = V(G)*V(G) +T*(I +Gr)-'(I - G*G)(I +Gr)*-'1Y.
Relation (4.35) implies that when S*S = T*T, the spaces Dv(G) and DG have the same dimensions and thus V(G) is isometric if and only if G is. This finishes the proof. In the 2 x 2 case another parametrization was derived in [4]. REMARK 4.5. By Theorem 4.4 we can reduce the existence of a co-isometric completion of the problem (1.2) to the existence of a strictly lower triangular co-isometry acting between R(') and Also, when S*S = T*T, the existence of a isometric completion of the problem (1.2) reduces to the existence of a strictly lower triangular isometry acting between R(1Y) and R(V). REMARK 4.6. There exists a unique solution to (1.2) if and only if 0 is the only strictly lower triangular contraction acting 7Z(1Y) -+ R(4)*). This can be translated in the following. If io denotes the minimal index for which'ioi,, 0, then there exists a unique solution if and only if 4ikk = 0 for k = io + 1, ..., n. REMARK 4.7. As in Remark 3.2 the upper triangularity of r characterizes the
central completion. For this one can simply use Theorem 2.1 and Lemma 4.3. Also the maximality of diag(4)ii4?;i) 1 or diag(W 1Yii) 1 characterizes the central completion (a maximum entropy principle). For a different analysis in the 2 x 2 case we refer to [4].
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95
REFERENCES [1] Gr. Arsene, Z. Ceau§escu, and T. Constantinescu. Schur Analysis of Some Completion Problems. Linear Algebra and its Applications. 109: 1-36, 1988. [2] Gr. Arsene and A. Gheondea. Completing Matrix Contractions. J. Operator Theory. 7 179-189, 1982.
[3] M. Bakonyi and H.J. Woerdeman. Positive Semi-Definite and Contractive Completions of Operator Matrices, submitted.
[4] M. Bakonyi and H.J. Woerdeman. On the Strong Parrott Completion Problem, to appear in Proceedings of the AMS.
[5] J.A. Ball and I. Gohberg. Classification of Shift Invariant Subspaces of Matrices With Hermitian Form and Completion of Matrices. Operator Theory: Adv. Appl. 19: 23-85, 1986. [6] J.P. Burg, Maximum Entropy Spectral Analysis, Doctoral Dissertation, Department of Geophysics, Stanford University, 1975. [7] T. Constantinescu, A Schur Analysis of Positive Block Matrices. in: I. Schur Methods in Operator Theory and Signal Processing (Ed. I. Gohberg). Operator Theory: Advances and Applications 18, Birkhauser Verlag, 1986, 191-206. [8] H. Dym and I. Gohberg. Extensions of Band Matrices with Band Inverses. Linear Algebra Appl. 36: 1-24, 1981. [9] C. Davis, W.M. Kahan, and H.F. Weinberger. Norm Preserving Dilations and Their Applications to Optimal Error Bounds. SIAM J. Numer. Anal. 19: 444-469, 1982. [10] C. Foias and A. E. Frazho. The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, Vol. 44. Birkhauser, 1990. [11] C. Foias and A. Tannenbaum A Strong Parrott Theorem. Proceedings of the AMS 106: 777-784, 1989.
I. Gohberg, M. A. Kaashoek and H. J. Woerdeman. The Band Method For Positive and Contractive Extension Problems. J. Operator Theory 22: 109-155, 1989. [13] I. Gohberg, M. A. Kaashoek and H. J. Woerdeman. The Band Method For Positive and Contractive Extension Problems: an Alternative Version and New Applications. Integral Equations Operator Theory 12: 343-382, 1989. [14] I. Gohberg, M. A. Kaashoek and H. J. Woerdeman. A Maximum Entropy Priciple in the General Framework of the Band Method. J. Funct. Anal. 95: 231-254, 1991. [15] D. Timotin, A Note on Parrott's Strong Theorem, preprint. [12]
Department of Mathematics The College of William and Mary Williamsburg, Virginia 23187-8795
MSC: Primary 47A20, Secondary 47A65
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
96
INTERPOLATION BY RATIONAL MATRIX FUNCTIONS AND STABILITY OF FEEDBACK SYSTEMS: THE 4-BLOCK CASE
Joseph A. Ball and Marek Rakowski
Abstract. We consider the problem of constructing rational matrix functions which satisfy a set of finite order directional interpolation conditions on the left and right, as well as a collection of infinite order directional interpolation conditions on both sides. We set down consistency requirements for solutions to exist as well as a normalization procedure to make the conditions independent, and show how the general standard problem of H°° control fits into this framework. We also solve an inverse problem: given an admissible set of interpolation conditions, we characterize the collection of plants for which the associated H°°-control problem is equivalent to the prescribed interpolation problem.
Key words: Lumped and generic interpolation, homogeneous interpolation problem, stabilizing compensators, 4-block problem, H°° control.
Introduction The tangential (also called directional) interpolation problem for rational matrix functions (with or without an additional norm constraint) has attracted a lot of interest in the past few years (see [ABDS, BGR1-6, BH, BRan, D, FF, Ki]); much of this work was spurred on by the connections
with the original frequency domain approach to H°°-control theory (see [BGR4, BGR5, DGKF, Fr, Ki, V]). The simplest case of the interpolation problem is of the following sort. We are given points z1, , xM , zM in some subset o of the complex plane E, nonzero 1 x in row vectors x1, and 1 x n row vectors yl, , yM and seek a rational m x n matrix function W(z) analytic on a which satisfies (0.1)
x;W(z;) = y;,
i = 1,
, M.
In the two-sided version of the problem, we are given additional points wl, , wN in o, nonzero n x 1 column vectors ul, , uN and m x 1 column vectors v1, , vN and demand in addition that W satisfy W(wj)uj = vi, j = 1,... N. (0.2) If for some pair of indices (i, j) it happens that z; = w in applications a third type of interpolation condition arises x;W'(f+j)uj = Pij whenever z; = wj =: fj (0.3)
Ball and Rakowski
97
and where pij is a given number. By introducing matrices
xi
yl
,B-=-
,B+=
AC= 2M
XM
YM [Wi
C- _ [ul ... UN], C+ _ [VI ... VN], A, = and r = [yij]1
7ij =
Pij
ifzi=wj
(wi - zj)-lziuj if zi #1 wj
the conditions (0.1)-(0.3) can be written in the streamlined compact form
(0.1')
1: Resz=zo(z - AC)-lB+W(z) = -BzoEo
(0.2')
> Res,_,z.W(z)C-(z - A,)-l = C+ zoEo
(0.3')
ERest=zo(z- At)-1B+W(z)C-(z- A,)-l = r zoEo
where Resz=z,X(z) is the residue of the meromorphic matrix function X(z) at zo. Interpolation conditions of higher multiplicity can be encoded by allowing the matrices AC and A to have a more general Jordan form. This is the formalism developed in [BGR4]. If o is either the unit disk or the right half plane, one can also consider the problem with an additional norm constraint
(0.4)
sup 11W(z)11 < 1 zEo
to arrive at a matrix version of the classical Nevanlinna-Pick or Hermite-Fejer interpolation problem.
In [BGR4] a systematic analysis of the problem (0.1')-(0.3'), with or without the norm constraint (0.4), is presented, including realization formulas (in the sense of systems theory) for the set of all solutions.
In [BR5], in addition to the lumped interpolation conditions (0.1)-(0.3) or (0.1')-(0.3') there was imposed a generic (or infinite order) interpolation condition
(P+ W)(j)(zp) = P(j)(zo) for j = 0,1,2,... for some zo E v, or equivalently
(0.5)
P+(z)W(z) = P-(z) for all z,
Ball and Rakowski
98
where P+ and P_ are given polynomial matrices of respective sizes K x m and K x n. In [BR5] the theory developed in [BGR4] for the problem (0.1')-(0.3') was extended to handle the problem (0.1')-(0.3') together with (0.5) (with or without the additional norm constraint (0.4)), with the exception of the explicit realization formulas for the linear fractional parametrization of the set of all solutions. The features obtained include: an admissibility criterion for data sets (C+, C_, A,, AS, B+, B_, r, P+(z), P_(z)) which guarantees consistency and minimizes redundancy in the set of interpolation conditions (0.1')-(0.3') and (0.5), reduction of the construction of the linear fractional parametrizer of the set of all solutions to the solution of a related homogeneous interpolation problem, and a constructive procedure for solving this latter homogeneous interpolation problem.
The associated homogeneous interpolation problem is of the following form. In general, we let R denote the field of rational functions and R"' denotes the set of m x n matrices over 7Z (i.e. rational m x n matrix functions). For a C E, R(a) is the subring of R consisting of rational functions analytic on a, and Rmxn(a) is the set of m x n matrices over R(a). From the data set w = (C+, C_, A,,, & B+, B_, r, P+(z), P_(z)) we construct an R(a)-module S C R('n+n) xl given by
(z - Ax)-lx +
[h+(z), : x E En-,
h+ E R"nx'(a),h_ E Znxl(a) such thatt
ERes,_Z"(z-At)-1[B+ B_] [h+(z)1 = rx} J
zo Eo
(0.6)
fl {r E R(m+n)x1 : [P+(z) P_(z)]r(z) = 0}.
Then the module form of the homogeneous interpolation problem is to find a rational (m + n) x (m - K + n) matrix function O(z) such that
S = OR(m-K+n)xl(a). More concretely, the condition (0.7) can be viewed as prescribing the zeros and poles of 0 on a (including partial multiplicities) from the local Smith form together with some additional directional
information, as well as prescribing a left kernel polynomial for 0. If the norm constraint (0.4) is also part of the original (nonhomogeneous) interpolation problem, then 0 is required to satisfy additional conditions
(0.8a)
O(z)'(Ir (D -I1)O(z) = I.-K ®-In for z E 80r
Ball and Rakowski
99
and
O(z)*(I- ® -II)O(z) < I,.-K ® -In for z E o.
(0.8b)
The linear fractional parametrization of the set of all solutions takes the form
W = (OiiQi + O12Q2)(O21Q1 +
where Q1 E
IZ(m-K)xn(a) and Q2
O22Q2)-1
E Rnxn(o) are appropriate parameters, and where
0 = 011
012 with 011 E 022 ]
021
R"`x(,n-K).
For more complete details we refer to [BR5]. The purpose of this paper is to consider the interpolation problem (0.1')-(0.3') and (0.5)
with an additional right generic interpolation condition
(0.10)
W(z)Q-(z) = Q+(z)
where Q_ and Q+ are given matrix polynomials of respective sizes n x L and m x L. Here we obtain a canonical extension of all the results in [BR5] to handle the additional interpolation condition (0.10). In this more general setting the relevant analogue of S in (0.6) is the 7Z(o)-module S C 7Z(m+n)X1 given by
S+ Q+ izLx1
(0.11)
IQ-]
where S is given by (0.6). The associated homogeneous interpolation problem is to find a rational
(m + n) x (m - K + n) matrix function 0 such that
(0.12)
S=0
[R(m-K+n-L)x1(o) RLx1
]
If the norm constraint (0.4) is included in the original interpolation problem, then 0 is also required to satisfy
(0.13a)
O(z)*(Ir (D -In)O(z) = Im-K ® -In for Z E ao
(0.13b)
O(z)*(Im ® -I4)0(z) < I,.-K ®-In-L ® 0 for z E o.
Ball and Rakowski
100
and the linear fractional parametrization has the degenerate form W = [O11Q1 + 012Q2 O13][O21Q1 + 022Q2 023]-1
where Q1 E 7Z(m-K)x(n-L)(a) and Q2 E R(n-L)x(n-L)(,,) are appropriate parameters and where 0 = 011 012 013] with 01, E IZ-x(m_K) and 012 E R+nx(n-L) 1
021
022
023 J
In addition we show here how the standard problem of H°° control theory (see [Fr]) fits into this framework; this also extends the work in [BR5] to incorporate the extra condition (0.10).
Specifically, given a rational matrix function P = [P11 P22] representing the plant for an H°°
problem, we show that the set of interpolation conditions (0.1')-(0.3'), (0.5), (0.10) are satisfied by the closed loop transfer function P11 + P12K(I - P22K)-1P21 if and only if the closed loop system with compensator K is internally stable. The characterization is in terms of null-pole data of a related matrix function P (a partial inversion or chain formalism transform of P). For the case considered here the more general chain formalism transform introduced in [BHV] is required. We also solve an inverse problem of describing which plants are associated with a prescribed set of
interpolation conditions, and thereby establish an equivalence between interpolation and feedback stabilization. It turns out that the set of interpolation conditions (0.1')-(0.3') corresponds to the 1block case, the set (0.1')-(0.3'), (0.5) corresponds to the 2-block case, and the general interpolation problem (0.1')-(0.3'), (0.5), (0.10) corresponds to the general 4-block case in the HO° theory. A significant limitation of the interpolation approach in the early development of the H°° theory was the lack of an interpolation theory incorporating generic interpolation conditions (0.5) and (0.10)
in addition of (0.1')-(0.3') (but see [Hu]). Part of the motivation of [BR5] and this paper is to address this situation. The paper is organized as follows. Section 1 recalls preliminaries concerning null-pole structure from [BGR4] and [BR1]-[BR3] which will be needed in the sequel. Section 2 formulates and solves the homogeneous interpolation problem of the type (0.12). Section 3 sorts out admissibility conditions on an interpolation data set (C+, C_, A,, At, B+, B_, F, P+(z), P_(z), Q+(z), Q_(z)) to guarantee consistency and minimize redundancy. In Section 4 we obtain the linear fractional parametrization for the set of all solutions and Section 5 delineates the connection with the H°°control theory.
1. Preliminaries. In this paper, we will use the concepts of the null-pole structure of a rational matrix function W over a fixed subset a of the complex plane C. They have been developed for a regular rational matrix function (that is, a rational matrix function which is square and whose determinant does not vanish identically) in [BGR1]-[BGR3]; for a comprehensive treatment see [BGR4]. These concepts have been generalized to an arbitrary rational matrix function in [BR1]-[BR3] (see also [BR4]). We recall now the basic definitions and facts which will be used later.
Ball and Rakowski
101
Let R denote the field of scalar rational functions, and let R(a) be the subring of R formed by functions analytic on a. Let IZ'nx" denote the space of m x n rational matrix functions, and let R"`x" (a) be the subspace of l?. formed by functions analytic on a. An observable pair of matrices (Cr, of sizes m x n and nx x n,, respectively, is said to be a right Dole pair for W over a if
(i) a(A.,) C a, (ii) for every x E Cn.x1 there exists w E Rnxl(a) such that
C,r(z - A,r)-lx - W(z)
(1.1)
(iii) for every W E Rnxl(a) there exists x E In-x1 such that (1.1) holds. A right pole pair (C.x,
can also be related to a canonical set of right pole chains or pole functions
for W(z) over a, and to a piece of a realization W(z) = D+C(z-A)-1B for W(z); this is discussed in [BR5]. For a more complete discussion, see [BGK] or [BGR4]. To discuss null or zero structure in the nonregular case, we need to introduce a definition. We define a real non-archimedean valuation I Iz=a of R by putting Iriz=a =
0,e,_n
if r =O
if r#0
where 77 is such that r(z) = (z-A)'7T(z) with r(z) analytic and nonzero at A. If x = (x1, x2,
, xn) E
R", where R" can be identified with Rnxl or R1xn, let IIxIIz=a = maX{IxiIz=a, IX21z=a, ... , IXnlz=a}.
IIz=a) is a non-Archimedean nQrmed vector space over the real valued field (R, I Subspaces A and a of (Rn, II I. IIz=a) are said to be orthogonal (see [M]) if (Rn, II
(1.2)
I==a)
IIz + yll==a = max{IIxIIx=a, IIiII:=a}
for all x E A and y = Q. We say that subspaces A and fl of Rn are orthogonal on a C C if (2.2) holds for all x E A, Y E fl, A E a. We say that an element x of 1Z" is orthogonal to a subspace of 0 of IV (resp. an element y E Rn) on a if Rx is orthogonal to fl (resp. to Ry) on a. The orthogonality on a of subspaces 1l and A of Rn has a straightforward characterization in terms of linear algebra. Let 11(A) (resp. A(A)) denote the subspace of 0;n formed by the values
at A of those functions in fl which are analytic at A. By Proposition 2.3 in [BR2], El and A are orthogonal on a if and only if fl(A) n A(A) = (0)
Ball and Rakowski
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for each A E a.
Let W E Rmxn, and let W°i denote the left kernel of W, that is
W°l={cPER1xm:WW =0}. A controllable pair of matrices (AS, Bt) of sizes nC x nC and nc x m, respectively, is said to be a left null pair for W over o (see [BR2]) if
(i) a(At) C a, (ii) for each x E C1xn< the function x(z - As)-1BC is orthogonal to W°i on a(A(), (iii) the function (z-A()-1BCW(z)h(z) is analytic on a whenever h E Rnx1(a) and W(z)h(z)
is analytic on a, (iv) the size of At is maximal subject to the above conditions. A left null pair can be related to a canonical set of left null chains or left null functions for W(z) over a (see [BR2], [BR3] for the nonregular case, [BGK], [BGR4] for the regular case) and to a piece of a realization W x (z) = D - DtC(z - A + BD#C)-1 BD2 for a generalized inverse W x (z) for W(z) (see [BCRR], [R] for the nonregular case and [BGK], [BGR4] for the regular case). Complete information regarding left null-pole structure of W E Rmxn over a is given by the null-pole subspace
S'(W) = WRnx1(a).
An indication of this statement is that the null-pole subspace is a complete invariant for right equivalence over W; more precisely, two rational m x n matrix functions (assumed for simplicity to have trivial right annhilators) W1 and W2 are related by W1 = W2Q where Q is analytic with nonsingular values on a if and only if S,(W1) = S0(W2). be a right pole pair and let (AC, Bt) be a left null pair for a function W E Let (Cr, R,"". Then there exists a unique matrix r (see [BR3]) such that
(1.3) S,(W) = (WRnx1) fl
h(z) : x E C'-x1, h E Rmxl(a)
and E Resz=zp(z - At)-1Bch(z) = 1'x} zoEor
The matrix r is called the coupling matrix associated with the right pole pair (C,, null pair (AC, BC) for W over a. The triple
and a left
(1.4)
is called a left null-pole trinle for W over a or a left a-spectral triple of W. More detailed motivation for these concepts and connections with more established notions in systems theory can be found in [BGR1] and [BGR4] for the regular case and [BR5] for the nonregular case.
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Let W E Rmxn Any matrix polynomial P. whose rows form a minimal polynomial basis for W°I (see [F]) is said to be a left kernel polynomial for W. The following proposition characterizes matrix polynomials P,, and data sets w arising as a left kernel polynomial and a left v-spectral triple for some rational matrix function. PROPOSITION 1.1. (cf. Proposition 4.1 in [BR2] and Theorem 3.1 in [BR3]) Let W E R,nxn be a
function with a left kernel polynomial P. and left Q-spectral triple w as in (1.4). Then w and P satisfy the following conditions:
(NPDSi) the pair (C,., A,r) is observable and v(A,r) C o, (NPDSii) the pair (AC, CC) is controllable and a(AC) C o, (NPDSiii) FA, - ACF = BCC,., (NPDSiv) P. has no zeros in E and the leading coefficients of the rows of P,. are linearly independent, (NPDSv) the function P,c(z)C.,(z - A,r)-1 is analytic on C, , A,.}, where the points A1, A2, , A, are distinct, then the pair (NPDSvi) if o(AS) = {AI, A2i
,1I
A21
is controllable.
Conversely, if {C,r, A,r, At, BC, r} is a collection of matrices and P,. is a matrix polynomial
such that (NPDSi)-(NPDSvi) hold, then there exists a rational matrix function W with a left aspectral triple w as in (1.4) and with a left kernel polynomial equal to P,,. The construction of such a function W is indicated in [BR2] (see also Section 6 in [BR3]).
Given a matrix polynomial P. and a triple w = ((C,r, A,r), (AC, Be), t), where C,r, A,r, At, BC, IF are matrices of appropriate sizes, we may associate with w and P. a linear space
(1.5)
S,(w, P,) = {h E Rmxl : P,.h = 0} fl {C,r(z - A,)-1x + h(z) : x E
h E Rmxl(a)
and > Res,=,.(z - At)-1BSh(z) = rx}. zoEo,
It follows from formulas (1.3) and (1.5) that if w is a left a-spectral triple and P. is a left kernel polynomial for a function W E 7V"xn, then S0(w, P,) = SS(W). We will use the notation S,(w, PK) and S,(W) interchangeably. We shall refer to the collection (w, P.) = (C,r, A,r, AC, BC, t, P,c(z)) as a rcom lete (left) null-pole data set for W over a.
Ball and Rakowski
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2. A homogeneous interpolation problem. In the previous section we saw that a complete null-pole data set (w, PK) over o for a rational matrix function W is a useful set of invariants for describing the null-pole subspace S0(W) = WRnx1(a) associated with W. However, for the 4-block interpolation problem, a more general submodule of R'n arises in a natural way. Specifically, assume that W in Rmxn has a block row decomposition W = [W1 W2] with W1 E Rmx(n-k) and W2 E Rmxk, and consider the R(a)-submodule of R,nxl
W1R(n-k)xl(a)+ W2Rkx1
Sa(W1,W2) =
We refer to S,(W1,W2) as the extended null-Dole subspace associated with the block row matrix function [W1 W2] over a. For our applications [W1 W2] has trivial right annhilator in Rnx1,
i.e., the columns of [W1 W2] are linearly independent over R. Hence we make the assumption on W = [W1 W2] that the decomposition in (2.1) is direct. Note that we can recover W2Rkx1 from S., (W1, W2) as W27Zkx1
= n rS,(W,,W2) rER
as uniquely determined by S,(W1,W2); all one can say is that it is a direct sum complement to n rS,(W1,W2) inside the R(a)- submodule S,(W1,W2). The submodule
W17Z(n-k)x1(a) is not
rER
In any case the direct sum decomposition of S,(W1, W2) can be viewed as an of lne version of the
Wold decomposition for an isometric operator on a Hilbert space (see [NF]). From another point of view, note that S,(Wi,W2) = S,(W1,W2) if and only if
[WiW2]_[W1W2][G F H] = [W1F+W2G W2H]
where F and F-1 are in see this, note that both
R(n-k)x(n-k)(a), G E
F-1 F 01 and [-H-1GF-1 0 H-1]
[G
H]
are in Rkxk Indeed, to
Rkx(n-k) and H and H-1
IFF
0
1
HI
map &(n-k)X1(a) ® Rkxl onto itself, and conversely, any multiplier in Rnxn with this property F Rkx(n-k) H±1 E Rkxk necessarily must be of the form IG ] with F±1 E R(n-k)x(n-k) (a), G E
R
In particular we may always choose H so that the columns of W2 = W2H form a minimal polynomial
basis for their span, namely, n rS,(W1iW2). Note that the zeros and poles of Wi = W,F+W2G rER
can be quite different from those of W1; the invariant is the left zero-pole structure in a subspace
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105
complementary to the R-span of the columns of W2. We may always choose G E Rkx(n-k) so that the span of the columns of Wi = W1 + W2G is orthogonal over the set of zeros and poles of Wi to the span of the columns of W2 (or, equivalently, of W2). Let us call any a-spectral triple w = ((C,,, (At, Bt), r) for Wi obtained from the pair (W1, W2) in this way a W2-normalized left a-spectral triple for W1. The R-span of S,(W1iW2), U rS,(W1iW2) = [W1 W2]Rnx1, can rEI.
be specified via a left kernel polynomial P,. for the function [W1 W2]. Given a block row matrix function W = [W1 W2], we have now introduced (1) a W2-normalized left a-spectral triple w for W1, (2) a left kernel polynomial P,. for [W1 W2] and (3) a polynomial matrix Q E RmxL whose columns form a minimal polynomial basis for the R-span of the columns of W2. Let us call the whole collection (w, P,c, Q) a complete extended null-pole data set for W = [ W1 W2]. The analysis above shows that two block row matrix functions [W1 W2] and [Wi W21 for which the associated extended null-pole subspaces are the same (S, (W,', W2) = S,(W1,W2)) share the same complete extended null-pole data sets. Moreover, we can recover the extended null-pole subspace S,(W1i W2) from the data set via the formula
(if E -R-x1 : P,cf = 0}n
{C,(z - A,)-1x + h(z) : x E Jn.x1 and h E IZ' (a) such that (2.2)
E Resz=,O (z - At)-1 BCh(z) = rx}) + QRkxl zoEor
For our application to interpolation problems in Section 4, we need to understand the inverse problem, namely: which data sets (w, P,,, Q) arise as the extended complete null-pole data set over
a of a block row W = [W1 W2] E Rnxn We view this problem as a homogeneous interpolation problem. The answer is given by the following result. THEOREM 2.1. Suppose (w, PK, Q) (where w = ((C,r, A,), (A(, Bs), F)) is the extended complete null-pole data set for a block row function W = [W1 W2] E Rmxn (where W1 E Rmx(n-k) and W2 E R'"xk). Then the data set (w, P,., Q) satisfies the following conditions: (NPDSi) the pair (C,,, A,) is observable and C a, (NPDSii) the pair (Ac, BS) is controllable and a(AC) C a,
(NPDSiii) rA,r - AtI' = BCC,, (NPDSiv) P. has no zeros in C and the leading coefficients of the rows of P. are linearly independent,
(NPDSv) the function P,,(z)C,,(z - Ax)-1 is analytic on C, (NPDSvi) if a(AC) = {Al, A2i ... , Ar), where the points A1, A2,
Ar are distinct, then the pair
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106
BS
AS
P.(A1
,\17
P, (A2
A21
a, l
P.(,4)
is controllable. (NPDSvii) Q has no zeros in C and the leading coefficients of the columns of Q are linearly indepen-
dent
(NPDSviii) the function (z - As)-1BcQ(z) is analytic on C (NPDSix) if v(A,) = {w1, w2, , w,}, then the pair w11
[Cr Q(wl) Q(w2) ... Q(wa)],
w21
is observable;
(NPDSx) P.Q = 0. Conversely, if {C,, A,, AC, Bt, r} is a collection of matrices and P,, and Q are matrix polynomials such that conditions (NPDSi) - (NPDSx) hold, then there is a block row matrix Q) is an extended complete null-pole data set for function W = [WI W2] E R7"x" such that (w, W over v. REMARK. Note that there is some redundancy in the conditions (NPDSi) - (NPDSx). In particular
(NPDSvi) implies the first part of (NPDSi) and (NPDSix) implies the first part of (NPDSi). Nevertheless, we list them in this form to make the connections with the 1-block case (NPDSi) (NPDSiii) and the 2-block case (NPDSi) - (NPDSvi) clear. PROOF: To prove the first statement, we note that condition (NPDSvii) follows from Proposition
3.18 in [BR2], and condition (NPDSix) follows from the fact that Q is analytic on C and the column spans of W1 and Q are orthogonal on a. We give now the constructive proof of the second statement. The construction is a modification of the construction in [BR2] (see also Section 6 in [BR3]).
Let {C A.,, Ac, B<, r} be a collection of matrices and let P and Q be matrix polynomials such that conditions (NPDSi) - (NPDSx) hold. Step 1 Using the results from [GK], find a regular rational matrix function H with the o-spectral triple w = ((Cr, A,), (At, Be), F). Find a Smith-McMillan factorization EDF of H and set Wl =
ED. Step 2 Let v be the largest geometric multiplicity of a pole of H in a, let p be the largest geometric multiplicity of a zero of H in o, and let p be the largest sum of the geometric multiplicity of a pole
Ball and Rakowski
107
and the geometric multiplicty of a zero of H at any single point of or. Let d; denote the ieh diagonal
entry of D. For i = q - µ + 1,11- i + 2, , v, let pi be the minimal degree monic polynomial such that if d,,,_,,+i has a zero point A E a of order k then pidi has a zero at A of order k. Define an m x q matrix polynomial Q = [qij] by
ifi=j <- rl- µ
if7-Am-µ 0,
otherwise,
W2=WWQ.
Step 3 F o r i = 1, 2, ... , it, let 4i be the (m - i - 1)th row of E-1, so that if H has a zero at , ¢,i} is a canonical set of left null functions for H at A. Note that the left null pair constructed from the functions 01, , ¢ is left-similar to the pair (AC, Be). We may, in fact, assume that both pairs are equal. Modify the , 0, as follows. Suppose that v(A,) U o(At) _ {A1, A2, , a,} and the largest functions 01, 02, multiplicity of a zero of H in a is µ, and consider the point Ai (i = 1, 2, ... , s). Let the geometric multiplicity of a zero of H at Ai be K. By condition (NPDSvi), the matrix a point A E Q of the geometric multiplicity K then {O1, 02,
qS1(Ai)
42(Ai)
Ai = 4µ(A1) P,i(Ai)
By conditions (NPDSviii) and (NPDSx), A;Q(AI) = 0. Since .i geometric multiplicity of a zero of H in a, there is a point A such that the matrix has full row rank.
is the largest
41(A)
02(A)
A= 4 (A) PK(A)
has full row rank and AQ(A) = 0. Hence we can add to Oj (j = K + 1, n + 2,
,
i) a function
p(z)c,
where c is a vector and p(z) is a scalar polynomial vanishing at Al, A2, , A. to the order j , so that , 46,) are such that the matrix the modified 0.+1, 46K+2, , 0µ (which are again called 0,i+1,
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42(A1)
Bi = 0µ(A1) PK(A$)
has full row rank and B;Q(Ai) = 0. In this manner we obtain functions 01, 02, , 0µ whose span is orthogonal to the row span of P. on a(A,) U a(A0) and such that the function
11(z)
(z) -
02(z)
Q(z)
µ(z)
vanishes on a(A,) U a(AC). Now it follows from the construction and condition (NPDSviii) that if ¢i is a left null function for H at Al, A2, , A, of orders 11 i 12, ,1 then 4i(z)Q(z) vanishes at
A, to the order at least ki where k3 > max{l,l;}. Suppose k, < oo. By condition (NPDSvii),
0i(z)Q(z) = (A(z- \i)k')vi(z)Q(z) = 0i(Z)Q(Z)-
where vi is analytic, and does not vanish, at A1, A2,
, A,.
Let ki = ¢i - ¢i (i = 1, 2,
, µ). Then the set {1/'1,''2, ,1/iµ} contains a canonical set of left null functions for H at each zero of H in a, the span of {1/'i, 02, ,1/iµ} is orthogonal to the row span of P. on a(A.,) U a(At), and i,biQ = 0 (i = 1, 2, ... , /1.). Extend the span of {1/'l, 02, ,1/iµ} to an orthogonal complement ° of the row span of PK in (Q01,a(A,) U a(A()), and project each column of W2 onto an orthogonal complement of the row span of Q in (PKr, o(A.,) U a(A()) along or + {column span of Q} to get W3.
Step 4 Multiply W3 on the right by a regular rational matrix function without poles or zeros in a(A,) U a(AC), so that the resulting function W4 has no zeros nor poles in a \ (a(A.,) U a(AC)). Step 5 Find a minimal polynomial basis {ul, u2, , ur} for an orthogonal complement of the column span of [W4 Q] in (P,r, a(A,) U a(AS)). Set
W = [W4 u1 u2 ... nr Q] The extended null-pole subspace SQ(w, PK, Q) given in (2.2) has a special relationship with the null-pole subspace S(w, PK) studied in [BR3] as the following result shows.
Ball and Rakowski
109
Q(z)) is a a-admissible extended null-pole data set and let S,(w, P,) given by (1.5) and S,(w, P', Q) given by (2.2) be the associated R(a)-modules. Then THEOREM 2.2. Suppose (w, Pc(z), Q(z)) = (C,r, A,,, AC, Bt, I',
s, (w, P,) n Rmxl(a) = s, (w, P,, Q) n Rmxl(a). PROOF: The containment C is trivial from the definition. Conversely, suppose that f E S, (w, P", Q)
is analytic on a. By (2,2)
f (z) = C,,(z - A,r)-lx + h(z) + Q(z)r(z)
(2.3)
where x E V-, h E R'n x1(a), r E Rkxl are such that P,cf = 0 and E ResZ=zo(z-At)-1BCh(z) _ zoEo
I'x. In general, if g E R' x' has partial fraction decomposition g = g_ +g+ where g_ E Ro xl(a`) (elements of Rmxl with all poles inside a and vanishing at infinity) and g+ E Rmxl(a), we define g_. Since C,,(z - A,r)-1x E 1 xl(ac) and h E R'"xl(a) in (2.2), we have
(2.4)
0=
C,r(z - A,r)-lx + (P,°Qr)(z)
But a consequence of (NPDSix) is that (2.4) can happen only if C,r(z - Acr)-lx = 0, Qr E Rmxl(a). From (NPDSi) and Lemma 12.2.2 in [BGR41, we conclude that x = 0. Hence (2.3) becomes
f (z) = h(z) + Q(z)r(z) where P,cf = 0 and E ResZ=ZO(z-A,r)-1B(h(z) = 0. Since Qr is analytic on a and by (NPDSvii) zoEr
Q has no zeros on a, we must have that r is analytic on a. Furthermore, from (NPDSviii) we see that
E Resz=zp(z-AS)-1BCQ(z)r(z) = 0. Then f = h+Qr in fact is an element of S,(w, P,,)nRmxl(a) zoEo as asserted.
3. Interpolation Problem. Let a be a subset of the complex plane C. We look for a rational matrix function W which is analytic on,a and satisfies the following five interpolation conditions. Let zl, z2i , zM be given (not necessarily distinct) points in or. Let xi and yj be prescribed 1 x m and 1 x n vector functions analytic at z1 (j = 1, 2, , M), and let k1, k2, , km be positive integers. We require that
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110
(3.1)
dzi 11(xi(z)W(z))LZj
- dii 11'(z)1Z=Zj
1, 2, - ,M.
for i=
,WN be given (not necessarily distinct) points on or. Let uj and vj be , IN prescribed n x 1 and m x 1 vector functions analytic at wj (j = 1, 2, , N) and let 11, 12, be positive integers. The second condition is that Let w1i w2,
di-1
ds-1
dz(W(z)ui(z))Lwj
for i=
dzf-1 fj(z) z=w;
and j = For each pair of points zi and wj such that zi = wj, we are given numbers -y fg(f =
1
df+1-1
1)!dzf+g-l(xf(z) W(z)u3s)(z))
(3.3)
z-
-z;
(f + g -
= 7fg,
where h(n)(z) is a polynomial obtained by discarding all but the first 71 coefficients in the Taylor expansion of a function h at zi; that is, if h(z) h{j}(z - zi)j-1 in a neighborhood of zi, h(n)(z) _ E.7=1 h{j}(z
- zi)j-1.
The fourth interpolation condition is as follows. We are given 1 x m and 1 x n rational vector functions pj+ and pj_ (j = 1, 2, , K) analytic on a and require that (3.4)
di-1
di-1
dzi-1(pj+(z)W(z)) z=a = dzt-1Pr(z) z= for all positive integers i and for at least one (and hence for all) A E a. Finally, we are given n x 1 and m x 1 rational vector functions qj- and qj+ (j = 1, 2,
, L)
analytic on a and demand that di-1 = dzt-1 qj+(z) z for all positive integers i and for at least one (and hence all) points A E a. di-1
dzi_1(W(z)gj-(z))
z=a
Below, we reformulate the conditions (3.1) - (3.5) and show when they are consistent. The combination (3.1) - (3.4) was considered in detail in [BR5]; we summarize this case (the so-called 2-block case) first. We note that the problem of finding a rational matrix function which is analytic on o and satisfies conditions (3.1) - (3.3) is called the two-sided Lagra ge- Syly s . r interpolation
problem. Its complete solution is presented in Chapter 16 of [BGR4]. The problem of finding a rational matrix function which is analytic on v and satisfies conditions (3.1) - (3.4) has been solved in [BR5].
Ball and Rakowski
111
3.1 2-block Interpolation Problem. In [BR5] the following general interpolation problem was considered. We are given a subset a of the complex plane IF and an interpolation data set
(3.6)
w = (C+, C-, A, AC, B+, B-, r, P+(z), P_(z))
E E Cnxnc,B+ E Fn,xm, A. E consisting of matrices C+ E B_ E (F' 1
minimal polynomial basis;
(IDSv) the function
[P+(z) P (z)] IN (zis analytic on C. (IDSvi) if
{A1,\2, A,.}, the pair AC P (al)
all
A2I
P+(,\2)
A,I
P+(A,-)
is controllable. A collection of data w in (3.1.1) satisfying (IDSi) - (IDSvi) is said to be a aadmissible interpolation data set. The problem then is to describe all rational matrix functions with W(z) analytic on a which satisfy the interpolation conditions
ResZO(z- A()-1B+W(z) = -B_
(3.1')
zo Eo
Resz_Z0W(z)C_(z - A,)-' = C+
(3.2')
zoEu
ResZ-ZO(z - A()-1B+W(z)C_(z - A,)-' = F
(3.3')
zoEo
and
(3.4')
P+W = -P_
Ball and Rakowski
112
We mention that the problem (3.1') - (3.4') is simply a more compact way of writing conditions (3.1) - (3.4). Indeed (3.1') is equivalent to (3.1) if we let, for each j = 1, 2,. .. , M,
1
(3.7)
y{1}
x{1} zt2}
z
zj
Ac, =
, 1
Bj+ =
zj
'
{2}
, Bj- = -
, {kj}
x{ki}
where f f0 denotes the ith coefficient in the Taylor expansion of a function f at zj, and set AC,
A
B _
ACS
B1+ B2+
BM+
AC,,,
B1B_ _
B2-
BM-
Conversely, if matrices AC, B+, B_ of appropriate sizes are given, there exists a nonsingular matrix
S such that SACS-1 is a block diagonal matrix with the diagonal blocks in lower Jordan form. After replacing At by SACS-1, B+ by SB+ and B_ by SB_, we can convert the interpolation condition (3.1) to the equivalent condition (3.1'). Similarly, if for each j = 1, 2, .. , N we let
Ci+ = [v,{1},(2}..
wj
and then set C+ = [C1+ C2 + ... CN+], C- = [Cl- C2_ ... CN-],
A, =
A,.
then (3.2') collapses to (3.2). Conversely, if the matrices C+, C_, A,r are given, we can find a nonsingular matrix S such that S-1A,rS is in Jordan form. After replacing A,, by S-1A,rS, C_ by C_S, and C+ by C+S, we can reformulate the interpolation condition (3.2') in the more detailed form (3.2).
Ball and Rakowski
113
A similar analysis gives an equivalence between (3.4) and (3.4'). Suppose first that z; = wj
and let F;j = [yf9] where yf9(f = 1, 2, k; and g = 1, 2, , !j) are numbers associated with z; and wj as in (3.3). Then the interpolation conditions (3.3) hold if and only if ResZ_Z, (z - AC:)-1B++W (z)Cj-(z - A*,)-1 = rij
If z; # to, let F,, be the unique solution of the Sylvester equation in X AC,X = B;_C3_ + B;+Cj+
where A,,; and As; are as in (3.7) and (3.8) (see [BGR4], Appendix A.1) and set r = []F;j] with 1 < i < M and 1 < j < N. Then the set of interpolation conditions (3.3) is equivalent to the single block interpolation condition (3.3'). We also mention that validity of the Sylvester equation FA., - ACF = B_C_ + B+C+ is a necessary and sufficient condition for the consistency of (3.1') - (3.3') (see [BGR4] or [BR5]). Conditions (3.3) can be recovered from (3.3') by multiplying both sides of the equality by non-
singular matrices S and T such that SACS-1 and T-1AcT are in lower and upper Jordan forms, respectively, and reading the numbers yf9 from the appropriate blocks of the matrix SFT. Finally, if P+ and P_ are rational matrix functions whose jth rows (j = 1, , K) are equal to pj+ and -pj_ respectively, then it is easy to see that conditions (3.4) are equivalent to (3.4'). Demanding that P+ and P_ are polynomials is with no loss of generality. The admissibility conditions (IDSi) - (IDSvi) guarantee consistency and minimize redundancies in the interpolation conditions. For complete details we refer to [BR5]. 3.2 The general 4-block case.
Let Q_ and Q+ be rational matrix functions whose j" columns (j = 1, 2, .. , L) are equal to qj_ and qj+, respectively. Conditions (3.5) are equivalent to
(3.5')
W(z)Q-(z) = Q+(z)
Similarly as with P+ and P_, we will assume first Q_ and Q+ are matrix polynomials and th columns of
(3.10)
form a minimal basis.
Ball and Rakowski
114
We consider now the consistency of condition (3.5'). It follows immediately from (3.5') that
R1xmQ+(z) C 7Z1xnQ-(z)
Also, since W is analytic on a, Q_ has no zeros in a. Indeed, if Q_ had a zero in a, then Q+ = WQ_
would have a zero in a, and so the rank of the matrix polynomial (3.10) would drop at some point of a, contradicting the fact that the columns of (3.10) form a minimal polynomial basis.
Condition (3.1) says that the first ki Taylor coefficients at zi of xi(z)W(z) coincide with the first ki Taylor coefficients at zi of yi(z). Therefore, by (3.5'), the first ki Taylor coefficients at zi of xi(z)Q+(z) coincide with the first ki Taylor coefficients at zi of y,,(z)Q_(z). Consequently, the function
(z-AC;)-1[Bi+ Bi-]
[Q+(zi)]
is analytic on 0' where At,, Bi+ and Bi_ are as in (3.7). Hence the function
(z- At)-1[B+ B-] I Q+(Z) is analytic on It. Suppose that a vector ui(wi) is in the column span of Q_(wi), where ui is one of the functions in (3.2). Then ui(wi) = Q_(wi)uo, and it follows from (3.2) and (3.5') that
Q+(wi)uo = vi(w,) So in this case conditions (3.2) and (3.5) are either redundant or contradictory. To exclude such situations we require that the matrix {u1(w1) ui, (w.i,) ... uik (w.ik) Q- (v'.i, )]
have full column rank whenever wi, = wi, = = wik. In terms of the data A,, C+, C_, this assumption is equivalent to the controllability of the pair
(3.11)
Q-(wi.)...
I [C-
Q-(w.ik)]I
where {wi wi..... wik } is the set of distinct points among (W1, w2, , wN}. We will call this assumption the consistency of right generic and right lumped interpolation conditions. Finally, in view of (3.4') and (3.5'),
Ball and Rakowski
115
P+Q+ = P+WQ-
_ -P Q_. So conditions (3.4') and (3.5') are consistent if [P+ P_] [Q+] = 0.
We summarize various consistency and minimality properties our interpolation data must have in the following definition. We will call the data
(3.12)
W= (C+, C-, An, AC, B+, B
r, P+(z), P_(z), Q-(z), Q+(z))
C_ E Tnxn-, A,r E Tn. Xn AS E Tn(xnC B+ E a a-admissible interpolation A&.ta et if C+ E Q:"/ xm, B_ E fnC xn, IF E Tnc xn. , P+(x) E IZKxm, P _(Z) E gKxn, Q_ E 1n, xL, and Q+ E lZnxL
are such that (IDSi) the pair (C_, A,r) is observable and a(A,r) C a; (IDSii) the pair (AS, B+) is controllable and a(A,r) C a; (IDSiii) rA,r - Act' = B+C+ + B_C_;
(IDSiv) P_Rnxl C P+1v Xl, P+(z) has no zeros in a and the rows of [P+(z) P_(z)] form a minimal polynomial basis; (IDSv) the function
[P+(z) P-(z)] IN (z is analytic on M. (IDSvi) if a(A,r) _ (Al, A2,
, A,.}, the pair
A1I A21
is controllable;
(IDSvii) 7ZlxnQ+(z) C R1xnQ_(z), Q_(z) has no zeros in a, and the columns of
rQ+(z)] Q-(z) form a minimal polynomial basis;
Ball and Rakowski
116
(IDSviii) the function (z - AC)-1[B+ B_]
is analytic on C. (IDSix) if a(A,) = {w1, w2i
[Q+(z)1
, w,}, then the pair
w1I
w21
[C- Q-(wl) Q-(w2) ... Q-(w,)],
is observable.
1
(IDSx) [P+ P_] [Q+] = 0.
Given a a-admissible interpolation data set, the interpolation problem is to find which m x n rational matrix functions W analytic on or satisfy conditions (3.1') - (3.5').
4. Parametrization of solutions. Theorems 4.1 and 4.2 below present our results on parameterization of solutions of the in-
terpolation problem (3.1') - (3.5') (respectively with or without an additional norm constraint). We first need the following observation. If w = (C+, C_, A,,, Ac, B+, B_, r, P+(z), P_(z), Q_(z), Q+(z)) is a a-admissible interpolation data set, then the collection of matrices
CD-
\([C+A-), (A[B+ B_]),r)
together with the matrix polynomials P,, = [P+ P_] and Q = - (NPDSx) in Theorem 2.1. Therefore there exists a rational
[01 03] = [011 012 1
021
022
R(m+n)xL and 01 =
:
[u1 021
IQQ+l
satisfy conditions (NPDSi)
blocfk
013 I in R(m+n)x(m-K+n) (where 01 E 023 J
012 ] with 011 E
Rmx(m-K), 03
=
[13]
row matrix function 0 = R(m+n)x(m-K+n-L)' 02
E
with 013 E RmxL) which
has (w, [P+(z) P_ (z)], Q+(z) IQ-WD as an extended complete null-pole data set over a. The 2-block vers ion of the following appears as Theorem 4.1 in [BR5].
THEOREM 4.1. Let a E 1 and let w as in (3.12) be a a-admissible interpolation data set. Then there exist functions W E Rmxn(a) which satisfy the interpolation conditions (3.1') - (3.5). Moreover,
if 0=
011
012
013 1
021
022
023
Ball and Rakowski
117
is a block row rational matrix function with extended complete null-pole data set over or equal to
(c, [P+(z) P_(z)], [11]) then a function W E 1Z'"' is analytic on a and satisfies interpolation conditions (3.1') - (3.5) if and only if W = [011Q1 + 012Q2 013] [021Q1 + 022Q2 023]-1 where Q1 E
R(m-K)x(n-L)(a) and Q2
R(n-L)x(n-L)
E
[021Q1 + 022Q2 023]
are such that
((n_L)x1() ® 7ZLx1)
= [021 022 0231
(7Z(m-K+n-L)x1(a) (D 7ZLx1) .
We note that condition (4.2) holds for a generic pair (Q1i Q2) of rational matrix functions
which are coprime over a. One simply must arrange that the n-dimensional 7Z-vector subspace Q1
Q2 0
0 0
I
7Znx1 is a-orthogonal to the right annhilator of the n x (m - K + n) rational matrix
function [021 022 023], an R-subspace inside R(m-K+n)xl of dimension m - K.
For the H°°-control problem, one takes a to be either the right half plane II+ = {z Re z > 0) or the unit disk D = {z : IzI < 1} and asks that W, in addition to meeting a set of interpolation conditions on a, satisfy IIWII00 := rmsupZEUIIW(z)II < y for some prescribed tolerance level y. Without loss of generality we assume that y has been normalized to y = 1. The following refinement of Theorem 4.1 gives the solution of this generalized Nevanlinna Pick problem (i.e. interpolation problem with norm constraint). The 2-block version of the following result is Theorem 4.2 in [BR5]. THEOREM 4.2. Let a be either the right half plane ll+ or the unit disk V and let w as in (3.12) be a a-admissible interpolation data set. Suppose that there exists a rational (m + n) x (m - K + n) block row matrix function
0(z) =
011(z) 012(z) 021(x) 022(x)
013(2) 023(Z)
(with 011 E Rmx(--rK), 012 E lZmx(n-L)) such that (a)
I
z) CJ , A", A, [B+ B_],1', [P+(z) P_ (z)], [Q+(' J
nu`ll-pole
I
is an extended complete
data set for 0 over a
(b) 0 is analytic on 8a (including at infinity if a = II+) and satisfies 0(z)*J0(z) _
i® -IL,ZEBa Where J=lm®-In,7=Im-KED -In-L Then the following are equivalent.
Ball and Rakowski
118
(i) There exist functions W E R"`x"(a) which satisfy the interpolation conditions (3.1') (3.5') and in addition satisfy IIWII0 < 1. (ii) 0 satisfies O(z)*JO(z) < j at all points z of analyticity in a. Moreover, if (i) and (ii) hold, function W E R'x" is analytic on a, satisfies the interpolation conditions (3.1') - (3.5') and has IIWII0 < 1 if and only if 1
(4.3)
W=
1011 H + 012
[021H + 022
0131
0231
ce[H]
for an H E R(m-K)x(n-L)(a) with IIHII0 < 1. REMARK. In fact the existence of a matrix function 0 satisfying conditions (a), (b) in Theorem 4.2 is also necessary for interpolants W with IIWIIoo < 1 to exist, but we do not prove this here.
To make Theorem 4.2 useful we need a systematic way of computing the rational matrix
function 0 appearing in Theorem 4.2. For practical purposes we need only compute a rational matrix function 0 of the form
I
0
I
0 0
71
72
73
0=0 0
where 71,72, and 73 are any rational matrix functions (of the appropriate sizes) with 73 invertible
and O is as in Theorem 4.2, since in such a case 0 and O induce the same linear fractional map go = Ge (see [BHV]). One can obtain such a matrix function 0 from a preliminary 0 constructed to meet the specifications in Theorem 4.1 as follows.
by Wij(z) = Oli(z)*Ol(z) -
define W(z) = [vvi(z)] 1
Partition W as a block 2-x 2 matrix function by W = r Wo
W3o
L W30
W33 JJ
where Wo = [Wij]1
Finally find a square outer matrix function Ro(z) such that V(z) = Ro(z)*jRo(z). Then
0=O
1
0
0 I
If 0 =
911
912
013
021
022
023
Ball and Rakowski
119
is the desired matrix function such that the associated linear fractional map Ge parametrizes interpolants with norm less than 1 as in Theorem 4.2. Hence, once one has constructed a 0 as in Theorem 4.1, one arrives at an appropriate 0 via one j-spectral factorization V = R*jR. For more complete details we refer to Theorem 4.3 in [BHV]. Alternatively, once one has a matrix function 0 meeting the specifications in Theorem 4.1,
one could transform to a plant P such that the transform P of P to the generalized chain formalism is equal to O (see Section 5). Then the interpolation problem with norm constraint IIWII. < 1 is equivalent to the H°°-problem for the plant P (with tolerance level 1); one can then use any of the known solutions of the H°°-problem (e.g. [DGKF], [G], [PAJ]) to produce the parameterizer 0 of contractive (or "bounded real" in engineering parlance) interpolants. Of course, it would be desirable to have the criterion for existence of interpolants W with IIWIk < 1 and the construction of the linear fractional map Ge expressed more directly in terms of the interpolation data w, as has been done for the 1-block case in [BGR4]; this together with a more explicit realization formula for 0 in terms of the data set a, is the remaining open problem in this area. / is a a-admissible extended nullAs was remar ked above, I w, [P+(z) P_(z)], Q+(z)
IQ-WD
`
pole data set whenever w =
\
, C_, Ax, A(, B+, B-, r, P+(z), P_(z), Q_(z), Q+(z)) is a a-admis-
sible interpolation data set. The converse statement is not true in general; the following theorem, which will be used in our characterization of stabilizable linear feedback systems in the next section, characterizes which a-admissible extended null-pole data sets
P_(z)],
arise
from a-admissible interpolation data sets in terms of the associated extended null-pole subspace S(C,, [P+ P_], {+]); see Theorem 4.3 [BR5] for the 2-block case and Theorem 13.2.3 in [BGR4]
Q-
for the 1-block case.
THEOREM 4.3. Let (w, P, Q) be a a-admissible extended null-pole data set of the form (w, P, Q) = ([C-+
J,
A,, As, [B+ B-], F, [P+(z) P (z)], [Q±(z)
J)
and set
w= (C+, C-, A,r, AC, B+, B-, r, P+(z), P_(z), Q-(z), Q+(z) )
.
Then the following are equivalent.
(i) w is a a-admissible interpolation data set.
(ii) The extended null-pole subspace S = S(w,P,Q) C following properties:
(a) Sn [mxio] CRmx1(a)e 0,
R(m-K+n)x1 (see (2.2)) has the
Ball and Rakowski
120
(b)
S n R(m+n)xl(a) = 0 ® Rnxl(a) where
([]) -
[0] b
PROOF: Suppose first that w is a a-admissible interpolation data set. By definition, any function f E S has the form
AZ) _
(z)=
[g+](z_A)_1X+Fh+(zl
[Q±(z,I
(z,+
r(z)
where x E en-, h+ E Rmx'(a),h_ E Rnxl(a),r E 7LLx1. Suppose that f_(z) = C_(z - Ax)-'x+
h_(z) + Q_(z)r(z) = 0. Let P°° denote the projection map P° : k --+ k- if k E Rnxl has partial fraction decomposition k = k_ + k+ where k_ has all poles in a and vanishes at infinity (i.e. k_ E 1Z0n11(a°)) and k+ E 1 nxl(a) Then from f_ = 0 we get that P°9(f_) = 0; since C_(z - A.,)-lx E Roxl(a`) and h_ E R"xl(a), we see that
C_(z -
0=
(4.4)
But now by condition (IDSix), (4.4) forces
C_(z -
(4.5)
0,
0.
By Lemma 12.2.2 in [BGR4], the first condition in (4.5) forces x = 0. Also, since by (IDSvii) Q_ has no zeros in a, the second condition in (4.5) forces r to be analytic on a. But then f+(z) = h+(z) + Q+(z)r(z) is analytic on a. Thus S satisfies conditions (ii-a) in Theorem 4.3. To ver(ii-b) we must show that for any given fE Rmxl(a) we can find f+ E R'"xl (a)
so that
[4.]
E S. But by Theorem 2.2, S,(w, P) n R('"+n)xl(a) = S5(C"a, P, Q) n R(m+n)x1(a),
Hence this result follows in the/ same way as for the 2-block case (see Theorem 4.3 in [BR5]). Conversely, suppose I w, [P+ P_],
which S = S, (, [P+ P_], [Q+
1\ I
[ti.])
is a a-admissible extended null-pole data set for
satisfies (u-a) and (ii-b). Thus
(z,EP
P_],
[4])
satisfies
J
(NPDSi)-(NPDSx) and we must verify that w satisfies (IDSi) - (IDSx). First note that (IDSiii), (IDSv), (IDSviii) and (IDSx) follow from their counterparts (NPDSiii), (NPDSv), (NPDSviii) and (NPDSx) respectively without use of any assumptions on S,
[P+ P_],
[+
) 1
.
Next, one can
use Theorem 2.3, assumption (i-b) and the same arguments as those for the 2-block case (see Theorem 4.3 in [BR5]) to deduce (IDSii), (IDSiv) and (IDSvi) from their counterparts (NPDSii), (NPDSiv) and (NPDSvi). It remains now only to verify (IDSi), (IDSvii) and (IDSix). Suppose now that (C_, A..) is not observable. Then by Lemma 12.2.2. in [BGR4] there is an x E 0;11" not zero with C_(z - A,,)-lx = 0 identically in z. From Theorem 4.5 (i) in [BR3], we can solve for h+ E Rmxl(a) so that
Ball and Rakowski
121
[C+(z - A) lx+h+(z)]
f(z)_
(4.6)
[C+1
is in S,(w, P) C S,(Co, P, Q) (here we use that (IDSvi) has already been verified). J Furthermore, since
([C+]
, A-/ 1
is observable by (NPDSi), and since C_(z - A.,)-lx = 0, by
Lemma 12.2.2. in [BGR4] again necessarily C+(z - A,)-lx is not identically zero. Then f in (4.6) is an element of S, (w, P, Q) in violation of (ii-a). Hence we must have (C_, A,) observable, i.e. (IDSi) holds.
Next suppose (IDSvii) is not true. Then we can find r E RLx' such that Q+r V Rmxl(a)
but Q_r E R°xl(a). Again by an argument based on Theorem 4.5 (i) in [BR3], we can then
produce an element h+ E Rmxl(a) so that_r[4] E S(, P) C S,(, P, Q). The n 1
f = [Q+] r - [Qh+r[Q+r0 h+] 1
is an element of S,(w, P, Q) in violation of (ii-a). Hence (IDSvii) must hold. Finally, suppose (IDSix) is violated. Then we can find x # 0 in 6;11- and r # 0 in RLx' so that h_(z) = C_(z - A,)-lx + Q_(z)r(z) E R"xl(a) From Theorem 4.5 (iii) in [BR3] we
can produce h+ E Rmxl(a) so that [h+] E S,(w, P) C S,(w, P,Q) where h_ = (I Also by Theorem 4.5 (i) from [BR3] there is g E R^xl(a) so that [C+] (z - Ate)-lx + 1g0 (z)] E S,(w, P) C S,(w, P, Q) (here we use that (IDSvi) has already been verified). But then
[h+(z;]
f= [C+] (z-A,)_lx+
+ [Q+(z)1 r(z)
_ [C+(z - A,)-lx + g(z) + Q+(z)r(z) - h+(z)] E Sq(w, P, Q) 0
Q_(z)r(z) is analytic on a and (NPDSix) is in force, we However, since x # 0, C_(z are guaranteed that C+(z - Ax)-lx + Q+(z)r(z) is not analytic on a. Hence f is an element of S,(', P, Q) in violation of (ii - a). Thus, (IDSix) must hold and Theorem 4.3 follows. The proof of Theorem 4.1 requires the following lemma, an adaptation of one of the main ideas in [BC].
LEMMA 4.4. Let a C Q, let., as in (3.12) be a a-admissible interpolation data set, and let 011
012
013]
021
022
023
Ball and Rakowski
122
be a block row rational matrix function with extended complete null-pole data set over v equal
to
(,[P+
P_],
[+])
where w is as in (4.1). Then a function W E W>' is analytic on v and
satisfies the interpolation conditions (3.1')-(3.5') if and only if
W(Z)
I
j [021 022 023]
\
(&(m-K+n-L) x1(a) ®RLx1)
J
C 0 ((m_K+n_L)X1(Y) ®RLxl) PROOF: As in the Proof of Theorem 4.3, one can show that
[021 022 0231 (R.(m_K+n_L)x1(o.) ®
RLxl
1
J Q_RLxl x E Cn.} + Rmxl(o)+
_ {C_(z -
= S, (((C-, A,,), (0, 0), 0) 1 0, Q-) Suppose W is analytic on v and satisfies the interpolation conditons (3.1')-(3.5'). By Lemma 4.4 in [BR5],
['']
({C_(z - A,)-lx : x E 4^`'} + Rmxl(O))
S"' (w, P) C Sg (W, P, Q)
= O(R(m-K+n-L)x1(c) ® RLx1) But by condition (3.5'),
[W]Q-_ [Q-I
and hence the inclusion (4.7) follows.
Conversely, suppose that the inclusion (4.7) holds. Then
[w] QLx1 c n re TEIIZ
(R(m-K+n-1(a)
RLx1)
/
= O(O ®RLx1)
=
[Q+]Lx1
and hence W satisfies (3.5'). Moreover, by an argument in the proof of Theorem 4.3 we know that
Ball and Rakowski
123
{C-(z - A,)-'x: x E C'-} n Q_7ZLxl = {0} Using also that Q_r = 0 implies that Q+t' = 0 (a consequence of (IDSvii)), we see that the inclusion (4.7) implies that
[ fl LC
(_z - A,r)-lx : x E
fin' } + 1nx1(p) 1
sa(w [P+ P_]J = 07Z(m-K+n)x1(a) 1
Now Lemma 4.4 from [BR5] implies that W is analytic on a and satisfies the remaining interpolation
conditions (3.1') - (3.4'). PROOF OF THEOREM 4.1. By Lemma 4.4 it suffices to show that W E Rmxn satisfies
[W]
I
[021 022 023] 1
R(m-K+n-L) x1(a) E)7ZLx1 1
J
C 0 (R(m-K+n-L)x1(a) ® RLxl) if and only if W has a representation of the form
W = [011Q1 + 012Q2 013][021`1 + 022Q2 023]-1
where Ql E 7Z(m-K)x(n-L)(a),Q2 E 7Z(,,-L)x(n-L)(a) are such that
[021Q1 + 022Q2 023] (4.10)
()Z(n-L)x1(a) q) RLx1)
= [021 022 023] (IZ(m-K+n-L)xl (a) ®RLx1
Suppose first the (4.9) and (4.10) hold. Rewrite (4.9) in the form IQ 1
[W]
[021Q1 + 022Q2 02310 Q2 0
Apply both sides to
01 0
I
7Z(n-L)xl(a) ® RLx1 and use (4.10) to get
[W]
[021 022 023] 01
(L)Ql
(Rm_1t+n_x1(a)e1Lx1)
= 0 Q2 0] ((n_L)x1(a)eLx1) 0
1
Ball and Rakowski
124
Since Q1 and Q2 are analytic on a, we have \\
I
0
RLx1
Q101(R(n-L)x1(Q) ® RLx1) C R(m-K+n-L) x1(a)
Q2 I
and (4.8) follows. Con versely, suppose (4.8) holds. Note that n r
J®
rER
[W]
[021 022 023] 1
R(m-K+n-L)x1(a)®
RLxl) = [W1 023RLX1 is a R-subspace of Rnxl of dimension L which (by (4.8)) is contained
in ( r0 (Pm_K+n_L)x1() rER
[13J
RLxl)
=
RLx1, also an R-subspace of dimension L. By
dimension count, I
Wl 023RLxl = 10131 RLx1
LIJ
1023
Next note that the quotient module [w] [021 022 023] 1
R(m-K+n-L)xl(a) ®RLxl/
is a free R(a)-submodule of
/
Oz37ZLx1
L
j
RLxl 0 (R(m-K+n-L)xl(a) ®RLxl/1 1/ 1013] L 023 q1,+
qn-L,+ qn_L,_
0
0
with n - L generators; we may choose n - L generators of the form 0 ql
and
then set
R(m-K)x(n-L)(,.)'
Q+ = [1+
qn_L,+J
Q- = lql,-
qn-L,- I E R(n-L)X(n-L)(a)
E
Then the left side of (4.8) has the representation
[W] I
[021 022 023]
Ql = 0 LQ2
LO
0J1 0
I
((m_K+n_L)X1(a) ®RLx1)
l
(R(n-L)xl(a) ®RLxll ``
JJ
[O11Q1 + 012Q2 0131 (i(n_L)x1(a) 021Q1 + 022Q2 023 J
RLx11 f/
From equality of the bottom components in (4.11) we deduce that the pair (Ql, Q2) satisfies (4.10) and then (4.11) can be rewritten as
Ball and Rakowski
125
[W] I (
4.12
O23](7Z(n-L)x1(a)
[O21Q1 + 022Q2
_ O11Q1 + 012Q2 013 O21Q1 + O22Q2 023
)
®
IZLx1)
J
7Z(n-L)x1 Q ®7ZLx1 ( )
1(
Also from (4.10) we see that the square matrix function [021Q1+022Q2 023] cannot have determinant vanishing indentically, and hence [O21Q1 +O22Q2 0231-1 exists. Now (4.12) leads immediately
to the representation (4.9) for W. The proof of Theorem 4.2 requires the following lemma. 011 LEMMA 4.5. Suppose o,, w and 0 = 1021
012 022
013 023
are as in the hypothesis of Theorem 4.2.
Then 0 satisfies
O(z)'JO(z) < j ® 0, z E a (where J = I,,, (D -I,, and j = I,,,-K ® In-L) if and only if [In-L 0][022 023] -1021 has analytic continuation to all of a. PROOF: By assumption O(z)*JO(z) = j ® -IL for z E 8a. In particular
[ 013 Hence
] [012 013] - 1 023
[2] [022 023] = 12. + [
that [022(z) 023(z)] is invertible
[022
In, Z E 8a.
023] _
[012 013] for z E 8a. As [022 023] is square, we conclude 012zJ E for
8a. Then for each fixed z we can rewrite the system of
equations
(4.13)
[011 012 0131 021 022 023 (z)
u
-_ [ v
I
w
yo
(for z E 8a) in the equivalent form I1 U11
LU21
(4.14)
U31
U1211II
U22 1 (x) U32
1 [wI u =
y
y°J
where
U11 = [012 013][022 023]-1
U12 = 011 - [012 013][022 023]-1021
[ U31 ] - [022 023]-1 [U231
- -[022
0231-1021
Ball and Rakowski
126
Note that O(z)*JO(z) = j E) -IL on Oa is equivalent to IIvII2 - IIwII2 = IIuII2 - IIy1I2 - IIy°II2
or equivalently, to
IIvII2 + IIy1I2 + IIy°II2 = IIwII2 + IIy1I2
whenever z E Oa and (u, y, y°, v, w) satisfies (4.13). Since (4.13) and (4.14) are equivalent, we see
that O(z)*JO(z) = j ® -IL on 8a is equivalent to U(z) = L
U11 U21
U12 U22
I31
U32
Similarly, the validity of O(z)*JO(z) < j ® 0 on a is equivalent to
(z) being isometric on 8a.
IIvII2 - IIw1I2 <_ IIuII2 - IIyII2,
or otherwise stated, to
IIvII2 + IIyII2 <_ IIwII2 + IIuII2
whenever z E a and (u, y, y°, v, w) satisfy (4.13). Since (4.13) and (4.14) are equivalent, we read off that this in turn is equivalent to [Im+n-L 0]U(z) being contractive on a. In particular -U22 = [In-L 0][022 023]-1021
must be contractive at all points of analyticity of a, and hence has analytic continuation to all of the set a. Conversely, to show that O(z)*JO(z) < j (DO on a is equivalent to showing that [Im+n-L 01
U(z) has contractive values at all points of analyticity in a, or, equivalently by the maximum modulus theorem, that [Im+n-L 0]U(z) _
[U2U11(Z)
1(z)
U12(Z)
U22(z)]
have analytic continuation to all of a. By assumption we know only that U22 = -[In-L 0][022 023]-1021
has such an analytic continuation. The fact that this forces U11, U12, and U21 to have such an analytic continuation as well relies on the special structof the extended null-pole subspace 0 (R(m-K+n-L)x1(a) RLx1) = S, I w, [P+, P ], given by Theorem 4.3. Details for the
[p])
1-block case (the case where K = 0 and L = 0) are givenin Theorem 13.2.3 in [BGR4]; we leave it to the reader to check that the general case here can be verified by analogous arguments.
Ball and Rakowski
127
PROOF OF THEOREM 4.2. Let a,w and O be as in the statement of Theorem 4.2. Suppose first
that O(z)*JO(z) < i ® 0(n-L)x(n-L) at all points z E a where O(z) is analytic and that H E R(m-K)x(n-L)(a) with IIHII00 < 1. Then
[021H + 022 0231 = [022 023] ([022 023]-1021[H 0]
(4.15)
+ In)
By Lemma 4.5 we know that [IT_L 0][022 023]-1021 is analytic on a and, from the proof of 023]_1021H + In_L is Lemma 4.5, that II[021 023]-'021(z)II < 1 for z E 8a. Hence t[In_L 00][022
invertible for all z E 8a and for all t with 0 < t < 1. Hence wno detl [In_L 0][022 023]-1021H +
In-L/ = wno det In_L = 0, where wno X is the change of the argument of X (z) along 8a. As we observed above that [In_L 0][022 023]-1021H + In-L is analytic on a, we conclude that
\
/l
det [In_L 0][022 023]-1021H + In_L I has no zero in a. From this we see that
([022 023]-1021[H 0] +
(7_I4x1(a) (D 7ZLx1) = 7Z(n-L)x1(0,) ®7ZLxl
In/
From (4.15) we conclude that
[021H + 022 023] = [022 023]
(4.16)
x 1(a) ®7ZL/ (l(n-L)
(7Z(n-L)x1(a) ® 7ZL
I.
On the other hand,
[021 022
(4.17)
023](7Z(m-K+n-L)x1(a)e
LI
= [022 023] [[022 023]-1021 In] ((m_K+n_L)x1(0)@L) .
But by Lemma 4.5 [In-L 0][022 023]-1021 is analytic on or, so
[[022023]-1021 In] (izm_K+n_Lx1 (a) ® 7ZLx1) (4.18)
= R(n-L)x1(0,) ®RLxl
Combining (4.17) and (4.18), we obtain
J
Ball and Rakowski
128
[021 022 023] (4.19)
(7(m_K+n_I4xi(a) ® RLx 1 1
7ZLx1 = [022 023] (i(n_L)x1(a) ®
\\ 1
Combine (4.19) and (4.16) to see that the pair (H, I) is an admissible parameter pair in the sense of Lemma 4.4. Hence, by Lemma 4.4 we know that W = [011H + 012 O13][O21H + 022 023]-1 is analytic on a and satisfies the interpolation conditions (3.1') - (3.5'). Moreover, for z E 8a,
W(z)*W(z) - I = [W(z)* I]J
[W;z]
H
_ *[0 I
J
O*,O
H*H - I 0
where P = [021H + 022 023]-1 principle, IIW(z)ll < 1 for z E a.
IJ
LO
0 -I ,P0
Hence IIW(z)ll < 1 for z E 8a. By the maximum modulus
Conversely, suppose that there exists a function W E Rmxn which is analytic on a, satisfies the interpolation conditions (3.1') - (3.5') and has IIWIIoo < 1. Then by Theorem 4.1 we R(n-L)x(n-L)(a) such that know that there are Q1 E R(m-K)x(n-L)(a) and Q2 E
[O21Q1 + O22Q2 023] (4.20)
= [021 022 023]
((n_L)x1(a) ® RLx1)
((m_K+n_L)X1 (a) ® RLx1 1
from which we recover W as
(4.21)
W = [011Q +O12Q2 013][021Q1 +022Q2 023]-1
An alternative way to write (4.21) is Q1
[ W0 = 0 Q2 0
0
0J 1]
where Y' = [O21Q1 + O22Q2 823123] Since IIWII. < 1 and 0 satisfies condition (b) in Theorem 4.2 on Oa by assumption, we have, for z E a,
Ball and Rakowski
129
0<1b'(W'W-I),b 1Q1
01 0
Q1
= I Q2 01* 0 O`J0 I Q2
J
_ [Qi 0
Qz 0
I0] \ 1®-IL /
_ [QiQ1 - Q2*Q2
(4.22)
0
IL -IL
[Q1 Q2 0
0] 0
I
1. '
Thus Q2(z) cannot have any null space for z E o, and hence, as it is square, Q2(z)-1 exists for z E 8a. Also (4.22) then implies that H(z) = Q1(z)Q2(z)-1 is well defined and satisfies IIH(z)II < 1 for z E 8a. Next note from (4.20) the inclusion
023](R(n-L)x1(a) ®RLx1)
[022 (4.23)
C [O21Q1+ 022Q2 023]
((n_L)Xi() ® RLx1)
We also know from the proof of Lemma 4.5 that [022 023] is invertible on 8a. Then (4.23) can be written in the form
R(n-L)x1(a) ®RLx1 (4.24)
C ([E)22 023]-1O21[Q1 01 + [Q2
IL] /
(,R(.-L)xl(a) ®7ZLx1).
Note that the n x n matrix function [022 023]-1021 [Q1 0]+ [ Q2
L
] has the form [
B 1L J
where
A = [In-L 0][022 023]-1O21Q1 + Q2 has size (n - L) x (n - L) and B = [0 IL][O22 023]-1O21Q1 has size L x (n - L). In this notation (4.24) assumes the form
R(n-L)xl(a) ®RLx1 C [A
(4.25)
Note that
R(n-L)x1(a)
0
ILJJ
C
RaB ((n_L)X1()LX1)
A1Z(n-L)x1(a).
Ball and Rakowski
130
A = ([In_L 0][022 023]-1O21H + I) Q2 where 11[022 O23]-'02I fl < 1 on 8v (from the proof of Lemma 4.5) and where IIH(z)II < 1 and Q2(z) is invertible on 80, from observations made above. Hence A(z) is invertible on 8z and wno det A(z) is well defined. The inclusion (4.25) gives us that
wno det A(z) := wno det ([J_L 0][022 0231-1 O21Q1 + Q2)
(4.26)
<_ 0.
On the other hand,
(4.27)
A(z) = ([In_L 0][022 023]-'021H+I/ JQz
As was observed above, II[In-L 0][022 023]-1O21HII < 1 on 8a. Then by a standard homotopy argument wno det (ti_. 0][022 023]-1O21H + I I = 0, so
wno det A = wno det Q2.
(4.28)
Since Q2 is analytic on a, by the argument principle we deduce
(4.29)
wno det Q2 > 0.
Combining (4.26), (4.28) and (4.29) leads to
(4.30)
wno det A = 0 = wno det Q2.
Condition (4.30) has several consequences. First of all, det Q2 has no zeros on a so Q21 and H := QIQ21 are analytic on a. Secondly, we deduce that equality must hold in (4.24) and hence in (4.23). This has a consequence that
[022 0231
((_14x1 (a) ®RLx11 fJ
= [021 022 023]
((m_K+n_L) xl(a)
RLx1
or equivalently
R(n-L) x1(a) ®RLxI
11
= [[022 0231-1021
(Pim_+n_>z1(o.) ® RLx1)
Ball and Rakowski
131
This condition in turn is equivalent to [In-L O][022 023]-1021 is analytic on a.
By Lemma 4.5 this last condition is equivalent to O(z)'J0(z) < j ® 0 for z E or. Thus we have verified that condition (ii) in Theorem 4.2 is necessary for interpolants W with IIWIIao < 1 to exist.
It remains only to verify that W has the form (4.3) for an H E 1 (,n-K)x(n-L)(a) with IIWIIoo < 1. We have already verified that H := QiQ21 is analytic on a with norm < 1 on 8a. By the maximum modulus theorem it follows that supIIH(z)II < 1. Moreover, from (4.21) we deduce zEa that
W = Q011 H + 012 013]
[Q2
0
)([021 H + 022 013] 1
Q2
J
IL ])-1
_ [011H + 812 013][821 H + 022 023]-1 as needed.
5. Interpolation and internally stable feedback systems. In this section we establish the connections between the interpolation theory presented in the previous sections and the problem of designing a compensator to stabilize a given plant in a standard feedback configuration which has been studied in the control literature (see [Fr], [DGKF]). We emphasize that the connection between internal stability and interpolation has been a recurring theme in the systems theory literature (see e. g. [YBL], [V], [Ki]). The matrix version of the result is usually derived via a coprime factorization of the plant and the Youla parametrization of stabilizing compensators (see [YJB]). Our contribution here is to relate the interpolation conditions
directly to the original plant P; the connection is given in terms of the extended complete null-pole data set of a related block row rational matrix function P. We also solve the inverse problem of
describing which plants P go with a prescribed set of interpolation conditions
5.1 Preliminaries on feedback systems. Suppose we are given a rational block matrix function P = [P11 P21
P12where P11,P12, P2211
P21, P22 have respective sizes nz x n,,,, nZ x nu, ny x n,,,, ny x nu, and or is a subset of the extended
complex plane COO. (For discrete time systems a is usually taken to be closed unit disk while for continuous time systems a is usually taken to be the closed right half plane including infinity.) The problem is to design a rational nu x ny matrix function K (the compensator) so that the closed loop system depicted in Figure 5.1 is intern y stable, a notion which we shall make precise in a moment. (Here we assume that all input-output maps are causal, linear, time invariant and finite dimensional, and that the Laplace transform has already been implemented so all input-output maps are represented as multiplication by rational matrix functions.)
Ball and Rakowski
132
Figure 5.1
In Figure 5.1, w,z,u,y are functions with values in
and Cnyx1, re-
spectively, which are analytic in some right half plane (for the continuous time case) or in some disk
centered at the origin (for the discrete time case); in our discussion here we also assume that all the functions are rational, although this assumption is not necessary. The configuration depicted in Figure 5.1 is equivalent to the system of algebraic equations
P11w+P12U=Z P21w+ P22u = y
Ky = u. In the control theory context, the function w is the disturbance or reference 2i", u is the control
", z is the error &i" and y is the measurement ai". The idea is to design a compensator K which computes the control signal u based on the measurement y so as to make the overall system E(P, K) : w -* z perform better. The standard H°°-control problem is to design K which minimizes the largest error z (in the sense of L2-norm) over all disturbances w of L2-norm at most 1, subject to the additional constraint that K stabilizes the system: min
K stabilizing
max 11z112.
11wll3<1
Here the L2-norm is over the imaginary line for continuous time systems and over the unit circle for discrete time systems. In this section we put aside the norm constraint (5.2) and analyze only the connection between the stability of the system in Fig. 5.1 and interpolation theory. To define precisely internal stability and the related notion of well-posedness for the system in Figure 5.1, following [Fr] we introduce auxiliary signals v1 E Rn-I' and v2 E RI-x' as in Figure 5.2.
Ball and Rakowski
133
Figure 5.2
The diagram in Figure 5.2 is equivalent to the system of equations
P11w+P12u=z P21w + P22u = y - V2
Ky=u-v1. Well_po edn .ss means that the system (5.3) can be solved for (z, u, y) in terms of (w, v1i v2) and that w z the resulting map v2 -is given by multiplication by a proper rational matrix function 11 LuJ (for the continuous time case) or by a rational matrix function 'H analytic at zero (for the discrete
time case). Internal stability means that in addition this function f is analytic on all of a. Thus internal stability amounts to the assertion that the output signal z and the internal signals u and y are uniquely determined and stable for any choice of stable disturbance signals w, v1, v2; for a more complete discussion we refer to [Fr] and [V]. By elementary algebra one can show that H is given by the explicit formula
(5.4)
71=
Pll + P12Ki-1P22 P12 + P12K0-1P22 P12KA-11 KL-1P21
I + KA-1P22
0-1P21
0-1P22
KA-1 p-1
J
where 0 = I - P22K. In particular the closed-loop transfer function TZ,,, from disturbance (and/or reference) signal w to error z is given by
T:w = P11 + P12K(I - P22K)-1P21 (5.5)
_: Pp[K].
Ball and Rakowski
134
A standard assumption in the literature is that P12 is injective (as a multiplication operinto 7Z"'x1) and that P21 is surjective (as a multiplication operator from 7Z^-xI ator from into 1V vx1); in particular, n,. < nZ and ny < n,,,. An equivalent assumption is that the linear fractional map K -p F [K] defined by (5.5) is injective. If P21 in fact is square and invertible, one can solve the system of equations
P11w+P12u=z P21w + P22u = y
for (z, w) in terms of (u, y); the result is
P11 u + P12l/ = Z
P21u + P22Y = W
where P __ I P12 - P11P211P22 L -P211 P22
P11P211 P211
In the language of circuit theory the transform P -+ P amounts to the transform from the scattering to the chain formalism (see [B]), and was the basis for the analysis in [BR5]. To remove the
assumption that P21 be invertible, we use here the generalized transform to the chain formalism introduced in [BHV]. Since P21 by assumption is surjective, we may add extra rows P21 to P21 so that the augmented matrix function I P21
is square and invertible. Define P22 of a compatible size arbitrarily
and set
2= P21
P22
PO 21
Po22
[uW
Here y° is to be considered as a physically meaningless fictitious signal created for mathematical convenience. Since I P21 I is invertible, we may solve the system of equations
P11w + P12u = Z
P21w + P22u = y P21w + P22u = y°.
Ball and Rakowski
135
for (z, w) in terms of (u, y, y°). The result is
P where P =
P11 P21
y°J - LW
P13 is given by P23
P12 P22
Al = P12 - P11 [ P21
(5.6a) _
[P12 P13] = P11 [
(5.6b)
P22 J
1
P21 P21
]
P21 = - Ep211-1
P 1[P22]
(5.6c)
and
_
(5.6d)
1 IIr
[P22 P23] =
P21
L P21
j
The system of equations (5.1) associated with the feedback configuration in Figure expressed in terms of P as
0
(5.7)
I
1-0
0 0
0 1 0 ]
I
K
w1 v1
] +
-I
v2
0 0
5.1 can be
Z P21
-I
P22
P23
K
0
y0
In particular the closed-loop transfer function Tzw from w to z works out to be 1
(5.8)
Tzw =
[PiiK+Pi2P13] [P2iK+P22P23]
.
We also remark that the assumptions on P imply that [P22 P23] is square and invertible and that
and that one can into always backsolve for P from P. For more complete details we refer to [BHV]. Below, in this section we shall say that a square rational matrix function W is biproper if W is analytic and invertible at infinity and we are in the continuous time case, or if W is analytic P is injective as a multiplication operator from
and invertible at zero and we are in the discrete time case. In addition we say that the square rational matrix function R is u-outer if both R and R-1 are analytic on o (where u is the closed right half plane in the continuous time case and the closed unit disk in the discrete time case).
Ball and Rakowski
136
5.2 Stability of feedback systems and interpolation In this section we delineate the connections between internal stability of a feedback system
in Figure 5.1 and the theory of null-pole structure and interpolation developed in the previous sections. We assume that we are given a rational matrix function P = 5.1 with P21 surjective and P12 injective, and then define P =
as in (5.6). We consider P as a block row matrix function
[13]) Pand apply the notions of extended null-pole P23
[u1
12
P21
P22
J as in Section
1P11
P12
P13
P21
P22
P23 P11 P12 P21 P22
with Pl =
and P2 =
structure for block row matrix functions
explained in Section 2. Define integers m, n, K, L by
M=nz,n=nw,K=nz-nut L=nw-ny, so that P has size (m + n) x (m - K + n) with P11 of size m x (m - K), P12 of size m x (n - L), etc. Let a be the closed right half plane (continuous time case) or the closed unit disk (discrete time case), and let ('D, P, Q) = (c'ir, A,, AC, BC, r, P, Q)I be an extended complete null-pole data
set over a for the block row matrix function P in the sense explained in Section 2. Then in fact (Co, P, Q) has the finer block structure \LC+J
(Col P,Q)=
,A,r,AC,[B+ B-],r,[P+ P ],
[p])
where C+, C_, A,r, AC, B+, B_, r are matrices of respective sizes m x n,r, n x n,r, n, x n., nC x nC, nC x
m, nC x is, and nC x n.,r, and where P+, P_, Q+, Q_ are matrix polynomials of respective sizes K x m,
K x is, m x L, n x L. The following is the main result of this section. P11 P21
THEOREM 5.1. Let P =
P12 P22
with P12 injective and P21 surjective be a rational matrix
function describing the plant in Figure 5.1, and define the rational block row matrix function P11
P12
P21
P22
equivalent:
Al of size (m + n) x (m - K + n) as in (5.6). :
Then the following are
P23
(i) P is stabilizable
(ii) The extended(mxl null-pole subspace S := PI R(m-&(m-L)x1(a) ® RLx1) Jl of P satisfies:
) CRmxl(a)0 and
0
(a) Sn (b)
(iii) If (, P, Q) =
.xr1Sn7 \ (m+n)x1(a)I =0ED 7ZnX1(a)where Poen-x-([a]) b
([g
[0].
= / 1 r 1\ J , A.,, A, [BB_], r, [P+ P_], [+ 1) is an. extended complete set
of null-pole dataover a for the block row matrix function P, then
Ball and Rakowski
137
w= (C+, C-, A.x, AC, B+, B-, r, P+, P, Q+, Q-) is a a-admissible interpolation data set (see (3.12)). Moreover, if P is stabilizable, the following holds true: (1) A rational matrix function K stabilizes P if and only if K has a a-coprime factorization
K=NKDK' with NK E
R(m-K)x(n-L)(,) and with DK E
[P21NK+P22DK,P23](7Z(n-L)x1(a)®RLX1
1
R(n-L)x(n-L)(a) a-biproper such that
[P21,P22,P23](R(m-x+n-L)xl(a)®RLx1).
=
W/1is
(2) An m x n rational matrix function the closed loop transfer function T_-,,, associated with a stabilizing compensator K for P if and only if W is analytic on or and satisfies the interpolation conditions (3..1')- (3.5') associated with the a-admissible interpolation data set w, and [0 In_L 0]P-L [1T] p is biproper. Here P-L is any left inverse of P and IF is any regular rational n x n matrix function such that
(R(n-L)x1({A}) ®RL/ = ['21 P22 P23J (R(m-K+n-L) x1({A}) ®RLx1) with
f 0o for the continuous time case 0
for the discrete time case.
As a corollary we obtain a solution of an inverse problem, namely: given an admissible interpolation data set, describe the plants P for which the closed loop transfer functions T.. associated with internally stabilizing compensators are characterized by the prescribed set of interpolation conditions (3.1') - (3.5'). In particular such plants P always exist; thus interpolation and stabilization of feedback systems are equivalent. COROLLARY 5.2. Let w =
(C+, C_, A,,, Ac, B+, B_, r, P+' P_, Q+, Q_ ) be a a-admissible interpo-
lation data set (where either or = II or a = D) and let P =
P11
P12
P22 2
be a rational matrix
function describing a plant as in Section 6.1. Then the following are equivalent: (i) The proper rational matrix function K stabilizes P if and only if W := P11 + P12K(I P22K)-1P21 is stable and satisfies the interpolation conditions (3.1') - (3.5') associated with w
.-
Ball and Rakowski
138
(ii) The transform 1
P11-Pll [P21J
1
[P22J
Pll
[P21
[P20121
P11
P2]
[P1]
of P to the generalized chain formalism has
B_],r,[P+
= ([C+1 as an extended complete set of null-pole data over a.
Corollary 5.2 follows immediately from Theorem 5.1 so we omit a formal proof. PROOF OF THEOREM 5.1. The equivalence of (ii) and (iii) is the content of Theorem 4.3. Hence we
need only verify (i) -#* (ii) together with the parametrization of stabilizing compensators (1) and of the associated closed loop transfer functions TZ,,, (2). Let K be a given compensator. As explained in Section 5.1, K stabilizes P if and only if one can solve the system of equations (5.7) uniquely for stable z, u, yl in terms of any prespecified stable w, v1, v2. In more concrete form, this means: given any stable h1, h2, h3 there must exist k1, k2, k3, k4 with k1, k2, k3 stable and unique such that 0 h1
(5.9)
1h, -Kh3
Note that {'2J
K2
1
J=
kl _P11k2 - P12k3 _P23k4 -P21k2 - P22k3 - P23k4
k2-Kk3
023] is square, so in fact, necessarily the pair (h1i h2-Kh3) must determine
k2, k3, k4 uniquely.
Now suppose that K with coprime factorization K = NKDK' is a stabilizing compensator for P. Then in particular we can solve (5.9) with h2 = 0, h3 = 0 and h1 arbitrary. From k2-Kk3 = 0 we get that necessarily k2 = NKg, k3 = DKg for some stable g. To simultaneously solve the second equation in (5.9), one must have such a stable g together with a k4 (not necessarily stable) so that
-hl = (P2lNK+'22DK)g+P23k4. From the first equation in (5.9), in addition we must have
(PliNK + PIZDK ) 9+P23k4
k1 E R"l(a)
As h1 is an arbitrary element of RnX1(a), we have thus verified condition (ii-b) in the statement of the theorem. To verify (ii-a), suppose that k2 E R(--K) xl(a), k3 E 7Z(n-L)X1(a), k4 E RLxl are such
that
Ball and Rakowski
139
P21k2 + P22k3 + P23k4 = 0.
Then the system of equations
0 = -P21k2 - P22k3 - P23k4
k2 - Kk3 = k2 - Kk3 uniquely determines k2 and k3, since well-posedness implies the regularity of the matrix function P21
[
-I
P22
K
P23 I
0J
, Since K is stabilizing for P , from the first equation in (5.9) we see that neces-
sarily P11k2 + P12k3 + P13k4 =: k1 E Rnx1(01)
This verifies (ii-a). Next we verify
[P21NK + P22DK P23] ((n_L)x1()
(5.10)
= [P21 P22 P231
® RLx1)
(IZ(m-K+n-L) x I(a) ®RLx1
To see this, choose h1 = 0 and let h2 and h3 be general stable functions in (5.9). From the last R(n-L) x1(a) with k2 = h2 + NKg, k3 = h3 + DKg. Now the equation in (5.9), there must be a g E second equation in (5.9) says there is some choice of such a g and a k4 E RLx1 such that (P21NK + P22DK)9 = P21h2 + '22h3 + P23k4 This verfies (5.10).
R(m-K)x(n-L) has a coprime Conversely, suppose S satisfies (li-a) and (li-b) and K E factorization K = NKD -1 such that the pair (NK, DK) satisfies (5.10). To show that K is sta-
bilizing for P we need to verify that (5.9) is solvable for k1 E Wnxl(a), k2 E R(m-K)xl(a), k3 E
R(n-L)x1(a) and k4 E 7ZLx1 for any h1 E Rnxl(a), h2 E
R(m-K)x1(a) and h3 E 7Z(n-L)x1(a).
By linearity it suffices to consider two special cases.
Case 1: h1 E Rnxl (a), h2 = 0, h3 = 0. By (ii-b) there are stable k2,k3 and a not necessarily stable k4 so that -hl = P21k2+P22k3+P23k4 and such that k1 := P11k2+P12k3+P13k4 is stable. Then (k1, k2, k3, k4) is the desired solution of (5.9).
Case 2: h1 = O,h2 E R(m-K)x1(a),h3 E R(n-L)x1(a) and k4 E RLxI so that
R(n-L)x1(a). By (5.10) we can find g E
Ball and Rakowski
140
-[P21h2 + P22h3] = (P21NK + P22DK)g + '23k4
Then k2 = h2 + NKg, k3 = h3 + DKg, k4 give a solution of the last two equations in (5.9). From (ii-b) combined with (ii-a) we see that also k1 := P11k2+P11k2+P23k4 must be stable as well. Then
(ki,kzksk4) gives the desired solution of (5.9) in this case. Hence K = NKDK1 is stabilizing for P as asserted. To complete the proof of Theorem 5.1 it remains only to verify (2). But this follows from the characterization (1) of stabilizing compensators and the characterization of the range of the linear fractional map (NK, DK) -. [P11NK + P12DK, P13][P21 NK + P22DK, P23]-1 for stable pairs
(NK, DK) satisfying (5.10) given by Theorem 4.1. The side condition that [0
0]P-L [1T]
'k
[ In_L 0 ] be biproper is imposed to restrict (NK, DK) to pairs for which K = NKDK1 is well defined
and proper. References
[ABDS] D. Alpay, P. Bruinsma, A. Dijksma, H.S.V. de Snoo, Interpolation problems, extensions of symmetric operators and reproducing kernel spaces I, in Topics in Matrix and Operator
Theory (ed. by H. Bart, I. Gohberg and M. A. Kaashoek), pp. 35-82, OT 50, Birkhauser Verlag, Basel-Boston-Berlin, 1991. [B] V. Belevitch, Classical Network Theory, Holden Day, San Francisco, 1968.
[BC] J. A. Ball and N. Cohen, Sensitivity minimization in an H°° norm: parametrization of all suboptimal solutions, Int. J. Control 46 (1987), 785-816. [BCRR] J. A. Ball, N. Cohen, M. Rakowski and L. Rodman, Spectral data and minimal divisibility of nonregular meromorphic matrix functions, Technical Report 91.04, College of William & Mary, 1991.
[BGK] H. Bart, I. Gohberg and M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Birkhauser, 1979
[BGR1] J. A. Ball, I. Gohberg and L. Rodman, Realization and interpolation of rational matrix Functions, in Topics in Interpolation Theory of Rational Matrix Functions (ed. I. Gohberg), pp. 1-72, OT 33, Birkhauser Verlag, Basel Boston Berlin, 1988. [BGR2] J. A. Ball, I. Gohberg and L. Rodman, Two-sided Lagrange-Sylvester interpolation problems for rational matrix functions, in Proceeding Symposia in Pure Mathematics, Vol. 51, (ed. W. B. Arveson and R. G. Douglas), pp. 17-83, Amer. Math. Soc., Providence, 1990. [BGR3] J. A. Ball, I. Gohberg and L. Rodman, Minimal factorization of meromorphic matrix functions in terms of local data, Integral Equations and Operator Theory, 10 (1987), 309348.
[BGR4] J. A. Ball, I. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, OT
Ball and Rakowski
141
45, Birkhauser Verlag, Basel-Boston-Berlin, 1990. [BGR5] J. A. Ball, I. Gohberg and L. Rodman, Sensitivity minimization and tangential Nevanlinna-
Pick interpolation in contour integral form, in Signal Processing Part If. Control Theory and Applications (ed. F. A. Griinbaum et al), IMA Vol. in Math. and Appl. vol. 23, pp. 3-25, Springer-Verlag, New York, 1990.
[BGR6] J. A. Ball, I. Gohberg and L. Rodman, Tangential interpolation problems for rational matrix functions, in Proceedings of Symposium in Applied Mathematics vol. 40, pp. 5986, Amer. Math. Soc., Providence, 1990. [BH] J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation, J. Operator Theory, 9, 1983, 107-142. [BHV] J. A. Ball, J. W. Helton and M. Verma, A factorization principle for stabilization of linear
control systems, Int. J. of Robust and Nonlinear Control, to appear. [BR1] J. A. Ball and M. Rakowski, Minimal McMillan degree rational matrix functions with prescribed zero-pole structure, Linear Algebra and its Applications, 137/138 (1990), 325349.
[BR2] J. A. Ball and M. Rakowski, Zero-pole structure of nonregular rational matrix functions,
in Extension and Interpolation of Linear Operators and Matrix Functions (ed. by I. Gohberg), OT 47, pp. 137-193, Birkhauser Verlag, Basel Boston Berlin. [BR3] J. A. Ball and M. Rakowski, Null-pole subspaces of rectangular rational matrix functions, Linear Algebra and its Applications, 159 (1991), 81-120.
[BR4] J. A. Ball and M. Rakowski, Transfer functions with a given local zero pole structure, in New Trends in Systems Theory, (ed. G. Conte, A. M. Perdon and B. Wyman), pp. 81-88, Birkhauser Verlag, Basel-Boston-Berlin, 1991. [BR5] J. A. Ball and M. Rakowski, Interpolation by rational matrix functions and stability of feedback systems: the 2-block case, preprint. [BRan] J. A. Ball and A. C. M. Ran, Local inverse spectral problems for rational matrix functions, Integral Equations and Operator Theory, 10 (1987), 349-415.
[CP] G. Conte, A. M. Perdon, On the causal factorization problem, IEEE Transactions on Automatic Control, AC-30 (1985), 811-813.
[D] H. Dym, J. contractive matrix functions, interpolation and displacement rank, Regional conference series in mathematics, 71, Amer. Math. Soc., Providence, R.I., 1989. [DGKF] J. C. Doyle, K. Glover, P. P. Khargonekar and B. A. Francis, State-space solutions to standard H2 and H°° control problems, IEEE Trans. Auto. Control, AC-34, (1989), 831-847.
[F] G. D. Forney, Jr., Minimal bases of rational vector spaces, with applications to multivariable linear systems, SIAM Journal of Control, 13 (1975), 493-520.
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[FF] C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Birkhauser Verlag, Basel-Boston-Berlin, 1990. [Fr] B. A. Francis, A Course in H°° Control Theory, Springer Verlag, New York, 1987. [GK] I. Gohberg and M. A. Kaashoek, An inverse spectral problem for rational matrix functions and minimal divisibility, Integral Equations and Operator Theory, 10 (1987), 437-465. [Hu] Y. S. Hung, H°° interpolation of rational matrices, Int. J. Control, 48 (1988), 1659-1713. [K] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, N. J., 1980. [Ki] H. Kimura, Directional interpolation approach to H°°-optimization and robust stabilization, IEEE Trans. Auto. Control, AC-32 (1987), 1085-1093. [M] A. F. Monna, Analyse non-archimedienne, Springer, Verlag, Berlin Heidelberg New York, 1970.
[McFG] D. C. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences Vol. 138, Springer-Verlag, New York, 1990.
[NF] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, American Elsevier, New York, 1970.
[R] M. Rakowski, Generalized Pseudoinverses of Matrix Valued Functions, Int. Equations and Operator Theory, 14 (1991), 564-585. [YBL] D. C. Youla, J. Bongiorno and Y. Lu, Single loop stabilization of linear multivariable dynamic plants, Automatica, 10 (1974), 151-173. [YJB] D. C. Youla, H. A. Jabr and J. J. Bongiorno, Modern Wiener-Hopf design of optimal controllers: I and II, IEEE Trans. Auto. Control, AC-291 (1977), 3-13. [V] M. Vidyasager, Control Systems Synthesis: A Factorization Approach, MIT Press, Cambridge, Mass., 1985.
Department of Mathematics Virginia Tech Blacksburg, VA 24061
Department of Mathematics Southwestern Oklahoma State University Weatherford, OK 73096
MSC: Primary 47A57, Secondary 93B52, 93B36
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
143
MATRICIAL COUPLING AND EQUIVALENCE AFTER EXTENSION H. Bart and V.E. Tsekanovskii
The purpose of this paper is to clarify the notions of matricial coupling and equivalence after extension. Matricial coupling and equivalence after extension are relationships that may or may not exist between bounded linear operators. It is known that matricial coupling implies equivalence after extension. The starting point here is the observation that the converse is also true: Matricial coupling and equivalence after extension amount to the same. For special cases (such as, for instance, Fredholm operators) necessary and sufficient conditions for matricial coupling are given in terms of null spaces and ranges. For matrices, the issue of matricial coupling is considered as a completion problem.
1
Introduction
Let T and S be bounded linear operators acting between (complex) Banach spaces. We say that T and S are matricially coupled if they can be embedded into 2 x 2 operator matrices that are each others inverse in the following way i
CT1 Tel
s2 Sl
(1)
This notion was introduced and employed in [7]. In the addendum to [7], connections with earlier work by A. Devinatz and M. Shinbrot [18] and by S. Levin [38] are explained (cf. [46]). For a recent account on matricial coupling, see the monograph [19]. Concrete examples of matricial coupling, involving important classes of operators, can be found in [2], [7], [8], [10], [19], [21], [24], [26], [33] and [45].
The operators T and S are called equivalent after extension if there exist Banach spaces Z and W such that T ® Iz and S ®Iw are equivalent operators. This means that there exist invertible bounded linear operators E and F such that
(0
IZ)=E(0 Iw )F.
(2)
144
Bart and 1 ekanovskii
Two basic references in this context are [22] and [23]. The general background of these papers is the study of analytic operator functions. Thus the relevant issue in [22] and [23] is analytic equivalence after extension, i.e., the situation where the operators T, S, E and F in (2) depend analytically on a complex parameter. Other early references with the same background are [14], [16], [30], [31], [32], [34], [41] and [42]. For a recent application of analytic equivalence after extension involving unbounded operators, see [35]. Ordinary analytic equivalence (without extension) plays a prominent role in [1], [29] and [37]. More references, also to publications not dealing with operator functions but with single operators, will be given in Section 3.
Evidently, operators that are equivalent after extension have many features in common. Although this is less obvious, the same conclusion holds for operators that are matricially coupled. The reason behind this is that matricial coupling implies equivalence after extension. For details see [7] and [19], Section III.4. The main point of the present paper is the observation that not only does matricial coupling imply equivalence after extension, in fact the two concepts amount to the same. The proof involves the construction of a coupling relation (1) out of an equivalence relation of the type (2). This is the main issue in Section 2. Section 3 contains examples. Two examples are directly taken from the literature; in the other three, known material is brought into the context of matricial coupling. Along the way, we give additional references. In Section 4, we specialize to generalized invertible operators. For such operators, matricial coupling is characterized in terms of null spaces and ranges. An example is given to show that the invertibility condition is essential. Things are further worked out for finite rank operators, Fredholm operators and matrices. For matrices, one has the following simple
result: If T is an mT x nT matrix and S is an ms x ns matrix, then T and S are matricially coupled if and only if (3) rank T - rank S = mT - MS = nT - ns. Section 4 ends with a discussion of matricial coupling of matrices viewed as a completion
problem: Under the assumption that (3) is satisfied, construct matrices To, T1, T2, So, S1 and S2 of the appropriate sizes such that (1) is fulfilled. Extra details are provided for the case when T and S are selfadjoint. A few remarks about notation and terminology. The letters 7Z and C stand for the real line and the complex plane, respectively. All linear spaces are assumed to be complex. The identity operator on a linear space Z is denoted by IZ, or simply I. By dim Z we mean
the dimension of Z. For two Banach spaces X and Y, the notation X Y is used to indicate that X and Y are isomorphic. This means that there exists an invertible bounded linear operator from X onto Y. If X is a Banach space and M is a closed subspace of X, then X/M stands for the quotient space of X over M. The dimension of X/M is called the codimension of M (in X) and written as codim M. The null space and range of a linear operator T are denoted by ker T and im T, respectively. The symbol ® signals the operation of taking direct sums, not only of linear spaces, but also of operators and matrices. Acknowledgement. The foundation for this paper was laid in May 1990 while the first author was visiting Donetsk (USSR) on the invitation of E.R. Tsekanovskii, the father of the second author.
Bart and Tsekanovskii
2
145
Coupling versus equivalence
We begin by recalling the notion of matricial coupling (cf. [7] and [19]). Let X1, X2,Yl and Y2
be (complex) Banach spaces. Two bounded linear operators T : Xl --i X2 and S : Yl - Y2 are said to be matricially coupled if they can be embedded into invertible 2 x 2 operator matrices
T
(
T2
So C
XlED Y2
Ti To
X2
S1
s2 S
@ Yi
X2 ®1'1,
(4)
X1 ED Yz,
(5)
involving bounded linear operators only, such that T
1
T2 l
(6)
(So S1
(T1 TO)
S2 S
The identity (6) is then called a coupling relation for T and S, while the 2 x 2 operator matrices appearing in (4) and (5) are referred to as coupling matrices.
Next, let us formally give the definitions of equivalence and equivalence after extension. The operators T : Xl -> X2 and S : Yl -& Y2 are called equivalent, written T - S, if there exist invertible bounded linear operators Vl : Xl -+ Yl and V2 : Y2 -* X2 such that T = V2SV1. Generalizing this concept, we say that T and S are equivalent after extension if there exist Banach spaces Z and W such that T ® Iz - S ® Iw. In this context, the spaces Z and W are sometimes referred to as extension spaces. Ordinary equivalence, of course, corresponds to the situation where these extension spaces can be chosen to be the trivial space. From [7] it is known that matricial coupling implies equivalence after extension (cf. [19], Section III.4). Our main result here is that the converse is also true. Theorem 1 Let T : Xl -+ X2 and S : Yl ---p Y2 be bounded linear operators acting between Banach spaces. Then T and S are matricially coupled if and only if T and S are equivalent after extension.
Proof. Assume T and S are equivalent after extension, i.e., there exist Banach spaces Z and W such that T ® Iz and S ®Ily are equivalent. Let Ell
E12
E21
E22
F = r Fit
F12
1\ F21
F22
E_
) Y2®W-)X2®Z
and
X1 ®Z -> Yl ® W
be invertible bounded linear operators such that T (D Iz = E(S ® Iw)F, i.e.,
TO 1_(E11 E12)(S 0 )\Fll F12 0
Iz
Ezl Ezz
0
Iw
Fzl F22
)
.
Bart and Tsekanovskii
146
Write the inverses E-1 and F-1 as
E-1=
E1 1)
Eis 1)
Eii 1)
Esz 1)
Fu 1)
Fist)
W
:X2ED
and
F1
F-1=
21
)
F22)
:Y1®W -+X1®Z.
1
A straightforward computation, taking into account the identities implied by the above set up, shows that the operators T
-Ell
F11
F12E21
XI (1) Y2
X2 ®Y1
and F121)E211)
-E(-1)
F111)
S
:X2®Yl -'X1®Y2
are invertible and each others inverse. Thus
T (F11
-Ell
(2 1) E21 1)
-El(1)
F12E21
F31 1)
S
is a coupling relation for T and S. This proves the if part of the theorem. For the proof of the if part, we could simply refer to [7] or [19]. For reasons of completeness and for later reference, we prefer however to give a brief indication of the argument. Suppose T and S are matricially coupled with coupling relation (6). Following [7], we introduce
EF= I
T2
Iy2
TS° S2
:Y2®X2-+X2®Y2,
T1
T2) 2
Then E and F are invertible with inverses
):°X2®Y2-Y2®X2,
S21 'Yl®X2IX1®Y2.
Bart and Tsekanovskii
147
A direct computation shows that T ® Iy, = E(S (D Ix,)F. Thus T ® Iy, and S T Ix, are equivalent. This completes the proof.
Of particular interest is the case when the operators T and S depend analytically on a complex parameter. Theorem 1 and its proof then lead to the conclusion that analytic matricial coupling amounts to the same as analytic equivalence after extension (cf. [7], Section 1.1 and [19], Section III.4).
Another remark that can be made on the basis of the proof of Theorem 1 is the following. Suppose T : Xl -& X2 and S : Y1 -+ Y2 are equivalent after extension, i.e., there
exist Banach spaces Z and W such that T ® Iz - S ® Iw. Then Z and W can be taken to be equal to Y2 and X2, respectively. Another possible choice is Z = Yl and W = X1 (cf. [7], Section I.1). Thus, if the underlying spaces X1iX2,Y1 and Y2 belong to a certain class of Banach spaces (for instance separable Hilbert spaces), then the extension spaces Z and W can be taken in the same class. Roughly speaking, equivalence by extension, if at all possible, can always be achieved with "relatively small" or "relatively nice" extension spaces. We conclude this section with some additional observations. But first we introduce
a convenient notation. Let T : X1 --+ X2 and S : Yl -+ Y2 be bounded linear operators acting between Banach spaces. We shall write T ti S when T and S are matricially coupled or, what amounts to the same, T and S are equivalent after extension. The relation - is reflexive, symmetric and transitive. This is obvious from the viewpoint of equivalence after extension. In terms of matricial coupling things are as follows. Reflexivity is seen from
-Ix2
T
0
C Ix,
-1
Ix
0
l
-
/
\ - IX2
T
Symmetry is evident from the fact that (6) can be rewritten as TT2 S2l 1_-(\0 (S ( S1 So )
T T1
Finally, if T ti S and S ; R, with coupling relations T Al
A2
Ao)
B,/
1
(
Bo B2 S
S
C21 1_ (Do Dl
CC1 Co)
),
D2 R
then T : R with coupling relation T
A2C2
1
( -C1A1 Co - C1AoC2) =
Bo - B1DoB2 -B1D1 D2B2
R
)
.
This can be verified by calculation.
The relation : implies certain connections between the operators involved. Those that are most relevant in the present context are stated in the next proposition.
Bart and Tekanovskii
148
Proposition 1 Let T : X1 -a X2 and S : Y1 -p Y2 be bounded linear operators, and assume T ^ S. Then kerT ^_- kerS. Also im T is closed if and only if im S is closed, and in that case X2/im T Y2/im S. All elements needed to establish the proposition can be found in [7], Section 1.2 and [19], Section III.4. The details are easy to fill in and therefore omitted. We take the opportunity here to point out that there is a misprint in [7]. On the first line of [7], page 44, the symbol B21 should be replaced by A12-
3
Examples
Interesting instances of matricial coupling can be found in the publications mentioned in the Introduction. These concern integral operators on a finite interval with semi-separable kernel, singular integral equations, Wiener-Hopf integral operators, block Toeplitz equations, etc.. Here we present five examples. In the first three, known material is brought into the context of matricial coupling. The fourth example can be seen as a special case of the Example given in [7], Section I.1, and the fifth summarizes the results of [7], Section IV.1 and [19], Section XIII.8.
Example 1 Suppose we have two scalar polynomials a(A) = Am +
a1A + a0,
b(A) = Am + bin_1Am-1 + ... + b1A + bo.
The resultant (or Sylvester matrix) associated with a and b is the 2m x 2m matrix
R=R(a,b)=
ao
a1
...
am_1
0
ao
...
am-2 am_1
0
...
bo
b1
0
bo
` 0 ...
0
... ... 0
1
0
ao
b,n_1
1
bm_2
bm-1
bo
...
0
0
1
a ,n_ 1
1
b,n_1
1f
1
The Bezoutian (or Bezout matrix) associated with a and b is the m x m matrix B = B(a, b) = (bit)
_1
given by a(.\)b(µ) - a(µ)b(,\)
A-µ
_
m
i,j=1
bi;Ai
Bart and Tsekanovskii
149
As is well-known, the matrices R and B provide information about the common zeros of a and b (see, e.g., [36], Section 13.3). Our aim here is to show that R and B are matricially coupled. Matrices are identified with linear operators in the usual way. It is convenient to introduce the following auxiliary m x m matrices: a1
a2
...' am_1
1
a2
a3
...
1
0
am_1
1
1
0
...
0
0
S(a)
ao
a1
0
ao
0
0
0
0
... ...
am-2 am_1 am_3 am-2
T(a) ...
ao
a1
0
ao
J=
Observe that R = R(a, b) can be written as (T(a) JS(a) T(b) JS(b)
R
(7)
From [36], Section 13.3 we know that
S(a)T(b) - S(b)T(a) = B,
S(a)JS(b) - S(b)JS(a) = 0.
Clearly J2 = Im, where Im stands for the m x m identity matrix. A simple calculation now shows that
T(a) JS(a)
0
T(b) JS(b)
-S(a)-1
Im
0
0
-1
=
0
0
Im
S(a)-1J
0
-S(a)-1JT(a)
S(b)
-S(a)
B
In view of (7), this is a coupling relation for R and B. By the results of Section 2, we have R®Im - BE) 12,,,, i.e. R and B are equivalent by two-sided extension involving the m x m and 2m x 2m identity matrix. In the present situation things can actually be done with one-sided extension. In fact, R - B ® Im. Details can be found in [36], Section 13.3 (see also the discussion on finite rank operators in Section 4 below).
Bart and hekanovskii
150
The equivalence after extension of the Bezoutian and the resultant already appears in [20]. For an analogous result for matrix polynomials, see [39]. It is also known that
the Vandermonde matrix and the resultant for two matrix polynomials are equivalent after extension (cf. [25]).
Example 2 This example is inspired by [40], Section 3. Let A : Z -i W, B : X Z, C : W -' Y, and D : X -' Y be bounded linear operators acting between Banach spaces. Then D + CAB is a well-defined bounded linear operator from X into Y. Put
M=
-Iw
0
A
C
D
0
:W®X®Z--+W®Y®Z.
B -IZ
0
Then D + CAB N M and the identity
D + CAB -C -Iy -CA AB
-Iw
0
IX
0
0
B
0
0
0
0
IX
0
0
-Iw
0
A
0
Iy
C
D
0
-Iz
0
0
0
-Iz
-A
_
is a coupling relation for D + CAB and M. This coupling relation implies that
D+CAB ®Iwexez-M®Ix.
D+CAB ®Iweyez-MED Iy,
Both equivalences involve two-sided extensions, but, as in Example 1, things can be done with
one-sided extension. Indeed, it is not difficult to prove that M is equivalent to D + CAB Iwez. The details are left to the reader. Example 3 Consider the monic operator polynomial
L(a) =
AA,+A0.
Here A0,. .. , An_, are bounded linear operators acting on a Banach space X. Put 1
0
I
0
0
I
... ... 0
...
0
CL =
0 0
:Xn -+ Xn.
.
0 0
0
0
...
0
-Ao -A, -A2 ... ...
I
-An-1
It is well-known that
L(a) ® IX"-, - AI - CL
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151
and, in fact, we have a case here of analytic equivalence after one-sided extension. Clearly it is also a case of linearization by (one-sided) extension. For details, see [3], [27], [28], [36] and [44].
Now, let us consider things from the point of view of matricial coupling. For
k=0,...,n-1, put
I
k-1
a'An_k_;
AkI +
Lk(A)
i=o
so Lo(a) = -I in particular. Then we have the coupling relation L(A)
Ln-1(A)
Ln-2(A)
0
0 0
I
-I
AI
-1
-AI
A21
...
... ...
L2(A)
L,(A)
Lo(A)
0 0 0
0 0 0
0 0 0
l -Jan-1I -an-2l -An-3I ... 0 0 0 0
I
0
0
AI
0
0
AI -I
...
...
-I Ao Al ...
0 0 0
0 0 0
0 0 0
0
AI
-I
An-3 An-2 JAI + An-1
showing that L(A) and AI - CL are matricially coupled. Note that this is a case of analytic matricial coupling.
D:Y-,Ybe bounded
Example 4Let
linear operators between Banach spaces. For A in the resolvent set p(A) of A, we put
W (A) = D + C(AIx - A)-1B. Assume that D is invertible and write A" = A - BD-1C. It is well-known that
(8)
W(A) ED IX -(aIX-Ax)ED Iy (9) and, in fact, we have another case here of analytic equivalence after (two-sided) extension. For details and additional information, see [4], Section 2.4. Considering things from the viewpoint of matricial coupling, we see that W(A) AIX - A" with coupling relation
-C(AIx - A)-1 \ - (AIx - A)-1B (AIx - A)' ) W(A)
I\
1
-(
Note that this is again a case of analytic matricial coupling.
D-1
D-1C
)
BD-1 AIx - Ax /
Bart and Tsekanovskii
152
An expression of the type (8) is called a realization for W. Under very mild conditions analytic operator functions admit such realizations. For instance, if the operator function W is analytic on a neighbourhood of no, then W admits a realization. For more information on this issue, see [4]. Whenever an operator function W can be written in the form (8) with invertible D, it admits a linearization by two-sided extension (9), and hence certain features of it can be studied by using spectral theoretical tools. Under additional (invertibility) conditions on B or C, even linearization by one-sided extension, i.e. analytic equivalence of the type W(\) ED I - MIx - A", can be achieved (cf. [4], Section 2.4; see also [31]).
Example 5 Let K : LP([O,oo),Y) -+ L,([O,oo),Y) be the convolution integral operator given by
[Kcp](t) = I k(t - s)cp(s)ds. 0
Here Y is a (non-trivial) Banach space, 1 p < no and k is a Bochner integrable kernel whose values are bounded linear operators on Y. The familiar Wiener-Hopf equation
W(t) -
J
k(t - s)W(s)ds = f (t),
t '= 0
involving Y-valued LP-functions cp and f can be written as (I - K)cp = f. Assume that the so called symbol W()i) = IY _ 1,00
e1 %tk(t)dt
admits an analytic continuation to a neighbourhood in the Rieman sphere C. of the extended real line 7Z,,.. Then W admits a realization
W(A) = Iy + C(.IX - A)-1B,
,E7Z C p(A).
In case Y is finite dimensional, the (state) space X can be chosen to be finite dimensional if and only if W is rational. For details, see [4], [19] and [28]. Suppose, in addition, that W takes invertible values on 1Z. In view of (9), this
means that the spectrum a(A') of A' = A - BC lies off the real line. Let P, respectively Px, be the Riesz projection corresponding to the part of Q(A), respectively o(A"), lying in the upper half plane, and put S = P" Ii P : im P --+ im P". So S is the restriction of P" to im P considered as an operator into im P'. Then
I - K : S.
(10)
For an explicit coupling relation between I - K and S we refer to [7], Section IV. 1 and [19], Section XIII.8. One of the (many) consequences of (10) is that I - K is invertible if and only if X = im P ® kerPx. For additional information, generalizations and related material, see [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [19], [21], [24], [26], [33] and [45].
Bart and Tsekanovskii
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153
Special classes of operators
Proposition 1 in Section 2 gives rise to the following question: Suppose T : X1 -+ X2 and S : Y1 -& Y2 have closed range, ker T ^' ker S and X2/im T ' Y2/im S. Does it follow that T ^ S? We shall see that without extra conditions on the operators involved, the answer to this question is negative (Example 6 below). But first we shall make clear that under additional invertibility assumptions it is postive. Let X and Y be Banach spaces, and let T : X - Y be a bounded linear operator. We say that T is generalized invertible if there exists a bounded linear operator T+ : Y -> X such that T = TT+T and T+ = T+TT+. In that case T+ is called a generalized inverse of T. Special instances are left invertible operators having left inverses (T+T = Ix) and right invertible operators having right inverses (TT+ = Iy). Note that T is generalized invertible if and only if there exists a bounded linear operator Tt : Y -+ X such that T = TTtT (take T+ = TtTTt to get a generalized inverse of T). Also T is generalized invertible if and only if ker T is complemented in X and im T is complemented in Y (cf. [43]). Theorem 2 Let T : X1 -. X2 and S : Y1 -> Y2 be bounded linear operators acting between Banach spaces, and assume that T and S are generalized invertible. Then T :, S if and only if ker T ker S and X2/im T Y2/im S. To place the result against its proper background, note that if two operators are matricially coupled (equivalent after extension) and one of them is generalized (left, right) invertible, then the other is generalized (left, right) invertible, too. For details, see [7], Section 1.2 and [19], Section III.4.
The only if part of Theorem 2 is covered by Proposition 1. Note that generalized invertible operators have closed range. The if part of the theorem can be proved by establishing
the following more detailed result. Let T+ : X2 -p X1 and S+ : Y2 - Y1 be generalized inverses of T and S, respectively. Then there exist bounded linear operators
T1: X1 -+ Y1, T2:Y2-+X2, S1 :Y1-+X1, S2 : X2 - Y2, such that 1
T T2 l C
T1 S+)
T+ S \ S2
S1
is a coupling relation for T and S. The argument is straightforward and uses the fact that, with respect to appropriate decompositions of the underlying spaces, the operators T, T+, S and S+ can be written in the form
T=I
0° 0T+0°1 0 I,
s=(0 0 l0
so)
'
S+ _r0 0 `l 0 Sot
Bart and hekanovskii
154
with To and So invertible. A complete proof is given in [13]. The only if part of Theorem 2 is true even without the generalized invertibility condition on T and S (cf. Proposition 1). For the if part, things are different. Counterexamples are easy to construct when one allows the spaces X1, X2, Y1 and Y2 to be different. The following example is concerned with the case when all these spaces are the same. It is inspired by an example given by A. Pietsch (see [43], page 366). The example also provides a negative answer to the question raised at the beginning of this section.
Example 6 Let 2. be the Banach space of all bounded complex sequences provided with the usual supremum norm, and let co be the subspace of PO consisting of all sequences converging to zero. Then co is a closed subspace of P0, but co is not complemented in 40 (cf. [17]). Put
X = Q. 6) co ® (P0/co), and introduce T : X -p X and S : X -ti X by stipulating T [(xl, x2, x3, ...), (yl, y2, y3, ...), i(zl, z2, z3, ...)] = = [(x2, x4, x6, ...), (0, 0, 0, ...),,c(xl, 0, x3, 0)x5, S [(X 1, x2, x3, ...), (yl, y2, y3,
...)] ,
.), k(zl, z2, z3, ...)] _
= [(xl,x2,x3,...),(0,0,0,"ic(zl,0,z3,0,z5,"')], where x :.C0 -, e0/co is the canonical projection of 20 onto 40/co. Then T and S are well-defined bounded linear operators on X. Since S is idempotent, the range of S is closed. Also im T = im S and so, in particular, X/im T X/im S. Analysis of the null spaces of
T and S shows that ker T
ker S too. However, in spite of all of this, T and S are not
matricially coupled. Indeed, the idempotent operator S is generalized invertible, but T is not, since ker T is not complemented in X. A bounded linear operator T acting between Banach spaces is called a finite rank operator if dim im T < oo. The number dim im T is then called the rank of T and denoted by rank T. Finite rank operators are generalized invertible. The following observations are pertinent to the topic of this paper. Details may be found in [13]. Let T and S be finite rank operators from a Banach space X into a Banach space Y. If rank T = rank S, then T - S. The converse is also true, but completely trivial. For Hilbert spaces, we have the following result. Let X and Y be infinite dimensional Hilbert spaces. Then T . S for all finite rank operators from X into Y. This is immediate from Theorem 2. Returning to the general situation, assume that T : Xl -, X2 and S : Yl --' Y2 are finite rank operators between Banach spaces. Suppose T - S and rank T > rank S. Let H be a finite dimensional space with dim H = rank T- rank S. Then T - S ® IH. Thus two finite rank operators that are equivalent after extension are equivalent after a one-sided extension involving a finite dimensional extension space. For other material on the reduction of extension spaces, see (15], Section 3.3 and [32], Section 5 (cf. also Theorem 3 below).
Extra details can also be obtained for Fredholm operators. Let X and Y be Banach spaces. A bounded linear operator T : X - Y is called a Fredholm operator if ker A
Bart and Tsekanovskii
155
has finite dimension and im S has finite codimension (in Y). Fredholm operators have closed range and are generalized invertible. If two operators are matricially coupled (equivalent after extension) and one of them is Fredholm, then the other is Fredholm too.
Theorem 3 Let T : X1 -p X2 and S : Y1 -i Y2 be Fredholm operators between Banach spaces. Then T ; S if and only if dim ker T = dim ker S, codim im T = codim im S.
(11)
Moreover, when T : S, and Z and W are Banach spaces, then T ®Iz ' S ® Iw if and only if X1®Z^-'Y1®W and X2®Z^-'Y2 W. The second part of the theorem tells us, for the Fredholm case, what freedom there is in choosing the extension spaces. The crux of the matter is this: Suppose T and S are Fredholm operators from a Banach space X into a Banach space Y. Then T - S if and only if (11) is satisfied (i.e. T ' S). It is a trivial matter to construct examples showing that the Fredholm condition in Theorem 3 is essential. For the (simple) proof of Theorem 3 and some related observations, see [13].
Finally, we specialize to (complex) matrices. As we shall see, some of the things discussed earlier can then be made more explicit. Matrices are identified with linear operators in the usual way. Theorem 3 immediately implies the following result. Let A be an mA x nA
matrix and let B be an mB x nB matrix. Then A' B if and only if rank A - rank B = mA - MB = nA - nB.
(12)
In the remainder of this section, we shall consider matricial coupling of matrices as a completion problem. The precise statement of the problem reads as follows. Given an MA x nA matrix A and an MB x nB matrix B such that (12) holds, construct matrices AO, A1i A2, Bo, B1 and B2 of the appropriate sizes such that the coupling relation
(AA2
_(Bo B1
\Ao/ B2 B is satisfied. These appropriate sizes are: nB X mB for Ao, nB x nA for Al Al
(13)
i mA X mB for A2, nA x mA for Bo, nA x nB for B1, and rn x mA for B2. In order to facilitate the discussion, we introduce the following notations
rA = rank A,
rB = rank B,
P=mA-rA, Q=nA-rA. We also let I. stand for the s x s identity matrix and O,,t for the s x t zero matrix. Now choose invertible matrices RA, CA, RB and CB (of the appropriate sizes) such
that
Bart and 1 ekanovskii
156
RAACA =IrA ® OP,q, RBBCB = OPA ® Ira.
(14)
Here it is used that p = mA - rA = mB - rB and q = nA - rA = nB - rB. The matrices RA and RB correspond to row operations, the matrices CA and CB to column operations.
It is easy to verify that the matrices appearing in the right hand sides of the identities (14) are matricially coupled with coupling relation IrA ORrA Oq,rA OrB,rA
OrA,q
OrA,P
O*A,rB
OP,q
IP
Iq OrB,q
Oq,P
OP,rB Oq,rB
OrB,P
IrB
IrA
_
Oq,*A
OP,rA
OrB,rA
OrA,P Oq,P
O*A,q
OrA,rB
Iq
Oq,*B
IP OrB,P
OP,q OrB,q
OP,rB I*B
From this one can obtain a coupling relation for A and B by multiplying from the left and the right by CA
0
RA
0
0
RBI
0
CBI
respectively. In fact, this gives a coupling relation (13) with, for instance, AO
= CB
( \
rB
)
( OrB,P IrB ) RB,
AI = CB ( I) (Oq,rA
Iq ) CA1.
Thus Ao is the product of the matrix obtained from CB by omitting the first q = nB - rB columns and the matrix obtained from RB by omitting the first p = mB - rB rows. Similarly, Al is the product of the matrix resulting from CB by omitting the last nB -q = rB columns and the matrix resulting from CAI by omitting the first nA - q = rA rows. Analogous descriptions can be given of A2, Bo, BI and B2 (cf. [13]).
We conclude this section with a remark on the case when both A and B are selfadjoint. To fix notation, let A be a selfadjoint m x m matrix and let B be a selfadjoint n x n matrix. We assume that rA - rB = m - n, so A * B. Here, as before, rA =rank A and rB =rank B. The selfadjoint matrices A and B can be brought into diagonal from with the help of unitary transformations. Using such diagonalizations instead of (14) and working along the same lines as above, one arrives at a coupling relation (13) such that the coupling matrices A2
(AAl Ao
'
B2 B (Bo BI/
(15)
are selfadjoint. It is also possible to describe the spectra of these matrices: If all ... , arA are the nonzero eigenvalues of A and #Ii ... , f3 are the nonzero eigenvalues of B, then the eigenvalues of the first matrix in (15) are a1i ... , arA, OT I, ...AD) -11 ... , -11 11 ... , 1,
where both -1 and +1 are repeated p(= m - rA = n - rB) times. To get these of the second matrix in (15), take reciprocals. We leave the details to the reader.
Bart and Tsekanovskii
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References [1]
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[2] Bart, H.: Transfer functions and operator theory, Linear Algebra Appl. 84 (1986), 33-61. [3] Bart, H., Gohberg, I., Kaashoek, M.A.: Operator polynomials as inverses of characteristic functions, Integral Equations and Operator Theory 1 (1978), 1-18.
[4] Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal Factorization of Matrix and Operator Functions, Operator Theory: Advances and Applications, Vol. 1, Birkhauser, Basel, 1979. [5] Bart, H., Gohberg, I., Kaashoek, M.A.: Wiener-Hopf integral equations, Toeplitz matrices and linear systems, in: Toeplitz Centennial, Operator Theory: Advances and Applications, Vol. 4, Birkhauser, Basel, 1982, pp. 85-135. [6] Bart, H., Gohberg, I., Kaashoek, M.A.: Convolution equations and linear systems, Integral Equations and Operator Theory 5 (1982), 283-340.
[7] Bart, H., Gohberg, I., Kaashoek, M.A.: The coupling method for solving integral equations, in: Topics in Operator Theory and Networks, the Rehovot Workshop (Dym, H. and Gohberg, I., eds.), Operator Theory: Advances and, Applications, Vol. 12, Birkhauser, Basel, 1984, pp. 39-73. Addendum: Integral Equations and Operator Theory 8 (1985), 890-891.
[8] Bart, H., Gohberg, I., Kaashoek, M.A.: Fredholm theory of Wiener-Hopf equations in terms of realization of their symbols, Integral Equations and Operator Theory 8 (1985), 590-613.
[9] Bart, H., Gohberg, I., Kaashoek, M.A.: Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, J. Funct. Analysis 68 (1986), 1-42. [10] Bart, H., Gohberg, I., Kaashoek, M.A.: Wiener-Hopf equations with symbols analytic in a strip, in: Constructive Methods of Wiener-Hopf factorization (Gohberg, I. and Kaashoek, M.A., eds.), Operator Theory: Advances and Applications, Vol. 21, Birkhauser, Basel, 1986, pp. 39-74.
'11] Bart, H., Gohberg, I., Kaashoek, M.A.: The state space method in analysis, in: Proceedings ICIAM 87, Paris-La Vilette (Burgh, A.H.P. van der and Mattheij, R.M.M., eds.), Reidel, 1987, pp. 1-16.
[12] Bart, H., Kroon L.G.: An indicator for Wiener-Hopf integral equations with invertible analytic symbol, Integral Equations and Operator Theory 6 (1983), 1-20. Addendum: Integral Equations and Operator Theory 6 (1983), 903-904. [13] Bart, H., Tsekanovskii, V.E.: Matricial coupling and equivalence after extension, Report 9170 A, Econometric Institute, Erasmus University, Rotterdam 1991.
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Boer, H. den: Linearization of operator functions on arbitrary open sets, Integral Equations and Operator Theory 1 (1978), 19-27.
[15] Boer, H. den: Block diagonalization of Matrix Functions, Ph. D. Thesis, Vrije Universiteit, Amsterdam, 1981.
[16] Boer, H. den, Thijsse, G. Ph. A.: Semi-stability of sums of partial multiplicities under additive perturbation, Integral Equations and Operator Theory 3 (1980), 23-42. [17] Day, M.M.: Normed Linear Spaces, 3rd ed., Springer, New York, 1973.
[18] Devinatz, A., Shinbrot, M.: General Wiener-Hopf operators, Trans. A.M.S. 145 (1969), 467-494.
[19] Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, Vol. I, Operator Theory: Advances and Applications, Vol. 49, Birkhauser, Basel, 1990.
[20] Gohberg,I., Heinig, G.: The resultant matrix and its generalizations, I. The resultant operator for matrix polynomials, Acta Sc. Math. 37 (1975), 41-61 (Russian). [21] Gohberg, I., Kaashoek, M.A.: Time varying linear systems with boundary conditions and integral equations, I, The transfer operator and its properties, Integral Equations and Operator Theory 7 (1984), 325-391. [22] Gohberg, I., Kaashoek, M.A., Lay, D.C.: Spectral classification of operators and operator functions, Bull. Amer. Math. Soc. 82 (1976), 587-589.
[23] Gohberg, I., Kaashoek, M.A., Lay, D.C.: Equivalence, linearization and decompositions of holomorphic operator functions, J. Funct. Anal. 28 (1978), 102-144. [24] Gohberg, I., Kaashoek, M.A., Lerer, L., Rodman, L.: On Toeplitz and Wiener-Hopf operators with contour-wise rational matrix and operator symbols, in: Constructive Methods of Wiener-Hopf factorization (Gohberg, I. and Kaashoek, M.A., eds.), Operator Theory: Advances and Applications, Vol. 21, Birkhauser, Basel, 1986, 75-125.
[25] Gohberg, I., Kaashoek, M.A., Lerer, L., Rodman, L.: Common multiples and common divisors of matrix polynomials, II. Vandermonde and resultant matrices, Lin. Multilin. Alg. 12 (1982), 159-203.
[26] Gohberg, I., Kaashoek, M.A., Schagen, F. van: Non-compact integral operators with semi-separable kernels and their discrete analogues: Inversion and Fredholm properties, Integral Equations and Operator Theory 7 (1984), 642-703.
[27] Gohberg, I., Lancaster, P., Rodman, L.,: Matrix Polynomials, Academic Press, New York, 1982.
[28] Gohberg, I., Lancaster, P., Rodman, L.,: Invariant Subspaces of Matrices with Applications, J. Wiley and Sons, New York, 1986.
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[29] Gohberg, I.C., Sigal E.I.: An operator generalization of the logarithmic residue theorem and the theorem of Rouche, Mat. Sbornik 84 (126) (1971), 607-629 (Russian); English. Transl., Math. USSR Sbornik 13 (1971), 603-625.
[30] Heinig, G.: Uber ein kontinuierliches Analogon der Begleitmatrix eines Polynoms and die Linearisierung einiger Klassen holomorpher Operatorfunktionen, Beitrage Anal. No. 13 (1979), 111-126.
[31] Heinig, G.: Linearisierung and Realisierung holomorpher Operatorfunktionen, Wissenschaftliche Zeitschrift der Technischen Hochschule Karl-Marx-Stadt, XXII, H. 5 (1980), 453-459.
[32] Kaashoek, M.A., Mee, C.V.M. van der, Rodman, L.: Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence, Integral Equations and Operator Theory 4 (1981), 504-547. [33] Kaashoek, M.A., Schermer, J.N.M.: Inversion of convolution equations on a finite interval and realization triples, Integral Equations and Operator Theory 13 (1990), 76-103.
[34] Kaashoek, M.A., Ven, M.P.A. van de: A linearization for operator polynomials with coefficients in certain operator ideals, Annali Mat. Pura Appl. (IV) 15 (1980), 329-336. [35] Kaashoek, M.A., Verduyn Lunel, S.M.: Characteristic Matrices and Spectral Properties of Evolutionary Systems, IMA Preprint Series no. 707, University of Minnesota, 1990.
[36] Lancaster, P., Tismenetsky, M.: The theory of matrices, Second Edition with Applications, Academic Press, Orlando, Fl., 1985. [37] Leiterer, J.: Local and global equivalence of meromorphic operator functions, I, Math. Nachr. 83 (1978), 7-29; II, Math. Nachr. 84 (1978), 145-170. [38] Levin, S.: On invertibility of finite sections of Toeplitz matrices, Appl. Anal. 13 (1982), 173-184.
[39] Lerer, L., Tismenetsky, M.: The Bezoutian and the eigenvalue-separation problem for matrix polynomials, Integral Equations and Operator Theory 5 (1982), 386-445. [40] Linnemann, A., Fixed modes in parametrized systems, Int. J. Control 38 (1983), 319-335. [41] Mee, C.V.M. van der: Realization and linearization, Rapport 109, Wiskundig Seminarium der Vrije Universiteit, Amsterdam, 1979.
[42] Mitiagin, B.: Linearization of holomorphic operator functions, I, II., Integral Equations and Operator Theory 1 (1978), 114-361 and 226-249. [43] Pietsch, A. Zur Theorie der o-Transformationen in lokalkonvexen Vektorraumen, Math. Nachr. 21 (1960), 347-369.
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[45] Roozemond, L.: Systems of Non-normal and First Kind Wiener-Hopf Equations, Ph.D. Thesis, Vrije Universiteit, Amsterdam, 1987. [46] Speck, F.-O.: General Wiener-Hopi Factorization Methods, Pitman, Boston, 1985.
H. Bart Econometric Institute Rotterdam Erasmus University P.O. Box 1738
3000 DR Rotterdam, The Netherlands V.E. Tsekanovskii 18 Capen Blvd Buffalo, N.Y. 14214 USA
MSC: Primary 47A05, Secondary 47A20
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
161
OPERATOR MEANS AND THE RELATIVE OPERATOR
ENTROPY
Jun Ichi Fujii
The notion of operator monotone functions was introduced by L"owner and that of operator concave functions by Kraus who is his student. Operator means were introduced by Ando and the general theory of them was established by Kubo and Ando himself. By their theory, a nonnegative operator monotone function is now considered as a variation of an operator mean. However this theory does not include the logarithm and the entropy function which are operator monotone and often used in information theory. These functions are operator concave and satisfy Jensen's inequality. So, considering operator means from the historical viewpoint, we shall introduce the relative operator entropy by generalizing the Kubo-Ando theory. Though its definition is derived from the Kubo-Ando theory of operator means, it can be constructed also in some ways. The relative operator entropy has of course some entropy-like properties.
1. INTRODUCTION The theory of operator means is initiated by T.Ando [2] and established by F.Kubo and himself [20] in connection with L6wner's theory for operator functions [19,21].
Roughly speaking, there are two origins of this theory. One is the parallel sum for positive semidefinite matrices which was discussed by W.N.Anderson et al [1,9] in electrical network
theory. The other is the geometric mean for positive sesquilinear forms which was discussed by W.Pusz and S.L.Woronowicz [23].
On the other hand, H.Umegaki [26] introduced the relative entropy for states on
Fujii
162
an operator algebra and it was developed by A.Uhlmann [25] by making use of the method
of Puss and Woronowicz. M.Nakamura and Umegaki [22] also introduced the operator
entropy for positive operators and showed that the entropy function i(x) = -z log a is operator concave. So E.Kamei and the author [11] defined the relative operator entropy
S(AIB) for positive operators as a generalization of the Kubo-Ando theory of operator means. For positive invertible operators A and B, it is defined by S(AIB) = A112log(A-1/2BA-1/2)A1/2.
In this note, we show the relative operator entropy has some properties like operator means and the relative entropy and it can be constructed in some ways which its origins reflect. As an application of S(AIB) to operator algebras, we see the relation between the relative operator entropy and Jones' index.
2. ORIGINS OF OPERATOR MEANS First we see the parallel sum A : B which corresponds with the impedance matrix of the parallel connection in electrical network theory. It was introduced first for positive semidefinite matrices in Anderson and Duffin [1] and second for positive operators
on a Hilbert space in Fillmore and Williams [9]. It is characterized by (A : Be, z) = inf { (Ay, y) + (Bz, z) I y + z = x }.
If A and B are invertible, then
A : B = (A-1 + B-1)-1 = A(A + B)-1B = A1/2(1 + A1I2B-1A1/2)-1A112 = B1/2(1 + B1/2A-1B1/2)-1B1/2.
In the theory of operator algebra, Pusz and Woronowicz [23] introduced the geometrical mean
for positive sesquilinear forms gyp, ,o on a vector space V: Put
H={aEV I cp(z,a)+1G(z,a)=0}. For a quotient map, V - V/JV, a H i, define an inner product on V/J1( by < i, y >= fo(z, y) +'RG(z, y).
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Then we have a Hilbert space ?f as the completion of V/Al and the derivatives A, B on
ii by < Ai, y >= rp(z, y)
< Bi, j >= O(x, y).
and
Since A and B commutes by A + B = 1, we can define a positive sesquilinear form by
V/R(21 y) _< A1/2B1/2i,y > . Then they showed that its definition does not depend on representations:
THEOREM(Pusz-Woronowicz). If there ezists a map, x ,--* i, onto a dense set of a Hilbert space /H with commuting derivatives C and D 9/1(x, x) _< Dil i >H,
cp(x, x) =< Cil i >H, then
(x, y) =< C1/2D1/2i l y >H.
More generally, if f (t, s) is a suitable (homogeneous) function (see also [24]), then one can define f (,p, 0) by y) =< f(C, D)z 19 >H
3. OPERATOR. MEANS AND OPERATOR. MONOTONE FUNCTIONS Seeing these objects, Ando [2] introduced some operator means of positive operators on a Hilbert space: ((
max (X > 0
(AX
geometric mean :
AgB
harmonic mean :
AhB - max {X > 0 I
B) >
I
(20
01,
2B) >
As a matter of fact, we have AhB = 2A : B and < AgBx, y >_ < Like numerical case, the following inequalities hold:
AhB
(Xx
B, >(z, y).
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164
In their common properties, note the following inequality called the transformer one:
T'(AmB)T < TAT m T'BT. If T is invertible then the equality holds in the above. Indeed, we have
T'(AmB)T < T'ATmT'BT = T'Ti-1(T'ATmT'BT)T-1T < T'(AmB)T. In particular,
AmB = Al/2(1 m A-1/2BA-1/2)A1/2 =
B112(B-1/2AB-1/2
m 1)B1/2
for invertible A and B. On the other hand, Lowner [21] defined the notion of operator monotone func-
tions. A real-valued (continuous) function f on an interval I in the real line is called operator monotone on I and denoted by f E OM(I) if
A < B implies f (A) < f (B) for all selfadjoint operators A, B with o(A), c(B) C I.
Then, for an open interval I, a function f is monotone-increasing analytic function and characterized in some ways: 1. Every Lowner matrix is positive semi-definite:
-a'
for
a1 < t1 < as < t2 < ... < a< tE I.
2. There esists an analytic continuation f of f to the upper half plane and Im j (z) > 0
for Imz>0. 3. f has a suitable integral representation, see also [2,5]. Now we see the general operator means due to Kubo and Ando [20]. A binary
operation m among positive operators on a Hilbert space is called an operator mean if it satisfies the following four axioms:
monotonousness: lower continuity:
A < C, B < D
AmB < CmD,
A. I A, B. I B : A,.mB. I AmB,
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transformer inequality: normalization:
T'(AmB)T < T'ATmT'BT,
and
AmA = A.
A nonnormalized operator mean is called a connection. For invertible A, we have
AmB = A""=f,,,(A-112BA-1/2)A1/2
(1)
and fn(z) = 1mz is operator monotone on [0, oo). (Note that fn(z) is a scalar since f,,,(z) commutes with all unitary operators by the transformer `equality'.) By making use
of an integral representation of operator monotone functions, we have a positive Radon measure A. on [0, oo] with
AmB = aA + bB +
(2)
fm-)
(tA) : B 1 + t dµm(t) t
where a = f,,,(0) = µ,,,({0}) and b = inftfm(1/t) = µ,,,({oo}). So the heart of the Kubo-Ando theory might be the following isomorphisms among them:
THEOREM (Kubo-Ando).
Maps m F+ f,,, and m -. µ,,, defined by (1)
and (2) give aoine order-isomorphisms from the connections to the nonnegative continuous
operator monotone functions on [0, oo) and the positive Radon measures on [0, oo]. If m
is an operator mean, then f,,,(1) = 1 and µ is a probability measure. Here f,,, (resp. µ,,,) is called the representing function (resp. measure) for m.
4.
OPERATOR CONCAVE FUNCTIONS AND JENSEN'S IN-
EQUALITY Like operator monotone functions, a real-valued (continuous) function F on I
is called operator concave on I and denoted by F E OC(I) if
F(tA + (1 - t)B) > tF(A) + (1 - t)F(B)
(0 < t < 1)
for all selfadjoint operators A, B with o(A), a(B) C I. (If -F is operator concave, then F is called operator convex.) Then, for an open interval I, a function F is concave analytic
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function and characterized by (see [5])
F'[a](z) = -
F(z)
_
F(a)
E OM(I)
(a E I).
a
Typical examples of operator concave function is the logarithm and the entropy function
n(z) - -z log z. In fact, Nakamura and Umegaki [22] proved the operator concavity of ,1 and introduced the operator entropy
H(A) - -A log A > 0 for positive contraction A in B(H) (see also Davis [7]). In the Kubo-Ando theory, the following functions are operator concave: f (z) _
lmz, f °(z) = zml E OC[0, oo) and Fn(z) - zm(l - z) E OC[0,1]. Moreover, Fm gives an bridge between OC[0, l]+ and OM(0, oo)+ via operator means, see [10]:
THEOREM 4.1. A map m'--' Fm defines an affine order-isomorphism from the connections to nonnegative operator concave functions on [0, 1].
One of the outstanding properties of operator concave functions is so-called
Jensen's inequality. For a unital completely positive map -t on an operator algebra and a positive operator A, Davis [6] showed
+(F(A)) < F(4k(A)) for an operator concave
function F. By Stinespring's theorem, a completely positive map is essentially a map
X F-4 C*XC. For a nonnegative function f, note that f E OM[0,oo) if and only if f E OC[0, oo) cf. [16]. So Jensen's inequality by Hansen [15] is
C-f(A)C < f(C-AC) for
D1CII < 1, A > 0.
For nonnegative f E OM[0, oo), there exists a connection m; f(z) Then, the transformer inequality implies
1mz.
C'f(A)C = C'(lmA)C < C*CmC'AC < 1mC'AC = f(C'AC). Hansen and Pedersen [16] gave equivalent conditions that Jensen's inequality holds:
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167
THEOREM(Hansen-Pedersen). For a continuous real function F on [0, a), the followings are equivalent: For 0 < A, B < a,
(1) C'F(A)C < F(C'AC) for JJCJJ < 1, (2) F E OC[0,a)
and
(3) PF(A)P < F(PAP)
F(0) > 0, for
every projection P,
(4) C'F(A)C + D'F(B)D < F(C'AC+D'BD) for C'C+D'D < 1. In the Kubo-Ando theory, for f (z) = lmz E OM(0, oo)+, the transpose is f °(z) = zml = z f (1/z). Adopting this definition for f E OM(0, oo), we have f °(z) _
-z log z = ,i(z) for f (z) = log z. In general, the transpose off E OM(0, oo) is just a function satisfying the above equivalent conditions (see [2,14,16]):
THEOREM 4.2.
f (z) E OM(0, oo) if and only if f °(z) E OC[0, oo) and
f°(0)>0. This theorem suggests that one can generalize the Kubo-Ando theory dealing with OM(0, oo)+, see [14].
5. RELATIVE OPERATOR ENTROPY Now we introduce the relative operator entropy S(AIB) for positive operators
A and B on a Hilbert space. If A and B is invertible, then it is defined as S(AIB)
A'I'log(A-1/2 BA- 1/2 )A'/2
= B1/2'7(B-1i2AB-1/2)B1/2.
The above formula shows that S(AIB) can be defined as a bounded operator if B is invertible. Moreover, S(AJB + e) is monotone decreasing as e 10 by log z E OM(0, oo). So, even if B is not invertible, we can define S(AIB) by (3)
S(AIB) = slim S(AJB+e)
if the limit exists. Here one of the existence conditions is (see [14]):
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168
THEOREM 5.1. The strong limit in (3) exists if and only if there exists c with
c < tB - (logt)A (t > 1). Under the existence, the following properties like operator means hold:
right monotonousness:
B < C = S(AI B) < S(AI C),
right lower continuity:
Bn J. B = S(AI Bn) I S(AI B),
transformer inequality:
T'S(AIB)T < S(T'ATIT'BT).
Conversely, if an operator function S'(AIB) satisfies the above axioms, then there exist f E OM(O, oo) and F E OC[O, oo) with F(O) > 0 such that
S'(AIB) =
A'/2f(A-1/2BA-1/2)A1/2 = B1/2F(B-1/2AB-112)B1/2
for invertible A, B > 0, so that the class of such functions S' is a generalization of that of operator means or connections, see [14]. In addition, the relative operator entropy has entropy-like properties, e.g.:
subadditivity:
S(A + BIG + D) > S(AIC) + S(CID),
joint concavity:
S(AIB) > tS(A1IB1)+(1 -t)S(A2IB2)
ifA=tAl+(1-t)A2 and B=tB1+(l-t)B2 for 0
-1(S(AIB)) < S(4;(A)I+(B))
for a normal positive linear map 4k from a W*-algebra containing A and B to a suitable
W*-algebra such that +(1) is invertible. In particular, we have
Peierls-Bogoliubov inequality:
cp(S(AI B)) < rp(A)(log p(A) - log ip(B))
for a normal positive linear functional ip on a W*-algebra containing A and B.
Now we apply S(AIB) to operator algebras. Let E be the conditional expec-
tation of a type III factor M onto a subfactor N, define a maximal entropy S( N ) as
S( N ) = sup{IIS(AIE(A))II I A E Mi }. Then, for Jones' index [M : Afl, we have (see [13])
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169
THEOREM 5.2. Let A( be a subfactor of a type III factor M. Then, S(N) = log[M :Al]. Here we recall the relative entropy S(iplo) for states ip, 0 on an operator algebra, cf., [27]. Derived from the Kullback-Leibler information (divergence): n PA, log
ti=I
Pw
for probability vectors p, q,
qw
Umegaki [26] introduced the relative entropy S(jpl r') for states (p, -0 on a semi-finite von Neumann algebra, which is defined as S(coI 4) = r(A log A - A log B)
where A and B are density operators of ip and b respectively, i.e.,
p(X) =,r(AX) and 1G(X) = r(BX). Araki [3] generalized it by making use of the Tomita-Takesaki theory, Uhlmann
[25] by the quadratic interpolation and Pusz-Woronowicz [24] by their functional calculus. These generalizations are all equivalent. The constructions of the last two en-
tropies are based on the Pusz-Woronowicz calculus:
Put positive sesquilinear forms
+(X,Y) _ cp(X*Y) and'Y(X,Y) = I/i(XY'), then Spw6o &)
(f log
)(1,1),
1(1,1).
Su('PI16) = -limo f
According to these definition, we see some constructions of S(AAB). Making
ase of the fact (zt - 1)/t J. log z as t j 0, we have (see [11,12])
Uhlmann type:
S(AIB) = s- lim
tjo
AgtB - A t
where gt is the operator mean satisfying lgtz = zt. Note that this formula gives an approximation of S(AIB). Putting 4(z, y) =< As, y > and %(z, y) =< Bz, y >, we have
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170
< S(AI B)z, y >= -(flog f)(z, y).
Pusz-Woronoiwicz type:
In [18], S.Izumino discussed quotient of operators by making use of Douglas' majorization
theorem [8] and the parallel sum, which is considered as a space-free version of the Pusz-
Woronowicz method. By making use of this, we also construct S(AI B): Let R = (A +
B)"2. Then, there exist X and Y with XR = A1/2 and YR = B1!2, which are uniquely determined by kernel conditions
ker R C ker X fl ker Y. Here X*X +Y*Y is the projection onto ran R and X*X commutes with Y*Y. Then, for F(z) = S(zll - z) = -z log(z/(1 - z)), we have
Izumino type:
S(AIB) = R(F(X*X))R.
Recently, Hiai discussed a bridge between the relative entropy and the relative
operator entropy. Note that if the density operators A and B commute, then S(j'I O) (= Spw(ipl O) = Su(so G)) = -r(S(AI B)).
Hiai and Petz [17] pointed out that the last term SBS('PI ,O) ° -r(S(AIB))
had already been discussed by Belavkin and Staszewski [4]. Hiai and Petz showed that S6vl1G) > SBS('PI'b)
for states on a finite dimensional C*-algebra. Furthermore, Hiai informed us by private communication that it also holds for states defined by trace class operators.
171
Fujii
REFERENCES [1] W.N.Anderson and R.J.Duffin: Series and parallel addition of matrices, J. Math. Anal. Appl., 26(1969), 576-594. [2] T.Ando: Topics on operator inequalities, Hokkaido Univ. Lecture Note, 1978.
[3] H.Araki: Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Univ., 11 (1976), 809-833.
[4] V.P.Belavkin and P.Staszewski: C*-algebraic generalization of relative entropy and entropy, Ann. Inst. H. Poincare Sect. A.37(1982), 51-58. [5] J.Bendat and S.Sherman: Monotone and convex operator functions, Trans. Amer. Math. Soc., 79 (1955), 58-71. [6] C.Davis: A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc., 8(1957), 42-44. [7] C.Davis: Operator-valued entropy of a quantum mechanical measurement, Proc. Jap. Acad., 37(1961), 533-538. [8] R.G.Douglas: On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc., 17(1966), 413-416. [9] P.A.Fillmore and J.P.Williams: On operator ranges, Adv. in Math., 7(1971), 254-281. [10] J.I.Fujii: Operator concave functions and means of positive linear functionals, Math. Japon., 25 (1980),453-461. [11] J.I.Fujii and E.Kamei: Relative operator entropy in noncommutative information theory, Math. Japon., 34 (1989), 341-348. [12] J.I.Fujii and E.Kamei: Uhlmann's interpolational method for operator means. Math. Japon., 34 (1989), 541-547. [13] J.I.Fujii and Y.Seo: Jones' index and the relative operator entropy, Math. Japon., 34(1989), 349-351.
[14] J.I.Fujii, M.Fujii and Y.Seo: An extension of the Kubo-Ando theory: Solidarities, Math. Japon, 35(1990), 387-396. [15] F.Hansen: An operator inequality, Math. Ann., 246(1980), 249-250. [16] F.Hansen and G.K.Pedersen: Jensen's Inequality for operators and Lowner's theorem, Math. Ann., 258(1982), 229-241. [17] F.Hiai and D.Petz: The proper formula for relative entropy and its asymptotics in quantum probability, Preprint. [18] S.Izumino: Quotients of bounded operators, Proc. Amer. Math. Soc., 106(1989), 427-435.
[19] F.Kraus: Uber konvexe Matrixfunctionen, Math. Z., 41(1936), 18-42.
[20] F.Kubo and T.Ando: Means of positive linear operators, Math. Ann., 248 (1980) 205-224.
[21] K.L6wner: Uber monotone Matrixfunctionen, Math. Z., 38(1934), 177-216. [22] M.Nakamura and H.Umegaki: A note on the entropy for operator algebras, Proc. Jap. Acad., 37 (1961), 149-154. [23] W.Pusz and S.L.Woronowicz: Functional calculus for sesquilinear forms and the purification map, Rep. on Math. Phys., 8 (1975), 159-170.
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[24] W.Pusz and S.L.Woronowicz: Form convex functions and the WYDL and other inequalities, Let. in Math. Phys., 2(1978), 505-512. [25] A.Uhlmann: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys., 54 (1977), 22-32. [26] H.Umegaki: Conditional expectation in an operator algebra IV, Kodai Math. Sem. Rep. 14 (1962), 59-85. [27] H.Umegaki and M.Ohya: Entropies in Quantum Theory (in Japanese), Kyoritsu, Tokyo (1984).
Department of Arts and Sciences (Information Science), Osaka Kyoiku University,
Kasiwara Osaka 582 Japan MSC 1991: Primary 94A17, 47A63 Secondary 45B15, 47A60
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
173
AN APPLICATION OF FURUTA'S INEQUALITY TO ANDO'S THEOREM
Masatoshi Fujii*, Takayuld Furuta ** and Eizaburo Kamei ***
Several authors have given mean theoretic considerations to Furuta's inequality which is an extension of Lowner-Heinz inequality. Ando discussed it on the geometric mean. In this note, Furuta's inequality is applied to a generalization of Ando's theorem.
1. INTRODUCTION. Following after Furuta's inequality, several operator inequalities have been presented in [1,3,4,5,6,9,12]. We now think that they have suggested us a new progress of Faruta's inequality. In particular, Ando's result in [1] has been inspiring us this possibility, cf. [3,5,6]. Here we state Furuta's inequality [7] which is the starting point in our discussion.
FURUTA'S ]INEQUALITY. Let A and B be positive operators acting on a Hilbert space. If A > B > 0, then (1)
(BT
APB')1/q > B(P+2r)lq
and
(2)
A(P+2*)/q >
(A*BPA')1lq
for all p, r > 0 and q > 1 with (1 + 2r)q > p + 2r . /
Fujii et al.
174
If we take p = 2r and q = 2 in (2), then we have Ap > (Ap/2BPAP/2)1/2
(3)
for all p > 0. From the viewpoint of this, Ando [1] showed that for a pair of selfadjoint operators A and B, A > B if and only if the exponential version of (3) holds, i.e., epA > (epA/2epBepA/2)1/2
for all p > 0. So we pay attention to the exponential order due to Hansen [10] defined by eA > eB, and introduce an order among positive invertible operators which is just opposite to the exponential one. That is, A >> B means log A > log B. We call it the chaotic order because log A might be regarded as degree of the chaos of A. Thus Ando's result in [1] is rephrased as follows :
THEOREM A. Let A and B be positive invertible operators. Then the following conditions are equivalent (a) A >> B. (b) The following inequality holds for all p > 0;
Ap > (AP/2BPAP/2)1/2, i.e., A-P g B' < I.
(3)
(c) The operator function G(p) = A-P g BP p > 0, where g is the geometric mean.
is monotone decreasing for
In this note, we first propose the following operator inequalities like Furuta's one, which are improvements of the results in the preceding note [6]
:
THEOREM 1. Let A and B be positive invertible operators. If A >> B, then (4)
)2'l(r+2r) > B2'
and (5)
A2r > (A'BPA')2rl(P+2r)
for allp,r>0. This is a nice application of Furuta's inequality and implies the monotonity of an operator function discussed in [3] (see the next section), which is nothing but an extension of Theorem A by Ando. As a consequence, we also obtain Faruta's inequality in case of 2rq > p + 2r under the chaotic order.
Fujii et at.
175
2. OPERATOR. FUNCTIONS. Means of operators established by Kubo and Ando [13] fit right in with our plan as in [4,5,6]. A binary operation m among positive operators is called a mean if m is upper-continuous and it satisfies the monotonity and the transformer inequality
T'(A m B)T < T* AT m T'BT for all T. We note that if T is invertible, then it is replaced by the equality
T'(A m B)T = T'AT m T'BT. Now, by the principal result in [13], there is a unique mean m, corresponding to the operator monotone function z' for 0 < a < 1;
1m,z = z' for z > 0. Particularly the mean g = m1/2 is called the geometric one as in the case of scalars. In the below, we denote m(1+,)/(P+,) by m(P,,) for all p > 1 and a > 0. Here we can state our recent result in [3], which is a nice application of Furuta's inequality.
THEOREM B. If A > B > 0, then (6)
M(p,r) = B -2, M(,,2,) Al
is a monotone increasing function, that is,
M(p+t,r+a)>M(p,r) for p > 1 and r, a, t > 0. On the other hand, we have attempted mean theoretic approach to Furuta's inequality in [2,8,11,12]. It is expressed as (7)
M(p,r) =
B-zr
m(P,2,) AP > B
and equivalently (7')
N(p,r) = A-2' M(,.2,) BP < A
under the assumption A > B, p > 1 and r > 0. However the argument in [12], [9] and [3] might say that the key point of Furuta's inequality should be seen as (8)
under the same assumption.
M(p, r) = B-z' M(,, 2,) AP > A
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176
Concluding this section, we state that (4) and (5) are rephrased to B-2?
(4')
m2a/(P+2r) AP > 1
and
A-2' m2,/(P+2,) BP < 1
(5')
respectively. If we take p = 2r in (5'), then it is just (3) in Theorem A.
3. FUR.UTA'S TYPE INEQUALITIES. In this section, we prove Furuta's type inequalities (4) and (5) in Theorem 1. We will use Ando's result (3). Moreover we need the following lemma on a mean m,, cf. [9].
LEMMA 2. Let C and D be positive invertible operators and 0 < a < 1. Then
(a)Cm,DDm1_,C, (b) D m, C = (D-1 and consequently
(c) C m, D = (D-1 m1_,
C-1)-1.
PROOF. The function fn(z) = 1 m z is called the representing function for a mean m and the map : m -+ fn is an affine isomorphism of the means onto the operator monotone functions by [13]. Therefore it suffices to check that zm,1=1m1_,zandIm,a=(1m,x-1)-1
for x > 0. Actually we have easily
z m, 1 = z(1 m, z-1) = z1-' = 1 m1_, z by the transformer `equality', and
1 m, z = x' = (z-')-1 = (1 m,
z-1)-1.
PROOF OF THEOREM 1. Assume that A >> B. Then it follows from Theorem A that C = AP > (AP/2BPAP/2)1/2 = D.
Suppose that 2r > p > 0 and take t > 0 with 2r = p(1 + 2t). Faruta's inequality (7') ensures that C > C-29 'n(2,2t) D2.
Fujii et al.
177
In other words,
C1+2t > (CtD2Ct)(1+2t)/(2+2t)
and so
A2' > (A' BP A') 2s l (p+2') .
Next we have to show the case where p > 2r > 0. Since B-1 >> A-', Theorem A also implies that
B-2' > (B-'A-2'B-+)1/2, that is, (B'A2'B')1/2 > B2r Again applying Furuta's inequality (7) to this, we have B-4't
and so
ln(2,2t) B'A2'B' > B2',
B-2t(1+2t) m(a 2t) A2a > I.
If we choose t > 0 with p = 2r(1 + 2t) since p > 2r > 0, then 1 - (1 + 2t)/(2 + 2t) _ 2r/(p + 2r) and consequently it is equivalent to
A-2' m2,/(P+2,) B' < I by Lemma 2 (c). This completes the proof.
The following corollary of Theorem 1 plays an important role in the next section.
COROLLARY 3.
If A >> B, then
(B'APB')'l(P+2r) > B'
(9)
forp > 0 and 2r > s >0, and As > (AP/2B2*AP/2)sl(P+2')
(10)
forallr>0 andp>s>0. 4. AN APPLICATION TO ANDO'S THEOREM. Finally we discuss a generalization of Theorem A by Ando [1]. Such an attempt has been done by [3], cf. also [9]. The purpose of this section is to complete it. A modification of Theorem B might be considered as in [3]. Let us define m'(P.,,t) = m'(t+s)/(P+,)
for p > t > 0 ands >0. Clearly m(P,s,1) = m(P s) .
Fujii et al.
178
THEOREM 4. If A >> B, then for a given t > 0 Mt(p, r) =
B-2r
m(P,2,,t) A"
is monotone increasing for p > t and r > 0.
PROOF.
First of all, we prove that for a fixed r > 0, Mt(p+s, r) > Mt(p, r) for p > s > 0. Putting m = m(P+,,27t), it follows from (10) that
Mt(p+s,r) = B-2r m
AP+'
= AP/2(A-P/2B-2rA-P/2 m A')AP/2 > m (AP/2B2'APl2)'l(P+2*))AP/2 = AP/2(A-P/2B-2'A-P/2)(P-t)l(P+2r)AP/2 AP/2((APl2B2'AP/2)-1
AP m(P-t)/(P+2r)
B-2r
= B-2r m(P,2r,t) AP. The last equality is implied by Lemma 2 (a). Next we show the monotonity on r. Putting m = m(P 2,+, t) for 2r > s > 0, it follows from (9) that
Mt(p, r + s/2) = B-'(B-' m B'APB')B-' > B-'((B'APB7)-'/(p+2?) m B'APB')B r = B-'(B'APB')(t+2r)l(P+2r)B-'
= Mt(p,r) As a result, Theorem A has the following generalization.
THEOREM 5. For positive invertible operators A and B, the following conditions are equivalent (a) A >> B. (b) For each fixed t > 0, Mt(p,r) > At for r > 0 and p > t.
(c) For each fixed t > 0, Mt(p, r) is a monotone increasing function for r > 0
and p>t. Finally, we mention that Furuta's inequality is extended to the following in the sense of (8). Actually, if we take t = 1 in (b) of Theorem 5, then we have :
COROLLARY 6. (8)
If A >> B, then (8) holds, that is,
M(p,r) = B-2' m(,,2,) AP > A.
j ujii et al.
179
REFERENCES [1] T.Ando, On some operator inequalities, Math.Ann., 279 (1987), 157-159. [2] M. Fujii, Furuta's inequality and its mean theoretic approach, J. Operator Theory, 23 (1990), 67-72. [3] M.Fujii, T.Furuta and E.Kamei, Operator functions associated with Furuta's inequality, Linear Alg. its Appl., 149 (1991), 91-96.
[4] M.Fujii and E.Kamei, Furuta's inequality for the chaotic order, Math.Japon., 36 (1991), 603-606.
[5] M.Fujii and E.Kamei, Furuta's inequality for the chaotic order, II, Math. Japon., 36 (1991), 717-722.
[6] M.Fujii and E.Kamei, Furuta's inequality and a generalization of Ando's theorem, Proc. Amer. Math. Soc., in press. [7] T.Furuta, A > B > 0 assures (B'APB')1/q > B(p+2r)/q for r > O, p > O, q > 1 with (1 + 2r)q > p + 2r, Proc.Amer.Math.Soc., 101 (1987), 85-88. [8] T.Furuta, A proof via operator means of an order preserving inequality, Linear Alg. its Appl., 113 (1989), 129-130. [9] T.Furuta, Two operator functions with monotone property, Proc.Amer.Math.Soc., 111 (1991), 511-516.
40] F.Hansen, Selfadjoint means and operator monotone functions, Math.Ann., 256 (1981), 29-35.
11] E. Kamei, Furuta's inequality via operator mean, Math.Japon., 33 (1988), 737-739. j12] E. Kamei, A satellite to Furuta's inequality, Math.Japon., 33 (1988), 883-886. 1i3] F.Kubo and T.Ando, Means of positive linear operators, Math.Ann., 246 (1980), 205-224. * Department of Mathematics, Osaka Kyoiku University, Tennoji, Osaka 543, Japan ** Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, Kagurazaka, Shinjuku, Tokyo 162, Japan 'R* Momodani Senior Highschool, Ikuno, Osaka 544, Japan
MSC 1991: Primary 47A63 Secondary 47B15
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
180
APPLICATIONS OF ORDER PRESERVING OPERATOR INEQUALITIES TAKAYUKI FURUTA
A > B > 0 assures (B.APBr)l/q > B(P+2r)/q for r > 0, p > 0, q > 1 with (1 + 2r)q > (p + 2r). This is Furuta's inequality. In this paper, we show that Furuta's inequality can be applied to estimate the value of the relative operator entropy and also this inequality can be applied to extend Ando's result. §0.
INTRODUCTION
An operator means a bounded linear operator on a complex Hilbert space. In this
paper, a capital letter means an operator. An operator T is said to be positive if (Tx, x) > 0 for all x in a Hilbert space. We recall the following famous inequality ; if
A > B > 0, then A' > B' for each a E [0, 1]. This inequality is called the LownerHeinz theorem discovered in [14] and [12]. Moreover nice operator algebraic proof was
shown in [16]. Closely related to this inequality, it is well known that A > B > 0 does not always ensure AP > BP for p > 1 in general. As an extension of this Lowner-Heinz theorem, we established Furuta's inequality in [7] as follows; if A > B > 0, then for each r > 0, (BrAPBr)1/q > B(P+2r)/ and
A(P+2r)/q > (ArBPAr)l/q
hold for each p and q such that p > 0,q > 1 and (1 + 2r)q > p + 2r. We remark that Furuta's inequality yields the Lowner-Heinz theorem when we put r = 0. Also we remark that although AP > BP for any p > 1 does not always hold even if A > B > 0,
Furuta's inequality asserts that f (AP) > f (BP) and g(AP) > g(BP) hold under the suitable conditions where f (X) = (B'XBr)1/9 and g(Y) = (ArYAr)1/9. Alternative proofs of Furuta's inequality are given in [4] [8] [9] and [13]. The relative operator entropy
for positive invertible operators A and B is defined in [2] by
S(A I B) = Al/2(logA-1/2BA-1/2)Al/2.
Furuta
181
In [11], we showed that Furuta's inequality could be applied to estimate the value of this relative operator entropy S(A I B). For example, let A,B and C be positive invertible operators. Then logC > logA > logB holds if and only if S(A_r
I Cp) > S(A-r I AP) >
S(A-r
I BP)
holds for all p > 0 and all r > 0. In particular logC > logA-1 > logB ensures S(A I C) > -2AlogA > S(A I B) for positive invertible operators A, B and C. In this paper, we shall attempt to extend this result by using Furuta's inequality. In [11], we showed an elementary proof of the following result which is an extension
of Ando's one [1]. Let A and B be selfadjoint operators. Then A > B holds if and only
if for a fixed t>0, Fe(p, r) = e-rB (erBepA erB) (t+2r)/(p+2r) e-rB
is an increasing function of both p and r for p > t and r > 0. In this paper, also by using Furuta's inequality we shall attempt to extend this result. §1.
APPLICATION TO THE RELATIVE OPERATOR ENTROPY
We shall show that Furuta's inequality can be applied to estimate the value of the relative operator entropy in this section. Recently in [2], the relative operator entropy S(A I B) is defined by S(A I B) = Al/2(logA-1/2BA-1/2)A1/2
for positive invertible operators A and B. We remark that S(A I I) = -AlogA is the usual operator entropy. This relative operator entropy S(A I B) can be considered as an extension of the entropy considered by Nakamura and Umegaki [15] and the relative entropy by Umegaki [17].
THEOREM 1. Let A and B be positive invertible operators. Then the following assertions are mutually equivalent.
(I) logA > logB. (IIo) Ap > (Ap/2BPAp/2)1/2 for all p > 0.
(II1) AP > (AP/2B8AP/2)P/(P+8) for all p > 0 and ails > 0.
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Furuta
(112) AP > (AP/2B90AP/2)p/(p+9°) for a fixed positive number so and for all p such that p E [0, po], where po is a fixed positive number. (113) AP0 > (AP0/2B8AP0/2)P0/(P0+9) for a fixed positive number po and for all s such that
s E [0, so], where so is a fixed positive number.
(IIII) logAP+9 > log(AP/2BBAP/2) for all p > 0 and all s > 0. (1112) logAP+9p > log(Ap12B90AP/2) for a fixed positive number so and for all p such that
p E [0, po], where po is a fixed positive number.
THEOREM 2. Let A and B be positive invertible operators. Then the following assertions are mutually equivalent.
(I) logC > logA > logB (IIo) (AP/2CPAP/2)1/2 > AP > (AP/2BPAP/2)1/2 for all p > 0.
(iii)
(AP/2C3AP/2)p1(P+9) > AP > (Ap12BsAP12)P1(p+9) for all
p J 0 and all s > 0.
(112) (AP/2C80AP/2)p/(P+s°) > AP > (APl2B90Ap/2)Pl(P+9°) for a fixed positive number so
and for all p such that p E [0, po], where po is a fixed positive number. (113) (AP-/2C3APo/2)p0/(p°+s) > Ap0 > (APO/2B8AP0/2)p°/(P0+9) for a fixed positive num-
ber po and for all s such that s E [0, so], where so is a fixed positive number. (IIII) log(AP12CsAP/2) > logAP+s > log(AP12B8AP12) for all p > 0 and all s > 0. (1112) log(AP12C3°AP12) > logAp+9° > log(AP/2Bs0AP12) for a fixed positive number so
and for all p such that p E [0, po], where po is a fixed positive number.
(IVi) S(A-P I Cs) > S(A-P I As) > S(A-P I Bs) for all p > 0 and all s > 0. (IV2) S(A-P I C'°) > S(A-P I As0) > S(A-P I B'°) for a fixed positive number so and for all p such that p E [0, po], where po is a fixed positive number.
COROLLARY 1 [11]. Let A, B and C be positive invertible operators.
If logC > logA-1 > logB, then S(A I C) > -2AlogA > S(A I B). In order to give proofs to Theorem 1 and Theorem 2, we need the following Furuta's inequaliy in [7].
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Vuruta
THEOREM A (Furuta's inequality). Let A and B be positive operators on a Hilbert space. If A > B > 0, then (BrAPBr')(1+2r)/(P+2r) > B1 +2r
(i)
and Al+2r > (ArBPAr)(i+2r)/(P+2r)
(ii)
hold for all p > 1 and r > 0.
LEMMA 1 [10].
Let A and B be invertible positive operators.
For any real
number r, (BAB)r = BA1/2(A1/2B2A1/2)r-1A1/2 B.
LEMMA 2. Let A and B be positive invertible operators. Then for any p, s > 0, the following assertions are mutually equivalent. (i)
AP > (AP/2B8AP/2)P1(8+P),
(ii)
(B8/2APB8/2)8/(s+P) > B8.
Proof of Lemma 2. Assume (i). Then by Lemma 1, AP > (AP/2B8AP/2)P1(8+P)
=
AP/2B8/2(B8/2APB8/2)-s/(8+P) B8/2AP/2,
that is, B-8 > (B8/2APB812)-s1(8+P)
holds. Taking inverses proves (ii). Conversely, we have (i) from (ii) by the same way.
Proof of Theorem 1. (I)
(IIo) is shown in [1]. (II1)
. (IIo) is obvious by As > (As/2B8As/2)1/2
putting s=p in (III). We show (IIo) . (III). Assume (IIo) ; for all s > 0. Then by (ii) of Theorem A, we have the following inequality (1) (1)
Ae(1+2t) > {Ast(A8/2B8A8/2)m/2Ast}(1+2t)/(m+2t)
form>1andt>0.
Putting m = 2 in (1), we have (2)
A8(1+2t) > {A8(t+1/2)B8A8(t+1/2)}(1+2t)/(2+2t)
fort>0.
Put p = s(1 + 2t) in (2). Then (1 + 2t)/(2 + 2t) = p/ (s + p), so we have
(3)
AP > (AP/2B8AP/2)P/(8+P) for all p and s such that p > s > 0,
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Furuta
because p = s(1 + 2t) > s. On the other hand (IIo) is equivalent to the following (4) by Lemma 2, (BP/2APBP/2)1/2 > B" for all p > 0.
(4)
Then appling (i) of Theorem A to (4), we have the following (5) (5)
{BPu(BP/2APBP/2)m/2BPu}(1+2u)/(m+2u) > BP(1+2u)
form > 1 and u > 0.
Put m = 2 in (5). Then we have {BP(u+1/2)APBP(u+1/2)}(1+2u)/(2+2u) > Bp(142u)
(6)
for u > 0.
Put s = p(1 + 2u) in (6). Then (1 + 2u)/(2 + 2u) = s/(p + s), so we have (B3/2APB8/2)3/(9+P) > B9 for all p and s such that s > p > 0,
(7)
because s = p(1 + 2u) > p. (7) is equivalent to the following (8) by Lemma 2 AP > (AP/2B3AP/2)P/(3+P) for all p and s such that s > p > 0.
(8)
Hence the proof of (III) is complete by (3) and (8).
(III) : (112) and (III)
(113) are obvious since (112) and (113) are both special
cases of (III). (III) : (IIII) and (112) = (1112) are obtained by taking logarithm of both sides of (III) and (112) respectively since logt is an operator monotone function. (IIII) (1112) is obvious since (1112) is a special case of (IIII). We show (1112) (I). Letting p = 0 in (1112), we have sologA > sologB, that is, (I). Finally we show (113)
(I). Assume (113). Then by Lemma 2, (113) is equivalent to the following (9) (B3/2AP0B9/2)9/(Po+9) > B3
(9)
holds for a fixed positive number po and for all s such that s E [0, so], where so is a fixed positive number. Taking logarithm of both sides of (9) since logt is an operator monotone function, we get log(B3/2APOB9/2) > (po + s)logB.
Letting s = 0, we have pologA > pologB, that is, (I). Hence the proof of Theorem 1 is complete.
We remark that equivalence relation among (I), (IIo) and (III) is shown in [11].
Proof of Theorem 2. (I) logA-1 > logC-I is equivalent to
. (III). The hypothesis logC > logA in (I), that is,
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Furuta
A-p > (A-p/2C-9A-p/2)p/(p+9) for all p > 0 and all s > 0
by (I) and (III) of Theorem 1. Taking inverses implies (Ap/2C9Ap/2)p/(p+9) > AP for all p > 0 and all s > 0
and the rest of (III) is already shown by (I) and (III) of Theorem 1. For the proof (III) (I) , we have only to trace the reverse implication in the proof (I) = (III). (III) is complete. By the same method as in the proof of So the proof of (I)
Theorem 1 and together wih the same technique as in the proof (I) .. (III) in theorem 2, we can easily obtain the equivalence relation among (I), (IIo), (III), (112), (113), (III1) and (III2).
(IIII)
.
(IV1). (IIII) is equivalent to the following inequalities A-p/2log(Apl2C9Ap12)A-p/2 > A-pl2log(Ap+9)A-p/2 >
A-p/2log(Ap/2B9Ap/2)A-p/2
for all p > 0 and all s > 0, equivalently S(A-p
I C9) ?
S(A-p
I A9) >
S(A-p
I B9)
for all p > 0 and all s > 0, which is just (IV1). (IV1)
. (IV2) is obvious since (IV2) is a special case of (IVI).
(IV2)
.
(I). Put p = 0 in (IV2). Then we have logC > logA > logB
since so is a fixed positive number. Hence the proof of Theorem 2 is complete.
Proof of Corollary 1. Corollary 1 easily follows by Theorem 2.
§2. APPLICATION TO SOME EXTENDED RESULT OF ANDO'S ONE Ando [1] shows the following excellent result.
THEOREM B [1]. Let A and B be selfadjoint operators. Then the following assertions are mutually equivalent.
(i) A > B (ii)
e-rA/2(erA/2erBerA/2)1/2e-rA/2 < 1 for all
r>0
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Furuta
(iii)
e-rA/2(erA/2erBerA/2)1/2e-rA/2
is a decreasing function of r > 0.
As an extended result of Theorem B, we shall show the following Theorem 3 which
includes Theorem 1 as a special case when t = 0.
THEOREM 3. Let A and B be positive invertible operators, then the following assertions are mutually equivalent.
(I) logA > logB. (IIo) For any fixed t > 0,
F(p, r) = B-r(BTAPBr)(t+2r)/(p+2r)B-r is an increasing function of both p > t
andr>0. (III) For any fixed t>0, ro>0, and po>0, F(p, ro) = B-ro(BroAPBro)(t+2ro)/(p+2ro)B-ro is an increasing function of p
such that p E [0, po] for p > t.
(112) For any fixed t > 0, ro > 0, and po > t, F(po, r) = B-r(BrAPOBr)(t+2r)/(po+2r)B-r is an increasing function of r such that r E [0, ro].
(IIIo) For any fixed t > 0, G(p, r) = A-r(ArBPAr)(t+2r)/(p+2r)A-r is a decreasing function of both p > t and
r>0. (III,) For any fixed t > 0, and r0 > 0, and po > 0, G(p, ro) = A-ro (Aro BPAro) (t+2ro)/(p+2ro) A-ro is a decreasing function of p such
that p C [0, po] for p > t.
(1112) For any fixed t > 0, ro > 0, and p0 > t, G(po, r) = A-r(ArBPOAr)(t+2r)/(po+2r)A-r is a decreasing function of r such that r E [0, ro].
(IV) For any fixed t > 0, and r > 0, log(BrAPBr)(t+2r)/(p+2r) is an increasing function of p for p > t.
187
Furuta
(V) For any fixed t > 0, and r > 0,
log(ArBPAr)(t+2r)/(p+2r) is a decreasing function of p for p > t.
Proof of Theorem 3.
(I)
. (Ho). Assume (I). First of all, we cite (10) by (I)
and (Ill) of Theorem 1 (10)
AP > (Ap/2B2rAp/2)p/(p+2r) for all p > 0 and all r > 0.
Moreover (10) ensures the following (11) by the Lowner-Heinz theorem (11)
As > (Ap/2B2rAp/2)s/(p+2r) for all p > s > 0 and all r > 0.
Then by (11), we have (BrAPBr)(P+s+2r)/(p+2r) = BrAp/2(Ap/2B2rAp/2)s/(p+2r)Ap/2Br
< BrAp/2 AsAp12 Br
=
by Lemma 1
by (11)
B*Ap+sBr.
So the following (12) and (13) hold for each r > 0 and each p > s > 0 , (12)
BrAP+sBr > (BrAPBr) (p+s+2r)/(p+2r)
and (13)
(ArBEAT)(P+s+2r)l(P+2r) >
ArBP+sAr.
(13) is an immediate consequence of (12) because logB-1 > logA-1 ensures that (A-rB-FA-r)(p+s+2r)/(P+2r) < A-rB-(P+s)A-r
holds for each r > 0 and for each p and s such that p > s > 0 . Taking inverses gives (13). As (t+2r)/(p+s+2r) E [0,1] since p > t > 0, (12) ensures the following inequality by the Lowner-Heinz theorem (BrAP+sBr)(t+2r)/(p+s+2r) > (BrAPBr)(t+2r)/(p+2r)
which implies the following results for a fixed t > 0 and r > 0 (14)
(BrAPBr)(t+2r)/(p+2r) is an increasing function of p > t,
and (15)
(ATBPAr)(t+2r)/(p+2r) is a decreasing function of p > t,
because (15) is easily obtained by (13) and its proof is the same way as in the proof of (14) from (12).
Furuta
188
Next we show the following inquality (16)
forr>s>0.
(16) (AP/2B2rAP/2)(t-P)/(2r+P) > (APl2B28AP/2)(t-P)/(2s+P)
By (15), we have
for2r>2s>t1>0.
(17) (AP/2B2rAP/2)(ti+P)/(2r+P) < (AP/2B23AP/2)(tl+P)/(29+P)
Put a = (p - t)/(p + ti) E [0, 1] since p > t > 0 and t1 > 0. By the Lowner-Heinz theorem, taking a as exponents of both sides of (17) and moreover taking inverses of these both sides, we have (16). Therefore for r > s > 0,
F(p,r) =
B-r(BrAPBr)(t+2r)/(P+2r)B-r
= AP/2(AP/2B2rAP/2)(t-P)/(P+2r)AP/2
by Lemma 1
> AP/2(AP/2B2sAP/2)(t-P)/(P+2s)AP/2
by (16)
=B-8 (B3APB8) (t+2s)/(P+28) B-9
by Lemma 1
=F(p, s), so we have (IIo) since F(p, r) is an increasing function of p > t by (14). So the proof of
(I) = (Ho) is complete. (II0) = (II1) and (IIo) = (112) are obvious since both (III) and (112) are special cases of (Ho). (II1)
. (I) . Assume (III). Then F(p,ro) > F(0,ro) with t = 0, that is,
B-ro(BroAPBro)2ro/(P+2ro)B-ro > B-r'B2roB-ro = I, equivalently, AP/2(AP/2B2rOAP/2)-P/(P+2ro)AP/2 > I
by Lemma 1
namely AP > (APl2B2roAP/2)Pl(P+2ro)
(18)
holds for all p such that p E [0, po] and a fixed r0 > 0. Taking logarithm of both sides of (18) since logt is an operator monotone function, we have (p + 2r0)logA >
log(AP/2B2roAP/2).
Letting p -p 0, we have logA > logB since ro is a fixed positive number. (112)
(I). Assume (112). Then F(po, r) > F(po, 0) with t = 0, that is,
B-r(BrAPoBr)2r/(po+2r)B-r > I, equivalently, (BrAPoBr)2r/(Po+2r) > B2r
(19)
for all r such that r E [0, r0] and a fixed p0 > 0. Taking logarithm of both sides of (19) since logt is an operator monotone function, we have log(BrAPOBr) > (po + 2r)logB.
Letting r -p 0, we have logA > logB since p0 is a fixed positive number. (I)
(IIIo). This is in the same way as (I) ==> (IIo).
(IIIo) : (IIII) and (IIIo)
(1112) are obvious since both (IIII) and (1112) are
special cases of (IIIo). (IIII) : (I) and (1112) : (I) are obtained by the same ways
as (III) = (I) and (112) : (I) respectively.
(II0)
(IV) and (IIIo)
(V) are
both trivial since logt is an operator monotone function. (IV)
(I). Assume (IV) with t = 0. Then log(BrAPBr)2r/(p+2r) > logB2r,
that is, log(BrAPBr) > (p + 2r)logB.
Letting r -p 0 and p = 1, we have logA > logB. (V)
(I). This is in the same way as (IV) ; (I).
Hence the proof of Theorem 3 is complete.
We remark that the equivalence relation between (I) and (II0) has been shown in [6] as an extension of [5,Theorem 1].
I would like to express my sincere appreciation to Professor T. Ando for inviting me to WOTCA at Sapporo and his hospitality to me during this Conference which has been held and has been excellently organized during June 11-14, 1991. I would like to express my cordial thanks to the referee for reading carefully the first version and for giving to me useful and nice comments.
References [1] T.Ando, On some operator inequality, Math. Ann.,279(1987),157-159. [2] J.I.Fujii and E.Kamei, Relative operator entropy in noncommutative information theory, Math. Japon.,34(1989),341-348. [3] J.I. Fijii and E.Kamei, Uhlmann's interpolational method for operator means, Math. Japon.,34(1989),541-547. [4] M.Fujii, Furuta's inequality and its mean theoretic approach, J. of Operator Theory,23(1990),67-72. [5] M.Fujii, T.Furuta and E.Kamei, Operator functions associated with Furuta's inequality, Linear Alg. and Its Appl., 149(1991),91-96. [6] M.Fujii, T.Furuta and E.Kamei, An application of Furuta's inequality to Ando's theorem, preprint. [7] T.Furuta: A > B > 0 assures (B'APB'')1/9 > B(p+2r)/q for r > O,p > 0,q> 1 with (1 + 2r)q > (p + 2r). Proc. Amer. Math. Soc., 101(1987),85-88. [8] T.Furuta, A proof via operator means of an order preserving inequality, Linear Alg. and Its Appl.,113(1989),129-130. [9] T.Furuta, Elementary proof of an order preserving inequality, Proc. Japan Acad.,65(1989),126. [10] T.Furuta, Two operator functions with monotone property, Proc. Amer. Math. Soc.,111(1991),511-516. [11] T.Furuta, Furuta's inequality and its application to the relative operator entropy, to appear in J. of Operator Theory. [12] E.Heinz, Beitragze zur St'rungstheorie der Spektralzerlegung, Math. Ann., 123(1951),415-438. [13] E.Kamei, A satellite to Furuta's inequality, Math. Japon,33(1988),883-886. [14] K.Lowner, Uber monotone Matrixfunktion, Math. Z., 38(1934),177-216. [15] M.Nakamura and H.Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad.,37(1961),149-154. [16] G.K.Pedersen, Some operator monotone functios, Proc. Amer. Math. Soc.,36(1972),309-310. [17] H.Umegaki, Conditional expectation in operator algebra IV, (entropy and information), Kadai Math. Sem. Rep.,14(1962),59-85
Department of Applied Mathematics Faculty of Science Science University of Tokyo 1-3 Kagurazaka, Shinjuku Tokyo 162 Japan
MSC 1991: Primary 47A63
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
191
THE BAND EXTENSION ON THE REAL LINE AS A LIMIT OF DISCRETE BAND EXTENSIONS, I. THE MAIN LIMIT THEOREM I. Gohberg and M.A. Kaashoek
In this paper it is proved that the band extension on the real line (viewed as a convolution operator) may be obtained as a limit in the operator norm of block Laurent operators of which the symbols are band extensions of appropriate discrete approximations of the given data. 0. INTRODUCTION
Let k be an m x m matrix function with entries in L2([-r,r]). An m x m matrix function f with entries in L1(R) fl L2 (R) is called a positive extension of k if
(a) f (t) = k(t) for -r < t < r, (b) I - f (A) is a positive definite matrix for each A E R.
Here f denotes the Fourier transform of f. If (b) is fulfilled, then (0.1)
(I - f(\))-1 = I - ry(a), \ E R
where y is again an m x m matrix function with entries in L1(R) fl L2(R). A positive extension f of k is called a band extension if the function -y in (0.1) has the following additional property: (c) y(t) = 0
a.e. on R\[-r, r].
It is known (see [7]) that the band extension may also be characterized as the unique positive extension f of k that maximizes the entropy integral E(f ), where (0.2)
E(f) = h
log ds2(2
o
27r
A,\)) 1
+
dA.
Gohberg and Kaashoek
192
The main aim of the present paper is to establish the above mentioned max-
imum entropy characterization of the band extension by reducing it to the corresponding result for the discrete case, which concerns Fourier series on the unit circle with operator coefficients. Our reduction is based on partitioning of operators and does not use the usual
discretization of the given k. Let us remark here that the maximum entropy principle for matrix and operator functions on the unit circle is well-understood and may be derived as a corollary of the abstract maximum entropy principle appearing in the general framework
of the band method ([12]). However, for the continuous case there are different entropy formulas ([2], [3], [5], [7], see also [6], [16]), and the maximum entropy principle does not
seem to'follow from the abstract analogue in the band method (see [12] for an example).
This paper consists of two parts. In the present first part we show that the band extension on the line may be obtained from discrete band extensions on the circle
by a limit in an appropriate norm. In this limit procedure the first step is to replace the given m x m matrix function k by a trigonometric polynomial with operator coefficients, namely n-1
I-
z"K"(n).
v=-(n-1)
Here n is a positive integer and Kim") is the operator on L2 ([0, n r]) defined by (K(n) cp)(t)
/'ar = J0 k(t - s + nr)cp(s) ds,
0
-r.
Note that the trigonometric polynomial (0.3) uses the information about the given data
on -T + n r < t < r, but not on -r < t < -T + r. The next step is to build the band n extension for the trigonometric polynomial (0.3), that is, we build an operator function 00
B(n)(z)
vB(n) V
with the following poperties
v = -(n - 1), ... , n - 1, 2) 1 - B(n)(z) is a positive definite operator on La ([0, -1-r]) for each jzj = 1, 1) BL n) = K,,
,
Gohberg and Kaashoek
193
3) (I - B(")(z))-1 is of the form
n=1
z"C(") . are Hilbert-Schmidt operators on
From the construction it follows that the operators B.
L2 ([0, hr]) and one can form a bounded linear operator f3(n) on Lz (R) such that (B
1
(. + LT))(t - -T),
!T < t <
T,
n
V, E L2 ([LT,
,n
1T]).
The main result is that the sequence of operators b(1), b(2).... converges in the operator norm to Lb, where b is a band extension of k and Lb stands for the convolution operator on LZ (R) defined by (LbW)(t) =
J
00 00
b(t - s)cp(s) ds,
t E R.
We prove this convergence also for some norms that are stronger than the operator norm.
This part of the paper is split into three sections (not counting the present introduction). The first section contains preliminary material and is of preparatory character. In the second section we recall the construction of the band extension both for the continuous and the discrete case. The main theorem and its proof appear in the third section. As a by-product we get the connection between discrete and continuous orthogonal polynomials for the positive definite case.
A few words about notation. An identity operator is denoted by I; from the
context it should be clear on which space it acts. By S2 we denote the class of Hilbert-
Schmidt operators, where the operators are assumed to act on a Hilbert space. For an invertible Hilbert space operator A we let A- denote the adjoint of A-1. The spectral norm of a matrix M is denoted by IIMII, i.e., IIMII is the largest singular value of M. 1. PRELIMANARIES AND PREPARATIONS
1.1 Operator Wiener algebra and block Laurent operators. Let H be a separable Hilbert space. By definition (cf., Section III in [11]) the operator Wiener algebra W(H; T) consists of all operator-valued functions G on the unit circle T such that (1.1)
G(z) = E z"G,,, V=-00
z E T,
Gohberg and Kaashoek
194
where each G,, is a bounded linear operator on H and 00
E IIGvll < oo.
(1.2)
"=-00
As usual, G is called the v-th Fourier coefficient of the function G. With the usual multiplication of operator-valued functions W(H; T) is a unital algebra, the unit being
given by the function E(z) = I of each z E T. Also, on W(H; T) there is a natural involution ', namely, for G as in (1.3) we have 00
(1.3)
G*(z) = E z"G"" = G(z)*,
z E T.
V=-00
Each G E W(H; T) defines in a canonical way a bounded linear operator
on ® 'H. Here ® 0H denotes the Hilbert space of all square summable sequences x = (xi)°O__00 with elements in H. Inner product and norm on E)' H are given by 00
00
(x, Y) _ E
IIxII = (E IIxjII2)1.
j=-00
j=-00
Now, let G E W (H; T) be given by (1.1), and define an operator M on ® 1 H by setting (1.4)
(Mx)i =
Gi-3xj,
i = 0, ±1, ±2, ... .
i=-0o
Then M is a well-defined bounded linear operator on ED' H. Instead of (1.4) we shall write M We call M the H-block Laurent operator with symbol G.
1.2 The Wiener algebra over the Hilbert-Schmidt operators. Let H be a separable Hilbert space. By W(S2, H; T) we denote the set of all operator-valued
functions F on the unit circle such that 00
(2.1)
F(z) =
z"F,,,
z E T,
v=-00
where F" is a Hilbert-Schmidt operator on H for each v and 00
(2.2a)
IIIFIII
E IIF"II < oc, 00
(2.2b)
IIF'"II2< oo.
IJIF1112:_
"=_00
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195
Here II 112 denotes the Hilbert-Schmidt norm (cf., Section VIII.2 in [8]). With the usual
multiplication of operator-valued functions W(S2, H; T) is an algebra (see Proposition 2.1 below). We shall refer to W(S2, H; T) as the Wiener algebra on T over the Hilbert-Schmidt
operators on H. From (2.2a) we see that W(S2,H;T) is contained in W(H;T). PROPOSITION 2.1. The Wiener algebra on T over the Hilbert-Schmidt
operators on H is a 2-sided ideal in the operator Wiener algebra W(H; T) and is closed under the involution *.
PROOF. If A is a Hilbert-Schmidt operator, then (see [8], Section VIII.2) the same is true for A* and IIA*II2 = IIAII2 From these facts it follows that W(S2iH;T) is closed under the involution *. To prove that W(S2, H; T) is a 2-sided ideal in W(H; T),
take F E W(S2iH;T) and G E W(H;T). Let F be as in (2.1) and Gas in (1.1). Then 00
(GF)(z) = G(z)F(z) =
zvSv,
z E T,
v--0o where 00
Sv = E Gv_kFk. Note that Gv_kFk is a Hilbert-Schmidt operator and IIGv-kF'khI2 <_ IIGv-khhhiFkhi2.
The sequence (IIFkII2) is bounded. Thus we can use (1.2) to show that the right hand side
of (2.3) converges in the Hilbert-Schmidt norm. So Sv is a Hilbert-Schmidt operator and 00
IISvII2 <_ E IIGv-khhhiFkII2.
k--oo
But then 00
vc-ao
00
hISvtl22
<
)2(E IIFkII2)2 < 00, k=-oo
where ry is equal to the left hand side of (1.2). Hence (2.2b) holds for GF in place of F. Since
G and F are both in W(H; T), the inequality (2.2a) also holds for GF in place of F. Thus
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196
GF E W(S2,H;T). In a similar way (or by duality) one proves that FG E W(S2iH;T). I-F(.), where F E W(S2, H; T), and assume
COROLLARY 2.2. Let
that G(z) is an invertible operator on H for each z E T. Then
(I - F(.))-1 - I E W(S2iH;T).
(2.5)
PROOF. Let R be the operator-valued function defined by the left hand side
of (2.5). By the Bochner-Phillips theorem, ([4], Theorem 1) we have R E W(H;T). Now
(I-R(z))(I-F(z)) = I for each z E T, and hence R+F-RF = 0. Since F E W(S2iH;T), Proposition 2.1 implies that the same holds true for RF. Thus F - RF E W(S2, H; T), and (2.5) is proved.
PROPOSITION 2.3. The algebra W(S2,H;T) is a Banach algebra with
respect to the norm (2.6)
IIFIIw := max{IIIFIII, IIIFIII2}.
PROOF. Recall that S2(H), the ideal of all Hilbert-Schmidt operators on H,
is a Hilbert space with respect to the Hilbert-Schmidt norm. It follows that the space 00
(2.7)
IF I F(z) = E z"F (z E T), F E S2(H)
is a Hilbert space with respect to the norm I
II
'
1
1
1 2.
(v E Z), 111FI112 < oo}
We also know that W (H; T) is a
Banach space with respect to the norm III III. Now, let (F(n)),°,°_1 be a Cauchy sequence in W(S2, H; T) with respect to the norm II IIw. Then the sequence (F(n))°°_1 has a limit
in W(H;T) and in the space (2.7). Obviously, both limits are equal. So, there exists F E W(S2iH;T) such that IIF(n) - FIIw goes to zero if n -* oo. Hence W(S2,H;T) is complete with respect to the norm II ' IIw.
It remains to prove that
II
' 11w is an algebra norm. Take F and G in
W(S2iH;T). Then G E W(H; T), and the arguments used in the proof of Proposition 2.1 (cf., formula (2.4)) imply that (2.8)
IIIGFIII2 <- IIIGIII ' IIIFI112 <- IIGIIwIIFIIw.
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197
Also, F E W(H; T) and hence IIIGFIII ! IIIGIII - IIIFIII !5 IIGIIwIIFIIw
(2.9)
Therefore IIGFIIw < IIGIIwIIFIIw
1.3 Block partitioning. By L2(R) and L2([0, a]) we denote the Hilbert spaces of square integrable C'°-valued functions on R and [0, a], respectively. Fix
o > 0. Often it will be convenient to identify operators on L2 (R) with operators acting
on 000 L2([0, a]), the Hilbert space of all square summable sequences with entries in L2([0, a]). To make this identification explicit we need a few auxiliary operators, namely: (3.1)
rli
: L2([0, a]) -* L2 (R),
(rli`G)(t) = S
V (t - aj), of
t <_ aU + 1), otherwise,,
0,
and (3.2)
Pi : L2 (R) -' L2([0, a]),
(p1 ')(t) =
ai),
0 < t < or.
Here j and i are arbitrary integers. We also need: (3.3)
JQ : L2 (R) -+ ® L2 ([0, a]),
Ji/i = (PiV))
The map Jo is a unitary operator. We call Jo the a-partitioner of L2 (R). Now, let T : L2 (R) -+ L2 (R) be a bounded linear operator. By definition,
the a-block partitioning of T is the double infinite operator matrix (T1)._
whose
(i, j)-th entry is given by Ti9 = piTrli
(3.4)
Note that the canonical action of the operator matrix (T1)° . _. on ® L2([O,a]) ,)recisely the operator J,TJ. 1
is
Let r = no, where n is a positive integer. With some obvious changes in notation one defines the a-block partitioning of a bounded linear operator T on L2([0, T])
to be the n x n operator matrix T11
...
T1n
Tnl
...
Tnn
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198
where Ti.i = piS1
i, j = 1,.. . , n. The operators rli and pi are defined as in (3.1) and
(3.2), except that in these definitions one has to replace L2 (R) by L2 ([O, r]). The operator
matrix in (3.5) is considered to be an operator on ED Lz ([O,a]), the Hilbert space direct sum of n copies of LZ ([O,a]).
1.4 Convolution operators. Let f be an m x m matrix function with entries in Ll (R). The convolution operator on LZ (R) associated with f will be denoted by L f. Thus (4.1)
t E R.
f (t - s)V(s) ds,
(L fcp)(t) = foo
We shall refer to L f as the convolution operator with kernel function f.
Let f be an m x m matrix function with entries
PROPOSITION 4.1.
in L1(R) fl L2(R). Then the a-block partitioning of the convolution operator L f is a L2 ([O, a])-block Laurent operator, F = (Fi_1)?=_,, with symbol in the Wiener algebra on T over the Hilbert-Schmidt operators on L2 ([0, a]) and 00
(4.2)
00
E IIFFII <-
IIf(t)Ildt,
v=_
IIF., V=_00
21_00
112
=aJ
00 oo
IIf(t)II2dt,
The proof of Proposition 4.1 is the same as that of Lemma 1.5.1 in [9] and,
therefore, it is omitted. Here we only note that for v = 0, ±1, ±2.... the operator F in Proposition 4.1 is the Hilbert-Schmidt operator on L2 ([0,a]) defined by (4.4)
(F,, )(t) = fo 0' f (t - s +
0 < t < a.
ds,
1.5 The algebra B(r). Throughout this section r > 0 is a fixed positive number. By B(r) we denote the set of bounded linear operators T on LZ (R) such that the r-block partitioning of T is a Lz ([0, r])-block Laurent operator with symbol in
W(S2iLz ([0,r]; T). Take T E 8(r), and let
be its r-block partitioning. We
define: (5.1a)
(5.1b)
I
IJ
IITi II < 00,
I ITI I I
IT1
112
00
1 1 T 11
' < 00,
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199
and
(5.2)
IITIIB(,) := max{IIITIII, IIITIII2}.
Formula (5.1a) implies that II - JIB(,) is stronger than the usual operator norm, i.e., (5.3)
T E B(T).
IITII <- IITIIB(,),
Given T E B(T), let FT denote the symbol of the T-block partitioning of T. Note that IITIIB(,) = IIFIIw, where II
denotes the norm on W(S2iLz ([O,o];T)
- 11w
introduced in subsection 1.2. So we know from Proposition 1.2.3 that B(T) endowed with the norm II
-
JIB(,) is a Banach algebra. In fact, B(T) and W(S2, L2 ([0, o]; T) are algebraically
and isometrically isomorphic, the isomorphism being given by the map T H FT. For later purposes we mention the following two inequalities (which follow from (2.8)): (5.4a)
I
(5.4b)
I ITSI 112 <_ I I ITI I I
I
1
1S1
1
12,
IIITSIII2 <- IIITIII2IIISIII,
T, S E B(T),
T, S E B(T).
Each T E B(T) can be represented in the form (Tcp)(t) =
(5.5)
a(t, s)cp(s) ds,
J
t E R,
where a(t +,r, s + T) = a(t, s) a.e. on R x R and (5.6)
f0'r
fl
IIa(t, 3)112 ds dt < oo.
Formula (5.5) holds true for each cp E LZ (R) with compact support. To obtain the representation (5.5), let
be the kernel function of the Hilbert-Schmidt integral
operator T (appearing in the r-block partitioning of T), and put it < t < (i + 1)T,
a(t, s) = a,_j(t - i2T, a - jr),
jr < s < (j + 1)T.
Then a has the desired periodicity , formula (5.6) holds true because of (5.1b), and we have the representation (5.5) because is the 7--block partitioning of T. We shall refer to a in (5.5) as the kernel function of T. For T as in (5.5) we have f, (5.7a)
IJIT1112= ( o
(5.7b)
i_: II a(t,8)112dsdt)
,
200
Gohberg and Kaashoek
PROPOSITION 5.1. Let T E 5(7). Put a = nr, and let the or-block partitioning of T. Then 0o
IIITIII2=(
(5.8a)
n-1 IITj")II2)3,
j=-oo i=0 00 n-1
(5.8b)
IIITIII2 = (E
IITij1 )II2)
i=-0o j=0
PROOF. Let a be the kernel function of T. We claim that (T,, cp)(t) =
o a(i,j)(t, s)cp(s) ds, J
0 < t < a,
0 < s < a,
where ai,j (t, s) = a(t + ia, 8 + ja). Indeed, for 0 < t < or we have (cf., Section 1.3):
a(t + ia,
(piT1]jcp(t) =
ds
roo
=
J
which shows that
0
(j+1)
a(t + io, s)cp(s - ja) ds
Li
a(t + ia, s + ja)cp(s) ds
has the desired form. Now, by (5.7a) 00
2 =r=-0o E
IIITIII2
oo
(f0'r
f
11«(t, s) II2 ds dt
rr
n-1 n-1
r=-oo i=0 j=0 n-1 n-1 00
r=-oo i=0 j=0 00 n-1 n-1
_
(r+l )r
rr+(j+1)a
(i+1)a
Jr +jo,
10
a
IIa(t,s)Ii2dsdt
a
f 0f Ilai,rn+j(t,8)Il2dsdt 0 112
r=-oo i=0 j=0 00 n-1
_ E E IITj")II2' j=-oo i=0
which proves (5.8a). In a similar, by (5.7b) in place of (5.7a), one derives (5.8b).
PROPOSITION 5.2. If or = I r, then B(a) C B(r).
be
201
Gohberg and Kaashoek
be the a-block partitioning PROOF. Take T E B(a), and let (S;_1)S of T. For each v E Z let T be the operator on L2 ([0, r]) whose a-block partitioning is equal to the following block Toeplitz matrix
... S("-1).+i
Si,,,
S("+1)n-1
Then (Ti_i)?R=_
,
. .
S"n
is the r-block partitioning of T. From these connections it is clear that
T E B(T). Moreover, we see that (5.9)
IITIIB(T) < nhITIIe(o).
II. BAND EXTENSIONS
11.1 The circle case. Consider the following trigonometric operator polynomial:
N-1
I - E z"F,,. "=-(N-1) The coefficients F_(N_1), ... , FN-1 are assumed to be bounded linear operators on the sep-
arable Hilbert space H. We shall call (1.1) a discrete operator band. To explain the word
!band", note the H-block Laurent operator with symbol (1.1) is a double infinite banded 'matrix of which all diagonals are zero except the 2N - 1 diagonals located symmetrically Around the main diagonal (the main diagonal included).
An operator-valued function I -
E W(H; T) is called a band extension
of the discrete operator band (1.1) if
(i) for IjI < N - 1 the j-th Fourier coefficient of B(.) is equal to Fi, (ii) I - B(z) is a positive definite operator for each z E T, (iii) for IjI > N the j-th Fourier coefficient of (I - B(.))-1 is equal to zero. If only (i) and (ii) are fulfilled, then I - B(.) is called a positive extension of (1.1). From {ll], Section 11.1, we know that a band extension exists if and only if the operator I - A,
202
Gohberg and Kaashoek
A
FO
F_1
F,
FO
... ...
F_(N_1) F_(N_2)
FN_1
FN_2
...
Fo
=
is a positive definite operator on HN, the Hilbert space direct sum of N copies of H. Furthermore, in that case (see [11], Theorem 11.1.2) there is precisely one band extension which may be obtained in the following way. Let X0, X1, ... , XN_1 be given by Xo
F0
X,
F,
XN-1
FN-1
and put X(z) = zjXi. Then I+X(z) is an invertible operator for each jzj < 1 and the operator function I - B(.), (1.4)
I - B(z) := (I + X(z))-'(I + Xo)(I + X(z))-1,
z E T,
is the band extension of (1.1). PROPOSITION 1.1. Let I - B(.) be the band extension of the discrete
operator band (1.1). If F1 is a Hilbert-Schmidt operator for each Ij I < N - 1, then B(.) is in the Wiener algebra on T over the Hilbert-Schmidt operators on H.
PROOF. Let A be as in (1.2). By our hypotheses, I - A is invertible. Put
A' = I - (I - A)-', and write A" as an N x N operator matrix with entries acting on H: A"00
...
A"0,N-1
Ax =
.
A"N-1,0
...
AxN-1,N-1 .
Let Xo, ... , XN_1 be the operators defined by (1.3). Then
Xi=Fi -
N-1
i=0,...,N-1,
i-o and thus, by the ideal property of the Hilbert-Schmidt operators, X0, ... , XN-1 are in S2.
Put X(z) = E 01 z'X,. We already know that I+X(z) is invertible for each z E T. Since
Gohberg and Kaashoek
203
E W(S2i H; T), we can apply Corollary 1.2.2 to show that the same holds true for
I - (I +
Next use (1.4) and Proposition 1.2.1 to obtain that
E W(S2, H; T).
There is an alternative way (cf., Theorem II. 1.2 in [11]) to construct the band
extension, namely replace (1.3) by Y_(N_1)
F-(N-1) =(I- A)
(1.5)
F-1
Y-
\
Y0
and X(z) by Y(z) = E° __(N_1)
I-
of the band (1.1) with
the coefficients are determined by
F0
Also one may describe all positive extensions
in W(S2,H;T) by a linear fractional map of which and Y(.). In fact, Theorems II.1.1 and 11.1.3 in
[11] remain valid if in these theorems the operator Wiener algebra W(H; T) is replaced by
the algebra
W(S2,H;T)={aE+FI a E C, FEW(S2,H;T)}. Here E is the unit defined by E(z) = I for each z E T.
11.2 The real line case. Throughout this section r > 0 is a fixed positive number and k is an m x m matrix function whose entries are in Ll([-r,T]). An m x m matrix function b with entries in L1(R) is said to be a band extension of k if
(i) b(t) = k(t) for -T < t < T, (ii) I - Lb is positive definite, (iii) the kernel function of I - (I - Lb)-1 is zero almost everywhere on R\[-T, T].
If only (i) and (ii) are fulfilled, then b is called a positive extension of k. Recall that Lb denotes the convolution operator with kernel function b (see Section 1.4). Condition (ii) implies that I- Lb is invertible. In that case I- (I - Lb)-1 is again a convolution operator
with kernel function -y, say. Condition (iii) requires that the support of ry is contained in the set [-T,T].
With the given function k we associate the following integral operator on
204
Gohberg and Kaashoek
Lz ([0,r]): (2.1)
(Ktp)(t) =
J
T
0 < t < T.
k(t - s)4p(s) ds,
From [7] we know that a band extension of k exists if and only if the operator I - K is a positive definite operator on Lz ([0,r]). Furthermore, in that case (see [7]) there is precisely one band extension which one may construct in the following manner. Let x be the solution of the equation: (2.2)
x(t) - jk(t - s)x(s) ds = k(t),
0 < t < r,
and put x(t) = 0 for t E R\[0,7-]. Then x is an m x m matrix function with entries in L1(R), the operator I + Lx is invertible and (I+ Lx)-1 = I+ L= x , where x x is an m x m
matrix with entries in L1(R) such that xx(t) = 0 for t E R\[0,oo). With x determined in this way one obtains the band extension b of k as the kernel function of the convolution operator Lb defined by
I - Lb := (I + Lx)-`(I + Lx)-1. PROPOSITION 2.1. Let b be the band extension of k. If the entries of k are in L2([-7-, r]), then b has entries in L1(R) fl L2(R).
PROOF. By our hypotheses, I - K is an invertible operator on LZ ([0"r]) and the right hand side of (2.2) is a function in Lz ([0,rr]). Thus x E LZ ([0,-r]). Since x(t) = 0 fort outside [0, z], we conclude that the entries of x are in L1(R) fl L2 (R). Let xjj
be the (i, j)-th entry of x, and let x' be the (i, j)-th entry of xx, where xx is determined by
LZx = (I + Lx)-1 - I. From the latter identity it follows that
=0,
i,j =I, ---'M-
V=1
Here * denotes the convolution product in L1(R). Now, recall that for f E L2(R) and g E L1(R), the convolution product f * g E L2(R). Thus L1(R) fl L2(R) is an ideal in
Gohberg and Kaashoek
205
L1(R) with respect to the convolution product. So we see from (2.4) that the entries of
x" are also in Li(R) fl L2(R). Put x#(t) = x"(-t)` for t E R. Then, by (2.3), we have
b=-(x#+x"+x**x"), and hence the entries of b are in L1(R) n L2(R).
There is an alternative way (cf., [7]) to construct the band extension b, namely, replace L. by Ly, where y is the solution of
y(t) -
0
k(t - s)y(s) ds = k(t),
-T < t < 0
and y(t) = 0 for t E R\[-T, 0]. Also (cf., [6] and [10]), all positive extensions f of k such
that the entries of f are in L1(R) fl L2(R) may be described by a linear fractional map of which the coefficients are determined by the functions x and y. In fact, the extension theorems in Section 1.4 of [10] remain valid if in these theorems the role of L1(R) is taken over by L1(R) fl L2(R).
III CONTINUOUS VERSUS DISCRETE
III.1 Main theorem. Throughout this section T > 0 is a fixed positive number and k is a given m x m matrix function with entries in L2([-T,T]). By K we denote the operator on L2 ([0, T]) defined by (1.1)
jk(t - s)(s) ds,
(Kw)(t) =
0 < t < r,
and we assume that I - K is positive definite. Put a = nT, with n a positive integer, and consider the following discrete operator band:
I-
n-1
ZvK,(,"),
v=-(n-1)
where for v = -(n - 1), ... , n - 1 the operator K(n) is the operator on L- Q0, a]) defined by (1.3)
(K,(,n)cp)(t) = J ,7 k(t - s + va)cp(s) ds, 0
0 < t < a.
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206
Put
A(n) _
Ko
K-1
Ki
Ka
K -1
Kn-2
... K((n-1) K(
-(i-2) KO(n)
Note that A(n) and the operator K in (1.1) are unitarily equivalent. In fact, A(n) is the a-block partitioning of the operator K. It follows that I - A(n) is a positive definite operator. The positive definiteness of I - K implies (see Section 11.2) that k has a band
extension on the real line, and (by Proposition 11.2.1) the entries of b are in
L1(R) fl L2(R). Similarly, since I - A(n) is positive definite, the discrete operator band (1.2) has a band extension I - B(n)(.) in W(L2 ([0, a]); T), and, by Proposition II.1.1, the
belongs to the Wiener algebra on T over the Hilbert-Schmidt operators
function
on L2 (0,a]). THEOREM 1.1. Let Lb be the convolution operator on LZ (R) whose kernel
function is the band extension b of k, and for n = 1, 2.... let f3(n) be the operator on LZ (R) whose a-block partitioning (a = -1,r) is the LZ ([0, a])-block Laurent operator with symbol B(n) (.), where I
is the band extension of the discrete operator band (1.2).
Then Lb, B(1), f3(2).... are in B(r), and limo JiLb - B(n)lI8(T) = 0 n-o
Recall that B(r) is the algebra of all bounded linear operators on L2 (R) such
that the r-block partitioning of T is the LZ ([0, r])-block Laurent operator with symbol in W(S2i LZ ([0, r]); T) (see Section 1.5). From Proposition 1.4.1 we know that Lb E B(r). The operators B(1), B(2), ... are in B(r) because of Proposition 1.5.2. The main result is
the limit formula (1.5), which we shall prove in the third subsection.
111.2 Main auxiliary theorems. Let r > 0 be a fixed positive number. Recall (see Section 1.5) that each operator T E B(r) may be represented as an integral operator. The expression T = Ta, means that a is the kernel function of T, i.e., T is represented as in formula (1.5.5). We say that the support of a belongs to the set V
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(notation: supp a C V) if a(t, s) = 0 almost everywhere on the complement of V. In B(T) we consider the following subspaces:
B1 ={TEB(r) IT=Ta, supp a C {(t,s)I t-82r}},
B2=IT EB(T)IT=Ta, suppaC{(t,s)I0
B(r) = B1+B2+B3+B4.
The corresponding projections are denoted by Q1i Q2, Q3 and Q4. So, for example, Q2 is
the projection of B(r) onto B2 along the other spaces in the decomposition (2.1). Recall (see Section 1.5) that the norm 11 - IIe(r) on B(r) is defined by (2.2)
IITIIB(T) = max{IIITI11, IIITII12},
where III' I I I and III' 1112 are given by (1.5.1a) and (1.5. 1b), respectively. From the expression
for 1111112 in formula (1.5.7a) it is clear that II I Q,.TIII2 5 II ITI I I2
(v = 1, 2, 3, 4).
LEMMA 2.1. For T E B2+B3 we have
IIITIII S vIIITIII2
PROOF. Let (T1_1)Z__. be the r-block partitioning of T. Since T E B2+B3i the r-block partitioning of T is tridiagonal, i.e., Ti.. = 0 for IvI > 2. Now use that IIAII < IIAII2 for any Hilbert-Schmidt operator A. It follows that IIITIII = IIT-1 II + IITo11 + IIT'i1I <- IIT-1112 +IIToII2 + IITi 112
Vr3-(IIT
1112 2
+IIToII2+IIT1II2)'l, 2
208
Gohberg and Kaashoek
which yields (2.4).
From (2.3) and Lemma 2.1 one sees that the projections Q2 and Q3 are bounded linear operators on B(r) with respect to the norm II IIB(o. Next, let a = nr. We also need the following subspaces:
Bi(n)=IT EB(r)I piTij1=0for(i,j)0{i-j>n}}, BZ(n)={TEB(r)I piTi1=0for(i,j)0{0 i - j > -(n - 1)}},
B4(n) = IT E B(r) I piT771 = 0 for (i, j) 0 1i - j < -n}}. Here pi and 77b are as in the first paragrapf of Section 1.3; in other words piT'77 is the (i, j)-th
entry in the a-block partitioning of T. We have the following direct sum decomposition:
B(r) = Bl(n)+B2(n)+Bd(n)+B3(n)+B4(n).
(2.5)
The corresponding projections are denoted by Q, (n), Q0 (n), Qd(n), Q0 (n) and Q4(n). We set
Q2(n)
Qd(n) + QO(n),
Q3(n)
Qd(n) + Q3 (n)
The following inclusions are valid: (2.7a)
B22(n) C B2i
(2.7b)
B2(n)+Bd(n)-}-83(n) C 82+B3.
B3(n) C B3,
By the expression for III . 1112 in formula (I.5.7a) we have
IIIQv(n)III2
<- IJIT1112
(v = 1,2,3,4,d)
for each T E B(r). From (2.8), the inclusion (2.7b) and Lemma 2.1 we conclude that the projections Q2(n),Q3(n) and Qd(n) are bounded linear operators on B(r) with respect to the norm 11 . JIB(,).
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209
LEMMA 2.2. For T E B(r), we have (2.9a)
lira IIIQa(n)TIII2 = 0, n-oo
(2.9b)
lim IIIQ2T - Q2(n)TI 112 = 0, n-oo
(2.9c)
lim IIIQ3T - Q3(n)TI112 = 0.
n-. oo
PROOF. We shall prove (2.9b); the other two limits are established in an analogous way. For R E B(r) we let Rij denote the (i, j)-th entry in the a-block partitioning of R. Here or = IT. We have n
(Q2(n)T)''
=
Tij, 0
otherwise.
Furthermore,
(Q2T)ij = (Q2(n)T)ij, for j # i and j # i - n. So, by Proposition 1.5.1, n-1
IIIQ2T - Q2(n)TIII2 = E{II (Q2T - T)iiII2 +
II(Q2T)i,i-n1122}.
i=0
Next observe that II(Q2T - T)ij112 5 IITijII2,
II(Q2T)ijll <_ IITijII2,
and thus
n-1
(2.10)
IIIQ2T - Q2(n)TIII2 5 E{IITiiII2 + IITi,i-nII2} = if Ila(t, x)112 dt ds. n
i=0
Here a is the kernel function of T and 92 = W U S2", where
S2'=U o{(t,s)ia<8<(i+1)a,
ia
0" = U o {(t, s) I is < 8 < (i + 1)a,
(n + i)a < t < (n + i + 1)a},
The total Lebesgue measure of 12 is equal to 2r2n-1 (and hence goes to zero if n -+ oo). Since IIa(., )112 is integrable over each compact subset of R x R, Lebesgue's dominated
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convergence theorem implies that the last term in the right hand side of (2.10) goes to zero, which proves (2.9b).
Let k be an m x m matrix function with entries in L2([-r, r]). Put k(t),
k(t) = 10,
(2.11)
-r < t < r, otherwise.
and let k be the convolution operator on L2 (R) with kernel function k; in other words,
K = L. From Proposition 1.4.1 we know that K E B(r). Since k(t) = 0 for t 0 [-7-,,r], we have
k = (Q2 + Q3)K.
(2.12)
Next, put a =
r, and let K(n) be the operator on L2 (R) whose a-block partitioning n is given by
Ii - jj < n-1,
K;)(n)
(2.13)
0,
otherwise,
where, for v = -(n - 1),... , n - 1, (K(n).)(t) = 104, k(t - s + va)cp(s) ds,
(2.14)
0
In other words:
(2.15)
K(n) = (Q2(n) + Qd(n) + Qs(n))K.
Now, use (2.12), (2.15) and apply Lemmas 2.1 and 2.2. It follows that (2.16)
limo liK - K(n)IIB(T) = 0.
PROPOSITION 2.3. The operators (2.17)
S : B(r) - B(r),
(2.18)
Sn : B(r) -, B(r),
S(X) := X - Q2(KX ),
X - Q2(n)(K(n)X),
are bounded linear operators on the Banach space B(r), and in the operator norm: (2.19)
n-mo JIS - Snjj = 0-
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211
PROOF. We already know that Q2 and Q2(n) are bounded as linear opera-
tors on B(r). Since B(r) is a Banach algebra, it follows that the operators S and Sn are bounded operators on B(r). To prove (2.19) it suffices to show that for each X E B(r) we have
IIQ2(KX) - Q2(n)(K(n)X)IIB(T) = 7(n)IIXIIB(r),
(2.20)
where 7(n) is a constant depending on n only such that 7(n)
0 if n -+ oo.
Recall that a = nr. In what follows we write Tij for the (i, j)-th entry in the a-block partitioning of T E B(r). As was shown in the proof of Lemma 2.2, for each T E B(r), we have n-1
IIIQ2T - Q2(n)TIII2 --
i=0
{IITiiII2 + IIT1,i-nII2}.
Let us apply this result to T = K(n)X. Note (see (2.13)) that
n-l+i
00
(K(n)X)ij = > (K(n))ivXvj =
K(") Xvj v=-n+l+i
V=_00
It follows that
n-l+i
II(K(n)X)iiII <
(
IIK(n) II2IIXvi112)2
v=-n+l+i n-l+i
(
n-l+i
v=-n+l+i
II Kin II2) (
v=-n+l+i
IIXviII2)
n-l+i
n-1
IIXviII2
IIK;n'Il2) ( v=-n+l+i
j=-n+1
In a similar way one shows that n-1
n-1+i
II(x(n)x)i,i-n112 < ( E IIKjn)II2)( E IIXv,i-n112). j=-n+l
v=-n+l+i
From Proposition 1.4.1 (applied to k) we see that n-1 j=-n+1
T
IIK;n)II < a
I
fT
Ik(t)II2 dt.
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212
Furthermore, n-1
n-1
n-l+i
IlXvill2
i=0 v=-n+l+i because of Proposition 1.5.1. Simalarly,
-
00
IIXv1II2 -111X11121
i=0 v=-00
n-1 n-1+i IIXv,i-n112 s IIIXIII2
i=0 v=-n+l+i So we have proved that
(2.21)
IIIQ2(x(n)x) - Q2(n)(x(n)x)1112 <-
2n
(f r IIk(t)II2 dt) IIIXIII2
Next, by (2.3) and inequality (I.5.4b), IIIQ2(KX)-Qz(K(n)X)1112 <- IIIKX -K(n)X1112 <- IIIK - K(n)111211IXI11, and thus
(2.22)
III Q2(KX) - Q2(K(n)X)1112 <- Ilk - K(n)II B(r)1II X II I
From (2.21) and (2.22) we see that
IIIQ2(KX) - Q2(n)(K(n)X)1112 < { FLn fT Ilk(t)112 dt) + II K - K(n)IIe(r)}IIX IIe(T).
Note that Q2(KX) - Q2(n)(K(n)X) E B2+B3i because of (2.7b), and thus we can apply Lemma 2.1 to show that
IIQ2(KX) - Q2(n)(K(n)X)IIB(r) <- VIIIQ2(KX) - Q2(n)(K(n)X)I112. It follows that (2.20) holds with 0 < 7(n) <
6n (f r Ilk(t)112 dt) Z + IIK - K(n)II e(r),
and hence ,(n) -* 0 (n -+ oo) by (2.16). Let K be the integral operator on L2 ([0,z]) defined by (2.23)
(Kcp)(t) -
J0
r k(t - s)cp(s) ds,
0 _< t <-'r.
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213
PROPOSITION 2.4. If the operator I - K is invertible, then the operator S on B(T) defined by (2.17) is also invertible.
PROOF. The proof of the invertibility of S is split into two parts. First we consider S on B2.
Part (a). Note that SB2 C B2. Define S : B2 -+ B2 to be the restriction of S to B2. Let Y E B2, and let y be the kernel function of Y. We know that y has its support
in the set
A2 = {(t,s) E R2 18 < t
f (t, s) = J
s < t < T + s.
k(t - u)x(u, s) du,
s
Thus the identity S(X) = Y implies that (2.24)
(2.25)
x(t, s) -
x t+s s
j
k(t - u)x(u, s) du
y(t, s), a.e. on
-u x u+s s du = t+s s jk(t f0'r
a.e. on 02.
Put (2.26a)
i(t, s) = x(t + s, s),
0 < t < T,
0 < s < T,
(2.26b)
y(t, s) = y(t + s, s),
0 < t < T,
0 < s < T,
The functions i and y are square integrable on [0,,r] x [0, T]. Let X and k be the integral
operators on L2 ([0,T]) with kernel functions i and y, respectively. Then (2.25) implies that (2.27)
(I - K)X = Y.
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214
Since I - K is invertible, it follows that k is uniquely determined by k, and therefore X is uniquely determined by Y. In other words, S is one-one.
The above arguments also show how to solve the equation S(X) = Y for a given Y in B2. Indeed, let y be the kernel function of Y, and define y by (2.26b). Then y is square integrable on [0, T] x [0,,r]. Let k be the corresponding Hilbert-Schmidt integral operator on L2 ([0, T]), and put
X := (I - K)-1Y.
(2.28)
We know that (I - K)-1 - I is a Hilbert-Schmidt operator on L-([0,T]), and hence the same is true for X. Let i be the kernel function of k, and define x : R2 -+ C by (2.26a) and the following rules: x(t, s) = 0,
(t, s) V 02i
x(t+T,s+T) =x(t,s), (t,s) E R2. Then x is uniquely determined by i, and there is a unique X E B2 such that x is the kernel
function of X. From (2.28) and the definition of x it follows that (2.24) is fulfilled, and hence S(X) = Y. We have proved that S is one-one and onto.
Part (b). Take Y E B(T), and let X be the unique solution in B2 of the equation
X - Q2(KX) = Q2(Y) + Q2{K(I - Q2)Y}. The result of Part (a) guarantees that such an operator X exists. Now define,
Z := (I - Q2)Y + X. Then a simple computation shows that S(Z) = Y. Indeed,
S(Z) = Z - Q2(KZ)
= (I - Q2)Y + X - Q2{K(I - Q2)Y} - Q2(KX) = (I - Q2)Y + Q2(Y) = Y.
It remains to prove that the equation S(Z) = Y has no other solution in B(T). To do this, assume that S(Z) = 0. Then (I - Q2)Z = 0 and Q2(Z) E B2 satisfies
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215
the equation X - Q2(KX) = 0. Thus, by the result of Part (a), we have Q2(Z) = 0, and therefore Z = 0.
111.3 Proof of the limit formula (1.5). In this subsection we prove formula (1.5). We continue to use the notations introduced in the previous subsection. Let S and Sn be the operators defined by (2.17) and (2.18), respectively. Let K be as in (2.23). According to our hypotheses the operator I - K is positive definite. In particular,
I - K is invertible, and so (by Proposition 2.4) the operator S is invertible. Proposition 2.3 implies that for n sufficiently large Sn is invertible and lim n-.oo
II
S-1
- Sn 1 II = 0.
Recall from Section 11.2 that I - Lb = (I + Lx)-'(1 + Lx)-1,
where x is the (unique) solution of the equation (11.2.2). In operator form (11.2.2) can be rephrased as
S(L1) = Q2K.
(3.3)
From Section II.1 we know that
I - B(n) = (I + X(n))-"(I + Qd(n)X(n))(I +
(3.4)
X(n))-1.
where X(n) E B2(n) and satisfies the equation (3.5)
Sn(X(n)) = Q2(n)K(n).
Indeed, (3.5) is just the analogue of (11.1.3) for the case considered here. Now apply (3.1).
It follows that IIL1 - X(n)JIB(r) <
IIS_' - Sn'II-IIQ2KIIB(T) + IISn'II-IIQ2K - Q2(n)K(n)IIB(T).
The right hand side of the above inequality tends to zero if n -+ oo (by (3.1) and (2.20)). Thus (3.6)
limo IIL1 - X(n)II B(r) = 0.
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216
Next, note that IIQd(n)X(n)IIB(r) 5 IIQd(n)LxIIB(r) +
VIILx - X(n)IIB(r).
Here we used that for each T E 13('r) II Qd(n)T II B(r) 5 f I I Qd(n)T I II2 <- V3I I ITI I I2 <- vIITIIB(r), I
because of (2.4) and the inequality (2.8) for Qd(n). By (2.9a) and (2.4) we have IIQd(n)Lx1IB(r)
0 if n - oo. Thus (3.6) yields II Qd(n)X (n)IIB(T) -* 0,
(n -, oo),
and therefore liynn III - (I + Qd(n)X (n))-1 JIB(r) = 0.
n
From (3.6), (3.8) and the identities (3.2) and (3.4) we conclude that
II(I - Lb)-1- (I and thus IILb
- B(n) II B(r)
0
(n
B(n))-1 II B(r)
(n --, oo),
0
oo).
111.4 Discrete and continuous orthogonal functions. Let k be an m x m matrix function with entries in L2([-r,T]), and let K be the operator on L2 ([0, T]) defined by (Kcp)(t) =
jk(t - s)V(s) ds,
0 < t < r.
In what follows we assume that I - K is invertible. The corresponding resolvent kernel is denoted by y(t, s), i.e., (4.1)
((I -
K)-' cp)(t)
= W(t) +
j7(t,s)2(s)ds,
0 < t < r.
By definition (see [14], [15]) the (first kind) orthogonal function associated with k is the entire m x m matrix function D(.) defined by (4.2)
D(A) = e'r-\(I +
J0
r y(t,
0)e-'at dt),
A E C.
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217
Next, let n be an arbitrary positive integer, and let
I-Kpn)
-K(i)
-K(n) 1
I - K(n) 0
-K(n) -(n-1) -K(n) -(n-2)
-K (n)
-K(-)
I - KOn)
be the n-block partitioning of I - K. In particular, K,(,n) (v = -(n - 1), ..., n - 1) is the operator on LZ ([0, nT]) defined by
(K,(n).)(t)
0
- s + nr)cp(s) ds,
n k(t 0
Since I - K is invertible, the n x n operator matrix in (4.3) is invertible. By definition (see [1], [13]), the orthogonal polynomial associated with (4.3) is the operator polynomial Pn(z) given by Pn(z) =
zn-1(I
+ Xo) + zi-2X1 + ... + Xn-1,
where
I + X0
X
1
I - KO(n) -K
-K(n)
I-
(n)
n
oo the n-th orthogonal polynomial P. converges
in a certain sense to the orthogonal function D associated with k. To make this statement precise, let Tn be the operator on L2 (R) whose n-block partitioning is the Lz ([0, in-TD-
block Laurent operator with symbol E o zjX;. Since k has its entries in L2Q-T, TI), the
operators K,n) (v = -(n - 1),. .. , n - 1) are Hilbert-Schmidt operators. We may rewrite (4.5) as
I -Kon) -K(n) (n) -Kn-1
-K(n) 1
I-
KO(n)
4722
-K (n) -(n-1)
(n) KO(')
-K
K(n)
I - K(n)
Kn
1
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218
It follows (by the same arguments as used in the proof of Proposition II.1.1) that X0,. .. , Xn_1 are Hilbert-Schmidt operators, and thus (cf., Section 1.5) the operators Tn belong to the algebra B(1-3,T). Let x,, be the kernel function of T,,. From (4.6) we see that xn is given by (4.7a)
xn(t,s
(4.7b) (4.7c)
0
tuku-a du
kt - s+ 10'r
xn(t,s) = 0,
0 < s < 1T,
t E R\[0,T],
n
Xn(t + 1 T, s + 1 T) = Xn(t, s), n n
Now, put
x(t)=
(t, s) E R.
(ry(t,0), 0
t E R\[0, T].
Then we see from (4.1) that (4.8)
x(t) = k(t) + j
ry(t, u)k(u) du,
0
and hence the formulas (4.7a) - (4.7c) suggest that for n --, oo the function xn(t, 8) converges to x(t - s). The precise result is the following.
PROPOSITION 4.1. Let T,T1,T2, ... be the integral operators on La (R) with kernel functions x(t - s), x1(t, s), x2(t, s), ... , respectively. Then (4.9)
11T - T(n)II e(,.) -, 0
(n -, oo).
PROOF. Note that T = L. Since x has its entries in L1(R) fl L1(R), we know (see Proposition 1.4.1) that T E B(T). Proposition 1.5.2 implies that Tn E B(T) for
n > 1. Next we use notations and results from the previous section. From the definition of x it follows that (4.10)
S(T) = Q2K.
Furthermore, because of (3.6), we have (4.11)
Sn(Tn) = Q2(n)K(n).
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219
But then, by the arguments used in the proof of the limit formula (1.5) (see Section 111.3),
we obtain (4.9).
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A. Atzmon, N-th orthogonal operator polynomials, in: Orthogonal matrixvalued polynomials and applications, OT 34, Birkhauser Verlag, Basel, 1988; pp. 47-63. D.Z. Arov and M.G. Krein, Problems of search of the minimum of entropy in indeterminate extension problems, Funct. Anal. Appl. 15 (1981), 123-126. J.J. Benedetto, A quantative maximum entropy theorem fro the real line, Integral Equations and Operator Theory 10 (1987), 761-779. S. Bochner and R.S. Phillips, Absolutely convergent Fourier expansions for non commutative rings, Annals of Mathematics 43 (1942) 409-418. J. Chover, On normalized entropy and the extensions of a positive definite function. J. Math. Mech. 10 (1961), 927-945. H. Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS 71, Amer. Math. Soc., Providence RI, 1989. H. Dym and I Gohberg, On an extension problem, generalized Fourier analysis and an entropy formula, Integral Equations Operator Theory, 3 (1980), 143-215.
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I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Volume I, Birkhauser Verlag, Basel, 1990. I. Gohberg and M.A. Kaashoek, Asymptotic formulas of Szego-Kac-Achiezer type, Asymptotic Analysis, 5 (1992), 187-220.
I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory, 22 (1989), 109-105.
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I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, The band method for
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positive and contractive extension problems: An alternative version and new applications, Integral Equations Operator Theory 12 (1989), 343-382. I. Gohberg, M.A. Kaashoek and H.J. Woerdeman, A maximum entropy priciple in the general framework of the band method, J. Funct, Anal. Anal. 95 (1991), 231-254.
[13]
[14]
I. Gohberg and L. Lerer, Matrix generalizations of M.G. Krein theorems oon orthogonal polynomials, in: Orthogonal matrix-valued polynomials and applications, OT 34, Birkhauser Verlag, Basel, 1988; pp. 137-202. M.G. Krein, Continuous analogues of propositions about orthogonal polynomials on the unit circle (Russian), Dokl. Akad. Nauk USSR, 105:4 (1955), 637-640.
[15]
M.G. Krein and H. Langer, On some continuation problems which are closely
related to the theory of operators in spaces II.. IV, J. Operator Theory 13 (1985), 299-417.
220
Gohberg and Kaashoek
[16]
D. Mustafa and K. Glover, Minimum entropy H. control, Lecture Notes in Control and Information Sciences 146, Springer Verlag, Berlin, 1990.
I. Gohberg Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel-Aviv University Ramat-Aviv, Israel
M.A. Kaashoek Faculteit Wiskunde en Informatica Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands
MSC: Primary 47A57, Secondary 45A10, 47A20
Operator Theory: Advances and Applications, Vol. 59 ® 1992 Birkhauser Verlag Basel
221
INTERPOLATING SEQUENCES IN THE MAXIMAL IDEAL SPACE OF H°° II Keiji Izuchi The objective of this paper is to study interpolating sequences in M(H°°) the maximal ideal space of H°°. Using the pseudohyperbolic distance, L. Carleson gave a characterization of interpolating sequences in D the open unit disk. But its condition does not characterize interpolating sequences in M(H°°). Carleson's condition has several equivalent conditions in D. It is studied the relations between these conditions and interpolating sequences in M(H°°).
1. INTRODUCTION Let H°° be the Banach algebra of bounded analytic functions on the open unit disc D with the supremum norm. We denote by M(H°°) the maximal ideal space of H°° with the weak-*topology. We identify a function in H°° with its Gelfand transform. We may consider that D C M(H°°), and by the corona theorem D is dense in M(H°°). The maximal ideal space M(L°°) of L°°(8D) may be considered as a subset of M(H°°). For points x and y in M(H°°), we define
p(x, y) = sup{Ih(x)I; h(y) = 0, h E H°°, IIhil < 1}. When z and w are points in D, p(z, w) = Iz - wI / 11 - awl. For a point x in M(H°°), we put P(x) = {y E M(H°°); p(y,x) < 1},
which is called a Gleason part. If P(x) = {x}, x is called a trivial point. Every point in M(L°°) is trivial. If P(x) # {x}, then x and P(x) are called nontrivial. In this case, there is a one to one continuous map Ls from D onto P(x) such that x = L=(0) and H°° o L. C H. Moreover if L= is a homeomorphism, P(x) is called a homeomorphic part. P(0) = D is a typical homeomorphic part. We put
G = {x E M(H°°); x is not nontrivial}. Then G is an open subset of M(H°°) (see [6]). For a sequence {zn}n in D with E"_11- Iznl < oo, a function
00 -zn z - zn
b(z) = II
_
n=1 IZnI 1 - ZnZ'
z E D
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222
is called a Blaschke product with zeros {zn}n. For a positive singular measure p on 8D, a function .e S[p](z)
I
= exp(- J e'9 - z dp(9))
is called a singular inner function. For a bounded measurable function F on 8D such that log IFI is integrable, a function
O[log IFl](z) = exp (f eie +
log I F(e`B)I d9/2ir)
z contained in H°°, and every function is called an outer function. These functions are f in H' is represented by f = O[log I f I] S[µ] b except a constant factor (see [5]). Let
fn = O[log Ifnl] S[µ.] bn be a function in H°O with IIfnII < 1. If En°_1 log lfnl is integrable, En '=l 1.4n(8D) < oo and lln 1 bn is still a Blaschke product, then we denote by II-1 1 fn the function O[En f log I fnl] S[E°°_1pn] II°_1 bn in H°°.
A sequence {xn}n in M(H°O) is called interpolating if for every sequence {an}n
of bounded complex numbers there is a function f in H°° such that f (xn) = an for all n. An interpolating sequence {Zn}n in D is characterized by Carleson [2] as follows; Zk
inf k
II
-
zn
n:n#k 1 - ZnZk
Let
I
> 0.
-Zn Z - Zn n
IZn)
1 - ZnZ
be a Blaschke factor. Then 1bn(zn) = 0, IjI'nII = 1 and inf k
II
n:n$k
p(zn, Zk) = infk I( n:k I'$
OGn)(Zk)
.
From this observation, we can consider similar conditions for sequences {xn}n in M(H°°) as follows: (A1) (A2)
inf
P(Xn, Xk) > 0k n:n$k II
There is a sequence {gn}n in H°° such that li9n1l < 1, gn(xk) = 0 if k # n, and inf Ign(xn)l > 0.
(A3)
There is a sequence { fn}n in H°O such that
IIfnII :5 1, fn(Xn) = 0, 11 In E H°°, and ikf (n:II
fn)(xk)
> 0.
In this paper, we shall study what kind of relations are there between interpolating sequences and conditions (A;), i = 1, 2, 3. First, we have
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223
FACT 1 (Proposition 1).
(A2) * (A1).
(A3)
By the open mapping theorem,
FACT 2. If {xn}n is interpolating, {xn}n satisfies (A2). In Theorem 1, we shall prove that the converse assertion of Fact 2 is true when each point xn is trivial. A sequence {xn}n in M(H°°) is called strongly discrete if there is a sequence of disjoint open subsets {Un}n of M(H°°) such that xn E U,,. Then we have
FACT 3. If {xn}n is interpolating, {xn}n is strongly discrete. A Blaschke product is called interpolating if its zero sequence is interpolating. For a function f in H°°, we put
Z(f) = {x E M(H°°); f (x) = 0}. Then we have
FACT 4 (see [8]). Let {xn}n be a strongly discrete sequence. If {xn}n C Z(b) for some interpolating Blaschke product b, then {xn}n is interpolating.
In [4], Gorkin, Lingenberg and Mortini proved that if {xn}n is a sequence in a homeomorphic part and satisfies condition (A1), then there is an interpolating Blaschke product b such that {xn}n C Z(b). In [7], the author got the same conclusion when {xn}n is a sequence in a homeomorphic part and satisfies condition (A2). Also in [8], the author studied a sequence whose elements are contained in distinct parts. In Theorem 2, we shall prove that if {xn}n is a sequence in G, then {xn}n satisfies condition (A3) if and only if {xn}n is strongly discrete and there is an interpolating Blaschke product b such that {xn}n C Z(b). And in Theorem 3, we show that there is a strongly discrete sequence {xn}n in a nontrivial part such that {xn}n satisfies condition (A1) but {xn}n is not interpolating.
2. CONDITION (A2) In this section, the following lemma plays an important role.
LEMMA 1 [9, Theorem 3.1]. Let b be a Blaschke product. Then there are Blaschke products b1 and b2 such that b = b1 b2 and bl(x) = b2(x) = 0 for every point x in M(H°°) with blp(,z) = 0. First we prove
PROPOSITION 1.
(A3)
PROOF. The implication (A3)
(A2)
.
(A1).
(A2) is trivial. To prove (A2) = (A1), it is
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224
sufficient to prove that if f E H°° satisfies 11f II : 1 and f (xn) = 0 for every n, then
0 nH P(x, x,,) > I f (x) I
for x E M(H°°).
Let f = O[log If I] S[p] b be a canonical factorization. We devide {xn}n into three parts: {(, },
=
{e,},
_ {xn; xn V {(,), and
{A3},
= {xn; f does not vanish indentically on P(xn)}.
0};
{xn; ODog If 11
0};
Let k be an arbitrary positive integer. We put f k = 0[k-'log I f I]
S[k-1 a].
Then fk E H°°, IIfkII < 1 and O[log If I] S[µ] = (fk)', hence fk((,) = 0 for every j. Therefore (1)
H1
P(x,(,) >_ Ifk(x)I1 = I(o[logIfI]S[i])(x)I
By [6, Theorem 5.3], there are interpolating Blaschke products b1i b2, ... , bk such that b = (Hj*=1 b,) B and b, (A,) = 0 for j = 1, 2, ... , k for some Blaschke product B. Then we have k (2)
k
k
H P(x,.)i) >_7=1 H Ib,(x)I = ,=1
(.,R1 bi)(x)
We apply Lemma 1 for B succeedingly. Then we have a factorization B = H1'=1B; such that BB(e;) = 0 for every i. Hence k
k
H P(x, e,) >_ (Jn1 B3)(x) = I B(x)I =1
.
By (2), we have k
H P(x, ,) P(x,'Xi) >_ Ib(x)I 3=1
Therefore by (1), k
H P(x,(.i)P(x,C.i)P(x,A.i) >_ If(x)I
7=1
Letting k --> oo, we have H00°=1 p(x, xn) > If (x)I I.
Now we prove the following theorem. The idea of this proof comes from Axler and Gorkin [1, Theorem 3].
THEOREM 1. Let {xn}n be a sequence of trivial points in M(H°°). {xn}n is interpolating if and only if {xn}n satisfies condition (A2).
Then
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Izuchi
PROOF. Suppose that {xn}n satisfies condition (A2). Then there is a sequence {Fn}n in H°° such that IIF,II < 1, (1)
F,(xk) = 0
if k # n, and
(2)
F,(xn)
for every n.
0
We may assume that F,(xn) > 0 for every n. Let {En}n be a sequence of positive numbers such that 00
H (1+en) < 00.
(3)
n=1
Take 0 < bn < 1 such that P(1 - bn, -1 + bn) = P(0' Fn(xn))
Here we note that if Fn(xn) closes to 1 then bn closes to 0. Let Fn = OnSnBn be a canonical factorization. For a positive integer k, we apply Lemma 1 for B. k-times succeedingly. As a result, we have a factorization Bn = Bn, Bn, Bn,. Since x j is a trivial point by our starting hypothsis, if Bn(x,) = 0 for some j then Bn, (xj) _ ... = Bnk(x,) = 0. Here we may assume that IBn7(xn)I < ... < IBnk(xn)I IBnt(xn)I
Then by (2), 11/k 1/k (0n Sn
hence
Bnk)(xn)
Fn(xn)
1/k
> Fn(xn),
1 (k -, oo). By (1), we have that (OnikSnikBnk)(xj) = 0
for j # n. Hence we can consider 0,1,/kSn/"Bnk instead of Fn, and so that we may assume moreover that (4)
bn < 1 - 1/
1 + 2En.
Let bn(z) -
z - (1 - bn)
zED.
1 - (1 - bn)z'
Then bn(0) = -1 + bn and bn(Fn(xn)) = 1 - 6n. Let hn(z) = bn o Fn(z)/(1 - bn),
z E D.
Then hn E H°°, 1 + Zen
(5)
IIhnII <
(6)
hn(xn) = 1, and hn(xk) = -1
by (4), and for every k with k # n.
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226
Now let
fn = ((1 + hn)/2)2
and
gn = ((1 - hn)/2)2.
Then fn and gn are contained in H°°, and by (5) Ifn1 +I9nI = (1+IhnI2)/2 < 1+En. Moreover by (6), (7)
fn(xn) = 1, and fn(xk) = 0
for k # n;
(8)
gn(xn) = 0, and gn(xk) = 1
for k # n.
Here let Gn = fng192
9n-1 Then by [1, Lemma 2] and (3),
II (1 + en) < 00 E IG, < n=1
(9)
on D.
n=1
Now we show that {xn}n is interpolating. Let {an}n be an arbitrary bounded sequence. Define
00
G= Ea,, Gn. n=1 By (9), we have G E H°°, and
G(xk) =
k-1
00
E1 anGn(xk) + akGk(xk) +(n
k+1 anGn)(xk)
00
ak + ( E anGn)(xk)
by (7) and (8).
n=k+1
is a function in H°° and
Here E 00
(n
k
00
IanGn)(xk)
=
(9192...9k)(xk)(n
=0
r+lanfngk+1'..gn-1)(xk)
by (8).
Hence G(xk) = ak for every k. This implies that {xn}n is interpolating. The converse is already mentioned as Fact 2. In the first part of the above proof, we show that if {xn}n is a sequence of trivial points, then there exists a sequence {Fn}n in H°° such that
IIFnII < 1, Fn(xk) = 0
if k # n, and
IFn(xn)I closes to 1 sufficiently for every n.
In the rest of the above proof, we prove that for a given sequence {xn}n in M(H°°), xn needs not to be trivial, if there is {Fn}n in H°° satisfying the above conditions, then {xn}n is interpolating. We note that this proof works for the spaces HaD on the other domains.
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Izuchi
PROPOSITION 2. Let H°° be the space of bounded analytic functions on the domain 12 in Cn. Let {xn}n be a sequence in M(H°°). If for every a with 0 < e < 1 there is a sequence {ff}n in H°° such that IIfnII < 1, fn(xk) = 0 for k # n and I fn(xn)I > e for every n, then {xn}n is an interpolating sequence. We have the following problem.
PROBLEM 1. Let {xn}n be a sequence of trivial points. (1)
If {xn}n is strongly discrete, is it interpolating ?
(2)
If {xn}n satisfies condition (A2), does it satisfy (A3) ?
We note that Hoffman proved in his unpublished note that if {xn}n is a strongly discrete sequence in M(L°°), then {xn}n is interpolating.
3. CONDITION (A3) In this section, we prove the following theorem.
THEOREM 2. Let {xn}n be a sequence in G. Then {xn}n satisfies condition (A3) if and only if {xn}n is strongly discrete and there exists an interpolating Blaschke product b such that {xn}n C Z(b).
To prove this, we need some lemmas. For a subset E of M(H°°), we denote by cl E the closure of E in M(H°°).
LEMMA 2 [5, p. 205]. Let b be an interpolating Blaschke product with zeros {zn}n. Then Z(b) = Cl {zn}n.
LEMMA 3 [6, p. 101]. If b is an interpolating Blaschke product, then Z(b) C G. Conversely, for a point x in G there is an interpolating Blaschke product b such that x E Z(b). For an interpolating Blaschke product b with zeros {zn}n, put
6(b) = inf k
II
n:n#k
p(zn, zk).
LEMMA 4 [6, p. 82]. Let b be an interpolating Blaschke product and let x be a point in M(H°°) with b(x) = 0. Then for 0 < a < 1 there is a Blaschke subproduct B of b such that B(x) = 0 and 6(B) > o-.
LEMMA 5 [3, p. 287]. For 0 < 6 < 1, there exists a positive constant K(b) satisfying the following condition; let b be an interpolating Blaschke product with zeros {zn}n such that 8(b) > 6. Then for every sequence {an}n of complex numbers with Ianl < 1, there
is a function h in H°° such that h(zn) = an for all n and IIhil < K(6).
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Izuchi
PROOF OF THEOREM 2. First suppose that {xn}n is strongly discrete and {xn}n C Z(b) for some interpolating Blaschke product b. Take a sequence {Un}n of disjoint
open subsets of M(H°°) such that xn E Un for every n. Let {zk}k be the zeros of b in D. For each n, let bn be the Blaschke product with zeros {zk}k fl Un. Since {Un}n is a 1bn is a subproduct of b. By Lemma 2, xn E cl {zk}k. Hence sequence of disjoint subsets, II-,b,, xn E cl ({zk}k fl Un), so that bn(xn) = 0. We also have IIn 7.7#
bi)(z)I > inf II p(z,, z;) = 6(b) > 0 i
77#$
for every z E {zk}k fl Un. Hence inf n
II b,)(xn) > 6(b). #n
Thus {xn}n satisfies condition (A3). Next suppose that {xn}n C G and {xn}n satisfies condition (A3). Then there is a sequence { fn}n in H°° such that JIfnII < 1, Hn 1 In E H°°, and (1)
fn(xn) = 0 andk inf (fn(xk)I > 6
for some b > 0. By considering {cn fn}n with 0 < cn < 1 and IIn 1 Cn > 0, we may assume
that (2)
Ilfnll < 1
for every n.
By (1), {xk}k is strongly discrete, hence we can take a sequence {Vk}k of disjoint open subsets of M(H°°) such that xk E Vk and (3)
inf { (n:n#k fn)(w)I;w E Vk} > 6.
Let K(6) be a positive constant which is given in Lemma 5 associated with 6 > 0. By (2), there is a sequence {En}n of positive numbers such that (4)
Ei
n=1
En < 6,
(5)
K(6) En < 6/2
(6)
IIfnII + K(b) En < 1
for every n, and for every n.
Since xn E G, by Lemmas 2 and 3 there is an interpolating Blaschke product bn with zeros {wn,,}, such that bn(xn) = 0 and (7)
{wn,,}, C Vn f1 D.
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229
By Lemma 4, we may assume that
5(bn) > b
(8)
for every n.
Since fn(xn) = 0, we may assume moreover that I fn(wn,,)I < En
(9)
for every j.
By considering tails of sequences {wn,,}, for n = 1, 2, ... , we may assume that En,, 1- I was I <
oo, that is, I10 , bn is a Blaschke product. By (3) and (7), for n # k we have Ifn(wk,,)I > b
for every j.
Hence by (5),
I fn(wk,,)I - K(b) En > 6/2
for every j.
Let take c > 0 such that
c(x-1) < logx
for5/2<x < 1.
Then c
(10)
(Ifn(wk,)I - K(b) E. - 1) < 109
(Ifn(wkj)I - K(b) En)
for n # k and j = 1, 2, .... Since 1-x < - log x for 0 < x < 1, we have 0 < 1 - Ifn(Wk,,) I < - 109 Ifn(wk3)I
(11)
forn
kandj=1,2,....
Now we shall prove that b = 110 , bn is an interpolating Blaschke product. By (8) and (9), there is a function gn in H°° such that gn(wn,,) = fn(wn,,) for every j and (12)
IIgnII < K(6)-Fn-
Then fn = gn + bnhn for some hn in H°°. By (6) and (12), IIhnII < 1. Hence
Ifn - 9nI < IbnI
(13)
on D.
Here we have the following for every j and k, I(n:11
bn)(wk,,)l
= n:H Ibn(wka)I
> n:#k Ifn(wk,) - gn(wk,)I
by (13)
> n:H k(Ifn(wk,,)I - K(b) En) exp log (Ifn(wka)I - K(b) En)}
by (12)
{n:E
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Izuchi
> exp{n:n#k c(Ifn(wka)I - K(b)En - 1)}
exp[-c(n:
> exp[-c n.
by (10)
1 - Ifn(wk,,)I)] exp(-cK(b) n. $k En) by (4) and (11) #k-log I fn(wk,,)I] exp(-cK(b) b)
#k E
Ifn(Wk,,) I)' exp (-c K(b) b)
(n:#k > be exp (-c K(b) b)
by (3) and (7).
Therefore we have b(b)
= inf 11 P(wn,,,Wki) kJ (n,i):(n,i)#(k,j) kJ
I(n:
#k bn)(wk,,)) :II
P(wk,i,Wk,,)
> be exp (-c K(b) b) inf b(bk)
> 6°+' exp (-c K(b) b)
by (8).
Thus b is an interpolating Blaschke product. Since bn(xn) = 0, b(xn) = (I1nn=1 bn)(Xn) = 0This completes the proof. In [8], the author actually proved the following.
_
PROPOSITION 3. Let {xn}n be a sequence in G such that P(Xn) flcl {xk}k#n for every n. If {xn}n satisfies condition (A2), then {xn}n satisfies condition (A3).
If {xn}n satisfies a more stronger topological condition, then we can get the same conclusion without condition(A2)-
PROPOSITION 4. Let {xn}n be a sequence in G. If cl P(xn)flci (Uk:k#n P(xk)) = 0 for every n, then {xn}n satisfies condition (A3). To prove this, we use the following lemma.
LEMMA 6 [8, Lemma 8]. Let x E G and let E be a closed subset of M(H°°) with P(x) fl E = 0. Then for 0 < E < 1, there is an interpolating Blaschke product b such that b(x) = 0 and Ibl > E on E. PROOF OF PROPOSITION 4. By our assumption, there is a sequence {Un}n of disjoint open subsets of M(H°°) such that P(xn) C Un for every n. Let {en}n be a sequence of positive numbers such that 0 < En < 1 and IIn 1 En > 0. By Lemma 6, there is an interpolating Blaschke product bn such that bn(xn) = 0 and Ibnl > En on M(H°°) \ Un. By considering tails of bn, n = 1, 2, ..., we may assume that IIn 1 bn E H°°. Since Un C M(H°°) \ Uk for k # n, we have II
k:k#n
bk I >k:k#n H
Ch
on D n Un for every n.
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Izuchi
Since xn is contained in cl (D fl Un), II
k.k#n
bk)(xn) > II
k=1
for every n.
Ek
Hence {xn}n satisfies condition (A3).
In [10], Lingenberg proved that if E is a closed subset of M(H°°) such that E C G
and HOE = C(E), the space of continuous functions on E, then there is an interpolating Blaschke product b such that E C Z(b). Here we have the following problem. PRPOBLEM 2. If {xn}n is an interpolating sequence in G, is cl {xn}n C G true ? If the answer of this problem is affirmative, we have that if {xn}n is interpolating
and {xn}n C G, then there exists an interpolating Blaschke product b such that {xn}n C Z(b). We have anothor problem relating to Problem 2. PROBLEM 3. Let {xn}n be a sequence in G. If {xn}n satisfies condition (A2), does {xn}n satisfy condition (A3) ?
By Theorem 2 and Lemma 3, it is not difficult to see that if Problem 3 is true then Problem 2 is true. We end this section with the following problem. PROBLEM 4. Let {xn}n be a sequence in G. If {xn}n satisfies condition (A2), is {xn}n interpolating ?
4. CONDITON (A1) In [6, p. 109], Hoffman gave an example of a nontrivial part which is not a homeomorphic part. We use his example to prove the following theorem. THEOREM 3. There exists a sequence {xn}n satisfing the following conditions. (i) (ii) (iii) (iv)
{xn}n is contained in a nontrivial part. {xn}n is strongly discrete. {xn}n satisfies condition (A1). {xn}n is not interpolating.
PROOF. We work on in the right half plane C+. Then S = {1 + nil, is an interpolating sequence for H°°(C+). Let b be an interpolating Blaschke product with these zeros. Let the integers operate on S by translation vertically. That gives a group homeomorphism of cl S; hk : cl S
CI S
hk(1 + ni) = 1 + (n + k)i.
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232
Let K be a closed subset of cl S \ S which is invariant under hl and which is minimal with that property (among closed sets). Let m E K. The sequence ink = hk(m),
k = 1, 2, .. .
is invariant under hl. Therefore K = cl {mk}k>N
(1)
for every N.
Let Lm be the Hoffman map from C+ onto P(m). Then Lm(1) = m and Lm(l + ik) = mk. Hence by (1), P(m) is not a homeomorphic part. Let xn = Lm(l + 1/n+in) for n = 1, 2,.... Then xn E P(m)
(2)
for every n.
We note that (3)
p(1 + 1/n + in, 1 + in) -p 0 (n --r oo).
Since {1 + in}n is an interpolating sequence in C+, {1 + 1/n + in}n is also interpolating. Hence 11 + 1/n + in}n satisfies condition(Al). Since Lm preserves p-distance [6, p. 103], {xn}n satisfies condition (A1).
(4)
Since b = 0 on K, b(xn) - 0 (n -> oo). But we have b(xn) # 0. Hence {xn}n is strongly discrete.
(5)
To prove that {xn}n is not interpolating, it is sufficient to prove that {xn}n does not satisfy (A2). Suppose that there exists gn in H°° such that IlgnII < 1, gn(xk) = 0 for k # n, and gn(xn) # 0. By (3), we have p(rnk, xk) -, 0 (k , oo). Hence by (1), gn = 0 on K. Therefore Ign(xn)I
p(xn, mn) --p 0 (n , oo)
This implies that {xn}n does not satisfy condition (A2).
REFERENCES [1]
S. Axler and P. Gorkin, Sequences in the maximal ideal space of H°°, Proc. Amer. Math. Soc. 108(1990), 731-740.
[2]
L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80(1958), 921-930.
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[3]
J. Garnett, Bounded analytic functions, Academic Press, New York and London,
[4]
P. Gorkin, H. -M. Lingenberg and R. Mortini, Homeomorphic disks in the spectrum of H°°, Indiana Univ. Math. J. 39(1990), 961-983.
[5]
K. Hoffman, Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs,
1981.
New Jersey, 1962. [6]
K. Hoffman, Bounded analytic functions and Gleason parts, Ann.
of Math.
86(1967), 74-111. [7]
K. Izuchi, Interpolating sequences in a homeomorphic part of H°°, Proc. Amer. Math. Soc. 111(1991), 1057-1065.
[8]
K. Izuchi, Interpolating sequences in the maximal ideal space of H°°, J. Math. Soc. Japan 43(1991),721-731.
[9] K. Izuchi, Factorization of Blaschke products, to appear in Michigan Math. J. [10] H. -M. Lingenberg, Interpolation sets in the maximal ideal space of H°°, Michigan Math. J. 39(1992), 53-63.
Department of Mathematics Kanagawa University Yokohama 221, JAPAM MSC 1991: Primary 30D50, 46J15
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
234
OPERATOR MATRICES WITH CHORDAL INVERSE PATTERNS*
Charles R. Johnson' and Michael Lundquist We consider invertible operator matrices whose conformably partitioned inverses have 0 blocks in positions corresponding to a chordal graph. In this event, we describe a) block entry formulae that express certain blocks (in particular, those corresponding to 0 blocks in the inverse) in terms of others, under a regularity condition, and b) in the Hermitian case, a formula for the inertia in terms of inertias of certain key blocks.
INTRODUCTION
For Hilbert spaces 9{i, i = 1, , n, let x be the Hilbert space defined by 7{ = 7{l ®. . . ®xn. Suppose, further, that A : J{ -, J{ is a linear operator in matrix form, partitioned as All A,2 ... Aln
A=
A21
Anl
...
Ann
in which Aid : x; -4 Xi, i, j = 1, , n. (We refer to such an A as an operator matrix.) We assume throughout that A is invertible and that A-' = B = [Bid] is partitioned conformably. We are interested in the situation in which some of the blocks Bid happen to be zero. In this event we present (1) some relations among blocks of A (under a further regularity condition) and (2) a formula for the inertia of A, in terlns of that of certain principal submatrices, when A is Hermitian. For this purpose we define an undirected graph G = G(B) on vertex set N = {1, , n} as follows: there is an edge {i, j}, i # j, in G(B) unless both Bid and Bpi are 0.
An undirected graph G is called chordal if no subgraph induced by 4 or more vertices is a cycle. Note that if G(B) is not complete, then there are chordal graphs *This manuscript was prepared while both authors were visitors at the Institute for Mathematics and its Applications, Minneapolis, Minnesota. 'The work of this author was supported by National Science Foundation grant DMS90-00839 and by Office of Naval Research contract N00014-90-J-1739.
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G (that are also not complete) such that G(B) is contained in G. Thus, if there is any symmetric sparsity in B, our results will apply (perhaps by ignoring the fact that some blocks are 0), even if G(B) is not chordal. A clique in an undirected graph G is a set of vertices whose vertex induced subgraph in G is complete (i.e. contains all possible edges {i, j }, i # j). A clique is maximal if it is not a proper subset of any other clique. Let C = G(G) = {al,. ,ay} be the collection of maximal cliques of the graph G. The intersection graph 9 of the maximal
cliques is an undirected graph with vertex set e and an edge between a; and a i # j if a; fl ai # 0. The graph G is connected and chordal if and only if 5 has a spanning tree 7 that satisfies the intersection property: a, fl aj C ak whenever ak lies on the unique simple path in 7 from a, to aj. Such a tree 7 is called a clique tree for G and is generally not unique [2]. (See [3] for general background facts about chordal graphs.) Clique trees
constitute an important tool for understanding the structure of a chordal graph. For example, for a pair of nonadjacent vertices u, v in G, a u, v separator is a set of vertices of G whose removal (along with all edges incident with them) leaves u and v in different connected components of the result. A u, v separator is called minimal if no proper subset
of it is a u, v separator. A set of vertices is called a minimal vertex separator if it is a minimal u, v separator for some pair of vertices u, v. (Note that it is possible for a proper subset of a minimal vertex separator to also be a minimal vertex separator.) If a, and ai are adjacent cliques in a clique tree for a chordal graph G then a, fl ai is a minimal vertex separator for G. The collection of such intersections (including multiplicities) turns out to be independent of the clique tree and all minimal vertex separators for G occur among such intersections. Given an n-by-n operator matrix A = (A,, ), we denote the operator submatrix lying in block rows a C N and block columns /3 C N by A[a, /3]. When the submatrix is principal (i.e. /3 = a), we abbreviate A[a, a] to A[a].
We define the inertia of an Hermitian operator B on a Hilbert space 3C as follows. The triple i(B) = (i+ (B), i_(B), io(B)) has components defined by i+(B) = the maximum dimension of an invariant subspace of B on which the quadratic form is positive. i_(B) = the maximum dimension of an invariant subspace of B on which the quadratic form is negative. And
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io(B) __ the dimension of the kernel of B (ker B). Each component of i(B) may be a nonnegative integer, or oo in case the relevant dimension is not finite. We say that two Hermitian operators B1 and B2 on 9C are congruent if there
is an invertible operator C : X -, X such that B2 = C*B1C.
According to the spectral theorem, if a bounded linear operator A : 7{ Hermitian, then A is unitarily congruent (similar) to a direct sum: U*AU =
A+
0
0
A_
0 0
0
0
0
W is
in which A+ is positive definite and A_ is negative definite. As i(A) = i(U*AU), i+(A) is the "dimension" of the direct summand A+, i_(A) the dimension of A_, and io(A) the dimension of the 0 direct summand, including the possibility of oo in each case. It is easily checked that the following three statements are then equivalent: I
(i) A is congruent to
0 0
0
0
0
0
-I 0
,
in which the sizes of the diagonal blocks are
i+(A), i_(A) and io(A), respectively; (ii) each of A+ and A_ is invertible; and
(iii) A has closed range. We shall frequently need to make use of congruential representations of the form (i) and, so, assume throughout that each key principal submatrix (i.e. those corresponding to maximal
cliques and minimal separators in the chordal graph G of the inverse of an invertible Hermitian matrix) has closed range. This may be a stronger assumption than is necessary for our formulae in section 3; so there is an open question here.
Chordal graphs have played a key role in the theory of positive definite completions of matrices and in determinantal formulae. For example, in [4] it was shown that if the undirected graph of the specified entries of a partial positive definite matrix (with specified diagonal) is chordal, then a positive definite completion exists. (See e.g. [6] for definitions and background.) Furthermore, if the graph of the specified entries is not chordal, then there is a partial positive definite matrix for which there is no positive
Johnson and Lundquist
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definite completion. (These facts carry over in a natural way to operator matrices.) If there is a positive definite completion, then there is a unique determinant maximizing one that is characterized by having 0's in the inverse in all positions corresponding to originally unspecified entries. Thus, if the graph of the specified entries is chordal, then the ordinary (undirected) graph of the inverse of the determinant maximizer is (generically) the same chordal graph. (In the partial positive definite operator matrix case such a zeros in-theinverse completion still exists when the data is chordal and is an open question otherwise.) This was one of the initial motivations for studying matrices with chordal inverse (nonzero) patterns. Other motivation includes the structure of inverses of banded matrices, and this is background for section 2.
If an invertible matrix A has an inverse pattern contained in a chordal graph G, then det A may be expressed in terms of certain key principal minors [1], as long as all relevant minors are nonzero:
fl det A[a] aEe
detA=
11
det A[a fl ,0]
{a,PIEE
Here a is the collections of maximal cliques of G, and T = (C, £) is a clique tree for G. Thus, the numerator is the product of principal minors associated with maximal cliques, while the denominator has those associated with minimal vertex separators (with proper multiplicities). There is no natural analog of this determinantal formula in the operator case, but the inertia formula presented in section 3 has a logarithmic resemblance to it. 2. ENTRY FORMULAE
Let G = (N, E) be a chordal graph. We will say that an operator matrix A = [AiJ] is G-regular if A[a] is invertible whenever a C V is either a maximal clique of G or a minimal vertex separator of G. In this section we will establish explicit formulae for some of the block entries of A when G(A-1) C G. Specifically, those entries are the ones corresponding to edges that are absent from E (see Theorem 3). LEMMA 1. Let A = [Ai3] be a 3-by-3 operator matrix, and assume that M1 = [Au A21
A121 A22 J ,
M
I A22
2 = L A32
A23 A33
and A22 are each invertible.
Then B = A-1 exists and satisfies B13 = 0 if and only if A13 = A12A22 lA23.
238
Johnson and Lundquist
Proof. Let us compute the Schur complement of A22 in A:
I (1)
-Al2Az2
0
I
0 0
0
-A32Az2
I
All A2, A3,
All - A12A22A21 0
A3, - A32 A22A21
0 A22 0
A12 A0 0 A22 A32
Al: 23
-A22 A2,
A33
0
I -A22 A23 0
I
A,3 - A12A22 'A23 0 A33 - A32A22 A23
If B = A-1 exists, then B11 (2)
B31
B13 B33 ]
All - A12A-22lA21 A,3 - Al2`922A23 1 [ A31 - A32 A22 A21
_1
A33 - A32A22 A23 J
and hence if B13 = 0, then necessarily we have A,3 = A12A22 A23. Conversely, if A,3 = A12A-lA23, then the (1,3) entry of the matrix on the right-hand side of (1) is zero. Note 22 that All -- A12'`122 A21 and A33 - A32A22A23 are invertible, because they are the Schur complements of A22 in M, and M2. Hence A is invertible, and by (2) we have B13 = 0. Under the conditions of the preceding lemma, if we would like B31 = 0 then we must also have A31 = A32 A22 A21. Notice that the graph of B in this case is a path:
G=10
0.
Suppose now that A = [A,,] is an invertible n x n operator matrix, that 1 < k < m < n, and that A-l = [B,,] satisfies B,j = 0 and Bj, = 0 whenever i < k and .7 > m. In this case B has the block form B12 B22
0 B23
B32
B33
I
in which B1, = B[11,... , k - 11], b22 = B[{k,... , m}] and b33 = B[{m + 1, ... , m}]. Let A = [A,,] be partitioned conformably. If in addition to the above conditions we also have that A[{1, ... , m}], A[{k,... , n}] and A[{k, ... , m}] are invertible, then we simply have the case covered in the preceding Lemma, and we may deduce that
A[{1,...,k- 1},{m+l,...,n}] = A[11,..., k - 1}, {k,... , m}] A[{k,... , m}]-'A[{k, ... , m}, {m + 1, ... , n}], with a similar formula holding for A[{m + 1, ... , n}, { 1, ... , k - 1}]. From this we may write explicit formulae for individual entries in A. For example, we may express any entry
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A,, for which i < k and j > m as (3)
A;, = A[{i},{k,...,m}] A[{k,...,m}]-'A[{k,...,m}, {j}].
There is an obvious similarity between this situation and that covered in Lemma 1, which one sees simply by looking at the block structure of A'. But there are also some similarities which may be observed by looking at graphs. In the block case we just considered,
the graph G(B) is a chordal graph consisting of exactly two maximal cliques, the sets al = {1, ... , m} and a2 = {k, ... , n}. The intersection /3 = {k,.. . , m} of al and a2 is a minimal vertex separator of G (in fact, the only minimal vertex separator in this graph). The formula (3) may then be written (4)
A13 = A[{i},/3]A[/3]-'A[/3,{j}].
Note now in the 3-by-3 case that the equation A13 = A12A22 A23 has the same form as (4) when we let /3 = {2}. In fact, since /3 = {2} is a minimal separator of the vertices 1 and 3
in the graph 1
2
3
w e see that {2} plays the same role in the 3-by-3 case as {k,. .. , m} does in the n x n case.
In Theorem 3 we will encounter expressions of the form (5)
A;3 = A[{i},/3k]A[Q1]-'A[/31,Q2]A[Q2]-' ...A[,am]-'A[Qm,{j}]
in which each Qk is a minimal vertex separator in a chordal graph. The sequence (01 , ... , /3m )
is obtained by looking at a clique tree for the chordal graph, identifying a path (ao, al .... , am ) in the tree, and setting /3k = ak_1 fl ak.
These expressions turn out to be the natural generalization of (4) to cases in which the graph of B is any chordal graph. In addition, the results of this section generalize results of [9] from the scalar case to the operator case. LEMMA 2. Let A : -C -* x be an invertible operator matrix, with B = A-'. Let G = (N, E) be the undirected graph of B. Let {i, j } V E and let /3 c N be any i, j separator for which A[,3] is invertible. Then
A;j = A[{i},Q]A[Q]-'A[,6, fill-
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Proof. Without loss of generality we may assume that /3 = {k,. .. , m}, with k < in, and that Q separates any vertices r and s for which r < k and s > m. Assuming then that i < k and j > in, we may write B as B11 B21 0
B12 B22 B22
0 B2S
B33
The result now follows from Lemma 1 and the remarks that follow it.
If G is chordal and i and j are nonadjacent vertices then an i, j clique path will mean a path in any clique tree associated with G that joins a clique containing vertex i to a clique containing vertex j. One important property of any i,j clique path is that it will "contain" every minimal i, j separator in the following sense: If (ao, ... , a,,, )
is any i, j clique path, and if /3 is any minimal i, j separator then /3 = ak_1 fl ak for some k, 1 < k < m. Another important property of an i, j clique path is that every set /3k = ak_1 flak, 1 < k < m, is an i, j separator. It is not the case, however, that every /3k is a minimal i, j separator (see [9]). THEOREM 3. Let G = (N, E) be a connected chordal graph, and let A : x H be a G-regular operator matrix. then the following assertions are equivalent:
(i) A is invertible and G(A-1) C G; (ii) for every {i, j } V E there exists a minimal i, j separator /3 such that Aid = A[10, 0] A[P]-1A[/3, {j}];
(iii) for every {i, j} V E and every minimal i, j separator /3 we have Aid = A[{i},/3] A[/3]-1A[/3, {j}];
(iv) for every {i, j } V E, every i, j clique path (ao, a1, ... , a,") and any k, l < k < m we have
Aid = A[{i}, Qk] A[Qk]-' A[Qk, fill, in which /3k = ak_1 flak; and
(v) for every {i, j }' E and every i, j clique path (ao, a1, ...
we have
Ai, = A[{i}, /31] A[,31 ]-1 A[Q1, / 3 2 ] ... A[#.]-'A[,6., {7 }],
in which /3k = ak_1 flak.
Johnson and Lundquist
241
Proof. We will establish the following implications:
(iv)
(iii) = (ii)
(i) 4=4' (iv)
(iv);
(v).
(iv) , (iii) follows from the observation that every minimal i, j separator equals Qk for some k, 1 < k < m. (iii) = (ii) is immediate.
For (ii) (iv), let {i, j) V E, and let (ao, a1, ... , a,,) be a shortest i,j clique path. We will induct on m. For m = 1 there is nothing to show, since in this case Ql = ao fl a1 is the only minimal i, j separator. Now let m > 2, and suppose that (iv) holds for all nonadjacent pairs of vertices for which the shortest clique path has length less than m. Since every minimal i, j separator equals /3k for some k, we have, by (ii), Aij = A[{i},/3k]A[/jk]-'A[/3k, {j}]
for some k, 1 < k < m. It will therefore suffice to show that for k = 1, 2, ... , m - 1 we have (6)
A[{i},Qk]A[Qk]-1A[,3k, {y }] = A[{i}, 8k+1]
A[,6k+11-1
A[Qk+1, {i}1
Let us first observe that f o r k = 1, ... , m - 1, (7)
A[Qk, {J }} = A[/3k, Nk+11
A[,6k+l]-1 A[Qk+1, {j }]
Indeed, suppose r E 13k. Then (ak, ak+1, ... , am) is an r, j clique path of length m - k, and by the induction hypothesis we may write Ark = A[{r},Qk+1] A[Qk+11-'A[Qk+1, {j}],
and equation (7) follows. A similar argument shows that for k = 2, ... , in we have (8)
A[{i}, Qk+11 = A[{i }, ,3k] A[,ak] -1 A[,3k, /3k+1 ]
By (7) and (8), both sides of (6) are equal to A[{i}, akl A[Qk]-1A[,8k,Qk+l]
A[Qk+l]-1A[Qk+l,
{j}],
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and hence (6) holds, as required.
(i) = (iv) follows from Lemma 2.
For (iv) = (i), let the maximal cliques of G be al, a2i ... , ay, p > 2. We will induct on p. In case p = 2 then the result follows from Lemma 1, so let p > 2 and suppose that the implication holds whenever the maximal cliques number fewer than p. Let 7 be a clique tree associated with G, let {ak, ak+1 } be any edge of T, and suppose the vertex sets of the two connected components of 7 - {ak, ak+l } are el = {al, ... , ak} and
(Let Gv be the subgraph of e2 = {ak+1, ... , ap}. Set V 1 = U jai and V2 = Up G induced by the subset V of vertices.) Since induced subgraphs of a chordal graph are k+jai.
necessarily chordal, Gv, and Gv2 are chordal graphs, and since (iv) holds for the matrix A, (iv) holds as well for A[V1] and A[V2]. By the induction hypothesis, A[V1] and A[V2] are invertible. Note also that V1 n V2 = ak n ak+1, which follows from the intersection property. Since A[V1 n V2] is invertible, we may now apply Lemma 1 to the matrix A (in which All is replaced by A[V1 \ V2], A22 by A[V1 nV2] and A33 by A[V2 \V1]), and conclude
that A-l [V1 \ V2i V2 \ Vl] = 0 and A-' [V2 \ V1, V1 \ V2] = 0. In other words, if we set B = A-l then Bid = 0 and Bpi = 0 whenever i E V1 \ V2 and j E V2 \ V1. Now if {i, j } ¢ E then ak and ak+1 may be chosen (renumbering the a's if necessary) so that i E V1 \ V2 and j E V2 \ V1. Hence it must be that Bii = 0 and Bpi = 0 whenever {i, j } V E.
For (iv) (v), let {i, j } ¢ E, and let (ao, a1, ... , a,,,) be any i, j clique path. First, we must observe that for any k,1 < k < m, A[Qk, {j}] = A[Qk,Qk+l]A[Qk+1]-lA[Qk+l, {]'}].
(9)
Indeed, by assumption, for any r E Qk we have A,., = A[{r},Qk+I]A[Qk+1]-'A[Qk+], {j}],
and (9) follows from this. By successively applying (9) we obtain Aij = A[{i},Q1]A[Q1]-1 A[Q1, {7}]
=
A[{i},Q1]A[/j1]-1A[Ql,Q2]A[Q2]-1A[Q2,
{j}]
= A[{i}, Q1]A[QI]-lA[Q1, Q2] ... A[Qm-1, Qm]A[Qm]-1A[Qm, {i}],
as required.
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F o r (v) = (iv), let {i, j } V E, and let (ao, ... , a,,) be an i, j clique path. Let r E ak-1, 1 < k < m. We may write [because of assumption (v)] Air = A[{i},, i]A[/31]-' ... A[Qk-1]-'A[/3k-1, jr)],
and because r E /3k ; r E ak we thus have (10)
A[{i}, O k] = A[{i}, /3,]A[/31]-' ... A[ak-1]-'A[Qk-1, 13k].
It may be similarly shown that (11)
A[Qk, {j }] = A[Qk, Qk+,]A[Qk+1]-' ...
A[Qm]-' A[Qm, {j}].
By using (10) and (11) we therefore obtain Ail = A[{i},01] ... A[Qk-1,Qk]A[$k]-'A[Qk,Qk+1] ... A[Qm, {j}] = A[{i},8k]A[FQk]-'A[ak, {j}1, as required.
3. INERTIA FORMULA
In [8], it was shown that if A E M,,(C) is an invertible Hermitian matrix and if G = G(A-') is a chordal graph, then the inertia of A may be expressed in terms of the inertias of certain principal submatrices of A. Precisely, let C denote the collection of maximal cliques of G, and let 7 = (C, E) be a clique tree associated with G. If G(A-') = G, then it turns out that (11)
i(A) =
i(A[c ]) aEC
i(A[a n ,Q]). (a,O)EE
It is helpful to think of (11) as a generalization of the fact that if A-' is block diagonal (meaning, of course, that A is block diagonal) then the inertia of A is simply the sum of the inertias of the diagonal blocks of A. To see what (11) tells us in a specific case, suppose
that A-' has a pentadiagonal nonzero-pattern, as in
X X X
XXXX
A-'- X X X X X
X X X X X X X
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The graph of A-1 is then
G=
which is chordal. The maximal cliques of G are al = (1,2,3], a2 = {2, 3, 4} and a3 = {3, 4, 5}, and the clique tree associated with the graph G is
Equation (11) now tells us that the inertia of A is given by i(A) = i(A[{1, 2,311) + i(A[{2, 3, 4}]) + i(A[{3, 4, 5)])
- i(A[{2, 3)]) - i(A[13, 4}]).
Thus, we may compute the inertia of A by adding the inertia of these submatrices:
X X X x X
X x x x x
X x x x x
A x x x x
X x x x x
and subtracting the inertias of these:
Our goal in this section is to generalize formula (11) to the case in which A = [A1 ] is an invertible n-by-n Hermitian operator matrix. We will be concerned with the case in which one of the components of inertia is finite, so that in (11) we will replace
iby i+,i_or i0. For a chordal graph G = (N, E), we will say that an invertible n-by-n operator matrix A is weakly G-regular (or simply weakly regular) if for every maximal clique or minimal vertex separator a both A[a] and A-'[a`] have closed range.
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X1®9C2 be represented by the 2-by-2 matrix
LEMMA 4. Let M : JC1ED X2
M-
IC
Dl
Suppose that A is invertible, and that
JJ
M-1 =
rP Ql
LR SJ
Then dim ker A = dim ker S.
Proof. Let x1i x2, ... , xn be linearly independent elements of ker A. Then
for 1 < k
[01
[Xk
0- k in which Ilk = CXk, k = 1,... , n. Since M is invertible it follows that y1, y2, LC DJ
, yn are
linearly independent. Observe not now that Yk i [0]
[Xkl C from which it follows that Ilk E ker S. follows now that dim ker S > dim ker A; by reversing the argument we find that dim ker A > dim ker S. Thus dim ker A = dim ker S. O S
IJt
LR
-
LEMMA 5. Let M : Wl ® JC2 --+ W1 ® JC2 be Hermitian and invertible, and
suppose that
M= I BA.
Bl. C
If i+(M) < oo, then i,,(A) < oo. Proof.
Clearly i+(A) < oc, so let n = i+(A). Let H be an invertible
operator for which H*AH = In ® -I ® 0, in which In denotes the identity operator on an n-dimensional subspace, and -I and 0 are operators on spaces of respective dimensions i_(A) and i,,(A). Then In Bl
01 [ O
1H0 J
[B*
CC]
1
Bj
We may reduce this further by another congruence: In Bl In
I
-I
I
-Bi B2 O I 0
In
0
-I
0
Bi Bz
0
0
B3
B3
S
'
0
B2 B3
B3
C
-I
0
B3
Bz
B3
C
In
I
-B1 B2
I
O
I
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246
in which S = C - BI BI + BZ B2. Hence
i+(M) = n + i+
O B3
and thus
But this implies that the zero block in this matrix must act on a space of finite dimension. Recalling that this dimension equals i,,(A), we obtain the desired conclusion.
The following Lemma generalizes a result of [5] to operator matrices from the finite-dimensional case (see also [8]).
LEMMA 6. Let M : x1 ®x2 -4 x1 ®x2 be Hermitian and invertible, with
M=
B] LBA.
and M-' = I Q RJ
If i+(M) < oo, and if A and R both have closed range, then
i+(M) = i+(A) + ia(A) + i+(R).
If io(A) = 0 then A is invertible and the result follows from the fact that R is the inverse of the Schur complement C - B*A-' B and that i+(M) = i+(A) + i+(C - B`A-'B). Hence, suppose that io(A) > 0. Since i+(M) < oo we have as Proof.
well from Lemma 5 that io(A) < oo.
Hence, let n = io(A), and let us consider the special case in which R = O. Since we require, by Lemma 4, that io(R) = io(A) = n, R must act on an n-dimensional space. Hence we have
M-' = P Q*
Q
0n1
where O denotes the zero operator on n-dimensional Hilbert space. By an appropriately chosen congruence of the form T1 = H ® I, we may reduce M to the form Ik
M1 = T1 MT1 = Bi
B1
-I B2*
On Bj
B3 B2
C
I
I
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247
where k = i+(A). With [1k
-B1
-I
T2 =
B2
0 I'
In
we then have
Ik
I
O O
-I
M2 =TZM1T2 =
On
B3
in which S = C - Bi B1 + B2 *B2. The matrix On
B3
[ B3
S
is an invertible operator on a 2n-by-2n Hilbert space, and in this case its inertia must be (n, n, 0). From the form of M2 we see that we must have
i+(M)=k+i+ CrBk B3
=k+n
SI)
= i+(A) + io(A).
Since i+(R) = 0, this last expression equals i+(A) + io(A) + i+(R)-
Now let us consider the general case, in which we make no assumption concerning the dimension of the space on which R acts. Choose an invertible matrix of the form T1 = I ®H so that T M 1 T1 has the form
P M1 1 =TiM-'T1 =
Q1
it
Q2
Qz Qg
Q3
On
in which 2 = i+(R) and n = io(R) [= io(A)]. Then with
I __
T2
-Qi it Qs
O ve obtain
S
MZ 1 = Tz M1 1T2 =
O Qg
00 it -I
Q3
On
248
Johnson and Lundquist
From the form of M2 1 , and by simple calculations, we find that M2 = TZ 1Ti 1 M(Ti 1)* (Ti'
has the form
AUU M =
lf2
it
O
O
-1 C2
BI*
for some operators B2 and C2. Hence we have
i+(M) (12)
=
i+(R)
+ t+
([B
82
1)
2
Observe that A I B2
B2
S
1
C2,
Q3
Q31'
0-
and thus by the special case we considered previously, t+
(13)
Q1) A
B2
B2 C2
= t+(A) + io(A).
Thus combining (12) and (13) we obtain
t+(M) = i+(A) + io(A) + i+(R), as required.
The following lemma will be used in the proof of the main result of this section. First, let G = (V, E) be any connected chordal graph, and let T = (C!, E) be any clique tree associated with G. For any pair of maximal cliques a and /3 that are adjacent in T, let 7a and ?s be the subtrees of T - {a, /3} that contain, respectively, a and /3, and let e0 and e be the vertex sets of Ja and ?p. Define Va\R =
( U 7) \ /, ry
o
with Vp\, defined similarly. LEMMA 7. [2] Under the assumptions of the preceding paragraph, the following hold:
(1) Va\# n V. = 0i (ii) (a n Q)C = Va\$ U Vp\ai
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249
and
(iii) a` is the disjoint union
U V VR\a,
aC =
QEadj a
in which adja={/3EC:{a,/3}El;}. We should note the following consequences of Lemma 7. Suppose B = [B;2] is
a matrix satisfying G(B) C G, in which G is a chordal graph, and let T be a clique tree associated with G. If {a, /3} is an edge of 7, then B[(a fl /3)`] is essentially a direct sum of the matrices B[Va\p] and B[Vp\a]. The reason for this is that there are no edges between vertices in Va\p and vertices in Vp\a, and hence B;; = 0 whenever i E Va\p and j E V. Similarly, if a is any maximal clique of G then B[ac] is essentially a direct sum matrices of the form B[Vp\a] as /3 runs through all cliques that are adjacent in 7 to a.
LEMMA 8. Let H = 7{, ® ®x,,, let A: 9{
X be an invertible operator matrix, and let G = G(A-1) be a connected chordal graph. If 7 = (C, E) is any clique tree associated with G, then
E dim ker A[a] _
(14)
oEe
dimker A[a fl /3]. {a,p}EE
Proof. Let us look first at the left-hand side of (14). by Lemma 4 and by .jemma 7 we have dim ker A[a] _ aEe
(15)
dim ker A-' [a`] aEC
_ 1: 1: dim ker A-' [Vp\a]. aEC QEadj a
On the other hand, by applying Lemmas 4 and 7 we may see that the righthand side of (14) is dim ker A[a fl /3] = {a,p}EE ' 16)
dim ker A-1 [(a fl )3)`] {a,p}EE
= E (dim ker A-' [Vp\a] + dim ker A-' [V,,\,3]) . {a,P}EE
Observe that with every edge {a, /3} of 7 we may associate exactly two terms in the rightmost expression of (15), namely dim ker A-' [Vp\a] and dim ker A-' [V,,,\,9]. But this just means that (15) and (16) contain all the same terms, and hence (14) is established.
Johnson and Lundquist
250
THEOREM 9. Let G = (N, E) be a connected chordal graph, let A = [A1 ] be an n-by-n weakly G-regular Hermitian operator matrix, and suppose that G(A-') C G. If i+(A) < oo, then for any clique tree 7 = (C, E) associated with G we have
i+(A) = 1] i+(A[a]) aEe
i+(A[a n Q]) (00)EE
Proof. Since i+(A) < oo, we must have i+(A[a]) < oo for any a C N, and by Lemma 5 we know that ia(A[a]) < oo for any a C N. By Lemma 6 we may write
E i+(A[a]) ace
-
i+(A[a n,81) {a,(3}EE
[i+(A)
=
-
i+(A-'
[a`]) - i0(A[a])]
aEe
(17)
aEe
[i+(A) -
i+(A-1
i+(A) - > a+(A) -
[(a n Q)`]) - io(A[a n f3])]
i+(A-' [a`]) aEe
+ j i+(A-' [(a n ,6)r]) - E io(A[a]) + {a,#}EE
aEe
io(A[a n fl]). {a,#}EE
The last two terms of the last expression in (17) cancel by Lemma 8, and the two middle terms cancel by an argument similar to that used in the proof of Lemma 8. Finally, since 7 has exactly one more vertex than the number of edges, the right-hand side of (17) equals i+(A). This proves the theorem.
Of course, a similar statement is true for i_(A), and the corresponding statement for i,,(A) is already contained in Lemma 8. ACKNOWLEDGEMENT
The authors wish to thank M. Bakonyi and I. Spitkovski for helpful discussions of some operator theoretic background for the present paper.
Johnson and Lundquist
251
REFERENCES 1.
W. Barrett and C.R. Johnson, Determinantal Formulae for Matrices with Sparse Inverses, Linear Algebra Appl. 56 (1984), pp. 73-88.
2.
W. Barrett, C.R. Johnson and M. Lundquist, Determinantal Formulae for Matrix Completions Associated with Chordal Graphs, Linear Algebra Appl. 121 (1989), pp. 265-289.
3.
M. Golumbic Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
4.
R. Grone, C.R. Johnson, E. Sa and H. Wolkowicz, Positive Definite Comple-
tions of Partial Hermitian Matrices, Linear Algebra Appl. 58 (1984), pp. 109-124. 5.
E.V. Haynsworth, Determination of the Inertia of a Partitioned Hermitian Matrix, Linear Algebra Appl. 1 (1968), pp. 73-81.
6.
C.R. Johnson, Matrix Completion Problems: A Survey, Proceedings of Symposia in Applied Mathematics 40 (American Math. Soc.) (1990), pp. 171198.
7.
C.R. Johnson and W. Barrett, Spanning Tree Extensions of the HadamardFischer Inequalities, Linear Algebra Appl. 66 (1985), pp. 177-193.
8.
C.R. Johnson and M. Lundquist, An Inertia Formula for Hermitian Matrices with Sparse Inverses, Linear Algebra Appl., to appear.
9.
C.R. Johnson and M. Lundquist, Matrices with Chordal Inverse Zero Patterns, Linear and Multilinear Algebra, submitted.
Charles R. Johnson, Department of Mathematics, College of William and Mary, Williamsburg, VA 23185,
Michael Lundquist, Department of Mathematics, Brigham Young University, Provo, Utah 84602,
U.S.A.
U.S.A.
MSC: Primary 15A09, Secondary 15A21, 15A99, 47A20
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
252
MODELS AND UNITARY EQUIVALENCE OF CYCLIC
SELFADJOINT OPERATORS IN PONTRJAGIN SPACES
P. Jonas, H. Langer, B. Textorius It is shown that a cyclic selfadjoint operator in a Pontrjagin space is unitarily equivalent to the operator A1, of multiplication by the independent variable in some space fl (0) generated by a "distribution" 0. Further, criteria for the unitary equivalence of two such operators A1,, A, are given.
INTRODUCTION
It is well-known that a cyclic selfadjoint operator in a Hilbert space is unitarily equivalent to the operator of multiplication by the independent variable in a space L2 (o') with a positive measure o,. In the present paper we prove a corresponding result for a bounded cyclic selfadjoint operator A in a Pontrjagin space: It is shown that A is unitarily equivalent to the operator A1, of multiplication by the independent variable in some space
fl (0), generated by a "distribution" 0 (which is a certain linear functional on a space of test functions, e.g. the polynomials in one complex variable). The class F of these "distributions" 0 is introduced in Section 1. We mention that, for an element 0 E F there exists a finite exceptional set 3 (¢)
such that ¢ restricted to C\s (0) is a positive measure on IR\s (0) (in the notation of Section 1.3, s (0) = s (cp) U oo (0), if 0 = W + ik is the decomposition (1.5) of 0 E F). In the exceptional points, 0 can be more complicated due to the presence of a finite number of negative squares of the inner product. In Section 2 the space fl (0) is defined and, by means of the integral representation of 0 (see Lemma 1.2), a model of fl (0), which is an orthogonal sum of a Hilbert space L2 (¢) with some measure o, and a finite-dimensional space, is given. In Section 3 the operator A1, of multiplication by the independent variable in fl (0) is introduced and represented as a matrix in the model space of Section 2. Thus it follows that each bounded cyclic selfadjoint operator in a Pontrjagin space is unitarily equivalent to such a matrix model. Naturally, this model is, in some sense, a finitedimensional perturbation of the operator of multiplication by the independent variable in L2 (o).
In Section 4 conditions for the unitary equivalence of two operators A1,, A for E F are given. For this equivalence it turns out to be necessary that the corresponding measures o, & are equivalent and, moreover, that the square root of the density dQ/d& has
Jonas et al.
253
"values" and, sometimes, also "derivatives" at the real exceptional points. This necessary condition for the unitary equivalence of Aqs and A is, in fact, necessary and sufficient for the unitary equivalence of the spectral functions of A0 and If c = 1, also a necessary and sufficient condition for the unitary equivalence of A# and A is given.
In this paper we restrict ourselves to a bounded selfadjoint operator in a Pontrjagin space. The case of a densely defined unbounded selfadjoint operator is only technically more complicated: The space of polynomials has to be replaced by another suitable set of test functions. Scalar and operator valued distributions have already played a role in the spectral theory of selfadjoint operators in Pontrjagin and Krein spaces e.g. in the papers [6], [7].
In order to find the model space fl (0) and the model operator A# we might also have started from the construction of the space fl (Q) and the operator AQ used in [11] for a function Q E N,, (with bounded spectrum). In this connection we mention that these functions of class N,, can be considered as the Stieltjes transforms of the elements 0 E F, the functions of the class P,, (see, e.g., [12]) with bounded spectrum are the Fourier transforms of these 0 (compare also [8]). It seems to be interesting to study corresponding models for the case that A is not cyclic but has a finite number of generating vectors, which in a Pontrjagin space can always be supposed without loss of generality. 1. THE CLASS F OF LINEAR FUNCTIONALS
1.1. Distributions of the class F(IR). By F(IR) we denote the set of all distributions cp on 1R. with compact support such that the following holds. (a) 'P is real, that is, cp has real values on real test functions.
(b) There exists a finite set s (,p) C IR (the case s (so) = 0 is not excluded) such that ,p restricted to IR\s (gyp) is a (possibly unbounded) positive measure, and s (,p) is
the smallest set with this property. For a E IR, the set of all cp E F(IR) with s (cp) = {a} is denoted by F (1R, a)
Let cp E F (IR). Asumme that n > 1, s (gyp) = {al, ... , an}, -oo < al < ... < < an < oo, and let ti, i = 1, ... , n - 1, be real points with ai < ti < ai+l, i = 17.. ., n -1, such that V has no masses in the points ti. We set Al (-oo,tl],D1 := (ti_liti],i = 2, ... , n -1, An := (tn-1i oo). A system of intervals Ai, i = 1,. . . , n, with these properties is called a (p-minimal decomposition of R. Let xo, be the characteristic function on IR of Lxi. Then xo;s° E F (IR, ai), i = 1,... , n, and n
'p f = E(xoi1P) f,
f EC°° (1R).
i=1
Here
denotes the usual duality of distributions and C°O functions on R.
If a E 1R and 1p E F (IR, a), the order i (rp) of sv is, as usual, the smallest n E No (= IN U {0}) such that cp is the n-th derivative of a (signed) measure on R. We denote by µo (a;,p), (µr (a;,p), µj (a;,p)) the smallest n E INo such that, for some measurer
254
Jonas et al.
on IR, the n-th derivative r(") of rr and cp coincide on (-oo, a) U (a, no) ((a, oo), (-oo, a), respectively). The numbers µo (a; gyp), µ,. (a; gyp), µ1(a; gyp) are called reduced order, right
reduced order, left reduced order, respectively, of V at a. Evidently, we have µo (a; ca) = max {µ,. (a; gyp), µ1(a; V)}. Some more properties of these numbers are given in the following lemma (compare [6; Hilfsatz 1,2]). LEMMA 1.1. If w E .F(IR,a), then the following statements hold. (i) jr (a; rp) (µ1(a; cp)) coincides with the minimum of the numbers n E INo such that
(t - a)"(p is a bounded measure on (a, oo) ((-oo, a), respectively).
(ii) (t -
a measure.
Here and in the sequel t denotes the function f (t)
t.
PROOF. (i) Choose a > IaI with supp V C (-a, a) and let n be such that (t-a)"p is a bounded measure on (a, a). If f is an element of
M
If E Co (IR)
:
supp f C (a, a),
sup I f (") (t) I < 1}, tE (a,a)
Taylor's formula implies sup{I(t - a) -"f (t) I : t E (a, a)} < 1. It follows that sup {IV f I
: f EM}=sup{I(t-a)' .(t-a)-"fI: f EM}
Then, by a standard argument of distribution theory, cp restricted to (a, a) is the n-th derivative of a bounded measure on (a, a), i.e. µ,. (a; gyp) < n.
It remains to show that with µ,. := µ, (a; cp), (t - a) µ'V I (a, a) is a bounded measure. To this end choose a nonnegative /3 E C°° (IR) equal to 0 on a neighbourhood of (-oo, 0] in JR and equal to 1 on a neighbourhood of [1, oo) in JR, and set ,Qk (t) := /3 (k (t-a) ), k E IN. A simple computation yields the uniform boundedness of the functions ( (t - a) k E IN, on (a, a). Then, since there exists a measure 'Po on JR with compact support such that V(µ') = p on (a, oo), it follows that
(t - a)"V - flk=gyp'(t-a)W'/3k = (-1)"Vo.((t
-ce)µ'/3k\(W). -a),U-V
The last expression is uniformly bounded with respect to k, hence (t is a bounded measure on (a, a). The claim about µ1(a; co) is proved analogously. (ii) Evidently, µ := µ (gyp) > µo (a; gyp). Then by (i) there exists a measure V o on JR with supp cpo C (-a, a) such that
(t - a)' ='po + 1=1
where ba is the 6-measure concentrated in the point a. Let I E IN and define fl,E(t) := (t - a + E)1 fort < a - e, fj,E(t) = (t - a - E)1 fort > a + s, fl,e(t) = 0 for t E (a - f, a + E). Then we have
(t - a)" V . (t - a)1 = lim(t - a)µcp - (t - a)-"f1+µ,e =
Jonas et al.
255
(t -
= lim E-OWO
a)-µfl+µ,E
= cp0
(t - a)1.
It follows that al = 0, 1 = 1, ... , s, and (ii) is proved.
1.2. Integral representations of the distributions of.F(IR). The distributions of the classes F (IR, a) can be represented by certain measures. Consider cp E F (IR, a) and set 1
k
I
ILO (a; W)
z (po(a cp) + 1)
if po (a; V) is even, if µ0(a; cp) is odd.
Then, by Lemma 1.1, the distribution (t - a)2kcp is the restriction to IR\{a} of a positive
measure o on IR with compact support and of{a}) = 0. If k > 1, then the function (t - a)-2 is not o-integrable, f (t - a)-2 do (t) = oo. tR
Indeed, otherwise (t - a) 2k-2cp would be a bounded measure on IR\{a}. Then, as 2k - 2 > µ0 (a; cp) -1 > 0, the distribution (t - a) µO (a;W) -1 cp would be a bounded measure on IR\{a}, which in view of Lemma 1.1 is a contradiction to the minimality of po (a; cp).
By the definition of o the distribution (t - a)2kcp - o is concentrated in the point a. If k = 0, define
i=0,1,...,
ifk>1, t -a 1
_
Ci -
for i = 0,...,2k - 1
((t - a)21cc - o) (t - a) i-2k
for i = 2k,2k + 1,...
By Lemma 1.1, ci = 0 if i is larger than the order of W.
In the sequel for a function f with n derivatives at t = a we use the notation f{a,0}(t)
:= f (t),
f{a,n}(t) :=
f (t)
n-1
- E i!-1(t - a)`f (i)(a) i=0
for n > 1, or, if a is clear from the context, shorter f {0} and f{n}, respectively. Then, for every f c C°°(IR), cp _ f{a,2k} = (t - a)2k p . (t - a)-2k f{a,2k} _ (1.1)
= o . (t - a)-2k f{a,2k} + ((t - a)2kp - 0,) . (t - a)-2k f{a,2k}) =
Jonas et al.
256
= o (t - a)-2k f{a,2k} + E ci i!-1 f (`) (a). i>2 k
For k > 1 we have 2k-1 fp
f = cp
i!-1(t - a)i f (i)(a)/ + f{a,2k} E i=o \
2k-1
1]
cii!-1 f(i)(a) +,p . f{a,2k}
i=o
The relations (1.1) and (1.2) imply the first assertion of the following lemma. LEMMA 1.2. If ro E
there exist numbers k,1 E INo, co, cl, ... , ci E
IR,cg # 0 if 1 > 1, and a measure o on IR with compact support, u({a}) = 0 and, if k > 0, f (t - a)-2d , (t) = oo, such that R
(1.3)
,P ' f
r
f{a,2k}
2kdQ (t) (t - a)-(t) +
R
ci
o0 f (i) (f E C (IP,)) i.-(a) 1
i=O
The numbers k,1, co, ... , cj and the measure a with the above properties are uniquely determined. Conversely, given k, 1, c0,.. . , ci, a with these properties then (1.3) defines a distribution W E .F(IR, a).
The uniqueness statement and the last assertion can be verified by an easy computation. REMARK 1.3. If co E F(IR, a) has the representation (1.3), then
ifk=Oork>Oand f It-al-1dc(t) =oo R 2k -1 Jfk>Oand f It - aI daft)
(a
1=
R
The order it (gyp) of p is the maximum of Fio (a; gyp) and 1. If /a'
n:=max{vEINo :0
m:=max{µEINo :0<µ<2k, J It -al-µda
Jonas et al.
257
then, according to Lemma 1.1,
µi(a;'p) = 2k - n, µ,.(a; cp) = 2k - m. 1.3. Linear functionals of the class F. In the sequel we need a class of functionals which is slightly larger than F(IR). Assume first /3i E C+ := {z : Im z > 0},
fl i # 13k if i # k, i, k = 1.... , m, vi E 0,1,... , vi - 1. We set B := {i31 , ... ,
and di) E C, i = 1, ...,m; j = For a function f which is locally
holomorphic on B we define
m V;-,
G(f) _
if(i)(ai))
i=1 j=O
The functional V, is considered as a linear functional on the linear space H (IR U B) of all locally holomorphic functions on IR U B. If for a function f E H (IR U B) we have f (z) = f (z) for all z E C such that z and z are in the domain of f, then 0 (f) is real. We denote the set of all linear functionals V, on H (IR U B) of the form (1.4) by .F(C\IR,B) and set .F(C\IR,0) = {0}. In a natural way.F(IR) is identified with a linear subspace of the algebraic dual space of H (IRUB), and we define F (C\IR) := UB.F (C\IR, B) and.F := UB (.F (IR) +.F (C\IR, B)), where B runs through all finite IR-symmetric subsets
of C\IR. If (1.5)
0 _ pp + 4 E F,
the minimal set B such that ik E F (C\IR, B) is denoted by a° (¢). every mials.
From well-known approximation results for holomorphic functions it follows that E F is uniquely determined by its restriction to the linear space P of all polyno-
2. THE PONTRJAGIN SPACE ASSOCIATED WITH ¢ E.F.
be an inner product space (see [3]), that is, C 2.1. Completions. Let (,C, By ic- (G) we denote is a linear space equipped with the hermitian sesquilinear form the number of negative squares of C or of the inner product [-, ], that is the supremum of the dimensions of all negative subspaces of C (for the notations see also [3], [2]). In what follows this number is always finite. Further, .C° := {x : [x, G] = {0}} is the isotropic subspace of (G, admits a and it is well-known that the factor space (,C/,C°, unique completion to a Pontrjagin space ([5, 2]), which is denoted by (,C/G°)" LEMMA 2.1. Let C be a linear space, which is equipped with a nonnegative inner product and suppose that on C there are given linear functionals V1.... , vn such that no linear combination of the vg's is bounded with respect to the seminorm . Further, let A = (ajk)i be a hermitian n x n-matrix which is nonsingular and has r. negative (and 12
Jonas et al.
258
n - r. positive) eigenvalues counted according to their multiplicities. Consider on C the inner product n
[x, y] _ (x, y) + E njkvk(x) Vj(y) (x, y E G). j,k=1
Then this inner product has rc negative squares on G. The completion of (,C/,C°, is the Pontrjagin space l ® A where f is the Hilbert space completion of G := G/G°iCo C-). More exactly, the mapping _ {x E C : (x, x) = 0}, and A:= (Cn, t :.C
is an isometry of (,C,
x -+ (i; vl (x), ... , vn (x))T
onto a dense subspace of f ® A, ker t = C°.
PROOF. Evidently, the mapping t is an isometry- of (C, into the 7r,.-space has a finite number (< rc) of negative squares. In order to prove 1{ ® A. Hence (C,
that the range of c is dense in ?f ® A we show that for each jo E {1, ... , n} there exists a sequence C .C such that 0, 0 if j E {1,...,n}\{j0} and 1, v - oo. Indeed, if for each sequence (y,,) the first two relations would imply vjo 0, then vjo would be a continuous linear functional on ,C with respect to the vjo semi-norm n t
This would imply a representation n
vj (y)vj (y E C),
vjo(y) = (y, e) + 3=1
I#io
with e E H and vj E C, which is impossible, as no nontrivial linear combination of the vj's is a continuous linear functional on (G, (-,-)12 ). The sequence
converges to fjo :_ (0;0,...,0,1,0,...,0)T E 7{ ®A,
where the 1 is at the jo-th component. It follows that for arbitrary x E C the element
bx-vl(x)fl-...-vn(x)fn= (i;0,...,Q)T is the limit in H ® A of a sequence belonging to t (G). Hence c (G) is dense in H ® A. The proof of the following lemma is straightforward and therefore left to the reader.
Jonas et al.
259
be an inner product space and
LEMMA 2.2. Let (,C,
.C = Cl[+]G2[+]...[+]C. be a direct and orthogonal decomposition. Then for the isotropic parts it holds ,C° = G°[+],CZ[+]...[+],C°,
and ,C/,C°
= Cl /L° [+]G2 /,C2 [+] ... [+],Cn /,C°.
If ic_ (C) < oo then a sequence in .C/G° is a Cauchy sequence if and only if for each k = 1,2,... , n the corresponding sequence of projections in Lk /Gk is a Cauchy sequence, moreover
(,C/,C°)
(,Cn/,Cn)- .
2.2. Inner products defined by the functionals of F. Let 0 E F be given and denote by P the linear space of all polynomials. On 'P an inner product by the relation [f,9]O 4)(f§) (f,9 E P),
is defined
where g (z) := g (z). This inner product can be extended by continuity to linear spaces which contain P as a proper subspace. To this end we observe the decomposition 0 = c P + 4, P E F (IR), 1' E F (C\IR), which implies
[f,9]o = [f,9]w + [f,gb, (f,9 E P). can be extended by continuity to C°° (IR) x H (co (4))) or to B2 (W) x H (ao (4))), where B2 (gyp) is the linear space of all functions f on IR such that Now
(i) f restricted to IR\s (vp) is gyp- measurable and f 1f12 dW < oo for each interval I I such that s (W) fl I = 0;
(ii) for some bf > 0, the restriction of f to {t E IR : dist (s (gyp), t) < bf} is a C°°- function. If s (cp) # , s (gyp) _ {a1, ... , an,}, let (oi)l 1 be a p-minimal decomposition of IR (see Section 1.1) such that ai belongs to the interior of A ; if s (cp) = 0 choose n = 1 and A l = R. Further, if 1 # 0, co (ibb) _ 1#11. * , Q,n01, ... , choose mutually disjoint neighbourhoods Uj of { 31, / }, j = 1,2,... , m, which do not intersect the real m axis, U U Uj. Define functions Xo; and Xj as follows: *
j=1 1
if zEL1
{ 0 if z E (IR\oi) U U,
_ Xj (z)
(1 ifzEUi {l 0
if z E (U Uk) U IR. k#j
Jonas et al.
260
If G,E is the linear subspace of B2 (cp) x H (co (0)) defined by
Li:_{Xo.f:f EB2('P) xH(ao(q))}, i=1,2,...,n, Ln+j :_ {Xjf : f E B2 ('p) x H(ao(q5))}, j = 1,2) ...,m, it follows that (2.1)
B2 ('P) x H (ao (qS)) _ 'Cl [+]G2[+J ...
[+1,Cn+m,
0, the last m terms do not and, if appear. If s (cp) = 0, the space C1 is just the space of all functions on IR which are square integrable with respect to the measure gyp, equipped with the scalar product of L2 (p). In that is we describe inner the following we study the other inner product spaces (,Cj, and for which s (gyp) = {a} or ao (V,) _ {/3,/3}. products The forms z/i c F (C\IR), ao (0) = 10,01 are easy to describe and wellwhere the sum is direct and
known.
PROPOSITION 2.3. Let i/i E .F(C\IR), ao (?i) = {/3,/3}, and Y-1
(2.2)
V' (p) _ E(dj j!-'p(j) (Q) + dj
j!-' p(j)
(Q))
(P E P),
j=o dj E CJ = 0,--.,-- 1; di_1 # 0. Then the mapping
t'k : P p -> (p(3),...,(v is an isometry of (P, (Ci2°, cu ), where
1)!-1 p(v-1)
(/3); p(Q),...,(v - 1)!-1p( 1)(Q))T onto the finite-dimensional nondegenerate inner product space
G:= [0G
G
0
J
'
do
d1
d1
d2
ds_1
0
G
and, hence, induces an isometric isomorphism of (P/Po, (C 2,' ')C2-). We have
...
dv-1
... .1
,) onto
r.v,:_ 'c- ((P, [., .10)) = v.
(2.3)
PROOF. If p, q E P it holds Ej!-'(dj (Pq)(j)()3)+dj(p9)(i)(Q))
[p,q] , = j=o v-1
v-1
E(p(µ)(Q) E dj i!-1(7 - .u) i=µ
µ=o v-1
7=µ
!-iq(j-a)(/3)
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261
which can be written as [p, 4]+G
= (Gt+GP, L,,4)C2
.
The matrix G has v positive and v negative eigenvalues (counted according to their multiplicities).
Evidently, in Proposition 2.3 the linear space P can be replaced by the linear space H := H ({i3, j3}) of all functions which are locally holomorphic on The identified in a natural way with (H/H°, [-, ],0), is finite-dimensional space (P/P°, denoted by fl (z/i).
cp E F (IR), s (cp) _ {a}, is more complicated;
The description of the forms it is given in the following section.
2.3. The spaces fl (p). In this subsection we suppose that rp E F(IR) and that s(W) consists of just one point a. We start with the LEMMA 2.4. Associate with p E F given as above the integers k, l E IN° and the measure a as in Lemma 1.2. Then no finite linear combination of the functionals P E) p -+ Pi = i!
-iP(i)
(a),
i = 0,1, ... ,
and, if k > 0,
PE) P - Pi :=
J R
(t -
a)-(zk-7)P{2k-j} (t) da (t)
j = 0, ... , k - 1,
,
is continuous with respect to the seminorm
P E)P -
(t - a)
2kIP{k}(t) 12
d0 (t)
PROOF. If the claim would not be true, there would exist an element v E L2 (u) N
k-i
i=o
j=0
and numbers yi, i = 0, ... , N, bj, j =0,...,k-1, such that E 1yi I + I lbjl 340 and
N (2.4)
k-1
E yiPi + E bjPi = i=1
j=°
(t - a)-kp{k}(t)v (t) do, (t)
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262
for p E P. For a polynomial p of degree < k-1 we have pi = 0, i = k, k + 1, ... , p{k} = 0 and P j ' = 0, j = 071.... , k - 1. Then (2.4) implies -ti = 0, i = 0,1, ... , k - 1.
Consider now polynomials p such that Pk = Pk+1 = ... = Pr p{2k-j} = P{k}7 max IN, 2k}. Then j = 0,1, ... , k - 1, and (2.4) implies k-1
E bj j=0
J
a)-kp{k} (i) (t
(t -
l
//
-
a)-k+j do, (t)
= 0 with r
_
]H
= J (t
-
a)-kp{k} (t) v (t) da (t).
JR
The set of polynomials (t - a)-kp{k} (t) admitted here is the set of all polynomials whose
derivatives vanish at t = a up to the certain order. As this set in dense in L2 (a), the relation (2.4) implies k[-1
j=0
bj (t -
a)-k+j
= v c L2 (a),
which, because of f (t - a)-2do (t) = oo, yields bj = 0, j = 0, ... , k - 1. Finally, put ]R
(t) := (t-a)-kp{k}(t). Then (')(a) = (k+n)!-1n!p(k+n)(a),n =0,1..., and N-k
N
E 7ipi = i=k
yk+nn! lp(n)(a) =
n=0
J(t)v(t) da (t). lR
As v has no concentrated mass at a it follows that ^yj = 0, j = k, ... , N. This proves the lemma.
In order to formulate the next theorem we need some more notations. Again with O E .P(IR), s (gyp) = {a} we associate k, l E INo, the measure a and also the numbers C 0 , . . . , ct E IR as in Lemma 1.2, and put r := max{0, l - 2k + 1}. If p E P, the numbers Pi and Pi are defined as in Lemma 2.4. Now a linear mapping t, from P into L2 (a) ®C2k+r is introduced as follows:
If1<2k-1 (that is r (2.5)
0):
GIp:P -+ ((t-a)
where Pj := P3. +
kp
ci+jPi,
k}iPO,...,pk-liPk-1,...,Pp)T j = 0, ... , k - 1;
if l > 2k: i
: p -- ((t - a)-kP{k};po,...,pk-1; Pk-1,... , Po)T,
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263
where now Pj := P +
ci+jpi, 7 = 0, ... , k - 1.
If 1 < 2k - 1 we denote by G the linear operator in L2 (u) ® C2k given by
if l > 2k, G is the linear operator
I
0.
0
...
...
.0
.
Cl-k
0.
.
.
1
Ck-1
Cl-1
1 . .
.
0
Ck
Cl
Co
Cl
Ck_1
Ck
Cl
0
(2.7)
G= 0
Cl-k 0
i
...
Cl_1
Cl
.
.
.
0
n
in L2 (Q) ® C2k+*
THEOREM 2.5. Consider V E J (IR) with s (sp) = {a}, and let the numbers k, l E INo, co, ... ,Cl E IR, cl # 0 if 1 > 0, and the measure o be as in Lemma 1.2. Then
(P,
is an inner product space with ,c,, negative squares, where ic,, := max (k, 2 (l + L
with
1)J
+6)
6=11 ifliseven,cl<0,1>2k-1, 10 otherwise.
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264
The linear mapping t,, generates an isometric isomorphism of the completion n (,P)
of (P/P°,
onto the 7r,,,. -space
(L2 (r) ® G2k+r (G
2k+. ).
)
in L2 (o) ®G2k+r
Here "generates" means that c,, induces a linear mapping of (P/P°, which extends by continuity to all of n (p).
PROOF. Assume first that 1 < 2k-1. Let p, q E P and define qi, Q;, Qi analogously to pi, Pi, Pi. In view of the relation
(Pq){2k}
(t)/ = p{k}(t)
q{k}(t) \
k-1
+E j=0(t - C,)3
l / +4 j
(Pjq{2k-j}(t)
P{2k-j}(t))
it follows that k-1
!
(t - a)-2kp{k}(t)q{k}(t) do(t) + E(Pi
[P, q] , =
+ g,PP) + E ci E Pµgv =
j-0 k-1
i-0
k-1
= J (t - a)-zkp{k}(t)q{k}(t) dO (t) + E E Qi ci+j Pj i=0 k-1
k-i
I
j=0
!
+E4i(E ci+jpj+F1)+EpjE ci+j4i+gj)P+`j) i=0
=
r
J
j=k
j=0
i=k
k-1
k-1
i=0
j=0
(t - a)-2k p{k} (t)q{k} (t) do, (t) + E qi E,,i+jpj
k-1
k-1
+ E vi Pi + E Pj Qj,
i=0
j=0
where we agree that c = 0 if v > 1. This relation can be written as [p, q], = (GiwP, tvq)L2(d') ®G2.
with the Gram operator G as in (2.6). Since (Gz, z) = 0 for all elements
(P, q E P)
µ+v=i
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265
`
c
2 = (Oi 0,...,DiSO,...,tk-1)T,
SO, ...,G-1 E C,
e.g. the minimax characterization of the negative eigenvalues of G implies that G has k negative eigenvalues, counted according to their multiplicities. On the other hand, ic,o = k
if1<2k-1(see(2.8)). According to Lemma 2.4, no linear combination of po, continuous with respect to the seminorm
p-
J
(t - a)-2k Ip{k}(t)
I2do,(t)
, pk-1, Po,
, Pk-1 is
,
and the claims of Theorem 2.5 in case I < 2k - 1 follow from Lemma 2.1.
Assume now that I > 2k. It follows as above that [p, q]w
= J (t - a)-2kp{k}(t)q{k}(t) du (t) IR 1-k
1-k
k-1
1
+EgiEci+j pj +Egi E ci+j Pi i=0
j=0
i=0
k-1
1
j=0
i=1-k+l
j=!-k+l
E ci+j 9i+Qj This relation can be written as [p, q]w = (Gtwp, t ,q) L3
OC2k+*
(p, q E P)
,with the Gram operator G in (2.7). The number of negative eigenvalues of G is equal to k + z if r is even and, if r is pgdd, r = 2p + 1, it is equal to k + p + 1 if c1 < 0 and k + p if c1 > 0. This is just the value *p (see (2.8)) in the case 1 > 2k. Now all assertions of Theorem 2.5 for the case I > 2k !follow as above from the Lemmas 2.1 and 2.4. Evidently, in Theorem 2.5 the linear space P can be replaced by C°° (IR) or B2 (V)-
2.4. The spaces fl (-0). Now let ¢ E F, 0 = p+tk with p E F (IR), 4' E F (C\1R). The considerations in Subsection 2.2, in particular (2.1), and the fact that all the inner ;product spaces on the right hand side of this relation have a finite number of negative squares (see Proposition 2.3 and Theorem 2.5) imply that (B2 (sp) x H (oo (0)), has a finite number of negative squares. We denote the completion of (BIB',[-, ]9s) where B := B2 (JP) x H (u° (-0)), by fl (0). On account of Lemma 2.2 this space decomposes as follows:
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266
(2.9)
n (0) = n (z&,') [+] ... [+] n (zoA'p)[+] n (;1 ') [+] ... [+] n (zmb).
is again a cp-minimal decomposition of IR and Xo..... Xi 7X,., are defined as in 2.2. Models of the spaces n(Xo; p), n (xj1/i), i = 1, ... 7711i = 1, ... IM) were given in Proposition 2.3 and Theorem 2.5, and a corresponding model of n (0) is a direct orthogonal sum of models of these types. The number of negative squares of n (0) is then, evidently, the sum of the numbers of negative squares of all the components on the right hand side of (2.9), which are given by (2.3) and (2.8), respectively. Here
Concluding this section we mention the following. The starting point of the above considerations was an inner product .]0 on the space P, given by some 0 E F, and it turned out that this inner product has a finite number of negative squares. We could have started from any inner product on P with a finite number rc of negative squares. Then, up to a positive measure near oo, the inner product is generated by some 0 E.P. Namely, for each sufficiently large bounded open interval A there exist a measure o,,,, on IR\A and a 0 E F which is zero on 1R,\A such that
(2.10)
[p,q] = [P,q]m+ f p(t)q()do.(t) (p,q E P). R\0
Here the measure o. is such that
f
jtjnda.(t) < o0
R\A
for all n E IN.
This follows immediately from the results of [13]. Observe that the hermitian sesquilinear form on P determines a sequence (sn) of "moments" (2.11)
8n := [tn,1], n = 0,1 ....
which belongs to the class HK, (see [13]). In general, even after the interval A has been chosen, neither 0 nor the measure o,,. in (2.10) are uniquely determined by More exactly, they are uniquely determined (after A has been chosen and we agree that o. does not have point masses in the boundary points of A) if and only if the moment problem for the sequence (an) from (2.11) is determined. 3. MODELS FOR CYCLIC SELFADJOINT OPERATORS IN PONTRJAGIN SPACES
3.1. The operator of multiplication by the independent variable in n (4'). Let 4, E F, 0 = cp +,/, with cp E F (]R),10 E F (C\1R.). On (P,
(or, what amounts to
Jonas et al.
267
the same, on B2(W) x H(oo(4)))) we consider the operator A0 of multiplication with the independent variable (Ao p) (z) := zp(z) (p E P). Evidently, [Aop, q] 0 = [p, Aoq]0 (p, q E P), hence Ao generates a hermitian operator AO in fl (q,). This operator A0 is continuous. In order to see this, let F be the order of the distribution cp and let 0 be an open interval with suppcp C A. Suppose again that 0 has
the form (1.4), and set v := max {v; - 1 : i = 1, ... , m}. Then the inner product and the operator A0 are bounded with respect to the norm IIPII
sup {Ip(`)(t)I : 0 < i < p,t E o} + max {Ip(h)(Q)I :,3 E oo(-0),0 _< k < v}
on P. A result of M.G.Krein (see [9], [4]) implies that A0 is bounded in fl (0) and, hence, can be extended by continuity to the whole space fl (0). The closure of A#, also denoted by A.,, is called the operator of multiplication by the independent variable in fl (4,). Consider now a decomposition (2.9) of the space f1 (q,). The operator A, maps each component on the right-hand side of (2.9) into itself. This implies:
PROPOSITION 3.1. Under the above assumptions the operator AO in fl (0) is the direct orthogonal sum of the operators
Azo;w E £(n(X&jW)), i = 1,2,...,n, AX,,, E £(fl(zj'0)), j = 1,2,...,m. Therefore, in order to describe the operator A0, it is sufficient to describe the ,perators A,o, cp E .F(IR;a) and Au, -0 E .F(C\IR), oo(1/i) = {Q,f3}. It is the aim of this .subsection to find matrix representations of A. and Ap in the model spaces of Proposition 2.3 and Theorem 2.5. For the sake of simplicity these matrix representations of A. and A* are denoted by the same symbols A,o and A0 (although they are in fact LA, 1.-1 with Dome isometric isomorphism t). THEOREM 3.2. Let V E F (IR), s (gyp) {a}, and suppose that k,1, o, co, ... , ct are associated with V according to Lemma 1.2. Then in the space L2 (o) OC2k+r, equipped
with the same inner product as in Theorem 2.5, the operator A,, admits the following matrix representation:
t
0
0
...
0
1
a
0
...
0
0
1
a
...
0
0
0
0
.
a
0
0
0
...
1
a
0 ,1 c 0 1 0
0
...
0
c2k_1
a
0
0
...
0
c2k_2
1
a
0
0
...
0
ck+1
0
{3.1)
0
1
-k columns -
if I < 2k - 1 ,
0
...
...
0
0
0
0
1
a
-k columns-
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268
t
0
...
0
1
a
...
0
0
0
0
a
1
0
...
1
0
...
0
1
a
0
...
0
0
1
0
0
0
...
1
a
... ...
0 0
Cl
0
0 0
0
0
...
0
0
(., 1)0
-k columns -
...
0
0
0
0
if I > 2k-1, a
...
1
0
-r columns -
...
0
0
1
a
-k columns-
In these matrices, all the nonindicated entries are zeros. The scalar product in L2 (o') is denoted by
PROOF. We assume, for simplicity, a = 0, and consider e.g. the case 1 < 2k - 1; if I _> 2k a similar reasoning applies. If P E P, with the mapping t, introduced in (2.5), we write iw(Awp) _ (t-k(tp){k};
o, 1, ... ,
k-I i Pk_1, .. , P1, Po)T
and express the components of this vector by those of t., (p). Evidently,
Po = 0,
Pj= Pi-1,
9 = 1,2,...,k - 1.
Further, (tp){n} = tp{n-1}
= tp{n} + tn(n - 1)!-1p(n-1)(0)
(n E IN),
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269
and we find (with evident notation)
p! =
f
/
r t-(2k-j-1) p{2k-j-1}(t)dC(t)
f
t
0,1, ... , k - 1, 1-1
1
Po=>2CA +Po=>2cjpj-1+Pi= E cj+lpj+P1 j=k
j=k
j=k-1
= Ckpk-1 + P1, t-1
t
P1 =
t-1
cj+1Pj-1 + Pi =
cj+1Pj + Pi
j=k
j=k
Cj+2Pj + P2
j=k-1
= Ck+lpk-1 + P2,
I
Pk-1 = E Cj+k-1Pj + Pk-1 j=k
Cj+k-lPj-1 + Pkr j=k
l-1
cj+kpj + Pk = C2k-lPk-1 + Pk j=k-1 where Pk := r t-kp{kl(t)d0(t) = (t-kp{kl, 1)0,.
It follows that A. has the matrix representation (3.1) with a = 0. The proof of the following theorem about the matrix representation of the operator rA#,'k E .P(C\IR), Co is similar but much simpler and therefore left to the keader.
THEOREM 3.3. Let 0 E ,F'(C\IR), ao('r&) = {(3, (i} and suppose that '0 has the form (2.2). Then in the space C2i, equipped with the same inner product as in Proposition 2.3, the operator A,, admits the matrix representation Q 1
0 0
.
0 (3 I
0
1
0
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270
3.2. Eigenspace, resolvent and spectral function of A.. In this section, if
E F(IR;a), for the model operator A, in L2(o) ®C2k+r from (3.1), (3.2) we study the algebraic eigenspace at a and the behaviour of its resolvent and spectral function near a. Without loss of generality we suppose that a = 0. First we mention that a = 0 is an eigenvalue of A,, with a nonpositive eigenvector and that A. has no other eigenvalues with this property. A maximal Jordan chain x0,017 ... , On of A. at a = 0 is given as follows:
If 1< 2k-1, thenm= k-1 and xk-1 = (0;0,...,0;1,0,...,0)T, xk_2 = (0;0,...,0;0,1,...0)T,
xo = (0;0,...,0;0,0,...,1)T
and the span of these elements is neutral. If 1 > 2k then m = 1- k + 1 and XI-k = (0;0,...,0;1,0,...,0;0,...,0)T, XI-k-1 = (0;0,...,0;0,1,...0;0....,0)T,
xk = (0;0,...,0;0,0,...,1;0,...,0)T,
xk_1 = (0.0
x0 = (0r .0v
0.0
s
0> .0>
0; C
s
0> .0>
c_
c_
)T
. 0sc1)T.
In the second case it holds 0
([xt,xi]).
.
k0 =
I
0
0 0 Cl
C2k+1 C2k+1
C2k J
where
(see (2.7)). Hence the elements xo,... , xk-1 span a neutral is nondegenerate. subspace, whereas on the span of xk,... , xl-k the inner product Next we consider the matrix representation of the resolvent of As,. For the sake of simplicity we write down the matrix (A, - zI)-1 if k = 2 and 1 < 2k - 1 = 3; its structure for arbitrary k and I < 2k - 1 will then be clear. This matrix is
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271
I
(t - z)-
z-2(t - Z)-1
z-1(t - z)-1
-z-1 -z-2
-z-1
0 0
-z-
(t - z)-
(t -
0
bll(z)
z)-1)o
b12(z) b22(z)
b2l(z)
where
(bi3(z))i
_
-C3
-c3z-2 + z-2 Pt -
Z-3 + z-3 f (t - z)-1do,(t)
[ -C3Z-4 - C2Z-3 + z-4 f (t - z)-1do,(t)
Z-3 - C2 Z-2 + z-3 f (t -C3
-1dr(t)
- z)-'du(t)
The growth of (A,, - zI)-1 if z approaches zero along the imaginary axis or, more generally, nontangentially, is given by the term b21(z).
In the general case the growth of (A,, - zI)-1 is also determined by the entry b on the second place of the last row. If 1 < 2k - 1, we have b (z) = -CkZ-k-1 -
Ck+lz-k-2
- ... - C2k-lz-2k + z-2k J(i - z)-1d?(t)
and, hence, for z = iy and y 10, Iy2k+lb(iy)I < c < oo.
Iy2k-lb(iy)I + oo,
li1 2k, we have b(z) = -ciz-1-1 this implies
Cl-1z-l
cl-k+lz-l+k-z _ z-2k r (t
- ... -
y1+1
J
- z)-1do, (t).
I b (iy) I-* I cll for y 10.
For sufficiently large exponents the powers of the model operatorA,, have a simple
matrix form which is independent of the numbers ci, j = 0,. .. ,1. If 1 > 2k, the operator A' with n > k + r is given by tn.
to-m
to-R-1
...
to-1
0
0
0
0
0
0
o /.'tn-1)o
0
0
0
0
0
tn-2)o
(1,tn-k-1)o (1,tn-k-2)o
,tn-k)o
(1,tn-2k)o
(1,tn-k),
(1,tn-k-1)o
...
(1,tn-2 )Cr
...
(1,tn-3)c,
...
(l,tn-k-l)o
/ (l,tn-2k+l)s
272
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If we agree that in the case I < 2k - 1 the third row and column in (3.3) disappear, then (3.3) gives also the matrix representation of A' for n > k, l < 2k - 1. Let E be the spectral function of A. and let L be an arbitrary interval with 0 0 L . Then making use of (3.3) and the fact that E (0) can be written as the strong limit of a sequence of polynomials of Aw for odd n, or applying the Stieltjes-Livshic inversion formula to the model operator of the resolvent of A,,, we obtain the following matrix representation for E (A):
t
XA
t
k+1
...
t
U
U
0
0
0
0
0
0
0
0
0
0
XA
(1,XAt-
-1)0
(',XAt-2)o
(1,XAt-k-2)o
(.,XAt-k)o
(1,XAt-2k)o
(1,Xat
XA
XA
(1,XAt-2)o
)o
(1'XAt-k-1)o
(1,XAt-S)., (1,XAt-k-1)Q
(1,XAt-2k+1)o
Again, if I < 2k - 1, the third row and column disappear. It follows that I IE (o)I I = 0 (f t-2kdo. (t)), A
if a boundary point of A approaches zero. The growth of I IE (0) I is determined by k and independent of 1. In particular, the point a = 0 is a regular critical point of A, ([10], [14]) if and only if k = 0. I
Evidently, by an appropriate choice of Q, k, l and cj, j = 0,.. . ,1, examples for selfadjoint operators in Pontrjagin spaces with different growth properties for the resolvent and the spectral function can be constructed.
3.3. Cyclic selfadjoint operators in Pontrjagin spaces. Let A be a bounded cyclic selfadjoint operator in the Pontrjagin space n. Recall that A is called cyclic, if there
exists a generating element u E n such that n = c.l.s.{A3u : j = 0,1,.. 11, If E F, then, evidently, the operator A0 is cyclic in n (¢) with generating element 1(or, more exactly, the element of n (0) corresponding to 1). The following theorem, which is the main result of this subsection, states that in this way we obtain all bounded cyclic selfadjoint operators in Pontrjagin spaces. In the following, "unitary" and "isometric" are always understood with respect to Pontrjagin space inner products.
THEOREM 3.4. Let A be a bounded cyclic selfadjoint operator in a Pontrjagin space (n, with generating element u. Then the linear functional
0:P3p-p[p(A)u,u]
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273
belongs to F and A is unitarily equivalent to the operator AO of multiplication by the independent variable in n (0). PROOF. If p CP, we have (3.4)
[p (A) u, u] = [p (A) E (a (A) n IR) u, u] + [p (A) E (o (A) \IR) u, u]
where, for a spectral set o of A, E (v) denotes the corresponding Riesz-Dunford projection. Further, [p (A) E (v (A) \IR) u, u] _
(3.5)
([p(A)E({)C3})u,u]+[p(A)E({0})u,u]) #Ea(A)nC+
and for 3 E Q (A) \IR there exists a vQ E IN such that (A - #1) °0 E ({/3}) = 0. Hence vµ -1
(3.6)
[p(A)E({(3})u,u] = >2 v!-1p(°)(3)[(A-fI)°E({,a})u,u] (p c P). V=0
Moreover, for v = 0,...,vf - 1 we have
[(A-/3I)°E{/3})u,u] = [(A-/3I)vE({(3})u,u].
(3.7)
From (3.5), (3.6) and (3.7) it follows that the functional
p-' [p(A)E(o, (A)\IR)u,u] belongs to .F(C\IR). Therefore, by (3.4), in order to prove that 0 E F it is sufficient to *how that the functional cP : P 3 p -[p (A) uo, uo],
uo := E (v (A) n IR) u,
belongs to P (IR).
Denote by po a definitizing polynomial of Ao := Aino, no := E (a (A) n IR) n I,with only real zeros, say a1, ... , a, (mutually different) of orders Xlonnegative on IR (see, e.g., [141). Let n
µ;
(PO lt) )-1 = >2 >2 cij (t - ai)-j,
t E IR\{a1 i ... , an}.
i=1 j=1
Then, for arbitrary p E P, (3.8)
p(t) =g(t;p) + po (t) h (t; p),
which is
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where n
µ;
g (t;P) := PO(t) E E cij (t - ai)-i(P (ai) + ... + (j - 1)!-,p U-1)(Ci) (t - ai)j-1), i=1 j=1 n
µi
cij (t - i)-jp{a:,.i}(t).
h(t;p) i=1 j=1
We choose a bounded open interval A which contains o (Ao ), denote by µ the maximum of the µi, i = 1, 2,. .., n, and consider the set S of all polynomials p such that
sup{Ip(k)(t)I:0
sup{ I[g(Ao;P)uo,uo]I : P E S} < oo.
As po is a definitizing polynomial of Ao, the inner product [po (Ao) , ] is nonnegative in flo. Evidently the operator Ao is symmetric with respect to this inner product. From the result of M.G. Krein used already above (see [9]) it follows that Ao induces a bounded selfadjoint operator in the Hilbert space which is generated in a canonical way from the inner product space (no, [po (Ao) , ]). Moreover, the spectrum of this operator is contained in A. Therefore the functional
P E) q - [po (Ao)q(Ao)uo,uo] can be written as f q (t) dµ (t) with a measureµ supported on A. Taylor's formula implies that the polynomials h (.; p), p E S, are uniformly bounded on A. Hence sup{ I [po (Ao) h (Ao; P) uo, uo]
P E S} < oo,
and from (3.8) and (3.9) it follows that sup{'P (p) : p E S} < oo.
This relation assures us that W is a distribution (for a similar reasoning cf. [7; proof of Theorem 1]). Moreover, if f c Co (IR) is nonnegative and ai V supp f for all i = 1,... n, then
4P-f=Pow-Polf?0, that is, V E F (IR).
In order to prove the unitary equivalence of A# and A, consider the isometric linear mapping Uo from (P, .]0) onto a dense subspace of n defined by
Uo P := P (A) u (PEP).
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Then U° (Asp) = Ap (A) u. Evidently, a polynomial p belongs to the isotropic subspace P° of (P, if and only if p(A)u = 0. Then, if Uo denotes the isometric bijection
p+P° -+ p(A) u into fl, we have of (P/P°, UUA0 = AUa on P/P°.
The extension U of Uo by continuity is an isometric isomorphism between fl(qS) and fl satisfying
UA, = AU, and the theorem is proved. UNITARY EQUIVALENCE OF CYCLIC SELFADJOINT
OPERATORS IN PONTRJAGIN SPACES
4.1. The general case. Recall that two selfadjoint operators A, A in Pontrjagin spaces fl and fl, respectively, are said to be unitarily equivalent if there exists a unitary operator U from fl onto fl such that (4.1)
Au = UA.
In this section we study the unitary equivalence of two cyclic selfadjoint operators A, A in Pontrjain spaces fl and fl, respectively. More exactly, we fix generating elements u, it of A and A, respectively. Then to A and A there correspond model operators AO, A in spaces F1(0), n(¢) for certain i, E .7 (see Theorem 3.4), and we express the unitary equivalence of A4, and A, in terms of 0, qS. Evidently, if A and A are unitarly equivalent, then ic_(fl) (0)U = UE (A) for all admissible intervals A (where E, t denote the spectral functions of A, A), o(A) = u (A), up(A) = vp(A) and the algebraic eigenspaces of A and A corresponding to the same eigenvalue are isometric. As for a nonreal eigenvalue )'° of A and A the unitary equivalence of the algebraic eigenspaces means just that the lengths of the Jordan chains coincide (recall that the algebraic eigenspace of A at \° consists of just one chain of finite length), we can suppose without loss of generality that A and A have only real Spectrum and, after suitable decompositions of the spaces fl and fl, that A and A have just one eigenvalue a with a nonpositive eigenvector, and that a = 0. So we are led to the following problem: Given gyp, cp E F (IR; 0). Find necessary and sufficient conditions for tlhe unitary equivalence of A,, and A0 in terms of W and 0. Here we suppose that gyp, 0 are
given by their representations according to Lemma 1.2, that is, ip is given by k,1 E INo i co, ... , ct E IR and a measure a,
0 is given by k,1 E IN°, c°, ... , cl E lR and a measure &, Where we always assume that these data have all the properties mentioned in Lemma 1.2.
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As k (k) is the length of the isotropic part of the Jordan chain of A,, (A,;) at 0, we have k = k. Further, if 1 > 2k, then 1- k + 1 is the maximal length of a Jordan chain of A. at 0, hence in this case I = 1. It follows that the sizes of the blocks of the matrices for A W and AO in (3.1) or (3.2) and for the Gram operators G,, and Go from (2.6) or (2.7) coincide:
(4.2)
Gv=
I
0
0
H1
0
0 H3 H2 0
Z
0
H3 Z 0
0 0
'
_ AV
t.
E
o
S1
0
J
0 0 S2
E' D C
0 0
0
Sl
Here we agree that in case r = 0 the third row and column disappear. The blocks in (4.2) can be read off from (2.6), (2.7), (3.1), (3.2) with a = 0; we only mention that D = 0 if r > 0. The corresponding blocks of AO are denoted by Al, Z etc. The operator U in (4.1) is partitioned in the same way: U = (U;;)',. As U maps the algebraic eigenspace of A. at 0 onto that of AO and these subspaces are given by the vectors with vanishing first and second block components, it follows that U13 = U14 = U23 = U24 = 0. Also the set of vectors with vanishing first, second and third block components is invariant under As,, and AO, hence U34 = 0.
Now we write the relation (4.1) for the block matrices of A,,, AO, U. Considering
hence U21 = 0 and, the components 21 on both sides it follows that . 1U21 = U21 similarly, U31 = 0. Then (4.1) turns out to be equivalent to the following relations: (4.3)
U11 t-,
U11E+U12S1,
(4.4) (4.5)
S1U22 = U22S1,
(4.6)
JU22 + . 2U32 = U32S1 + U33J,
(4.7)
S2U33 = U33S2,
(4.8)
E'U11 +. 1U41 = U41t +U44E',
(4.9)
E'U12 + DU22 + CU32 + S1U42 = U41E + U42S1 + U43J + U44D,
(4.10)
CU33 + S1U43 = U43S2 + U44C,
(4.11)
S1 U44 = U44 S1
As U is isometric with respect to the inner products on L2 (v) ® G2k+r and L2 (8-) ® G2k+r, generated by the Gram operators G,e and GO, respectively, we have also (4.12)
U*G, U = Ge,
which is equivalent to (4.13)
U11U11 =I,
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277
(4.14)
UilU12+U41ZU22 =0,
(4.15)
U12U12 + U22(H1 U22 + H3U32 + Z U42) + U82(H$ U22 + H2U32) + U42Z U22 = H1,
(4.16)
U22 (H3U33 + Z U43) + U332H2U33 = H3,
(4.17)
U22Z U44 = Z,
(4.18)
U33H2U33 = H2
The relations (4.3) and (4.13) imply that the operators of multiplication by the independent variable in L2 (o) and L2 (&) are unitarily equivalent. As is well-known this means the equivalence of the measures o and & and that U11 is the operator of multiplication by a &-measurable function y such that W 2 = do/d& ([1]). Next we consider (4.5). It implies that U22 is a Toeplitz matrix of the form Li
0 uo
Uk-2 Uk-1
Uk-3 Uk-2
'Lo
...
0 0
0 0
,ui EO2,
Further, writing U12 = (vk vk-1 tion (4.4) is equivalent to
uo
0
u1
uo
...
j =0,...,k-1.
v1) with vi c L2 (&), j = 1, 2, ... , k, the rela-
vl = t-1 (7 -'too), 62 = t-1 (v1 - ul),..., vk = t-1
(4.19)
This implies
7 (t) - (120 + 7lt + ... + uk-ltk-1) = tkvk (t)
(4.20)
e-a.e. That is, if k > 0 the function y(E L2(&)) has a well-defined "value" uo at t = 0 and also "derivatives" up to the order k -1. That uo, u.1, ... , iLk-1 are uniquely determined by the relations (4.19) or by (4.20) follows from the condition f t-2d&(t) = 00. Similary, making use of (4.11) and (4.8) and putting U41 = ((., v1)
......
and
vo
0
...
0
0
ui
uo
...
0
0
'+dk-2
uk-3 uk-2
... ...
TO 'tl1
0
,u.iEC,
U44 =
uk-1
j=0,...,k-1,
T0
ve find
V1 = t-1(7-1 - uo), v2 = t-1(vl - ul),..., Vk = t-1(Vk-1 - uk-1)
vk)o))T
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278
and all these functions belong to L2(r). It is easy to see that the relation (4.17) is now automatically satisfied. In particular, we have 1i # 0 and
Then by (4.20) there exists a polynomial p of order < k - 1 with real coefficients and p(0) = IuoI such that t-A`(Iry"I -p) E L2(&). In particular, we have
t-1(I7I -'1o) E L2(&).
(4.21)
We summarize some of the above results to a necessary condition for unitary equivalence of A,, and A0. Here a (and correspondingly ic) are given by (2.8). THEOREM 4.1. Let cp, cp E FIR; 0) with representation according to Lemma 1.2. If the operators A. and A0 are unitarily equivalent, the following statements hold: (i) sc = ic, k = k, and, if 1 > 2k, 1 = 1.
(ii) The measures a and Q are equivalent; if k > 0 there exists a polynomial p of order (dc/d&) 21 the < k - 1 with p(O) 0 and real coefficients such that with function
t-'`(9 - p) belongs to L2(&)-
The meaning of the necessary conditions (i), (ii) in Theorem 4.1 is enlightened by the following result, which, in fact, contains Theorem 4.1.
THEOREM 4.2. Let (p, cp be as in Theorem 4.1 and denote by E, E the spectral functions of As,, A0, respectively. Then the conditions (i), (ii) in Theorem 4.1 are necessary and sufficient for the unitary equivalence of the spectral functions E and E. PROOF. 1. Assume that E and t are unitarily equivalent. Then, evidently, r. = ic. The number k (k) coincides with dimension of the isotropic part of the closed linear span
,C(o) (G(o)) of all ranges of E(A) (E(0)), where A is an arbitrary interval with 0 g A. Therefore k = k. The orthogonal companion £(io)] (,C1()') of £(o) (G(o)) is the algebraic eigenspace of A,, (A0) at 0. Then, in view of the results of Section 3.2, we have l > 2k if and only if 1 > 2k, and in this case 1 = 1. Hence the condition (i) holds.
In order to prove (ii) set n := 2(k + r) + 1 with r = max{0,1- 2k + 1}. Then A 1 is nonnegative with respect to the Pontrjagin space inner product, and An = f tndE(t) (cf., e.g., [7; 3.3]). A similar relation holds for A. Therefore the operators An and An are unitarily equivalent. Assume that k > 0 (if k = 0, a similar, much simpler reasoning applies). We write An in a form similar to that of Av in (4.2) (see (3.3)): t is replaced by
t"., E =
(tn_p
... t"-1), Si = J = S2 = C = 0, E' _
(('.t,.-1)v
...
and
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279
(l,tn-k-1)o (1,tn-k-2)o
(1,tn-k)o (1,tn-k-1)o
(1 to-1) (1
D= (l,
to-2k)o
to-2k+1) (l
o
...
(1
to-2)
o
0
to-k-1)
o
Let U = (Ui;)i be a unitary operator from (L2(0') ®C2k+*,(G.., (L2(o-) ®
(4.22)
C2k+*,
)L,(o)®Qri2k+*) onto
such that
A! U = UAn
As above it follows that U13 = U14 = U23 = U24 = U21 = U31 = 0. The relation (4.22) is equivalent to the following: (4.23)
t"U11 = Ullt",
(4.24) InU12 + EU22 = U11E ,
(4.25)
E'U11 = U41tn + U44E',
(4.26)
E'U12 + DU22 = U41E + U44D.
By (4.23) and (4.13) the measures o and o are equivalent, and U11 is the operator of multiplication by a &-measurable function y such that IiI2 = do/d&. We write U12 = (vk 6k-1 . . . vl), U22 = (uij)o-1. Then (4.24) is equivalent to t"'U1 + t"-ku0,k_1 + (4.27)
... + t"-luk_l,k_1 = itn-1,
t"v2 + tn-ku0,k_2 + ... + t"-luk_1,k-2 =
to-kuo,0
tn'vk +
yt"-2,
+... + t"-l'flk-1,o = 1't"-k.
In view of f t-2 d&(t) = oo the first equation of (4.27) gives
vl = t-1(7 -
uk-l,k-1), uo,k-1 = ... = Uk-2,k-1 = 0.
The second equation gives
f12 = t-1(vl - uk-1,k-2), Uk-2,k-2 = uk-1,k-1, u.0,k-2 = ... = uk_3,k_2 = 0.
Pursuing this computation we obtain relations similar to (4.19) with u0 = uk-l,k-1, uk_1 = uk_1,o. Hence the condition (ii) is satisfied.
,
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2. Assume now that (i) and (ii) are fulfilled. It is sufficient to prove that there exists a unitary operator U such that with the integer n defined above the relations (4.22) and (4.12) hold. Let first k > 0.
With p(t) = io + ult + ... + vk :=
t-k(i - P), vj := t-i(ujt' + .
Then the operators U11 =
we associate functions vl, ... , vk E L2(&)
uk_ltk-1
.
.+
uk_1tk-1
+ tktlk), 7 = 1,..., k - 1.
U12 := (flk ... vl) and
U22 =
uo
0
ul
?o
uk-1
uk-2
satisfy the relations (4.13), (4.23) and (4.24). From (ii) it follows that there exists a (uniquely determined) polynomial q(t) = no + ult + ... + uk_1tk-1 with real coefficients and uo 0 0 such that the function
t-k(9-1 - q)
Vk
belongs to L2(0'). It is easy to see that the coefficients of p and q satisfy the relations
j (4.28)
uouo = 1, L uiuj-i = 0,
j = 1,...,k - 1.
i=0
Define functions v1,. .. , vk_1 E L2 (o) by
vj := t-'(ujt' +...+uk_1tk-1 +tkvk), j = 1,...,k - 1. Then with the operators U41 :=
v1)o
...
vk)o)T,
uo
0
...
0
ul
UO
...
0
uk-1
uk-2
...
no
U44
the relations (4.28) are equivalent to (4.17) and (4.14), (4.25) and (4.26) are satisfied. In order to prove Theorem 4.2 for k > 0 it remains to find operators Us2, U33, U42 and U43 which fulfil the relations (4.15), (4.16) and (4.18). Since by (i) the nondegenerate
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281
forms (A2 -, -) and (A2 -, -) have the same signatures, there exists a matrix U33 which satisfies
(4.18). We set U32 = 0. Consider the equation (4.15): U22Z U42 + U42 Z U22 = Al - U12 U12 - U22 Al U22 =: S.
Evidently, the operator U42 :_
}(U22Z)-1S satisfies this relation. If we choose
U43 := (U22Z)-1(H3 - U22A3U33),
then the equation (4.16) is fulfilled.
If k = 0, then the operator
U=
[U11 0
U33 0
with U11 and U33 as defined above has the required properties. This completes the proof.
REMARK 4.3. It is easy to see that for k = k = 0 the conditions (i) and (ii) in Theorem 4.1 are also sufficient for the unitary equivalence of A,, and A0. Necessary and sufficient conditions for the unitary equivalence of A,, and AO can be given also under other additional assumptions. However, a complete treatment of the relations (4.3) - (4.11) and (4.13) - (4.18) seems to be complicated. In the following subsection we consider the case rc = 1.
4.2. The case is = 1. Let cp, 0 E F (IR; 0) be such that the numbers rc, is given by (2.8) are one. By Remark 4.3 we can restrict ourselves to the case k = k = 1. THEOREM 4.4. Let W, 0 E F (IR, 0), s; = k = 1 (see (2.8)) and k = k = 1.
If l = 1 = 2, then the operators A, and A0 are unitarily equivalent if and only if (i) the measures o and & are equivalent,
(ii) with y := (du/d&) s there exists a nonzero real number 9o such that the function t -p t-1 (g (t) - go) belongs to L2 (&), (iii) I90I26
= C2 -
If 1,1 < 1, then AW and A0 are unitarily equivalent if and only if condition (i) and the following condition are satisfied.
(iv) There exists a complex function ' 5E L2 (&) with Iry12 = du/d& and a nonzero number 'o jsuch that the function t -> t-1 (ry" (t) - io) belongs to L2 (&) and
14.29)
X370 +
Ji_1
(7 (t) -'Yo) d& (t) = cl''. 1 + I t-1(7 (t)-1 - %-') do (t).
282
Jonas et al.
PROOF. If k = 1 and r < 1, then for the blocks in (4.2) we have S1 = S2 = Sl =
S2 = 0 and (if r = 1) J = J = 1. Consider first the case l = 2, that is r = 1 (or c2 > 0). As above the relations (4.3), (4.4) and (4.13) imply (i) and (ii), and (4.5) - (4.11) become (4.30)
U22 = U33-
(4.31)
(U11', 1) & = U41t - +U44(., 1) 0,
(4.32)
(U12., 1) o + C2U32 = U411 + U43,
(4.33)
C2U33 = U44C2.
Further, (4.14) - (4.18) are equivalent to (4.34)
U11 U12 + U41 U22 = 0,
(4.35)
U12U12 + U2260U22 + U3261U22 + U42U22 + U22C1U32 + U32C2U32 + U2*2U42 = Co,
(4.36)
U22C1U33 + U32C2U33 + U22U43 = Cl,
(4.37)
U22 U44 = 1,
(4.38)
U33C2U33 = C2
The relations (4.30) and (4.38) give U22U22C2 = c2. As above (see (4.21)) we find U22U22 = 14012 which proves (iii). U11
Assume now that 1 = l = 2 and the conditions (i), (ii) and (iii) hold. Then, if o with v1 := t-1 \(4-1 go), U22 = U33 := 90, U41 U12
go 1), U44 := go 1 the conditions (4.3), (4.4), (4.13) and (4.30), (4.31), (4.33), (4.34), (4.37) and (4.38) are fulfilled. There remain (4.32), (4.35) and (4.36) to be satisfied which give three equations for the three numbers U32, U42 and U43. The relations (4.32) and (4.36) lead to the following system of equations for (real) U32, U43: J U12d& + 6U32 =
J
vlda + U43,
U2261U33 + U326U33 + U22U43 = Cl-
Its determinant of the coefficients of U32, U43 is 62
det 1 CU33
-1 1 U22 J
= 2c29o 0 0,
hence U32 and U43 are uniquely determined. Finally U42 follows from (4.35), and A, and A0 are unitarily equivalent.
If r = 0 (or c2 = 0), in the block matrices in (4.2) the third rows and columns disappear and D = cl, D = cl. Assume that Aw and A0 are unitarily equivalent. Then,
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283
as above, the relations (4.3) - (4.5), (4.13) and (4.17) give (i) and the first part of (iv); and in view of (4.8) and (4.17) the relation (4.9) is equivalent to (4.29). Let now (i) and (iv) be fulfilled. Then if U11
U41 := (-,vi), with v1 = t-1
c, U12 := t-1 (7 - 'ro), U22 = Yo,
(y-1 - ry"o 1), U44 = ry"o
the relations (4.3), (4.4), (4.13), (4.8), (4.9), (4.14) and (4.17) are satisfied. Then, choosing U42 so that (4.15) is satisfied (observe that U22 # 0), we obtain the equivalence of A. and A0. The theorem is proved. REFERENCES:
[1] ACHIESER, N.I.; GLASMANN, I.M.: Theorie der linearen Operatoren im Hilbertraum, Akademie-Verlag, Berlin 1960. [2] AZIZOV, T.J.; IOHVIDOV, I.S.: Foundations of the theory of linear operators in spaces with an indefinite metric, Nauka, Moscow 1986. [3]
BOGNAR, J.: Indefinite inner product spaces, Springer-Verlag, Berlin-HeidelbergNew York 1974.
[4] DIJKSMA, A.; LANGER, H.; DE SNOO, H.: Unitary colligations in Krein spaces and their role in the extension theory of isometrics and symmetric linear relations in Hilbert space, Functional Analysis II, Proceedings Dubrovnik 1985, Lecture Notes in Mathematics, 1242 (1987), 1-42.
[5] IOHVIDOV, I.S.; KREIN, M.G.; LANGER, H.: Introduction to the spectral theory of operators in spaces with an indefinite metric, Akademie-Verlag, Berlin 1982.
[6] JONAS, P.: Zur Existenz von Eigenspektralfunktionen mit Singularitaten, Math. Nachr. 88 (1977), 345-361. [7] JONAS, P.: On the functional calculus and the spectral function for definitizable operators in Krein space, Beitrage Anal. 16 (1981), 121-135. [8] JONAS, P.: A class of operator valued meromorphic functions on the unit disc.I, Ann.Acad.Sci.Fenn.Ser.A I (to appear).
[9] KREIN, M.G.: On completely continuous linear operators in functional spaces with two norms, Zbirnik Prac' Inst. Mat.Akad. Nauk Ukrain RSR, No.9 (1947), 104-129 (Ukrainian).
[10] KREIN, M.G.; LANGER, H.: On the spectral function of a selfadjoint operator in a space with indefinite metric, Dokl.Akad.Nauk SSSR 152 (1963), 39-42. [11] KREIN, M.G.; LANGER, H.: Uber die Q-Funktion eines ir -hermiteschen Operators im Raume flu, Acta Scient.Math.(Szeged) 34 (1973), 191-230.
[12] KREIN, M.G.; LANGER, H.: Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raum flu zusammenhangen. I. Einige Funktionenklassen and ihre Darstellungen, Math.Nachr. 77 (1977), 187-236.
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[13] KREIN, M.G.; LANGER, H.: One some extension problems which are closely connected with the theory of hermitian operators in a space fl,,. III. Indefinite analogues of the Hamburger and Stieltjes moment problems, Part (I): Beitrage Anal. 14 (1979), 25-40; Part (II): Beitrage Anal. 15 (1981), 27-45. [14] LANGER, H.: Spectral functions of definitiziable operators in Krein spaces, Functional Analysis, Proceedings Dubrovnik, Lecture Notes in Mathematics, 948 (1982), 1-46.
Acknowledgements. The first author thanks the TU Vienna for its hospitality and financial support. The second author expresses his sincere thanks to Professor Ando for giving him the possibility to take part in the Workshop.
P. JONAS Neltestraile 12 D-1199 Berlin Germany
H. LANGER Techn. Univ. Wien Inst.f.Analysis,Techn.Math. and Versicherungsmathematik Wiedner Hauptstrafie 8-10 1040 Wien Austria
B. TEXTORIUS Linkoping University Department of Mathematics S-581 83 Linkoping Sweden
AMS classification: Primary 47 B50; secondary 47A67, 47A45
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
285
THE von NEUMANN INEQUALITY AND DILATION THEOREMS FOR CONTRACTIONS
Takateru Okayasu
In this paper we shall prove that, if S1, ,T are sets of commuting , S,,, and T1, contractions on a Hilbert space, both satisfy the von Neumann inequality "in the strong sense", each S, double commutes with every Tk, and, S1i , Sm generate a nuclear C*algebra, then the set Si,-.. , Sm, T1, , T. satisfies the von Neumann inequality "in the strong sense". This gives a new condition for a set of contractions to admit a simultaneous strong unitary dilation.
1. The von Neumann inequality and strong unitary dilation It is well-known that any contraction T on a Hilbert space satisfies the so-called von Neumann inequality:
IIp(T)II S IIpII = sup Ip(eie)I 0<e<2,r
for any polynomial p in one variable. It is also well-known, as the Sz.-Nagy strong unitary
dilation theorem, that any contraction T on a Hilbert space f admits a strong unitary dilation, that is, there exist a Hilbert space K D ?-l and a unitary operator U on K such that
T' = PUm l7{ (m > 0), where P is the projection onto 11. These matters are considered to be same; and rest on the fact that the linear map ql such that
q(p+q)=p(T)*+q(T),
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286
p, q polynomials in one variable, of the C*-algebra C(T) of all complex-valued continuous
functions on the torus T into the C*-algebra B(1f) of all bounded linear operators on 11, is completely positive.
For a set of commuting contractions, T1,..., T, to satisfy the von Neumann inequality in
the strong sense, and to admit a simultaneous strong unitary dilation, are closely bound up in each other. Actually we are able to state that these two conditions are equivalent:
Theorem 1. Let T1,
, T be commuting contractions on a Hilbert space 1{. If the
inequality II(Pi,(T1,...
,T,,))II <_ II(Pii)II =
sup
O<e,, <2
II(Pi,(eie',...
,e'e"))II
(to which we refer as the von Neumann inequality in the strong sense) holds for any m x m matrix (pij ) , pij polynomials in n variables, then the set of contractions T 1 ,--- , T admits
a strong unitary dilation, in other words, there exist a Hilbert space IC D fl and commuting
unitary operators U1i
, U on 1C, such that
Tim' ...Tn"=PUm'...U, "Ifl (m1,...,mn>0), where P is the projection onto 11; and vice versa. Ando's theorems [1], [2] give central cases where the (equivalent) conditions inTheorem 1
are fulfilled. One of them asserts that any pair of commuting contractions admits a strong
unitary dilation, and the other that any triple of commuting contractions, one of which double commutes with others, admits also a strong unitary dilation. These matters then show that any pair of commuting contractions, and, any triple of commuting contractions, one of which double commutes with others, admits the von Neumann inequality in the strong sense.
On the other hand, some examples (Parrott[6], Crabb-Davie[4], and Varopoulos[11]) show that the n variable version of the von Neumann inequality IIP(T1,...
Tn)II <_ IIPII =
sup
IP(e'e=,... ,eie")I
0<ek <2T
, Tn commuting contractions, n > 3; and hence T1, cases cannot admit strong unitary dilation. fails to be valid, T1i
, Tn in those
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287
We give here a sufficient condition for commuting contractions to satisfy the von Neumann inequality in the strong sense:
Theorem 2. Let S1,
, T. be sets of commuting contractions, both satisfy S.; T1, the von Neumann inequality in the strong sense, and S; double commute with every Tk. , T satisfies the If Sl, , S. generate a nuclear algebra, then the set S 1 , . . , Sm, Tl, von Neumann inequality in the strong sense. ,
A nuclear algebra means a C*-algebra A such that, for any C*-algebra B, the *-algebraic
tensor product A O B of A and B has a unique C*-norm (See [7]). A GCR-algebra (=a postliminal C*-algebra), which must be nuclear [10], means a C*-algebra A such that, for any *-representation -7r of A, the von Neumann algebra generated by the image 7r(A) of A
by ir is of type I; a GCR-operator, besides, means an operator T such that the C*-algebra
generated by T is a GCR-algebra. Normal operators, compact operators, and isometries, are GCR-operators [5].
, T be a set of commuting contractions Corollary. Let S be a GCR-contraction, T1, which satisfies the von Neumann inequality in the strong sense, and S double commute
with every Tk. Then one concludes that the set S, T1,
, T satisfies the von Neumann
inequality in the strong sense.
This generalizes an earier result due to Brehmer-Sz.-Nagy (See [9], I), that a triple of commuting contractions, one of which is an isometry double commutes with others, admits a strong unitary dilation.
2. Canonical representation of completely contractive maps We recall several notions on maps on operator spaces. An operator space means a subspace which contains the identity element (denoted by 1)
of a unital C*-algebra, and an operator system a self-adjoint operator space. A linear map 0 of an operator space S into another is said to be unital if 0(1) = 1 holds; contractive, positive if 114(x)II <_ IlxII (x E S),
¢(x)>0 (0<xES)
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288
holds, respectively; completely contractive, completely positive if the tensor product 0 0 id," of 0 and the identity map id," of the m x m matrix algebra Mm is contractive, positive, resectively, for any m > 1.
It is fundamental that any unital contractive map of an operator system into another is positive, that a positive map 0 of an operator system into another is bounded (in fact,
A into a C'-algebra B is
11011 < 2f1¢(1)11), and that a positive map 0 of a
completely positive if either A or B is abelian. The Steinspring theorem [8] asserts that any unital, completely positive map 0 of a unital C'-algebra A into the C`-algebra B(7{)
on a Hilbert space 7{ has a canonical representation, namely, there exist a Hilbrt space (unique up to unitary equivalence) IC 3 7{ and a *-representation 7r of A on IC such that
7r(A)f is dense in K and
¢(x) = Pir(x)I7-l (x E A), where P is the projection onto 11. Now we want to give a proof of Theorem 1. In it, Arveson's extension theorem [3] (See [7]) is essential; it asserts that, any unital completely contractive map of an operator space
S into an operator space T extends to a completely positive map of any C`-algebra A D S
into a C`-algebra B 3 T. Proof of Theorem 1. Assume that T1i
, T are commuting contractions on a Hilbert
space 9-t and satisfy the von Neumann iequality in the strong sense.
By assumption the linear map
0 : p - p(T1, ... ,T.), of the operator space P(T") of all polynomials in n variables e'el, , e'6", on T" T x ... x T, into B(71), is unital and completely contractive. Then it extends to a unital, completely positive map of the C`-algebra C(T") of complex-valued continuous functions
on T", into B(7f). Therefore, there exist a Hilbert space IC D 71 and a `-representation it of C(T") on IC such that
4,(f) = PT(f)I71 (f E C(T")), P the projection onto 7{. Put Uk = 7r(e'Bk) (k = 1, operators and satisfy that
, n). Then U1i
Ti' ... Tn n = PUi"1 ... Un ^ I7i (ml, ... , m" > 0)
, U" are unitary
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289
Conversely, let K be a Hilbert space D 7{ and U 1 ,
, U,,
be commuting unitary opera-
tors on /C such that
7-,r^1..Tn " = PUi I ...Un "I7{ (1111i... ,mn > 0), P the projection onto 7f. Consider the *-homomorphism ¢ of the *-algebra of all polynomials in variables eie', e-ie', ... , eie", a-'9", on Tn, to B(7{) such that O(eiek) = Uk
for k = 1,
and
4)(e-1Bk) = Uk ,
, n. We can see that it is bounded and satisfies the inequality I1n(U1,U1,...'Un,Un)II
11O(n)II =
Ilnll
for any p. Therefore, by the Stone-Weierstrass argument, it extends to a *-representation of QT n). So, 0 is completely contractive. Consequently, we have II(pij(T1,... ,Tn))II
II(pij(U1,... ,Un))II
®id.)((pij))II
= II((0
II(pij)II sup
II(pij(e1e',... ,e'B"))II,
o<e,, <2x
for any m x m matrix (pij ), pij polynomials in variables e'B',... , e'0", which completes the proof.
3. An effect of generation of nuclear algebras Next, we will give a
Proof of Theorem 2. It is sufficient to find a unital completely contractive map of P(Tm+n) into B(7{), which maps each variable ei9', eiek to S,,Tk, respectively.
We already have, via Theorem 1, unital completely contractive maps 01 of P(T') into B(7{) so that 4,1(eiei) = Sj, 02 of P(Tn) into B(7{) so that 4)2(ei°k) = Tk. According to Arveson's extension theorem, 01 (resp. 02) extends to a unital completely positive map W1 (resp. 02) of C(Tm) (resp. C(T")) into A (resp. B), where A (resp. B) is the C*-algebra
Okayasu
290
generated by Si,
,
It can be seen that the tensor product 1' 1®112
5,,, (resp. T1i
of 1/il and 02 is a unital, completely positive, and so completely contractive, map, of the
C'-tensor product QT n) ® QTn) of C(Tm) and C(T") (which can be thought of as C(Tm+")) into the minimal C`-tensor product A 0 B of A and B, i.e., the completion of A 0 B under the operator norm II II considered on A 0 B (which is known to be the smallest among all C*-norms on A O B [10]). Hence, the tensor product 010 02 of 01 and
02 , the restriction of 01 0 2 to P(Tm+") (identified with P(Tm) 0 P(T")), is a unital completely contractive map of P(Tm+") into A O B. Consider then the *-representation 0 of A 0 B on f such that
¢(X ® Y) = XY (X E A, Y E B). Since the operator norm 11 I Ion A®B coincides, by assumption, with the (largest) C* -norm II
II, defined by the identity IIV& = sup{II7r(V)II : it is a *-representation of A O B},
for each V E A O B (See [7]), we have the inequality
IIEXkYkII <_ II>X®®Ykli, = IIEXk ®Ykii, k
k
k
for Xk E A and Yk E B. This shows that 0 may extend to a *-representation of the C`-algebra A ® B. Hence, as above, ¢ is completely contractive. It is obvious, on the other
hand, that 0 is unital, so, the composition 0 o (01 ®02) of 0 and ¢® ®02 is unital and completely contractive; and maps each variable e'0', e'8" to S,,Tk, respectively. Now the proof is complete.
References 1. T. Ando, On a pair of commuting contractions, Acta Sci. Math. 24(1963),88-90. 2. T. Ando, Unitary dilation for a triple of commuting contractions, Bull. Acad. Polonaise Math. 24(1976), 851-853. 3. W. B. Arveson, Subalgebras of C*-algebras, Acta Math. 123(1969), 141-224.
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291
4. M. J. Crabb and A. M. Davie, von Neumann's inequality f or Hilbert space operators, Bull. London Math. Soc. 7(1975), 49-50 5. T. Okayasu, On GCR-operators, Tohoku Math. Journ. 21 (1969), 573-579.
6. S. K. Parrott, Unitary dilations for commuting contractions, Pacific Journ. Math. 34(1970), 481-490.
7. I. Paulsen, Completely bounded maps and dilations, Pitman Res. Notes Math. Ser. 146, 1986.
8. W. F. Steinspring, Positive functions on C'-algebras, Proc. Amer. Math.
Soc.
6(1955), 211-216.
9. B. Sz.-Nagy and C. Foia§, Harmonic analysis of operators on Hilbert space, Amusterdam- Budapest, North-Holland, 1970.
10. M. Takesaki, On the cross-norm of the direct product of C`-algebras, Tohoku Math. Journ. 16(1964), 111-122.
11. N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operator theory, Journ. Funct. Analy. 16(1974), 83-100.
Department of Mathematics Faculty of Science Yamagata University Yamagata 990, JAPAN
MSC 1991: Primary 47A20, 47A30; Secondary 46M05
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
292
INTERPOLATION PROBLEMS, INVERSE SPECTRAL PROBLEMS AND NONLINEAR EQUATIONS
L. A. Sakhnovich
The method of operator identities of a type of commutation relations is shown to be useful in the investigation of interpolation problems, inverse spectral problems and nonlinear integrable equations.
Suppose that the operators A, S, P1, P2 are connected by the relation AS - SA* = i (P1 Pz + P2 Pi )
(1)
where G1 and H are Hilbert spaces, dim G1 < oo,
S = S';
A, S E {H, H};
P1, P2 E {G1, H}
and {H1, H2} is the set of bounded operators acting from H1 to H2. We also introduce the operator J E {G, G} where G=G1ED G1,
J=[0
E1 J
Formula (1) is a special case of the operator identity of the form
AS-SB=H1H2
(2)
which is a generalization of the commutation relations and it also generalizes the wellknown notion of the node (M. S. Livsic [1] and then M. S. Brodskii [2]). The identities of the form (2) proved to be useful in a number of problems (system theory [3], factorization problems [3], interpolation theory [4], the method of constructing the inverse operator T = S-1 [5], the inverse spectral problem [3] and theory of nonlinear integrable equations [6]). There are close ties between all these problems and corresponding results.
293
Sakhnovich
In the present paper we shall consider three of these problems: interpolation problems, inverse spectral problems and nonlinear integrable equations. 1. Let £ be a collection of monotonically increasing operator-functions and
r(u) E {G1, G1 } is such that integrals
Sr = f00(E - uA)-1Pz[dr(u)JPz (E Zr
uA.)-1
(3)
dr(u)
f
(4 )
1+uz
converge in the weak sense. Then the integral
P1,r = - i
J
[A ( E
- u A ) -1 +
1
+ uz E] -P 2 d r(u)
(
5)
also converges in the weak sense. Let us introduce the operators P = P1,r + i(P2a + F01 /2)
S = Sr + FF*,
(6)
where
a=a*, Q>0 and operator F from {G1, H} is defined by the equality AF = P2Q1/z.
(7)
Now we shall formulate the interpolation problem which is generated by operation identity (1) [4].
It is necessary to describe the set of r(u) E £ and a = a`, Q > 0 such that the ,given operators S, P1 admit the representation S = S,
P1 = P.
(8)
Let us note that according to (3) the necessary condition of the formulated problem is the inequality
S>0.
(9)
As an example we shall consider the bounded operator S in the space Lz(0, w) of the form
(Sf)(x) =
d
f 0
f (i)s(x - t) dt.
(10)
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294
Then the equality
((AS-SA')f)(x)=iJW f (1) [M (x) + N (t)] dt
(11)
0
is valid. In equality (11)
N(x) = -s(-x),
M(x) = s(x),
0<x<w
and the operators A, A* are defined by the equalities
(Af)(x) = ifx f(t)dt,
(A*f)(x) _ -iJW f(t)di. x
Formula (11) is a special case of the operator identity of form (1), where
(11g)(x) = M(x)g,
(P29) (x) = g,
dim G1 = 1,
9 E G1,
and g are constants. The corresponding interpolation problem has the form. It is necessary to describe the set of r(u) E l; which gives the representation ,o
(sf,f)
00
f (x)e-'ux dx
2
fo
dr(u).
If the operator S has the form
(Sf)(x) =
I0
f (t)ic(x - t) dt
we come to the well-known Krein problem: it is necessary to describe the set of r(u) which gives the representation Kc(x) = J700 e1xu dr(u).
In our approach to the interpolation problem we use the operator identity and operator form of Potapov inequality [4]. Operator identity (11) gives a tool for constructing the inverse operator T = S. The operator T can be found in the exact form by means of the functions N1(x), N2(x) which are defined by the relations SN1 = M,
SN2 = 1.
We have proved that the knowledge of N1, N2 is that minimal information which is necessary for constructing T 1512. Let us consider the inverse spectral problem which is connected with operator identity (1).
295
Sakhnovich
THEOREM. Suppose that the following conditions hold: I. S is positive and invertible. II. There exists a continuous increasing family of orthogonal projections PC, 0 <
(< w, Po = O, P, = E such that A'Pc = PCA*Pc.
III. The spectrum of A is concentrated at zero. Then the following representations hold z) =
fexp [izJ do-, (t)]
where
w((, z) = E + izJ II`S 1(E - zA()-1 PdH, II = [P1, P2],
Ac = PcAPC.
Sc = PcSPC,
If o1(x) is absolutely continuous, then oi(x) > 0 and dw dx
= izJoi(x)w(x, z),
w(0, z) = E.
(12)
"anonical system (12) corresponds to operator identity (1). We shall introduce the main definitions. Let us denote by the space of vectorfunctions with the inner product (g, h) = LW h* (x) [dol (x)] 9(x)
We define the function
F(u) = fc w*(x, u) [dol(x)] 9(x) _ [ f2(u) ] and the operator
V9=f2 A monotonically increasing m x m matrix-function r(u), -oo < u < oo will be called a Spectral function of system (12) if the operator V maps L2(o1) isometrically into L2(7-).
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296
THEOREM. The set of r(u) which are solutions of the interpolation problem and the set of spectral functions of the canonical system coincide. The following results give a method for solving the inverse spectral problem
Let us suppose that the operators A and P2 are fixed. In this way we define the class of canonical systems (12). Then let us suppose that the spectral data of system (12) are given, i.e. the spectral function -r(u) and the matrix a are known. Using the interpolation formulas we have:
Pl = - i S=
/
f : [A E - uA -1 + 1 + u2 E] P dr(u) + W2, )
(
c'o
2
(E - uA)-1P2[dr(u)]P2 (E - uA`)-1
( 13 )
(14)
Il=[P1,P2],
(15)
These formulas (13)-(15) give the solution of the inverse spectral problem. If (Af)(x) = i fo f (t) dt, P2g = g we come to the well-known inverse problems for the system of Dirac type. In the general case when
(Af)(x)=iWJx f(t)dt we come to the new non-classical inverse problem. The necessity of investigating nonclassical problems is dictated both by mathematical and applied questions (interpolation theory, the theory of solitons). Under certain assumptions formulas (13)-(15) give the solution of the inverse spectral problem in the exact form [3]. Let us introduce an analogue of the Weyl-Titchmarsh function v(z) for system (12) with the help of the inequality E,,, iv
zwxz
E
x w x z)
000
1
-iv(z)
dx < no.
The matrix function v(z) belongs to the Nevanlinna class, i.e. v(z) - V* (Z)
>0,
Im z > 0.
i
The connection of v(z) with the spectral data r(u) and a is the following [3] V(Z)=a+ J
00
(u 1
z
1+ ua
dr(u).
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297
We have considered the case when 0 < x < oo. As in the case of Sturm-Liouville equation, the spectral problems on the line (-oo < x < oo) can be reduced to the problems on the half-line (0, 00) by doubling the dimension of the system. The problem on the line contains the periodical case. 3. The method of inverse scattering problems is effectively used for investigating the nonlinear equations (Gardner, Kruskal, Zabuski, Lax, Zaharov, Shabat [8]). The main idea comes to the following. The nonlinear equation is considered
together with the corresponding linear system. The evolution of scattering data of the linear system is very simple. Then by using the method of inverse problem the solution of nonlinear system can be found. The transition from the inverse scattering problem to the inverse spectral probem removes the demand for the regularity of the solution at the infinity and permits to :onstruct new classes of exact solutions for a number of nonlinear equations [7]:
Rt = 2(Rsx - 21R12R)
Rt = - 1 Rxxx + 02
V
8x81 =
2I RI2Rx
4 sh cp.
(NS)
(16)
(MKdV)
(17)
(Sh-G)
(18)
These equations have found wide applications in a number of problems of mathematical physics.
The corresponding linear system has the form 8w
8x
= izH(x, t)w,
w(0, t, z) = E,,.
(19)
On the case of Sh-Gordon equation we have [8]
H(x, i) _ 1exp[W(O'
exp[ a(x, t) - X0(0, t)]
0
t) - cp(x, t )]
0
(20) J J
Let vo(z) = v(0, z) of corresponding system (19), (20) be a rational function of z: i.e. N
v0(z) = i - E,33k,O/(z + iCfk o). k=1
1'hen v(t, z) is also a rational function N
v(t, z) = i - E Qk(t)/[Z + iak(t)] k=1
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298
Let us write down v(t, z) in the form v(t, z) = =P20, Ez)/P1(t, iz)
where
N
N
P1(t, z) = JJ[z - ak(t)],
P2(t, z) = JJ[z - vk(t)]
k=1
k=1
Let us introduce
Q(z) = P1(t, z)P2(t, -z) + Pl(t, -z)P2(t, z). It is essential that unlike P1(t, z), P2(t, z) the coefficients Q(z) do not depend on t. It means that the zeros of the Q(z) do not depend on t either. Let the inequality wi # wk be true when j k. Let us number the zeros of Q(z) in such a way that Re wi > 0 when 1 < j < N. The solution of the Sh-Gordon equation which corresponds to the rational vo(z) is as follows ,p(x,t) = 21nI 81(x,t)/b2(x,t)I
where k
19
bl(x,t) = det {w N
1
gtkchgjJ k
62(x, t) = det{wN atk sh11j }1<j,k
%(x,t) = wix+t/wi +cj
.
This result was obtained jointly with post-graduate Tidnjuk [7]. The corresponding solution R(x,t) of equations (16), (17) has the following form
R(x,t) = -2(-1)NO1(x,t)/z 2(x,t), 1
1
...
1
W1
W2
...
w2N
...........................
A1(x,t) =
N-2
W1
N-2
W2
71
72
W171
W272
...
...
N-2
W2N
72N W2N72N
...........................
wr71
'w2 72
...
w2nr72N
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299
02(x,1) =
1
1
W1
W2
... ...
W2N
wl
N-1
W2N-1
...
N-1 w2N
71 W171
72 W272
...
72N W2N72N
1
................................. W1
71
W2 -172
...
w2N-1
72N
where 7k = ck exp 2(wkx - S2kt),
f?k = -iwk
(NS),
(MKdV).
S2k = wk
4. If wj = T j-, aj,o = -a- then the corresponding solution R(x, t) of MKdV is real. All the real singularities of R(x, t) are poles of the first order with the residues +1 or -1. When t - ±oo, the solution R(x, t) is presented by a sum of simple waves N
R(z, t) E R (x, t),
t -> ±oo
(21)
j-1
R (x, t)=2(-1)'wj/sh[2(wjx-w,t+cj )].
(22)
The considered nonlinear equations do not have N-soliton solutions. The constructed solutions are similar to the N-soliton solutions. The behaviour of the singularities of the solutions is analogous to the behaviour of the humps of the N-soliton solutions and can be interpreted in the terms of a particles system. We have proved that the corresponding particles system is a completely integrable one with the Hamiltonian
H=21
N Ez
(23)
pj
j-1
where
Pi = wj2
(MKdV),
1 pj = -w2
(Sh-G)
(24)
W)
pj = 2 Im wj
qj =pjt+cj.
(NS)
(25) (26)
The variables pj, qj are variables of the action-angle type. It follows from formulas (21), (22) for MKdV that pj coincides with the limit velocity of the wave. The same situation is in Sh-G and NS cases.
Sakhnovich
300
MKdV
w1=0.3
w2=0.5
w3=0.7
w.4=1
a4 = 0.35
a2 = 0.45
a3 = 0.55
a4 = 0.65
c=k>0
Fig. 1
301
Sakhnovich
Sakhnovich 302
Sakhnovich
303
It also follows from formulas (21), (22) that the lines of singularities have N asymptotes (MKdV)
wax -wit+c} = 0,
1 < j
t
Let us number these asymptotes when t - -oo by the order of velocity values and consider the same lines of singularities when I --. +oo. Then the corresponding asymptotes are again
ordered by the velocity values but in the opposite direction (Fig. 1). It means that the particles exchange their numbers. In the case of Sh-Gordon equation the situation is the same (Fig. 2). The particles exchange their numbers even if there is no crossing (Fig. 3). The particles can be of two kinds: plus particles if the corresponding residues are +1, and minus particles if they are -1. The particles of the same kind don't cross. In Fig. 3 the particles are of the same kind. 5. The considered equations generated self-adjoint spectral problems. If we study the equations Re = 2 (Rxx + 21R12R) 1
3
Rt = - Rxxx - 2IRI2 Rx
(NS)
(27)
(MKdV)
(28)
(sin-G)
(29)
4 O2
_ 4 sin cP
8xot =
then the nonself-adjoint spectral problems correspond to them. The analogue of the WeylTitchmarsh function for this case was introduced and the analysis of the equations of the form (27)-(29) was done by A. L. Sakhnovich [9], [10].
REFERENCES 1. M. S. Livsic, Operators, oscillations, waves (open systems), Amer. Math. Soc., 1966.
2. M. S. Brodskii, Triangular and Jordan representations of linear operators, Amer. Math. Soc., 1971. 3. L. A. Sakhnovich, Factorization problems and operator identities, Russian Math. Surveys, 41 no. 1 (1986), 1-64. 4. T. S. Ivanchenko, L. A. Sakhnovich, An operator approach to the investigation of interpolation problems, Dep. at Ukr. NIINTI, N 701 (1985), 1-63.
Sakhnovich
304
5. L. A. Sakhnovich, Equations with a difference kernel on a finite interval, Russian Math. Surveys 35 no. 4 (1980), 81-152. 6. L. A. Sakhnovich, Nonlinear equations and inverse problems on the semi-axis, Preprint, Institute of Math., 1987. 7. L. A. Sakhnovich, I. F. Tidnjuk, The effective solution of the Sh-Gordon equation, Dokl. Akad. Nauk. Ukr. SSR Ser. A, 1990 no. 9, 20-25. 8. R. K. Bullough, P. I. Caudrey (Eds), Solitons, New York, 1980. 9. A. L. Sakhnovich, The Goursat problem for the sine-Gordon equation, Dokl. Akad. Nauk. Ukr. SSR Ser. A 1989, no. 12, 14-17. 10. A. L. Sakhnovich, A nonlinear Schrodinger equation on the semi-axis and a related inverse problem, Ukrain. Math. J. 42 (1990), 316-323. Sakhnovich, L. A. Odessa Electrical Engineering Institute of Communications Odessa, Ukraine MSC 1991: 47A62, 35Q53
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
305
Extended Interpolation Problem in Finitely Connected Domains SECHIKO TAKAHASHI
This paper concerns the matrix condition necessary and sufficient for the existence
of a function f, holomorphic in a finitely connected domain and having IfI < 1 and finitely many first prescribed Taylor coefficients at a finite number of given points. In a simply connected domain, some transformation formulas and their applications are given. The results of Abrahamse on the Pick interpolation problem are generalized to the above extended interpolation problem.
Introduction Let D be a bounded domain in the complex plane C, whose boundary consists of a finite number of mutually disjoint analytic simple closed curves, and let 13 be the set of functions f holomorphic in D and satisfying If I < 1 in D. In this paper we consider the following extended interpolation problem: Let z1i z2,
, zk be k distinct points in D and, for each point z;, let
c10,... , yin; -1 be n; complex numbers. For these given data, find a function f E 13 which satisfies the conditions
(El)
f (Z)
n;-1
= E c;a(z - z;)a + O((z - z;)n')
(i = 1, ... , k).
a=o
In the Part I, we introduce, as powerful tools for the studies of this problem (EI), the Schur's triangular matrix 0 used by Schur in [13] and our rectangular matrix M, which made it possible to unify Schur's coefficient theorem and Pick's interpolation theorem (Takahashi [14]). We give important transformation formulas which express the changes of these matrices under holomorphic transformations in terms of the transformation matrices. In the Part II, we recall our main results obtained in [14] and [15] for the problem (EI) in the case where D is the open unit disc. As an application of these results and the transformation formulas, we give a criterion matrix of the extended interpolation problem in the case where D is a simply connected domain in the Riemann sphere having at least two boundary points and the range W is a closed disc in the Riemann sphere. When W contains the point at infinity, we have of course to modify the conditions (EI) appropriately and the solutions may have poles.
Takahashi
306
In the Part III, we show that the results of Abrahamse in [1] on the interpolation problem in finitely connected domains can be extended to our extended interpolation problem. PART I. MATRICES AND TRANSFORMATION FORMULAS
§1. Matricial Representation of Taylor Coefficients. (1) Schur's Triangular Matrix A. To a function f(z) = E00 Ca(Z - zo)a a=0
holomorphic at zo and to a positive integer n E N, we assign a triangular n x n matrix
I
0(f;zo;n =
C1
CO
Cn-1
Cl
Let g(z) be another function holomorphic at zo, we see immediately
0(f + g; zo; n) = 0(f; zo; n) + z(g; zo; n), 0(fg; zo; n) = i(f; zo; n) ' o(g; zo; n) = 0(g; zo; n) 0(f; zo; n), 0(1; zo; n) = In (the unit matrix of order n).
(2) Rectangular Coefficient Matrix M. To a function 00 (0)6
F(z,() _ E as,6(z - zo)a(C a,9=O
holomorphic w.r.t. (z, C) at (zo, (o) and to (m, n) E N x N, we associate an m x n matrix a00
...
a0n-1
M(F; zo, Co; m, n) _ am-10 am-ln-1 For another function G(z, () holomorphic w.r.t. (z, C) at (zo, (o), we have
M(F + G; zo, Co; m, n) = M(F; zo, Co; m, n) + M(G; zo, Co; m, n)
Moreover, for functions f (z) and g(C), holomorphic at zo and Co respectively, we have the useful product formula
(PF) M(f F9; zo, Co; m, n) = 0(f ; zo; m) ' M(F; zo, Co; m, n) . 0(g; Co; n)*, where by 0* ='; we mean the transposed of the complex conjugate of A. This product formula can be established by a direct calculation.
307
Takahashi
§2. Transformation Matrix and Transformation Formulas The transformation formula for the matrix M which we established in [14] is the pivot of our present studies. For a transformation z = V(x) holomorphic at xo with zo = W(xo) and for m E N, we define the transformation matrix S (co; x0; m) as follows: Write
AX) = zo + (x - xo)'PI(x), -(Dm =
xo; m),
M((z - zo)a( _- (o)o'; zo, (o; m, m) (a =0'... 'M - 1) and put
The matrix EM(') is the m x m matrix whose (a + 1, a + 1)-entry is 1 and the all
other entries are 0. V = I.. and 4);, = -tm 1 m (a = 1, 2,
). The matrix
52 is of the form 1
0
c
Q(,P; xo; m) =
c2
0
*
...
*
cm-1
where c = c,1(xo) = W'(x0). If c # 0, then Sl(W;xo;m) is an invertible matrix. If w(x) is the identical transformation, then W1(x) = 1, 4)m = I,,, and hence Q(V; x0; m) = Im. In terms of this transformation matrix S2, we showed in [14] THEOREM 1 (TRANSFORMATION FORMULA FOR M). Let F(z, () be a func-
tion holomorphic w.r.t. (z, Z) at (zo, (o). Let z = p(x), C = V,(e) be functions holomorphic at xo,lo, with zo = w(xo),Co = respectively. Put G(x,f) = F(,p(x),'+k(e)) Then, for (m, n) E N x N, we have M(G; xo, Co; m, n) = fZ(co; xo; m) . M(F; zo, Co; m, n) - Q(0; Co; n)*
Takahashi
308
As an application of the preceding transformation formula, we obtain THEOREM 2 (TRANSFORMATION FORMULA FOR 0). Let 00
f(z) = > ca(z - zo)a a=0
be a function holomorphic at zo and let W be a function holomorphic at x0 with zo = V(x0). Set 00
g(x) = f(o(x)) = > da(x - xo)a.
a=0
Then we have for n E N Co
(1)
SI(y; xo; n) = 1l(W; xo; n)
do
d1
d0
cl
Co
Cn_1
do
co
dl
c1
do-1
Cn- 1
cl
i
(2)
PROOF. Consider at (zo, 0) the function Fo(z, )
= 1-
r 00
(zl
zo)(=
O(z - zo)a(
and F(z, () = f (z)Fo(z, (). By definition we see M(Fo; zo, 0; n, n) = In and by (PF) in §1 M(F; zo, 0; n, n) = A(f; zo; n). Applying the transformations z = w(x) and ( = S to Fo and F, we have F(w(x), () = g(x)Fo(cp(x), () and hence, by the above transformation formula for M, the first relation SI(W; xo; n)0(.f; zo; n) = 0(g; xo; n)Q(co; xo; n).
Comparing the first columns of both sides of this equality, we see the relation (2) hold.
309
Takahashi
PART II. Disc CASES
§3. Main Theorems in the Unit Disc. In this section, we state the main results obtained in [14] and [15]. We assume D is the open unit disc {z : Izi < 1} and consider the extended interpolation problem (EI). Write Cio
C [Cl Cio
Cinj_l
'''
C= Ci1
Cio
rll " ' rlk .. ....
1
r ij =M ( 1 - zC ;zi,zj;ni,nj Aij=r ij-
,
r=
rk1
"'
rkk
All
...
Alk
Akl
...
Akk
A=
Then we have
matrix of order n1 + A criterion matrix of the problem (EI).
+ nk, which is called
Let £ denote the set of all solutions of (EI) in B. THEOREM 3 (EXTENSION OF THE THEOREMS OF CARATHEODORY-SCHUR
AND PICK). There exists an f E £ if and only if A > 0 (positive semidefinite). THEOREM 4 (UNIQUENESS THEOREM OF SOLUTIONS). For the problem
(EI), the following conditions are equivalent: (a) The set £ consisits of a unique element. (b) Some finite Blaschke product of degree r < n1 +
+ nk is in E. (c) A > 0 and det A = 0. If one of, therefore all of, these conditions are satisfied, then r = rank A. The proof of these theorems given in [14] was based on Marshall's method in [9], which makes use of Schur's algorithm.
Takahashi
310
In the case where the solution is not unique, that is, where A > 0 (positive definite), the following theorem, which may be proved as Corollary 2.4 in Chap.I
of the textbook of Garnett [7], shows that the problem (EI) has an infinite number of solutions. THEOREM 5. Suppose A > 0.
(a) Let zo E D, zo 0 z; (i = 1,
,
k). The set
W(zo) _ { f(zo) : f E £}
is a nondegenerate closed disc in D. (b) For each z; (i = 1,
,
k), the set
W'(Z;) = { f("i)(z;) : f E £} is a nondegenerate compact disc in C. In [15], we showed that if A > 0 then we have a bijective mapping ir
: B -k £
such that there exist four functions P, Q, R, and S holomorphic in the unit disc D and satisfying 7r(g) = Pg + Q
Rg+S
and Rg + S * 0
(`dg E B).
Let H°° denote the Banach algebra of bounded holomorphic functions f in D with the uniform norm IIf III = sup{ If (z)I : z E D}. The following Theorem 6, whose the first part (a) is due to Earl [4], can be derived immediately from Theorem 3 and Theorem 4. THEOREM 6.
(a) Among the solutions of (EI) in H°°, there exists a unique solution of (EI) of minimal norm. This unique solution is of the form mB, where
m = inf{ IIfI1.: f is a solution of (EI) in H°°} and B is a Blaschke product of degree < ni +
+ nk - 1-
(b) Conversely, if B is a Blaschke product of degree < nl + + nk - 1 and if cB (c E C) is a solution of (EI) then cB is the unique solution of minimal norm of (EI) in H°°.
Takahashi
311
§4. Criterion Matrix in Simply Connected Domains By virtue of the transformation formulas, we show in this section that our preceding results can be extended to the case where the source domain D is a simply connected domain in the Riemann sphere having at least two boundary points and the range W is a closed disc in C or a closed half plane in C. The case where W contains oo will be treated in the next section. Let z1i Z2,- , zk be distinct points in D and for each zi let cio, , cini-1 be ni complex numbers. Our present problem is to find a holomorphic function f in D such that f (z) E W for any z E D and f satisfies the conditions n; -1
(El)
f (z) = E cia(z - zi)° + O((z - Z,)"') °-o
,
k)
where if zi = oo for some i then we replace z - zi by 1/z. For a moment, we assume cio E W (i = 1, , k), which simplifies the statement. We shall later remove this assumption. We ask for a criterion matrix of this problem.
Let Do be the open unit disc in C, 'p : D -a Do be a conformal mapping, and O(w) = pw + q (p, q, r, and s: complex numbers with ps - qr = 1) rw+s be a linear fractional transformation which maps the interior of W onto Do. Put xi = 'p(zi) (i = 1, , k). Because of the presence of oo, we consider the transformation t(z) = 11z. As in [14], it is convenient to use the notion of local solution. A local solution of (EI) is by definition a function f, holomorphic in some neighborhood of the finite set {zl, z21 , zk} and satisfying the conditions (EI). The formulas in (1) of §1 and Theorem 2 show that a function f is a local solution of (EI) if and only if g = 0 o f o V-' is a local solution of the extended interpolation problem n; -1
dia(x - xi)° + O((x - x,)"')
g(x) =
(EI)o
(i = 1, ... , k),
°=0
whose coefficients are given by
d;l
d;o
din;-1
...
= Di = Sl; 1(rCi + sInc)-1(pC; + qln; )Q; dil
di0
where f1i = II('p; zi; ni) if zi # oo, S2i = fl('p o t ; 0; ni) if zi = oo, Ci is the triangular matrix defined from cio,
, cini -1 as in (1) of §1, and In, is the unit
Takahashi
312
matrix of order n;. The matrix S2; is clearly invertible. The matrix rC; + sl,,; is invertible since its entries on the diagonal are equal to rciO + s, which is not zero by assumption. Let 1
1 - xe
and put r;i = M(Go; x1, xi; ni, ni ) The matrix A(O) _
and
AM
A,°) ... A(O) .....
Aik Akk
is the criterion matrix of the problem (EI)° for B, defined in §3. Now, we define Fo(z, () = Go(V'(z), 5p(()) =
1
,
1 - O(zMO
r;i = M(Fo; z;, zj; n;, n,) ,
rl, ... r=
C=
......... ...
rkl
r,k
rkk
where if z; = oo and zi # oo then we replace FO and r;i by Fo(z,() = Go(,p(1/z), v(()) r;i = M(Fo; 0, zi; n;, ni)
and
respectively ;
if z; # oo and zi = oo then we replace FO and r;i by Fo(z, () = Go(w(z),
r;i = M(Fo; z;, 0; n;, ni)
and
respectively ;
and if i = j and z; = co then we replace FO and r;i by and Fo(z, () = Go('P(l/z), v(1/()) respectively. r;i = M(Fo; 0, 0; n;, ni) Write
1 - v'(w)r/i(v) = (rw + s)-'(rv + s)-*K(%b, v)
313
Takahashi
with
K(w,v) = awv+,Qw +/jv+y, Iri2 - 1pl2
: real<0,
=rs - pq, y
=1812-Ig12
:real,
1/312-ay=1. It is easy to see W = {w : K(w, w) > Q. As we pointed out in [14], if g is a local solution of (EI)° and if we put 1 - g(x)g(0
1-xz then we have A= M(G; x;, x,; n;, nj). Put f =,O-1 o go w
and
F(z, ) = G(w(z), w(())
Then F(z, )
Q1z) + 3f(C+ y = Fo(z, C) of(z)f(C+ (rf(z) + s)(rf(() + s)
The matrix M(F; z;, zi; n;, nj) is by Theorem 1 equal to Q; A;O) S1 , with appropriate change as above in the presence of oo, and, on the other hand, it is written in the form R; A,3 R, where A21 = ac, I';3 C; + QC; I',; + art1 C; + yr,, , Ri = (rC1 + sI.;)-1
Write
All
...
Alk
Akl
...
Akk
A=
S2=
,
Ilk
R= Rk
Then the matrix S1 and R are invertible and we have n A(O) ft' = RA R*.
This shows that A(°) > 0 if and only if A > 0 and that rank A(°)= rank A. Thus, we may adopt the Hermitian matrix A as criterion matrix of the problem (EI) for functions with values in W:
Takahashi
314
THEOREM 7. Let the notations and the assumption be as above. There exists a holomorphic function in D, having its values in W and satisfying (EI), if and only if the Hermitian matrix
A=acrc*+fcr+Qrc*+yr is positive semidefinite. Such a function is unique if and only if A > 0 and
detA=0. Note that the constants a, 0, and y depend on p, q, r, and s and that the matrix r depends only on gyp, zi and ni (i = 1, , k). In the case where W is the closed unit disc, that is, for the extended interpolation problem (EI) in 8, the criterion matrix reduces to
A=rsince we have then a=-1,/3=Dandy=1. We point out that, if the source domain D is an open disc or an open half plane in C and is defined by Ko(z, z) > 0, where
Ko(z,() =aoz(; +foz+/3o(+yo (ao and yo are real), then, as in Pick [11], we may replace the definition above of Fo by 1
Fo(z,() =
Ko(z,[;)
Finally, let us remove the assumption cio E W (i = 1, , k). If there exists a solution f of the problem, then cio = f(zi) E W. Conversely, suppose A > 0. The (1,1)-entry of the matrix
Aii = aciriiC; + /3Cirii + Qriicj + yrii is
1
io
io
to
y
1 - IW(zi)I2
As Jcw(zi)I < 1 and Aii > 0, we have K(cio,cio) > 0, which shows cio = 0-1(dio) E W. Thus we have removed the assumption.
315
Takahashi
§5. Criterion Matrix for Meromorphic Functions. Let D be a simply connected domain in the Riemann sphere having at least two boundary points and let W = {w : 1w - al > p} be a closed disc, including oo, , zk be k distinct points in the Riemann sphere (a E C, p > 0). Let z1i z2, c,n; _1 be n; in D. For each zi, let m; be a nonnegative integer and let c,o, complex numbers (n, > 1). Assume cio # 0 if m; > 0. The problem in this section is to find a meromorphic function f in D with values in W, which satisfies the conditions -(El)
nf(z) =
(z
- z,) Mi
(1 cicr(z - zi )a +0((Z - z,)n' ))
(i
a=0
where, if z, = oo, then z - z, is replaced by 1/z. We ask for a criterion matrix for this problem. Note that if m, > 0 then the order m, of pole of f at z, and , c,n, _ 1 of the Laurent expansion of f at z, are the first n, coefficients c,o, prescribed. For this purpose, as in the preceding section §4, we consider a conformal mapping V
: D -'Do
of D onto the open unit disc Do, the linear fractional mapping
W -+ Do
defined by
0(u)) =
p
w-a
the function 1
Fo(z,0) =
1 -'p(z)'P(S)
and the matrices
1'i, = M(Fo; z,, zj; m, + ni; mj + nj) with appropriate replacement as in §4 if oo presents.
Now, for n E N, we introduce the standard n x n nilpotent matrix
1
Takahashi
316
where n is a positive integer. Then N," = 0 (zero matrix). We define N,° = In (unit matrix). For each zi, put Cio
ni -1
Ci =
CV=0
Cia Nmai+ni =
...
0
0
Cini-1... Cio
m;
Ti = Nmi+ni ,
(i=1,...,k).
Ri=Ci - aT,
If mi > 0 then the diagonal entries of the triangular matrix Ri are all equal to cio, which is not zero by assumption, and hence Ri is invertible. If mi = 0, we may assume for a moment as in §4 that cio # a, so that Ri is invertible. This assumption can be removed as in the final part of §4. A meromorphic function f in D with values in W is transformed by z/i into a holomorphic function P
g(z) =
f(z)-a
in D with IgI < 1. On the other hand, writing fo(z)
f(z) =
(z - zi)mi
and P(z - zi)mi
g(z) -
fo(z) - a(z - zi)mi'
we see easily that the conditions (EI) for f are transformed into the conditions n i-1
(EI)# g(z) _ (z - zi)m' (> dia(z - zi)a + O((z - zi)ni ))
(i = 1, ... , k)'
a=0
where, if zi = oo, then z - zi is replaced by 1/z and the coefficients dia are defined by the relations ni -1
dia Nm;+n = PRi -'Ti. a=0
Takahashi
317
Denoting this matrix pRi 1Ti by Di and setting
... [h1;k] .......... A#
A# = rii - DirijD!
and
A# _
A#
k1
A#
...
,
A#
kk
we observe by Theorem 7 that the criterion matrix for the problem (EI)# in S is A# and we have
R;A#Rj _ (Ci - aTi)ri,(C, - aTj*) - p2TirijT,* = Ciri1C,* - aTirijcj - aCirijT; + (1a12 - p2)Tiri,T,*. Write
C [Cl
r11
..
rlk
rkl
..
rkk
r=
C=
,
R1
T1
T=
,
R=
Tk
Rk
and define
A=CrC*-aTrC*-aCrT*+(1a12-p2)TrT*. It should be noted that W is expressed by
ww-aw-aw+(1a12-p2)>>-0. Then we have A = R A# R*, where R is an invertible matrix. It follows that A > 0 if and only if A# > 0 and that rank A = rank A#. Theorem 7 yields thus THEOREM 8. Let the notations and the assumption be as above. There exists a meromorphic function f in D with values in W, which satisfies the conditions 1 mi (EI) f(z) = (z - zi)
(E cia(z - zi)° + O((z - zi)n' ))
(i = 1, ... k)
°=o
if and only if the Hermitian matrix A is positive semidefinite. Such a function is unique if and only if A > 0 and det A = 0.
Takahashi
318
PART III. DOMAINS OF FINITE CONNECTIVITY
Let D be a bounded domain in the complex plane whose boundary 8D consists of m + 1 pairwise disjoint analytic simple closed curves yi (i = 0, 1, , m). In this part, we generalize the results of Abrahamse [1] on the Pick interpolation
problem in D, replacing this problem by our extended interpolation problem (EI) and introducing appropriate matrices. The proof proceeds as that of Abrahamse. We point out that this Part III gives another proof of the main theorem of the Part II in the unit disc.
§6. Preliminaries. We consider the harmonic measure dw on 8D for the domain D and for a fixed point z' E D. In terms of dw, we define the norm 1If llp (1 < p:5 oo) of complexvalued measurable functions f on 8D :
IlfliP= (f DIfIPdw)°
(1
Ilf Ilc = ess.sup if I
(w.r.t. dw),
aD
and we have the Banach spaces LP = LP(8D, dw).
Let A = {A = (A1,
, A,,,)
:
Ai E C, IAi I = 1 (i = 1,
,
m)} be the
m-torus. In order to clarify the basic branch of multiple-valued modulus automorphic functions in D, we consider as in Abrahamse [1] m pairwise disjoint analytic cuts bi (i = 1, , m), which staats from a point of yi and terminates at a point of ^to. The domain Do = D \ (U; ` 1 bi) is thus simply connected. For A = (A1,... , Am) E A, let HA(D) denote the set of complex-valued functions f in D such that f is holomorphic in Do and that, for each t E bi fl D,
f (z) tends to f (t) when z E Do tends to t from the left side of bi and f (z) tends to Ai f (t) when z E Do tends to t from the right side of bi. We can easily verify that if one miltiplies by Ai 1 the values of f on the right side of bi then the function thus modified is holomorphic at every point of bi fl D. We define the canonical function VA in HA(D) in the usual following way:
For each i = 1, , m, let vi be the harmonic function in the neighborhood of D = DUBD such that vi = 1 on yi and vi = 0 on the other ryi (j 9k i, 0 < j < m), and let vi be the conjugate harmonic function of vi in Do. For t E bi fl D (j = 0, , m), vi(t) stands for the limit of vi(z) when z E Do tends to t from the left side of bi. There exist m real numbers L1, , Sm such that VA(z) =
expX .i(vi(z) + i vi(z))) ic1
Takahashi
319
belongs to HA(D) (see Widom [16]). We see that VA can be continued analytically across the boundary 8D, in the usual sense except at the end points of the cuts b, and in the following sense at an end point t of b; : multiplying by A 1 the values of VA on the right side of bi, one can continue analytically across 8D the function
thus modified in the neighborhood of t. For a function f in D, f E HA(D) if and only if f V;-' is holomorphic in D. Clearly, I VA l is single-valued in D and can be extended to a continuous function in a neighborhood of D, which has no zeros there.
Let Ha denote the space of all functions f in HA(D) such that 1f J2 < u in D for some harmonic function u in D. It is known that any function f in Ha admits nontangential limits f *(t) at almost all t E 8D (w.r.t. dw). Via f F-+ f *, the space Ha can be viewed as a closed subspace of the Hilbert space L2. Thus Ha is a Hilbert space with the inner product
(f, g)=laD f*g`dw
(f,g E Ha).
From now on, no distinction will be made between a function f in Ha and its boundary function f* in L2. If A = (1, ,1) is the identity of A, then HA(D) is the space of holomorphic functions in D and Ha is the usual Hardy space H2 (D).
It is easy to see that, for A E A and C E D, the mapping f H f (C) is a bounded linear functional on H2, so that we have a unique kA< E H2 such that f(C) = (f, k,\<)
(for every f E Ha).
We write kA(z,() = kA((z)
and
kA( ,() = kAc.
The properties of kA in the following Lemma 1 are known (see Widom [16]).
LEMMA 1. ForAEA, zEDandCED, we have kA(z, () = (kA(, (), kA( , z)) = kA((, z) When A is fixed, kA(z, C) is holomorphic w.r.t. (z, Z) in Do x Do. It is continuous on A x Do x Do as a function of (A, z, C). The function IkA(z, ()I is single-valued and continuous on A x ((D x D) U (D x D)) with its appropriate boundary values.
320
Takahashi
LEMMA 2. Let A be fixed in A. For to E OD and (o E D, there exist a neighborhood U1 of to and a neighborhood of U2 of Co in D such that the function VA(z)-1 ka(z, () VA(() -1 can be extended to a function holomorphic w.r.t. (z, in U1 x U2.
Roughly speaking, Lemma 2 says that ka(z, () can be continued analytically across the boundary as a function of two variables (z, (). The presence of Va is only to simplify the statement concerning the cuts b;. This Lemma 2 seems to be well known, but we could not find it in an explicit form in the bibliographies, so that we shall give its proof later in §8.
Let a, R be nonnegative integers. For a holomorphic function f(z) we shall denote the a-th derivative off by f (a). For a function F(z, () holomorphic w.r.t. Oa+,e F
(z,
the notation F(a'fl) will stand for
8za
-
,
although this is a slight abuse
of the notation.
(z, () is well Let A be fixed in A. It is obvious that the derivative defined and holomorphic w.r.t. (z, in D° x D°. By Lemma 2, the function VA(z)-1 k(a'P)(z, () Va(() -1 can be extended to a function holomorphic w.r.t. (z, in a neighborhood of (D x D) U (D x D). Fort E bi and C E Do i k("6) (t, () is defined to be the limit of k("') (z, () when z E D° tends to t from the right side of bi, and so on. The function I k,, (z, () I can be considered as a function single-valued and continuous on (D x D) U (D x D). k(",6)
Though the following lemmas are valid for the points on the cuts, we shall restrict ourselves to D° in order to simplify the statement. This will be sufficient to apply them later.
LEMMA 3. For ( E Do, k("'16)(, () belongs to Ha fl L. We have
(f E Ha)
f(a)(() = (f, k(°'a)( ,()) and
kAa'P)(z,
() =
ka°'a)(
,
z))
(z E Do). a
In fact, the first assertion is obvious. Since ka°'a)(t, () = a(a ka(t, () is bounded on (8D \ U', j) x (Do fl D') for any relatively compact domain D' in D, we have ack
a(a f (() = & f D f (t)ka(t, () &v(t) = Cfe ka°'a)(, ())
Takahashi
321
and the second equality, replacing f by ka°'0)(, ).
We denote by H°°(D) the space of bounded holomorphic functions in D and we regard it as a closed subspace of L°°. LEMMA 4. Let f E HOO(D). Put F(z,() = f(z)ka(z,()f(()
and
Then F and G are holomorphic w.r.t. (z, to Ha for any ( E D°. We fave F(",16)(,, ()
G(z,() = ka(z,()f(().
in D° x D° and G(°",6)(, () belongs
= (G(°,a)( , (), G(°'")(, z))
(z E D°, ( E
D0)
In fact, we have F ",a)(z,C) µ=0 i'=0
)(C)
µ
and CG(°,M)(,
(), G(°'")(, z))
f("-v)(x)
(ka°'y)(,C),
µ µ=0 v=0
ka°,µ)( z)>
.
Lemma 3 shows Lemma 4.
Let Pa denote the orthogonal projection of L2 onto H. LEMMA 5. Let f E H°°(D) and put G(z, () = ka(z, () f ((). Then we have Pa(f k(°'")( , ()) = G(°'")( , () To prove it, let cp E H. By Lemma 3 we observe
(w, PA(fk(°'")(,C)))
fk(°'")(,C))
_ (fV, k(°'")(,C)) _ (f(a)(")(C) _ v=0
()f("-")(C) v=0
=,G(°'")(,C)),
ka°'°)(,C))
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322
which shows Lemma 5.
We shall denote by S1 the orthogonal complement of a subspace S in L2. LEMMA 6 (ABRAHAMSE [1]). Let w be an invertible function in L°°. If f is in (w H2(D))1 n L°° then there exist some A E A, g E Ha and h E (w H2)1 such that f = g-h
and
ill = IgI2 = Ih12
(a.e. on 8D).
LEMMA 7 (ABRAHAMSE [1]). The linear subspace (H2)1 n L°° is dense in
(H2)1.
§7. Main Theorem in Finitely Connected Domains. , zk be k distinct points in a domain D in C, bounded by m +
Let z1i z2,
1 disjoint analytic simple closed curves. For each zi, let cio, , ci,i _1 be a sequence of ni complex numbers. Our present extended interpolation problem (EI) is to find a holomorphic function f in D, satisfying If < 1 and the conditions n;-1 (EI)
.f(z) = E cia(z - zi)° + O((z - zi)n') a=0
(i = 1,...
k).
For each element A of the m-torus A, let ka be the kernel function introduced in the preceding section. We define the following matrices for i, j = 1, ,k:
Ci =
C [Cl
cio
C=
,
a
rid ) = M (ka; zi, zi; ni, ni) , =I',;)-Ci'Lij)'Cj,
A;;l A11
AkI
...
A('\)
Akb
In terms of these matrices Aa (A E A), we have
Takahashi
323
THEOREM 9 (EXTENSION OF THE THEOREM OF ABRAHAMSE). The prob-
lem (EI) admits a solution f with If I < 1 in D if and only if the matrix Aa is positive semidefinite for each A in A. The solution is unique if and only if the determinant of Aa is zero for some A E A.
PROOF. We may assume without loss zi E Do = D \ (U0 bi). Let via be complex numbers (i = 1, , k ; a = 0, , ni - 1) and set [[k ni -1
K =i=1La=0 E l;iakao'a)
(1)
zi)
By Lemma 3, we have k
n;-1 n,-1
IIKII2 =
faZja kaR)(zi, zj)
i,j=1 a=0 p=0
Let f E H00 (D). Put, for z E Do,( E Do,
F(z,() =. f(z)ka(z,()f(() By Lemma 5,
G(z,() = ka(z,()f(()
and
k ni-1
P,.(fK)
( , Sia G ( 0 , 01zi)
i=1 a=0
It follows from Lemma 4 that k
IIPa(7K)II2 =
ni-1 nj-1 6iaSj6F(a,9)(zi,zj)
i,j=1 a=0 9=0
On the other hand, we have IIPa(fK)1122 <- II7K1122
- IIfII00IIKll2
and hence k
n;-lni-1
(
`2)
i,j=1 a=0 8=0
t t
F
k(Aa,0)lzi,
Iloo
zj) -
F(a,p)(zi,
z j))
0.
Now, assume that f satisfies If I < 1 and the conditions (EI). Then, by the product formula (PF) in §1 and the definition of F, we see that kaa'I9)(zi,z,) -
Takahashi
324
F(a'.f) (zi, z,) is the (a+ 1, Q+1)-entry of the matrix of A; ). This implies A), > 0 for each A E A.
To prove the converse, assume AX > 0 for each A E A and take a polynomial O(z) which satisfies the conditions (EI). We may find such a polynomial by the method of indeterminate coefficients. Let w(z) be the polynomial w(z) _ (z - zl)"' (z - zk)"k. It is easy to see that the subspace wHa is the orthogonal complement in Ha of the subspace Ma spanned by the functions {kao'° ( zi)} (i = 1,... k ; a = Q,... , ni - 1). Thus we have
Ha=Ma ®wHa. Let f be in L°° fl (wH2(D))1. By Lemma 6, there exist some A E A, g E Ha and h E (w Ha)1 such that
f = gh
and
i l l = igI2 = Ih12
(a.e. on 8D).
The function K = P,\(h) is in Ma and hence it can be written in the form (1). Since A,, > 0, we have .fdw
IIKII2 by (2). Hence,
=II
ghd,
I
= I < K, c9 >I = I <
g >1 !5 11 PA
11 2119112
< IIKII2119112 < IIhuI2 II9II2 = Ill Iii.
By the theorem of Hahn-Banach, there exists a function 0 in L°° such that II0II.,, < 1 and that for each f E L°° fl (wH2(D))1 we have
LD
W.fdw=J
fdw.
8D
This shows, by virtue of Lemma 7 and the relations H2(D) = M,., wH2(D) and M.,, C HOO(D), where Al is the identity of A, that z/i - 0 is orthogonal to (wH2(D))1 in L2, that is, it belongs to wH2(D). Therefore, z/' is a solution of the problem (EI) with IikI < 1, which completes the proof of the first part of the theorem. To prove the uniqueness assertion, it suffices to follow the proof of Abrahamse [1], using instead of his Lemma 6 in [1] the following lemma which will be deduced immediately from Lemma 1 and Cauchy's integral formula. The details will not be carried out here.
LEMMA 8. Let (zo, (0) be in Do x Do. Let a and 6 be two nonnegative integers. Then the mapping A r-+ ka°`''si(zo, (o) is continuous on the m-torus A.
325
Takahashi
§8. Proof of Lemma 2 It is known that, for a fixed C, the function kA(z, () of z can be continued across the boundary 8D. The problem is to find, for a relatively compact neighborhood U2 of Co in D, a connected neighborhood U1 of to common to all C E U2 such that, multiplying if necessary by )`t 1 the values on the right side of the cut 8 , we may continue the function VA(z) to a function holomorphic and invertible in U1 and that, for any fixed C E U2, the function VA(z)-1 ka(z, () VA(() -1 of z may be extended to a function holomorphic in U1. If we find such a neighborhood U1, then it will follow from the theorem of Hartogs [8] that the function thus extended to U1 for each C E U2 is holomorphic w.r.t. (z, ) in U1 x U2, since the original function is holomorphic w.r.t. (z, Z) in (U1 fl D) x U2. This will complete the proof of Lemma 2. Now, we reduce by means of Va to the case without the periods A but with a slightly modified measure M
dµ(t) = exp
(3)
2Civi(t)) dw(t). i=1
The kernel function k(z,() of H2(D) w.r.t. dµ, which satisfies by definition
f(C) =
(4)
.
8D
f(t) k(t,() dp(t)
(f EH 2 (D)),
has the relation ka(z,() = VA(z) k(z,() VA(()
(see Widom [16]), so that it suffices to prove the Lemma 2 for k(z, C).
Let g(z, z*) be the Green function of D with its pole at the reference point z* and let g(z, z*) be its harmonic cunjugate. Put G(z) = g(z, z*) + i g(z, z*). Then we have dw(t) =
(5)
dt .
The function G' is single-valued and holomorphic in D except at the single pole z*. It can be continued analytically across the boundary 8D by virtue of the reflection principle. It has m zeros zi , , z,*n in D but it does not vanish on OD.
For f E H2(D) we have
f(C) =
2xi laD t (tC dt
(C E D)
Takahashi
326
(see Rudin [12]), so that (4) yields
(f EH 2 (D))
d t)) dt = 0
fD f (t) (27ri t 1 - k(t,
T his shows that there exists a unique N E H°°(D) such that (6)
N(t)
21ri t
- (t,() ddtt)
1
(t E OD)
(see Rudin [12]). The function
_
1
1
z-S
M(z)
- N(z)
is holomorphic in D except at the single pole C. is a constant # 0 on y;, we have by Assume to E yj. As (3), (5), and (6)
k(t, () = cj M(t) G'(t)-1
(t E 7,),
where ci is a constant # 0. The function Lj = cjMGi-1 is meromorphic in D and its poles are at most at ( , z , . . . , zn,. Since P = Lj + k( , () is real and Q = Lj - k( , () is purely imaginary on -y the functions k( , () and Lj can be cuntinued analytically across yi by the reflection principle as well as P and Q. Let U2 be a relatively compact neighborhood of Co in D. Since the z, are independent of C, we can find a neighborhood U1 of to, which is symmetric w.r.t.
yi and contains no z; or no points of U2. Then P and Q, and hence Li and k( , (), can be extended to holomorphic functions of z in U1 for any C E U2, which completes the proof of Lemma 2.
References [1] M. B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math. J. 26 (1979), 195-203. [2] L. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, McGraw-Hill, New York, 1973. [3]
C. Caratheodory, Uber den Variabilitiitsbereich der Fourier'schen Konstanten von positiven harmonischen F'unktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217.
[4]
J. P. Earl, A note on bounded interpolation in the unit disc, J. London Math. Soc. (2) 13 (1976), 419-423.
Takahashi
[5] [6]
[7]
327
S. D. Fisher, Function Theory on Planer Domains, Wiley, New York, 1983. P. It. Garabedian, Schwarz's lemma and the Szego kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhangiger Verii.nderlichen, insbesondere fiber die Darstellung derserben durch Reihen, welche nach Potenzen einer Veranderlichen fortschreiten, Math. Ann. 62 (1906), 1-88. [9] D. E. Marshall, An elementary proof of Pick-Nevanlinna interpolation theorem, Michigan Math. J. 21 (1974), 219-223. [10] R. Nevanlinna, Uber beschrankte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser A 32 (1929), No 7. [11] G. Pick, Uber die Beschriinkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23. [8]
[12] W. Rudin, Analytic functions of class Hp, Trans. Amer. Math. Soc. 78 (1955), 46-66.
1. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschriinkt sind, J. Reine Angew. Math. 147 (1917), 205-232. [14] S. Takahashi, Extension of the theorems of Caratheodory-Toeplitz-Schur and Pick, Pacific J. Math. 138 (1989), 391-399. [15] S. Takahashi, Nevanlinna parametrizations for the extended interpolation problem, Pacific J. Math. 146 (1990), 115-129. [16] H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3 (1969), 127-232. [13]
Department of Mathematics Nara Women's University Nara 630, Japan
MSC 1991: Primary 30E05, 30C40 Secondary 47A57
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
328
ACCRETIVE EXTENSIONS AND PROBLEMS ON THE STIELTJES OPERATOR-VALUED FUNCTIONS RELATIONS
E. R. Tsekanovskii
Dedicated to the memory of M. S. Brodskii, M. G. Krein, S. A. Orlov, V. P. Potapov and Yu. L. Shmul'yan This paper presents a survey of investigations in the theory of accretive extensions of positive operators and connection with the problem of realization of a Stieltjes-type
operator-valued function as a linear fractional transformation of the transfer operatorfunction of a conservative system. We give criteria of existence, together with some properties and a complete description of a positive operator.
In this paper a survey of investigations in the theory of accretive extensions of positive operators and connections with the problem of realization of a Stieltjes-type operator-valued function as a linear fractional transformation of the transfer operatorfunction of a conservative system is given. We give criteria of existence together with some properties and a complete description of the maximal accretive (m-accretive) nonselfadjoint extensions of a positive operator with dense domain in a Hilbert space. In the class of m-accretive extensions we specialize to the subclass of 9-sectorial extensions in the sense of T. Kato [24] (in S. G. Krein [29] terminology is regularly dissipative extensions of the positive operator), establish criteria for the existence of such extensions and give (in terms of parametric representations of 9-cosectorial contracitive extensions of Hermitian contraction) their complete description. It was an unexpected fact that if a positive operator B has a nonselfadjoint m-accretive extension T (B C T C B*) then the operator B always has an m-accretive extension which is not 0-sectorial for any 9 (0 E (0, 7r/2)). For Sturm-Liouville operators on the positive semi-axis there are obtained simple formulas which permit one (in terms of boundary parameter) to describe all accretive and 0-sectorial boundary value problems and to find an exact sectorial angle for the given value of the boundary parameter. All Stieltjes operator-functions generated by positive Sturm-Liouville operators are described. We obtain M. S. Livsic triangular models, for m-acretive extensions (with real spectrum) of the positive operators with finite and equal
Tsekanovskii
329
deficiency indices. In this paper there are also considered direct and inverse problems of the realization theory for Stieltjes operator-functions and their connection with 9-sectorial extensions of the positive operators in rigged Hilbert spaces. Formulas for canonical and generalized resolvents of 9-cosectorial contractive extensions of Hermitian contractions are given.
Note that Stieltjes functions have an interesting physical interpretation. As it was established by M. G. Krein any scalar Stieltjes function can be a coefficient of a dynamic pliability of a string (with some mass distribution on it). §1. ACCRETIVE AND SECTORIAL EXTENSIONS OF THE POSITIVE OPERATORS, OPERATORS OF THE CLASS C(9) AND THEIR PARAMETRIC REPRESENTATION. Let A be a Hermitian contraction defined on the subspace Z(A) of the Hilbert space 55.
DEFINITION. The operator S E [fj, 9j] ([55,55] is the set of all linear bounded operators acting in 55) is called a quasi-selfadjoint contractive extension (qsc-extension) of
the operator A if
SDA, S*JA,
IISII<<1
Z(S)_15. Self-adjoint contractive extensions (sc-extensions) of the Hermitian contraction were investigated, at first, by M. G. Krein [27], [28] in connection with the problem of
description and uniqueness of the positive selfadjoint extensions of the positive linear operator with dense domain. It was proved by M. G. Krein [27] that there exists two extreme sc-extensions A,, and AM of the Hermitian contraction A which are called "rigid" and "soft" sc-extensions of A respectively. Moreover, the set of all sc-extensions of the Hermitian contraction A consists of the operator segment [A,,, AM]. Let B be a positive closed Hermitian operator with dense domain acting on the Hilbert space 55. Then, as it is well known [27], an operator A = (I - B)(I + B)-' is a Hermitian contraction in $ defined on some subspace ;t)(A) of the Hilbert space 55. Let A,, and AM be "rigid" and "soft" sc-extensions of A. Then (see [27]) the operator B,, = (I - A,,)(I + Am)-1 is a positive self-adjoint extension of B and is in fact the extension obtained, at first, by K. Friedrichs in connection with his theorem that any positive operator with dense domain always has a positive self-adjoint extension. Besides, as it proved was in [27], the operator BM = (I - AM)(I + AM)-1 is also a positive self-adjoint extension of B. We tall the )perators By and BM the K. Friedrichs extension and the M. Krein extension respectively. DEFINITION. Let T be a closed linear operator with dense domain acting on the Hilbert space 55. We call T accretive if Re(Tf, f) > 0 (Vf E Z(T)) and m-accretive if it does not have accretive extensions.
T§ekanovskii
330
DEFINITION. We call the m-accretive operator T 0-sectorial if there exists 0 E (0,x/2) such that
ctg0llm(Tf,f)I <- Re(Tf,f)
(Vf E V(T)).
(1)
DEFINITION. An operator S E [15,15) is called a O-cosectorial contraction if there exists 0 E (0, x/2) such that
2ctgOIIm(Sf,f)I <_ IIfII2
- IISf1I2
(Vf E 15)
(2)
Note that inequality (2) is equivalent to
IIS±ictgOIII <
1+ctg20.
(3)
Denote by C(O) the set of contractions S E [b, 15] (0 E (0, x/2)) satisfying (2) (or (3)) and let C(x/2) be the set of all self-adjoint contractions acting on $. It is known [43] that if T is a O-sectorial operator, then the operator S = (I - T)(I + T)-1 is a 0-contraction, i.e. S belongs to C(O). The converse statement is also valid, i.e. S E C(O) and (I + S) is invertible, then the operator T = (I - S)(I + S)-' is a 9-sectorial operator. THEOREM 1. (E. R. Tsekanovskii [40], (41]) Let B be a positive linear closed
operator with dense domain acting on the Hilbert space $. Then the operator B has a nonselfadjoint m-accretive extension T (B C T C B*) if and only if the K. Fl-iedrichs extension Bµ and the M. Krein extension BM do not coincide, i.e. Bµ L BM. If Bµ # BM, then 1. for every fixed 0 E (0, x/2) the operator B has nonselfadjoint O-sectorial exten-
sion T(BCTCB*); 2. the operator B has a nonselfadjoint m-accretive extension T (B C T C B*) that fails to be 0-sectorial for all 0 E (0, x/2). The description of all 0-sectorial extensions of the positive operator B can be obtained via a linear fractional transformation with the help of the following theorem on parametric representation. THEOREM 2. (Yu. M. Arlinskii, E. R. Tsekanovskii [7], [8], (91) Let A be a Hermitian contraction defined on the subspace D(A) of the Hilbert space S5. Then the equality S = 2(AM + Aµ) + 2(AM - Aµ)' '2X (AM - Aµ)1 /2 (4)
hekanovskii
331
establishes one-to-one correspondence between qsc-extensions S of the Hermitian operator A and contractions X on the subspace 91o = 99(AM - A,,). The operator S in (4) is a 0cosectorial qsc-extension of the Hermitian contraction A if the operator X is a 0-cosectorial contraction on the subspace 91o.
Let A E [(A),55] be a Hermitian contraction, A* E [5,7)(A)] be the adjoint f- = (I - AA*)1/215 e 9 and 91 = $ e V(A). Let PA, Pte, PP be orthoprojectors onto V(A), 91, G respectively. We will define the contraction Z E [9,91] in the form
of the operator A. Denote G = (I -
Z(I - AA*)1121 = PPAf
(f E Z(A))
and let Z` E [91, 9] be the adjoint of the operator Z. THEOREM 3. Let A be a Hermitian contraction in 15 defined on the subspace V(A). Then the equalities
A,, = APA + (I - AA*)1/2(Z*Pn - Pc(I - AA* )1/2) AM = APA + (I - AA`)1/2(Z*Pt + Pr(I - AA*)' /2) are valid. The equality AA = AM holds if and only if C = {0}. Theorem 3 was established by Yu. M. Arlinskii and E. R. Tsekanovskii [8]. In terms of the operator-matrix "rigid" and "soft" extensions were established by Azizov [1] and independently by M. M. Malamud and V. Kolmanovich [25]. The M. Krein extension BM of the positive operator B with dense domain was described at first by T. Ando and K. Nishio [4], later by A. V. Shtraus [39]. In terms of the operator-valued Weyl functions [17] and space of boundary values operator BM was described by V. A. Derkach, M. M. Malamud and E. R. Tsekanovskii [18]. Contractive extensions of the given contrac-ion in terms of operator-matrices were investigated by C. Davis, W. Kahan and H. Wein)erger [16], M. Crandall [15], G. Arsene and A. Gheondea [5], H. Langer and B. Textorius [34], Yu. Shmul'yan and R. Yanovskaya [38]. Theorems 1 and 2 develop and reinforce investigations by K. Friedrichs, M. G. Krein, R. Phillips [27], [28], [37] and give the solution of the T. Kato problem [24] on the existence and description of 0-sectorial extensions of the positive operator with dense domain. Note that m-accretivity (0-sectoriality) of an operator T is equivalent to the fact that the solution of the Cauchy problem 1
+Tx = 0
dt 1 X(0) = xo
(xo E 7(T))
hekanovskii
332
generates a contractive semigroup (a semigroup analytically continued as a semigroup of contractions into a sector I it/2 - 9 of complex plane) [29]. Now, we consider some applications of Theorem 2 to the parametric representation of the 9-cosectorial qsc-extensions of a Hermitian contraction. Let S be a linear bounded operator with finite dimensional imaginary part acting on the Hilbert space $. Then as it is known [32] r
Im S =
(, 9.)7ap9p a,p=1
where J = [jop] is a self-adjoint and unitary matrix. Consider the matrix function V(A) given by
V(A) = [(Re S - .1I)-19a, 9p)]
THEOREM 4. In order that the linear bounded operator S with finite dimensional imaginary part be a contraction, it is necessary and, for simple' S, sufficient that the following conditions are fulfilled: 1) V(.\) is holomorphic in Ext[-1, 1]
2) the matrices V-'(-1) = (V(-1 - 0))-1, V-'(1) = (V(1 + 0))-', (V-'(-1) V-'(1))-'/2 exist and the matrix
Kj = (V-'(-1) -V-'(1))-1/2{2iJ+V-'(-1)+V-'(1)} X (V-'(-1) -
(5) V-1(1))-1/2
is a contraction.
The contraction S is 9-cosectorial (belongs to the class C(9), 9 E (0, it/2)) if Kj of the Arm (5) is a 9-cosectorial contraction (belongs to the class C(9), 9 E (0, 7r/2)). Moreover the exact value of the angle 9 is defined from the equation
IIKj ±ictgOIII2 = 1+ctg29. This theorem was obtained by E. R. Tsekanovskii [42], [43]. For the operator with one-dimensional imaginary part another proof was given by V. A. Derkach [20]. EXAMPLE. Let a(x) nondecreasing function on [0, e]. We consider the operator
(Sf)(x) = a(x)f (x) + i fe f(t) dt
(f E L2[0, e])
acting on the Hilbert space L2 [0, 2]. It is easy to see that
(In, Sf)(x) =
2
f l f (t) dt = (f, 9)9 0
(9(x) = 1/v)
(6)
'The operator S is called simple if there exists no reducing subspace on which one induces a self-adjoint operator.
Tsekanovskii
333
(Re Sf)(x) = a(x)f(x) + 2
f f(t) dt - f f(t) dt.
From simple calculations
t
2 f a(t)t
V(A) = ((ReS - UI)-19,9) = tg
.
(7)
(x E [0, e], f E L2[0, e])
(8)
Set a(x) _- 0 and consider the operator
(Sof)(x) = i
f
I f (t) dt
x
From (6) and (7) it follows that V(1) = - tg(e/2), V(-1) = tg(e/2). As J = I and an operator So is simple, we shall find all e > 0 for which So is a 9-cosectorial contraction. For this, in accordance with Theorem 4 (see relation (5)), the number
K=Kj=
2i + V-1(1) + V-1(-1)
V-1(-1) - V-1(1)
has to satisfy the inequality
2ctg9IImKI<1-IKI2. Exact value of 9 can be calculated from the formula
ctg9 =
1
- IKI2
2IImKI'
The operator So is a 9-cosectorial contraction if and only if 0 < e < 7r/2, moreover, an exact value of 9 is equal to a (0 = e). From this and the definition of the class C(9) (0 E (0,1r/2)), we obtain the inequality ctg e
f
2
t f (t) dt
< f t If(t)I2dt -
ft
2
dx 0
(de E [0, 7r/2], V f E L2[0, e] ).
Moreover, the constant ctge can not be increased so that for all f E L2 [0, e] as mentioned above, the inequality is valid. With the help of Theorem 4 there was established a full description of positive and sectorial boundary value problems for Sturm-Liouville operators on the semi-axis, at first (see also [22], [26]). Let J5 = L2[a, oo], 1(y) = -y" + q(x)y, where q(x) is a real locally aummable function. Assume that a minimal Hermitian operator
By = 1(y) = -y" + q(x)y (9)
y'(a) = y(a) = 0
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334
has deficiency indices (1, 1) (the case of limiting Weyl point). Let cok(x, A) (k = 1, 2) be the solutions of the Cauchy problems l(oo) = A 1 (a, A) = 0 (a, A) = 1
l(V2) = A 2 coi(a, A) _ -1 W2(a, A) = 0.
Then, as it is known [35], there exists Weyl function m,,.(A) such that w(x, A) = W2(x, A) + moo(A)Vi(x, A) E L2[a, oo].
Consider a boundary problem
Thy = l(y) = -y" + q(x)y 1 y'(a) = hy(a).
(10)
THEOREM S. (E. R. Tsekanovskii [42], [43]) 1. In order that the positive Sturm-Liouville operator of the form (9) have nonselfadjoint accretive extensions (boundary problems) of the form (10), it is necessary and sufficient that m,,.(-O) < oo. 2. The set of all accretive and 0-sectorial boundary value problems for Sturm-Liouville operators of the form (10) is defined by the parameter h, belonging to the domain indicated in (11). Moreover, as 1) h sweeps the real semi-axis in this domain, there results all accretive self-adjoint boundary value problems; 2) h sweeps all values not belonging to the straight line Re h = -mom(-0) and h # h, then there results all 0-sectorial boundary value problems (0 E (0, 7r/2)); moreover, the exact value of 0 is defined as it was pointed out in (11); 3) h sweeps all values with h # h and belonging to the straight line Re It = -mom(-0), then there results all accretive boundary value problems which are not 0-sectorial for any 0 E (0, 7r/2).
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335
Thus, Theorem 5 indicates which values of boundary parameter h correspond to the semigroup generated by the solution of the Cauchy problem +Thx=0
dt
(xo E i(Th))
X(0) = xo
in the space L2[0, oo], being contractive (Re h > -m,,(-0)), and for which h it can be analytically continued as a semigroup of contractions into a sector I arg (I < ir/2 - 9 of the complex plane. At the same time, it helps to calculate 9. Also the value of the parameter h in (11) (Re h > -m,(-0)) determines whether this semigroups of contractions can not be analytically continued as a semigroup of contractions into any sector I arg < e of the complex plane (Re h = -mom(-0)) (Im h # 0). Note that the M. Krein boundary value problem for the minimal positive operator B of the form (9) has the form (as it follows from (11))
BMy = -y" + q(x)y y'(a) + mo,(-0)y(a) = 0
(x E [a, oo])
and the K. Friedrichs boundary value problem, as is well known, coincides with Dirichlet problem
By = -y" + q(x)y S
a) = 0
(x E [a, oo]).
y(a)
`Consider a Sturm-Liouville operator with Bessel potential
By = -y" + 1
v2 - 1/4 x2
y
(x E [1, oo], v > 1/2)
y'(1) = y(1) = 0
in the Hilbert space L2[1, oo]. In this case the Weyl function has the form [35]
-v:
X)
iY (vfA-)
are Bessel functions of the first and second genus, m.(-0) = v.
where §2.
STIELTJES OPERATOR-VALUED FUNCTIONS AND THEIR RE-
ALIZATION.
Let B be a closed Hermitian operator acting on the Hilbert space 15, B* be the
adjoint of B, Z(B) = 15, 91(B) C 15o = Z(B). Denote Sj+ = Z(B*) and define in 15+ scalar product (f, g)+ = (f, g) + (B*.f, B`9)
(f, 9 E s5+)
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336
and build the rigged Hilbert space y1+ C 15 C fj_ [14]. We call an operator B regular, if an operator PB is a closed operator in 150 (P is an orthoprojector 15 to 15o) [6], [46]. We say that the closed linear operator T with dense domain in $ is a member of the class SZB, if
1) T 3 B, T* 3 B, where B is a regular closed Hermitian operator in f). 2) (-i) is a regular point of T. The condition that (-i) is a regular point in the definition of the class 1B is non-essential. It is sufficient to require the existence of some regular point for T. We call an operator A E [J5+,55+] a biextension of the regular Hermitian
operator B if A 3 B, A* 3 B. If A = A*, then A is called a selfadjoint biextension. Note that identifying the space conjugate to $ f with $:F gives that A* E [15+,J5-]. The
operator Bf = Af, where £(B) = If E b+ : Af E 15} is called the quasi-kernel of the operator A. We call selfadjoint biextension A strong if f3 = B* [45], [46]. An operator A E [f7+,S7_] is called a (*)-extension of the operator T in the class 11B if A 3 T 3 B, A* D T * D B where T (T*) is extension of B without exit of the space f5; moreover, A is called correct if AR = (A+ A*)/2 is a strong selfadjoint biextension of B. The operator T of the Class SZB will be associated with the class AB if
1) B is a maximal common Hermitian part of T and T*; 2) T has correct (*)-extension. The notion of biextension under the title "generalized extension", at first, was investigated by E. R. Tsekanovskii [45], [46]. (There the author obtained the existence, a parametric representation of (*)-extensions and self-adjoint biextensions of Hermitian operators with dense domain.) The case of biextensions of Hermitian operators with nondense domain was investigated by Yu. M. Arlinskii, Yu. L. Shmul'yan and the author [6], [45], [46]. Consider a Sturm-Liouville operator B (minimal, Hermitian) of the form (9) and an operator Th of the form (10). Operators Ay =
-y" + 4(x)y +
A`y =
-y" + 4(x)y +
h [hy(a) - y'(a)][µb(x - a) + b'(x - a)] (12) 1
h
[hy(a) - y'(a)][µb(x - a) + b'(x - a)]
for every µ E [-oo, +oo] define correct (*)-extension of Th (Th) and give a full description of these (*)-extensions. Note that b(x-a) and b'(x-a) are the b-function and its derivative respectively. Moreover,
(y, µb(x - a) + b'(x - a)) = µy(a) - y'(a)
(y E r7+).
337
Tsekanovskii
DEFINITION. The aggregate 0 = (A, r7+ C 17 C 9j_, K, J, £) is called a rigged operator colligation of the class AB if
1) J E [£, £] (£ is a Hilbert space), J = J = J-1; 2) K E [£,1j]; 3) A is a correct (*)-extension of the operator T of the class AB, moreover, (A A*)/2i = KJK*; 4) !l(K) = R(Im A) +,C, where C = Sj e 150 and closure is taken in fj_.
The operator-function We(z) = I- 2iK*(A- zI)-1KJ is called a M. S. Livsic characteristic operator-function of the colligation 0 and also M. S. Livsic characteristic operator-function of operator T. The operator colligation is called M. S. BrodskiiM. S. Livsic operator colligation. In the case when T is bounded, we obtain the well-known definition of the characteristic matrix-function [13], [32] (with M. S. Brodskii modification) introduced by M. S. Livsic [32]. The other definitions, generally speaking, of unbounded
operators were given by A. V. Kuzhel and A. V. Shtraus. For every M. S. BrodskiiM. S. Livsic operator colligation we define an operator-function
Ve(z) = K*(AR - zI)-1K. The operator-functions Ve(z) and We(z) are connected by relations Ve(z) = i[Wo(z) + I]-1[We(z) - I]J We(z) = V+ iVe(z)J]-1 [I - iVe(z)J].
(14)
(15)
The conservative system of the form
(A - zI)x = KJcp_
(16)
2iK*x
where x E f)+, Vf E £, V_ is an input vector, 'p+ is an output vector, x is the vector of inner state, will be associated with each operator colligation. It is easy to see that the transfer mapping of such a system S(z) ('p+ = S(z)cp_) coincides with W0(z). After D. Z. Arov [2], in case when J # I, we call the above mentioned system a passage system. When J = I, the above mentioned system is called a scattering system. Many problems for systems with distributed parameters, and a scattering problem as well, are packed into this scheme and were investigated by M. S. Livsic and W. Helton [23], [33]. The Dperator colligation 0 will be called accretive if Re(Af, f) > 0, V f E $+ and dissipative if
J = I. An operator-function V(z) E [E, E], where £ is a finite-dimensional Hilbert space, will be associated with the class S of the Stieltjes operator-functions if 1) V(z) is holomorphic in Ext[0, oo]; 2) Im V(z)/ Im z > 0;
3) Im[zV(z)]/Imz > 0. .t was established by M. G. Krein that any Stieltjes operator-function V(z) has an integral 'epresentation
V(z) ='Y+ ,j
dG(x)
(_t > 0)
0
where G(t) is a nondecreasing operator-function and f 000(t + 1)-1dG(t) < oo.
(17)
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338
DEFINITION. We call the operator-functions V(z) E S, acting on the Hilbert space C (dim £ < oo), realizable if, in some neighbourhood of (-i), V(z) can be represented in the form
V(z) = i[We(z) + I]-'[We(z) - IIJ
(18)
where We(z) is a characteristic operator-function for some rigged accretive and dissipative colligation of the class AB (We(z) is a transfer mapping of the some scattering system 9).
Thus, realization problem for Stieltjes operator-function is a problem on representation of this operator-function in the form of linear fractional transformation of the transfer mapping of some conservative scattering system (16), the main operator A of which is accretive. DEFINITION. The Stieltjes operator-function V(z) E [C, £] (dim £ < oo) will be said to be a member of the class S(R) of Stieltjes operator-functions, if for an operator y in (17) the equality
7f =0
(f E£.L)
is valid on the subspace £j = If E £ : f 00(dG(t) f, f )e < oo}. THEOREM 6. Let 0 be a rigged accretive colligation of the class AB (dim £ < oo). Then the operator-function Ve(z) of the form (14) belongs to the class S(R). Conversely, if V(z) acts on a finite-dimensional Hilbert space £ and belongs to the class S(R), then V(z) is realizable.
Thus, we specialize to the subclass S(R) in the class S which can be realized. In this case, when dim £ = 1, the operator-function
V(z)e = I y +
J
dG(z) I e
(e E £,
J
00
dG(t) < oo, -t > 0)
does not belong to the class S(R) and, therefore, is not realizable. We define three subclasses in the class S(R): 1) An operator-function V(z) of the class S(R) will be a member of So(R) if in the integral representation (17) j(dG(t)f, f )e = oo
(Vf E £, f
0).
2) An operator-function V(z) of the class S(R) will be a member of S1(R) if in the integral representation (17) j(dG(t)f, f )e < oo
(V f E C)
and?=0. 3) An operator-function V(z) of the class S(R) will be a member of So1(R) if £z 96£.
'I?;ekanovskii
339
THEOREM 7. Let 0 be an M. S. Brodskii-M. S. Livsic rigged accretive colligation of the class AB, where B is a positive operator with dense domain and dim C < oo. Then Ve(z) of the form (14) belongs to the class So(R) and X)(T) # V(T`). Conversely, assume V(z) E [£,£] (dim£ < oo) belongs to the class So(R). Then V(z) can be realized, moreover, B has dense domain and Z(T) # D(T*).
The direct statement in this theorem was established by V. A. Derkach and the author [19]. Theorem 7 belongs to I. N. Dovzhenko and E. R. Tsekanovskii [21]. The regular positive operator B acting on the Hilbert space 15 is called an R-operator [46], if its semideficiency indices (deficiency indices of PB) are equal to 0. THEOREM 8. Let O be an M. S. Brodskii-M. S. Livsic rigged accretive colligation of the class AB, where B is a positive R-operator with domain not dense. Then
Ve(z) of the form (14) belongs to the class SI(R) and D(T) = V(T`). Conversely, if V(z) E SI(R), V(z) E [£,£] (dim£ < oo), then V(z) can be realized, moreover, B is the positive regular R-operator with domain not dense and D(T) = D(T'). Note that the criteria of the equality V(T) = X)(T*) was established at first by A. V. Kuzhel in terms of characteristic matrix-functions introduced by himself [30], and the obtained results add to and make more precise these investigations for this class of operator-functions acting on a finite-dimensional Hilbert space. The analogous theorem may be formulated for operator-functions of the class Sol(R). Consider the following subclasses of the class So(R): 1) The operator-function V(z) E [£, £] (dim £ < oo) belongs to the class So (R) if V(z) E So(R) and
f 0
t
(dG(t)f, f )e = oo
(bf 54 0, f E £).
2) The operator-function V(z) E [£, £] (dim £ < oo) belongs to the class S01(R) if V(z) E So1(R) and (dG(t) f, f )e < oo
J
(t1 f E £).
t 3) The operator-function V(z) E [£,£] (dim£ < oo) belongs to the class SOM0 1(R) if V(z) E So(R) and for the subspace
£Z ={f E£:J"o1(dG(t)f,f)e
340
T§ekanovskii
THEOREM 9. Let 0 be an M. S. Brodskii-M. S. Livsic rigged accretive colligation of the class AB, dim £ < oo and let BM be the M. G. Krein extension of B being quasi-kernel for AR (AR J BM). Then an operator-function Ve(z) of the form (14) belongs to the class So (R). Conversely, if V(z) E [£, £] (dim £ < oo) belongs to the class So (R), then V(z) is realizable, moreover, AR includes BM as a quasi-kernel (AR D BM). The analogous theorems (direct and inverse) may be formulated for the classes S01(R) and Soo1(R). It may be established that operator-functions of the class So(R) can not be realized with the help of K. Friedrichs extension Bµ, when A D B.
DEFINITION. Let B be a positive operator in 55, and T be an m-accretive extension of B (B C T C B*). The operator T is called extremal if the operator X is unitary for the operator S = (I - T)(I + T)-1 in the parametric representation (4). As follows from (11) the Sturm-Liouville operator Th of the form (10) will always be extremal if the boundary parameter h lies on the straight line Re h = -mom(-0). Let 0 be a rigged M. S. Brodskii-M. S. Livsic operator colligation of the class AB, dim£ < oo and We(z) be its M. S. Livsic characteristic operator-function. Consider an operator-function
Q(z) = i[We'(-l)We(z) +I]-1[We1(-1)We(z) - I]J
(19)
where we assume that the right part (19) is defined. Let
KJ = [Q-1(-oo) - Q-1(-0)]-1/2{2iJ + Q-'(-oo) + Q-'(-0)} x [Q-1(-o0) - Q-1(-0)]-1/2
(20)
Recall, further, that operator-functions Ve(z) and We(z) are connected by relation (15). THEOREM 10. Let 0 be an M. S. Brodskii-M. S. Livsic rigged accretive colligation of the class AB, dim £ < oo, D(B) = S7 and let T be an a-sectorial (a E (0,x/2)) operator. Then an operator-function Ve(z) of the form (14) belongs to the class So(R) and an operator Kj of the form (20) is an a-cosectorial contraction. Conversely, if V(z) E [£, El
(dim£ < oo) belongs to the class S0(R), an operator KI of the form (19), (20) is an a-cosectorial contraction (a E (0, x/2)), then V(z) can be realized by the M. S. BrodskiiM. S. Livsic rigged accretive operator colligation of the class AB, dim£ < oo, D(B) = fj and T will be an a-sectorial operator. Theorems 8,9 were established by I. N. Dovzhenko and E. R. Tsekanovskii [21], Theorem 10 belongs to E. R. Tsekanovskii and is published at first. It may be shown that if 0 is a rigged accretive operator colligation of the class
AB, dim£ < oo, )(B) = $ and T is an extremal operator, then Ve(z) of the form (14) belongs to the class So(R), K1 of the form (19), (20) is unitary.., Conversely, if V(z) E [C, £]
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341
(dime < oo) belongs to the class S0(R), KI of the form (19), (20) is unitary, then V(z) can be realized by a rigged accretive operator colligation of the class AB, D(B) = 15 and T is an extremal operator. This fact was established by I. N. Dovzhenko and the author [21].
Consider an operator Th (Im h > 0) of the form (10) and let A be a correct (*)-extension of the operator Th. Then an operator A has the form (12) and
A - A' 2i
- (,
v)v,
(y, v) =
V =
(Im h)1/2 I µ - hl
(hn h) 12
Ip-hl
[,ub(x -a)+b'(x-a)] (y E 15+)
[µy(a) + y'(a)]
Let e= C1,K{c}=cv (c E C'). Then 0 = (A,15+ C 5CS7_,K,I,CI)will bean M. S. Brodskii-M. S. Livsic operator colligation. The M. S. Livsic characteristic function will be (21) We(z) - µ - h m,,.(z) + h
µ - h m,,.(z) + h' Characteristic function for a Sturm-Liouville operator of the form (10) without a phase factor (µ - h)/(µ - h) was investigated by B. S. Pavlov [36] in connection with scattering problems. We'll describe all phase factors in (21) under which Ve(z) = i[We(z) + I]-1[W9(z) - I] will be Stieltjes function (we assume, of course, that a minimal operator B of the form (9) is positive and m00(-0) < oo). The set of real p and non-real h satisfying the inequalities µ
(Im h)Z
mom(-0) + Re h +
Re h
(22)
Re h + m,.(-0) > 0 gives a complete description of all values of the parameters µ and h in (21), for which Ve(z) is a Stieltjes function. This fact was established by the author and published at first. As it was shown by I. N. Dovzhenko, the operator-function Ve(z) belongs to the class So if and only if the inequality (22) for parameter p holds with equality. Note that realization theory (and applications) for arbitrary rational matrixfunctions was developed by H. Bart, I. Gohberg, M. Kaashoek [12]. A realization theory for a very general class of transfer functions has recently been given by G. Weiss [47]. §3. M. S. LIVSIC TRIANGULAR MODEL OF THE M-ACCRETIVE EXTENSIONS (WITH REAL SPECTRUM) OF THE POSITIVE OPERATORS.
Let T be a nonselfadjoint m-accretive operator of the class AB, where B is a positive closed operator with dense domain having deficiency indices (r, r) (r < oo). Assume T has only real spectrum. We say that an operator T satisfying these conditions is in the class A,+. (B). As it is known [46], an operator T always has correct (*)-extensions.
Tsekanovskii
342
Let A be one of them. Let's include this extension A in the rigged operator colligation 0 for which a channel operator is invertible (it is always possible to do the same as in the case with the bounded operator) [13]. Let We(z) be the M. S. Livsic characteristic operatorfunction of this colligation (operator T). Then, as it was established by the author in [44], We(z) has the following regularized multiplicative representation
We(z) = t-.o0 lim
J
2i
exp 0
H*(x)11(x)J
a(x) - z
dx J (23)
E
x
o
J
exp 2i L
dx JWe ( ) a(x) _ JH*(x)H(x) J
where 1) a(x) is a non-decreasing function on [0, oo]; 2) H(x) E [£, £], dim £ = r, H(x) is invertible on a set of full measure, sp H*(x)ll(x) = 1; 3) if L(x) = fo H*(t)H(t) dt then limZ-.(L(x)W,V) = oo (V W # 0, cp E £). An operator T E A+r is called prime if clos. span {91, : z E p(T)} = fj
where p(T) is the set of regular points of the operator T, 91 is a defect subspace of B. The operator Tro = TI15o, where Sjo = clos. span 191- : z E p(T)} is called the prime part of T.
With the help of (23) consider in L' [0, oo] the operator
(Tf)(x) =a(x)f(x)+2iH*(x)JH(t)f(t)dt
(24)
where
Z(T)
f E LE[0, oo] : f f E L'[0, oo]}.
(25)
THEOREM 11. Let T E A,+. (B) and let T be a prime operator. Then T is unitarily equivalent to the prime part of the M. S. Livsic triangular model of the form (24), (25). The triangular model for an operator T E Ai (B) has the form
(T f)(x) = a(x)f(x) + 2i
J0
"o f (t) dt J
(J = ±I).
An operator T of the form (24), (25) is a-sectorial (prime part) if and only if an operator Kj of the form (19), (20) is an a-cosectorial contraction (a E (0, x/2)). An operator T (prime part) is extremal if Kj of the form (19), (20) is unitary.
This theorem was established by E. R. Tsekanovskii [44]. For the class of operators considered here, the model of the form (24), (25) is simpler than the model obtained earlier in another way by A. V. Kuzhel [30].
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343
In the case when T has a complete system of root subspaces, the factorization and model were obtained with the help of the same method by Yu. M. Arlinskii [3]. The resolvent of M. S. Livsic triangular model has the form
f(z) - 2i ((T - zI)-'f)(x) = a(x) -z
H(X)W *(X,2) J W(t'z)H*(t) f(t) dt
a(t) - z
a(x) - z
where
W(t, z) =
Jo
exp L-2i
II (t)H(t)i a] z
(
§4. CANONICAL AND GENERALIZED RESOLVENTS OF QSC-EXTENSIONS OF HERMITIAN CONTRACTIONS. Let A be a Hermitian contraction defined on the subspace D(A) of the Hilbert space 55. Let A,, and AM be rigid and soft sc-extensions of A. We consider the completely indeterminate case, i.e. we assume that ker(AM - A,) _ D(A). Let 55 C 55 and let S be a qsc-extension of Hermitian contraction A with exit in F) (A acts on 15). The operator-function Rz = P(S-zI)-' 115, where P is the orthoprojector from $ onto 55, is called a generalized resolvent. The resolvent, according to the definition, is canonical if fj = 55. Denote
Rµ = (A,, - zI)-1, R'' = (AM - zI)-1, +I)]I9,,
Q,,(z) = [(AM - Aa)1"2R=((AM - A,,)'"2
QM(z) = [(AM - A,)1/2RM((AM - A,,)1/2 + I)]I,,
91 = 55 e D(A).
Denote, also, 4(O) (O E [0, x/2)) the set of linear bounded operators Y acting on ¶11 and satisfying the condition
IIYf112+ctg01 Im(Yf,f)I
(f EOT).
The condition Y E s (O) is equivalent to the condition X = 2Y - I E C(O) (if 0 = 0 then 4i(0) is a class of non-negative operators in 01). Let A(O) be a set of points z in the complex plane satisfying the condition I sinO z ± i cos 01 < 1.
THEOREM 12. 1) If z E Ext A(O), then the equality RS = Rµ - R"(AM - A,,)1/2Y[I + (Q,(z)
- I)Y]-'(AM - A,,)1/2R"
establishes one-to-one correspondence between the set of canonical resolvents of qsc-exten-
sions S in the class C(O) (O E [0, x/2)) of the Hermitian contraction A and the constant operators Y of the class 1(0).
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344
2) If z E Ext A(6), then the equality
RS = R`ti + RM(AM - A,,)'/2Y[I - (QM(z) + I)Y]-'(Am - A,.)1/2RM establishes one-to-one correspondence between the set of canonical resolvents of qsc-exten-
sions S in the class C(6) (6 E [0, it/2)) of the Hermitian contraction A and the constant operators Y of the class 3) If z E Ext A(7r/2), then the equality
R= = R:" - Rz(AM - A,,)11'2Y(z)[I + (Q, (z) - I)Y(z)]-'(Am -
(26)
establishes one-to-one correspondence between the set of generalized resolvents in the class C(ir/2) of the Hermitian contraction A and the set of operator-functions Y(z) holomorphic
in Ext A(a/2) and in the class $(a/2). The constant operator-function Y(z) = Y in (26) corresponds to the canonical resolvents and only to them. This theorem was obtained by Yu. M. Arlinskii and E. R. Tsekanovskii [9], [10] and generalized some investigations by M. G. Krein and I. E. Ovcharenko [28] about generalized resolvents of sc-extensions of Hermitian contractions. Generalized resolvents of contractive extensions of an arbitrary contraction (not necessarily Hermitian) were investigated by H. Langer and B. Textorus [34]. The problem of describing generalized resolvents of qsc-extensions of Hermitian contractions not only of the class C(ir/2) (as in Theorem 12), but also of the class C(6) (6 E [0, it/2)) is open. Acknowledgements. The author is grateful to T. Ando, S. Belyi and the referee for helpful suggestions and aid.
REFERENCES 1.
2. 3.
4. 5.
6.
Azizov, T. Ya., On extension of positive operators, 17th Voronezh Winter School, Preprint VINITI, N4585, 1984, pp. 5-7. Arov, D. Z., Passive linear steady-state dynamical systems, Sibirsk. Mat. Zh. 20 (1979), 211-228. Arlinskii, Yu. M., A triangular model of an unbounded quasi-Hermitian operator with a complete system of root subspaces, Dokl. Akad. Nauk Ukrain. SSR Ser. A 11 (1979), 883-886. Ando, T., Nishio, K., Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J. (2) 22 (1970), 65-75. Arsene, G., Gheondea, A., Completing matrix contractions, J. Operator Theory 7 (1982), 179-189. Arlinskii, Yu. M., Tsekanovskii, E. It., The method of rigged spaces method in the theory of extensions of Hermitian operators with a non-dense domain, Sibirsk. Mat. Zh. 15 (1974), 243-261.
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7.
8.
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Arlinskii, Yu. M., Tsekanovskii, E. R., Nonselfadjoint contracting extensions of a Hermitian contraction and theorem of M. G. Krein, Uspekhi Mat. Nauk 37 no. 1 (1982), 131-1312. Arlinskii, Yu. M., Tsekanovskii, E. R., Sectorial extensions of positive Hermitian operators and their resolvents, Akad. Nauk Armyan. SSR Dokl. 79 no. 5 (1984), 199-203. Arlinskii, Yu. M., Tsekanovskii, E. R., Quasiselfadjoint contractive extensions of a Hermitian contraction, Teor. Funktsii Funktsional. Anal. i Prilozhen. 5Q. (1988), 9-16. Arlinskii, Yu. M., Tsekanovskii, E. R., On resolvents of m-accretive extensions of symmetric differential operator, Math. Phys. Nonlinear Mechanics 1 (1984), 11-16.
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Arlinskii, Yu. M., Tsekanovskii, E. R., Generalized resolvents of quasiselfadjoint contracting extensions of a Hermitian contraction, Ukrain. Mat. Zh. 35 (1983), 601-603. Bart, H., Gohberg, I., Kaashoek, M., Minimal factorization of matrix and operator-functions, Operator Theory: Advances and Applications, 1, Birkhauser, Basel, 1979.
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Brodskii, M. S., Triangular and Jordan representations of linear operators, Nauka, Moscow, 1969.
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Berezanskii, Yu. M., Expansions in eigen functions of selfadjoint operators, Naukova Dumka, Kiev, 1965. Crandall, M., Norm preserving extensions of linear transformations on Hilbert spaces, Proc. Amer. Math. Soc. 21 (1969), 335-340. Davis, C., Kahan, W., Weinberg, H., Norm-preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal. 19 (1982), 445-469. Derkach, V. A., Malamud, M. M., On the Weyl function and Hermite operators with lacunae, Dokl. Akad. Nauk SSSR 293 (1987), 1041-1046. Derkach, V. A., Malamud, M. M., Tsekanovskii, E. R., Sectorial extensions of
a positive operator, and the characteristic function, Dokl. Akad. Nauk SSSR 298 (1988), 537-541. Derkach, V. A., Tsekanovskii, E. R., Characteristic operator-functions of accretive operator colligations, Dokl. Akad. Nauk Ukrain. SSR 1986 no. 8, 16-19. Derkach, V. A., Tsekanovskii, E. R., On the characteristic function of a quasiHermitian contraction, Izv. Vyssh. Uchebn. Zaved. Mat. 1987 no. 6, 46-51. Dovzhenko, I. N., Tsekanovskii, E. R., Classes of Stieltjes operator functions and their conservative realization, Dokl. Akad. Nauk SSSR 311 (1990), 18-22. Evans, W. D., Knowles, I., On the extension problem for accretive differential operators, J. Funct. Anal. 63 (1985), 276-298.
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Helton, J. W., Systems with infinite-dimensional state space: the Filbert space approach, Proc. IEEE 64 (1976), 145-160. Kato, T., The perturbation theory for linear operators, Springer-Verlag, New York, 1966.
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Kolmanovich, V. Yu., Malamud, M. M., Extensions of sectorial operators and dual pairs of contractions, Preprint VINITI, N4428, 1985. Kochubey, A. N., Extensions of a positive definite symmetric operator, Dokl. Akad. Nauk Ukraine. SSR Ser. A 1979 no. 3, 168-171. Krein, M. G., Theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications, Mat. Sbornik 20 (1947), 431-495. Krein, M. G., Ovcharenko, I. E., Q-functions and sc-resolvents of non-densely defined Hermitian contractions, Sibirsk Mat. Zh. 18 (1977), 1032-1056. Krein, S. G., Linear differential equations in Banach space, Nauka, Moscow, 1967.
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Kuzhel, A. V., The reduction of unbounded non-selfadjoint operators to triangular form, Dokl. Akad. Nauk SSSR 119 (1958), 868-871. Kuzhel, A. V., Conditions for the equality DA = DA. for unbounded operators, Uspekhi Mat. Nauk 16 no. 3 (1961), 189-190. Livsic, M. S., On spectral decomposition of linear non-selfadjoint operators, Mat. Sbornik 34 (1954), 145-199. Livsic, M. S., Operators, oscillations, waves. Open systems, Nauka, Moscow, 1966.
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Langer, H., Textorius, B., Generalized resolvents of contractions, Acta Sci. Math. (Szeged) 44 (1982), 125-131. Naimark, M. A., Linear differential operators, Akademie-Verlag, Berlin, 1960. Pavlov, B. S., On a selfadjoint Schrodinger operator, Problems of Mathematical Physics, Leningrad University, Leningrad, 1966, pp. 102-132. Phillips, R., Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193-254. Shmul'yan, Yu. L., Yanovskaya, R. N., Blocks of a contractive operator matrix, Izv. Vyssh. Uchebn. Zaved. Mat. 1981 no. 7, 70-75. Shtraus, A. V., The extensions of semi-bounded operator, Dokl. Akad. Nauk SSSR 211 (1973), 543-546. Tsekanovskii, E. R., Nonselfadjoint accretive extensions of positive operators and the Friedrichs-Krein-Phillips theorems, Funktsional. Anal. i Prilozhen. 14 no. 2 (1980), 87-88. Tsekanovskii, E. R., The Friedrichs-Krein extensions of positive operators, and holomorphic semigroups of contractions, Funktsional. Anal. i Prilozhen. 15 no. 4 (1981), 91-92.
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Tsekanovskii, E. R., Characteristic function and description of accretive and sectorial boundary value problems for ordinary differential operators, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1985 no. 6, 21-24. Tsekanovskii, E. R., The characteristic function and sectorial boundary value problems, Trudy Inst. Mat. (Novosibirsk) 7, Issled. Georm. Mat. Anal., 1987, pp. 180-194. Tsekanovskii, E. R., Triangular models of unbounded accretive operators and the regular factorization of their characteristic operator functions, Dokl. Akad. Nauk SSSR 297 (1987), 552-556. Tsekanovskii, E. R., Generalized selfadjoint extensions of symmetric operators, Dokl. Akad. Nauk SSSR 178 (1968), 1267-1270. Tsekanovskii, E. R., Shmul'yan, Yu. L., The theory of biextensions of operators in rigged Filbert spaces. Unbounded operator colligations and characteristic functions, Uspekhi Mat. Nauk 32 no. 5 (1977), 69-124. Weiss, G., The representation of regular linear systems on Filbert spaces, Internat. Ser. Numer. Math., 91, Birkhauser, Basel, 1989, pp. 401-416.
Donetsk State University Universitetskaya 24 240055 Donetsk, Ukraine MSC 1991: Primary 47A20, Secondary 47A48, 47B25, 47B44
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
348
Commuting Nonselfadjoint Operators and Algebraic Curves
Victor Vinnikov As was discovered by M.S.Liv"sic, methods of algebraic geometry play an important role in the theory of commuting nonselfadjoint operators. Using further geometrical ideas, we construct triangular models for pairs of commuting nonselfadjoint operators with finitedimensional imaginary parts and a smooth discriminant curve. The characteristic function of a pair of commuting nonselfadjoint operators turns out to be a function on the discriminant curve, and the reduction of the pair of operators to the triangular model corresponds to the canonical factorization for semicontractive functions on a compact real Riemann surface.
1
Commuting Nonselfadjoint Operators and the Discriminant Curve
Our objective is an investigation, up to the unitary equivalence, of a pair A1, A2 of linear bounded commuting operators in a Hilbert space H (dim H = N < oo). We assume Al and A2 have finite-dimensional imaginary parts: dim G = n < oo, where G is the so-called nonhermitian subspace: G = (A1 - Ai)H + (A2 - A2)H. As was first discovered by Liv"sic [11,12], a certain real algebraic curve, the discriminant curve, plays a prominent role in all the investigations. The subspace G is clearly invariant under the operators Al-At, A2-A2, AlA2A2Ai, A;A1 - A1*A2. We can therefore (after choosing an orthonormal basis in G) define the following hermitian matrices of order n: 1
01= i(A1-Ai)I
G,
02 = a (A2 - Ai) I G, (A1A2
= z - A2Ai) I G, gout = 1(A2A1 - AiA2) I G ryln
1
We define a polynomial in two complex variables yi, y2 f (Y1, y2) = det(y102 - y2a1 + Itin)
(1.2)
349
Vinnikov
We assume f(yl,y2) 0 0. f(yl,y2) is a real polynomial of degree at most n. f(yl,y2) is called the discriminant polynomial of the pair A1, A2, and the real algebraic curve X of degree n in the complex projective plane, whose affine equation is f (yl, y2) = 0, is called
the discriminant curve. A (self-adjoint) determinantal representation of X is a n x n matrix of linear expressions in the affine coordinates Y1, Y2, with hermitian coefficients, whose determinant gives the equation of X. In particular we call Y1°2 - Y20`1 + _'n the input determinantal representation of X corresponding to the pair A1, A2. We have thus associated to the pair Al, A2 the discriminant curve and its input determinantal representation. The first evidence to the importance of the discriminant curve is given by the following generalization of the classical Cayley-Hamilton Theorem. Recall first [10] that the so-called principal subspace H = span {A;'Ak2(G)}kl k2-o = span {Ak'Ak2(G)}k,,k2=o
reduces Al and A2 and the restrictions of Al and A2 to the orthogonal complement He H are selfadjoint operators, so that it is enough to consider the restrictions of Al and A2 to H.
Theorem 1.1 (Livgic [11]) f (A1i A2) I H = 0. The joint spectrum of the operators A1, A2 is the set of all points A = (al, a2) E C2 such that there exists a sequence vm(m = 1, ...) of vectors of unit length in H satisfying
lim (Ak - AkI)Vm = 0 (k = 1,2)
(1.3)
(It has been shown by A.Markus that for a pair of commuting operators with finitedimensional imaginary parts this is equivalent to more general definitions of the joint spectrum due to Harte [4] and Taylor [16].) It follows that the joint spectrum of the operators A1, A2 (restricted to the principal subspace H) lies on the discriminant curve. We have obtained for the pair A1, A2 the following objects, which are clearly unitary invariants: the discriminant curve X; its input determinantal representation yla2Y2a1 + 71 n (up to the simulataneous conjugation of al, a2i ryln by a unitary matrix); and
the joint spectrum, which lies on X. We consider now the inverse problem. Suppose we are given a real projective plane curve X of degree n, a determinantal representation y1a2 - y20-1 + y of X, and a subset S of affine points of X, which is closed and bounded in C2, and all of whose accumulation points are real points of X (those conditions are always satisfied by the joint spectrum of a pair of commuting operators with finite-dimensional
imaginary parts (see [1])). We want to construct (up to the unitary equivalence on the pricipal subspace) all pairs of commuting operators with discriminant curve X, input determinantal representation yla2 - y2a1 +,y and joint spectrum S. We shall present a complete and explicit solution to this inverse problem under the assumption that the curve X is irreducible and smooth and possesses real points (the last is merely a technical condition). The solution will yield triangular models for pairs of commuting nonselfadjoint operators with finite-dimensional imaginary parts and a smooth $iscriminant curve. In the special case when one of the operators is dissipative (say a2 > 0)
Vinnikov
350
and the Hilbert space H is finite-dimensional, the inverse problem has been solved by Livsic in [11].
The main tools in the construction of triangular models are Livsic theory of commutative operator colligations [11] and our description of determinantal representations of real smooth plane curves [18]. The proof that every pair of commuting nonselfadjoint operators is unitarily equivalent to a triangular model is based on the factorization theorem for the characteristic function (see [12]) of the pair of operators. We prove it by showing that the characteristic function is in fact a function on the discriminant curve; this ties the theory of commuting nonselfadjoint operators with the function theory on a real Riemann surface. For more details and complete proofs see the forthcoming papers [19,20,21].
Y2°1 +Y
Before continuing we return to (1.1) and consider the polynomial det(y,o2 out )
Theorem 1.2 (Liv"sic [11])
y2°1 + yin) = det(y1cr2 - Y20'1 + -fout
We may say that the pair A,, A2 determines a "transformation" of the input determinantal representation ylo2 - y2a1 + yin of the discriminant curve into the output yout It will turn out that this transformation determinantal representation Y20r1 + can be recovered from the joint spectrum, and from the transformation one can recover the operators A1, A2 themselves.
2
Determinantal Representations of Real Plane Curves
We recall now briefly from [18] the description of determinantal representations of real smooth plane curves. See e.g. [3] for background algebro-geometrical details. Let X be a real projective plane curve of degree n. Two determinantal representations U = y1Q2 - Y20-i + y, U' = yia'2 - y2Qi + y' of X are called (hermitely) equivalent if there exists a complex matrix P E GL(n, C) such that U' = PUP`. We want to describe equivalence classes of determinantal representations of X. Let U be a determinantal representation of X. For each point x on X consider cokerU(x) = {v E (C")` : vU(x) = 0} (we write elements of C" as column n-vectors and elements of (C" as row n-vectors). It can be shown that if x is a regular point of X then dim coker U(x) = 1. Assume now X is a smooth curve. It follows that cokerU is a line bundle on X ; more precisely, we define coker U to be the subbundle of the trivial bundle of rank n over X, whose fiber at the point x is coker U(x). Clearly, if two determinantal representations U, U' of X are equivalent, then the corresponding line bundles coker U, coker U' are isomorphic. Conversely it turns out that if the line bundles corresponding to two determinantal representations of X are isomorphic, then the determinantal representations are equivalent up to sign. The description of determinantal representations has been thus reduced to the description of certain line bundles on X.
X is a compact Riemann surface of genus g, where g = (n - 1)(n - 2)/2. Choosing a canonical integral homology basis on X and the corresponding normalized basis
Vinnikov
351
for holomorphic differentials, we obtain the period lattice A in C9. The Jacobian variety J(X) = C9/A; it is a g-dimensional complex torus. The Abel-Jacobi map p associates to every line bundle L on X a point µ(L) in J(X). Furthermore the isomorphism class of L is determined by two invariants: the degree deg L of L, an integer, and the point µ(L) in
J(X). Some important geometrical properties of the line bundle can be expressed analytically in terms of the corresponding point in the Jacobian variety through the use of the Riemann's theta function 0(z). 0(z) is an entire function on C9 determined by the period lattice A. 0(z) is quasiperiodic with respect to A: when z is translated by a vector in A, 0(z) is multiplied by a non-zero number, so that we can talk about the zeroes of 0(z)
on J(X). It can be shown that if L = coker U, where U is a determinantal representation of X, then deg L = -n(n - 1)/2. One can determine necessary and sufficient conditions on a line bundle L of degree -n(n - 1)/2 to be the cokernel of a determinantal representation of X, and translating them into conditions on the corresponding point in the Jacobian variety yields
Theorem 2.1 (Vinnikov [18]) X possesses determinantal representations. There is a one-to-one correspondance between equivalence classes, up to sign, of determinantal rep-
resentations U of X and points ( of J(X) satisfying C + Z = e and 0(C) # 0.
The
correspondance is given by C = u(coker U(n - 2)) + ic.
The use of the twisted line bundle cokerU(n - 2) instead of cokerU and the translation of the point in J(X) by the so-called Riemann's constant k are technical details. e E C9 is a half-period (2e E A) explicitly determined by the topology of the set of real points XR, C X. Note that since X is a real curve, the period lattice A is invariant under complex conjugation, and the conjugation descends to J(X) = C9/A, so that the equation C + C = e makes sense there. We can also introduce an additional invariant, the sign of a determinantal representation U of X, which equals ±1 and distinguishes between the
representations U and -U. A complete study of the set of points in J(X) described in Theorem 2.1 appears
in [18]. This set is a disjoint union of certain g-dimensional "punctured" real torii, the "punctures" coming from the points C with 0(C) = 0. We consider here the simplest non-trivial example of real smooth cubics - n = 3, g = 1 (see [17]). A real smooth cubic X can be brought, by a real projective change of coordinates, to the normal form: Y2 - yi(y1 + 01)(Y2 + 02) = 0
(2.1)
(affine equation), where 01, 02 are two distinct numbers different from 0. We assume 01, 02 E R, so that the set XR of real points of X consists of two connected components. Since the genus g = 1, X is homeomorphic to a torus. This homeomorphism is given explicitly through the parametrization of X by elliptic functions:
yl = ' Y2 = p'(u)
92
B 1
3
(2.2)
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352
Here p(u) is the Weierstrass p-function with periods 1, r, the number r (3`r > 0) depending on 91i 02. Since 91, 02 E R, actually r E M. The point u varies in the period parallelogram, with vertices 0, 1, r,1 + r, say. A complete set of non-equivalent determinantal representations of X is given by
d+y2 q+yl
P
U=e -d+y2 a+yl q + yi
q=
01+92-e 2
0
,P=0102-
0
-1 01+02-e 01+02+3e 2
2
e=±1,eER,dEiR,d2=-e(-a+01)(-e+02) Note that (-e, -d) is an affine point on X. Choosing a suitable homology basis, the period lattice A of X in C is spanned
by 1, r, and J(X) = C/(Z + rZ) is simply the period parallelogram with opposite sides identified (and is isomorphic to X itself by (2.2)). The point ( corresponding to the representation U of (2.3) under the correspondance of Theorem 2.1 is (= v +'+, where v is the point in the period parallelogram corresponding to the point (-e, -d) on X under the parametrization (2.2). The condition 9(() # 0 gives v 0 0 (mod 1, r)
(2.4)
which is equivalent to (-e, -d) being an affine point on X, and (+ = 0 gives
v + v - 0 (mod 1, r)
(2.5)
which is equivalent to e E R,d E iR, in accordance with (2.3). So for smooth cubits Theorem 2.1 is the correspondance between non-equivalent determinantal representations (2.3) and points v in the period parallelogram satisfying (2.4)-(2.5). The set of such points consists of two connected components - a circle To and a punctured circle T1 (see Fig. 2.1). The sign of the representation (2.3) is E. It can be shown that a determinantal representation corresponds to a point v in To if and only if its coefficient matrices admit a positive definite real linear combination, i.e. after a suitable real projective change of coordinates we have, say, o2 > 0-
1/2
Figure 2.1: {v:v0- 0,v +1Y
1
0}=ToUT1
vnnikov
3
353
Commutative Operator Colligations
Operator colligations (or nodes) form a natural framework for the study of nonselfadjoint operators. We first recall (see [10]) the basic definition of a colligation for two operators. A colligation is a set
C=(A1,A2,H,4',E,al,a2)
(3.1)
where Al i A2 are linear bounded operators in a Hilbert space H, 4' is a linear bounded mapping from H to a Hilbert space E, and al, a2 are bounded selfadjoint operators in E, such that (3.2) 1(Ak - Ak) 4'`aA (k = 1,2)
We shall always assume that dim E = n < oo, while dim H = N < oo. We shall also assume that kera1 fl kera2 = 0. A colligation is called commutative if A1A2 = A2A1. A colligation is called strict if 4'(H) = E. If A1, A2 are commuting operators with finite-dimensional imaginary parts in a Hilbert space H, then (Ak - Ak) = PGakPG (k = 1, 2)
(3.3)
where G = (A1 - Ai)H + (A2 - A2)H is the nonhermitian subspace, PG is the orthogonal projection onto G, and ak = (Ak - Ak) I G (k = 1, 2). So the pair A1, A2 can always be embedded in a strict commutative colligation with E = G, 4' = PG. If C = (A1, A2, H, 4', E, al, a2) is a strict commutative colligation, there exist selfadjoint operators yin, gout in E such that 1(A1A2 - A2Ai) = 4,* 184,, (A2A1 - A*1A2) _
4.y utit
As evidenced in Section 1, the operators yin,7out play an important role, but the condition
4'(H) = E is too restrictive. The elementary objects - colligations with dimH = 1 are not strict when dim E > 1. We introduce therefore the notion of a regular colligation [8,11].
A commutative colligation C = (A1, A2, H, 4', E, a1, a2) is called regular if there
exist selfadjoint operators yin, -/out in E such that a14A2 - a24'Ai = -yln4,, a14'A2 - a24'A1 = -/out = yin + i(al$4''a2 - a24'4'a1) yout4p,
(3.5)
Actually, it is enough to require the existence of one of the operators yin, -/out,
the other one can then be defined by the last of the equations (3.4) and will satisfy the
Vinnikov
354
corresponding relation. Strict colligations are regular. For a strict colligation the operators Tin, lout are defined uniquely, but for a general regular colligation they are not, so we will include them in the notation of a regular colligation and write such a colligation as
C = (A,,A2,H,,,D,E,oi,
0'2,
,yin, ,Y out )
Regular commutative colligation turn out to be the proper object to study in the theory of commuting nonselfadjoint operators. As in Section 1 we define the discriminant polynomial of the regular colligation C f (y1, y2) = det(y102 - y20`1 + Tin)
(3.6)
and (assuming f (yl, y2) 0) the discriminant curve X determined by it in the complex projective plane. We have again the Cayley-Hamilton theorem
f(A1,A2) I H = 0
(3.7)
k2_o = span where H = span -o is the principal subspace of the colligation, so that the joint spectrum of the operators A1, A2 (restricted to the principal subspace) lies on the discriminant curve. Finally, det(yicr2 - Y2o1 + Tin) = det(yl62 - Y2°1 + ryout)
(3.8)
so that the discriminant curve comes equipped with the input and the output determinantal representations. We formulate now the inverse problem of Section 1 in the framework of regular
commutative colligations. We are given a real projective plane curve X of degree n, a of X, and a subset S of affine points of X, which determinantal representation yla2-y201+7 is closed and bounded in C2, and all of whose accumulation points are real points of X. We want to construct (up to the unitary equivalence on the principal subspace) all regular commutative colligations with discriminant curve X, input determinantal representation Y102 - Y201 + 7, and operators A1, A2 in the colligation having joint spectrum S (since Ql, 0'2,7 are given as n x n hermitian matrices we identify the space E in the colligation with C"). It is easily seen that the solutions of this problem that are strict colligations give the solution to the original problem of Section 1 (up to the equivalence of determinantal representations). Our solution of the inverse problem will be based on a "spectral synthesis", using the coupling procedure to produce more complicated colligations out of simpler ones.
Let C' _ (A', A', H1,111 El a, 7 0`2)
C" _ (A',', A2', H",'i", E, or,, Q2) be two colligations. We define their coupling
C = C'vC"=(A1,A2,H,4,E,al ,02)
355
Vinnikov
where
H Ak =
H1 ED H11,
(k = 1, 2), Ak' \ iV"+0k4' Ak O /
(the operators being written in the block form with respect to the orthogonal decomposition
H = H'(D H"). It is immediately seen that C is indeed a colligation. However, if C' and C" are commutative, C in general is not. Assume now C', C" are regular commutative colligations:
C= C"
=
(All', A121,
H") V17 E, al, Orz, '1' "in,Y/Iout
Theorem 3.1 (Livi;ic [11], Kravitsky [8]) The coupling in, out C=C'VC"=(A,,A2iH,I,E,ol,U2, 0'2,Y -Y
are given by (S. 8), yin = 7tin, gout = 7"out is a regular commutative colligation if and only if y'out = 7i,in where A,, A2i H,
This theorem illustrates aptly the meaning of the input and the output determinantal representations. Note that H" C H is a common invariant subspace of A1, A2. Conversely, if H" C H is a common invariant subspace of the operators A1, A2 in a regular commutative colligation C, we can write C = C' V C", where C', C" are regular commutative colligations called the projections of C onto the subspaces H = H e H", H" respectively [11,8].
4
Construction of Triangular Models: Dimensional Case
Finite-
We shall start the solution of the inverse problem for regular commutative colligations by investigating the simplest case when dim H = 1 and the joint spectrum consists of a single (non-real) point. Let X be a real smooth projective plane curve of degree n whose set of real points XR, # 0, and let y10`2 - y201 + y be a determinantal representation of X that has sign e and that corresponds, as in Theorem 2.1, to a point (in J(X). Let A = (A1, A2) be a non-real affine point on X. We identify the space H in the colligation with C, so that the operators A1, A2 in H are just multiplications by A,, A2, and the mapping 4 from H to E is multiplication by a vector 0 in Cn. We want to construct a regular commutative colligation C = (Al, A2, C, 0, Cn, 0i, Q2, It, Y)
(4.1)
Vinnikov
356
The colligation conditions (3.1) and the regularity conditions (3.4) are (X1a2 - A2Q1 + 7)¢ = 0
(4.2)
23`ak = O*ak, (k = 1, 2) 'r ='Y + i(al'4`0'2 - a20q'0`1)
(4.3)
(Aicr2 - A2o1 + 7)4' = 0
(4.5)
(4.4)
Let v E coker (A1Q2 - A20`1 + 7). It is easily seen that Al
V02V* = £'A2 V0'1v.
(4.6)
Therefore we can normalize v so that 0 = v' satisfies (4.3) if and only if 3`Ai v61v*
=
££A2
vo2v'
>0
(4.7)
In this case we define' by (4.4), and (4.5) follows. The one-point colligation (4.1) has thus been constructed. Note that we get at the output a new determinantal representation y10`2 - y2a1 + ry of X.
It is a fact of fundamental importance that the positivity condition (4.7) can be expressed analytically.
Theorem 4.1 The condition (4.7) is satisfied if and only if e
0 E a,a
B
> 0. In this case
the new determinantal representation y1Q2 - y2°1 + y defined by (4.2)-(4.4) has sign a and corresponds to the point S = S + A -A in J(X).
In the expressions like (+ A -Z we identify the point A on X with its image in J(X) under the embedding of the curve in its Jacobian variety given by the Abel-Jacobi map u. 9[(](w) is the theta function with characteristic (; it is an entire function on C9 associated to every point Sin J(X). 9[(](w) differs by an exponential factor from 9((+w). Therefore 9[(](0) # 0 always by Theorem 2.1; on the other hand, if the positivity condition of Theorem 4.1 is satisfied, 9(() = 9(( + A - Z) # 0, again in accordance with Theorem 2.1. Finally, E(x, y) is the prime form on X: it is a multiplicative differential on X of order -2, 2 in x, y, whose main property is that E(x, y) = 0 if and only if x = y. See [2] (C1((a 7 or [13] for all these. Note that each factor in the expression OEA,a is multi-valued, depending on the choice of lifting from J(X) to C9, but the expression itself turns out to B
be well-defined.
In the special case when, say, 02 > 0, XR divides X into two components X+ and X_ interchanged by the complex conjugation and whose affine points y = (y1, y2) satisfy ££ y2 > 0 and `aye < 0 respectively. The "weight" e o E Zx is positive on X+ and negative on X_, the sign e = 1 and the positivity condition of Theorem 4.1 becomes
AEX+or£A2>0(see [11]). As an example, let X be the real smooth cubic (2.1). Let y1a2 - Y2°1 + It be equivalent to the representation (2.3) of sign e corresponding to the point v in the period parallelogram (v 0 0, v + Ti - 0), and let the point A on X correspond to the point
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a in the period parallelogram under the parametrization (2.2). The region for a where the positivity condition of Theorem 4.1 is satisfied depends on the component of v (see Fig. 2.1). If v E To, the admissible region is always a half of the period parallelogram; if v E T1, the admissible region consists of two bands (or rather annuli) whose width depends on v; see Fig. 4.1-4.2 where the complementary admissible regions are depicted for E = 1, E = -1. If a is in the admissible region, the representation Y102 - y2°1 + y is equivalent to the representation of the form (2.3) corresponding to the point v - v + a - a in the period parallelogram. T
T
E=1
-v/2 + T
E=1
r/2
E=1
T/2
-v/2 + T/2
E=-1 1
1
1
Figure 4.1 : Admissible regions, v E To
Figure 4.2 : Admissible regions, v E T1
Using the coupling procedure we can solve now the inverse problem for regular commutative colligations with a finite-dimensional space H. Let X be a real smooth projective plane curve of degree n whose set of real points XR # 0, and let y1a2 - yza1 + 7 be a determinantal representation of X that has sign e and corresponds to a point (in J(X). Let 0) = (A(I'), A2'))(i = 1, .... N) be a finite sequence of non-real affine points on X. Assume that iO[( + E.=1(A(s) - T(i5)](A('+1) C
- Ti+1))
0[( + E'=JAW - A(i))](0)E(\(i+1) ,\(i+1))
> 0 (i = 0, ... , N - 1)
(4.8)
The conditions (4.8) turn out to be independent of the order of the points AM,_, A(^') If all the points are distinct, (4.8) can be rewritten, using Fay's addition theorem [2,13], in the matrix form
- A9)
E (_iok](A(i)
>0
(4.9)
We write down the system of recursive equations: (a1i)o2 - 2')Q1 + 7)W(') = 0,
7(i+l)
2s,\k') (k = 1, 2), = 7(') + i(110(1Yi).0'2
7(1) = 7
(4.10)
for i = 1,.. . , N. It follows from Theorem 4.1 that this system is solvable (uniquely up to multiplication of 00) by scalars of absolute value 1) and for each i y1o,2 - Y201 + 7(i) is a determinantal representation of X that. has sign e and corresponds to the point
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( + E'-'(AU) - T(T) in J(X). For each i C(') = (A('), A2'), C, 0('), Cn, a1, 627 7('), 7(i+1)) is a one-point (as in (4.1)) regular commutative colligation, and we can couple them by Theorem 3.1.
Theorem 4.2 Let A(')(i = 1,... , N) be a finite sequence of non-real affine points on X satisfying (4.8), and let 7('), 0(') be determined from (4.10). Then CN,,D, Cn, 0.1,
C = (A,, A2,
a2, 7, 7)
(4.11)
is a regular commutative colligation, where 0
Ak1)
x(2)
Ak
= ,i iO(N)"akO(2)
0(1)
_ (N+1)
...
...
0
...
0
...
iO(N)'ak,(N-1)
(k = 1, 2),
O(N))
(4.12)
The joint spectrum of Al i A2 is {.X(') };V 1, and the output determinantal representation y10'2 - y20`1 + ' of X has sign e and corresponds to the point ( in J(X), where N (4.13) i=1
We call the solution (4.11) of the inverse problem the triangular model with discriminant curve X, input determinantal representation Y10"2 -Y20`1 +7 and spectral data a(')(i = 1, ... , N). The reordering of the points ,X(N) leads to a unitary equivalent triangular model. Furthermore, the triangular model is the unique solution of the inverse problem.
Theorem 4.3 Let C = (A1i A2, H, C", a1, 0*2, 7, y) be a regular commutative colligation with dimH < oo and with smooth discriminant curve X that has real points. Let .(')(i = 1, ... , N) be the points of the joint spectrum of A1, A2 (restricted to the principal subspace H of C in H). Then A(') are non-real affine points of X satisfying (4.8) and C is unitarily equivalent (on the principal subspace H) to the triangular model with discriminant curve X, input determinantal representation y1a2 -Y2a1 +7 and spectral data A(')(i = 1, ... , N).
In the special case when one of the operators A1i A2 is dissipative, say a2 > 0, the conditions (4.8) reduce to a2') > 0(i = 1, ... , N) (see the comments following Theorem 4.1); Theorems 4.2-4.3 have been obtained in this case by Liv"sic [11]. The proof of Theorem 4.3 is based on the existence of a chain H = Ho D H1 3 3 Hnr_1 3 HN = 0 of common invariant subspaces of A1i A2 such that dim(Hi_1 eHi) = 1(i = 1,. .. , N) (simulataneous reduction to a triangular form; we assume for simplicity H = H). Projecting the colligation C onto the subspaces H,_1 a Hi, we represent C as the coupling of N one-point (as in (4.1)) colligations, which forces it to be unitary equivalent to the triangular model.
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The conditions (4.8) determine all possible input determinantal representations (if any) of a regular commutative colligation with the discriminant curve X and operators A1, A2 having the joint spectrum A(1),. .. , A(N) (on the principal subspace of the colligation). F o r example, let X be the real smooth cubic (2.1), and let At'), ... , A(N) be non-real affine points on X corresponding to the points a('),... , a(cv) in the period parallelogram under the parametrization (2.2). Assume m among those points lie in the upper half of
the period parallelogram: z < aai`l < &T, and k lie in the lower half: 0 <
z
(m + k = N). Let y1a2 - Y201 + y be the input determinantal representations of a regular commutative colligation with the discriminant curve X and operators A1, A2 having the joint spectrum X('), ... , )(N). y1Q2 - y2a1 + y is equivalent to the representation (2.3) of sign a corresponding to the point v in the period parallelogram (v 0, v + v - 0). We may take v E To (arbitrary) if and only if k = 0 (e = 1), or m = 0 (e = -1) (see Fig. 4.1). We may take v E T1 if and only if k
a(N) + 2m2
(e=1),or
-ail)
sa(N) +
2
-r <
r 2
M+1
2
(4.14)
(4.15)
(e = -1) (see Fig. 4.2). Since 0 < sv < sT, (4.14) implies that da(1) +... + Fia(N) >
2m
2k - 1
(4.16)
while (4.15) implies that
Ora(1) +... + aa(N) < m
1 Orr
(4.17)
If we have N = m + k (m, k # 0) points in the period2parallelogram that satisfy neither (4.16) nor (4.17), they can't be the joint spectrum of a pair of operators in a regular commutative colligation with the discriminant curve X. In the case of real smooth cubics one can also write down explicitly the solution of the system of recursive equations (4.10) and the corresponding matrices (4.12) using Weierstrass functions.
5
Construction of Triangular Models: General Case
The solution of the inverse problem for regular commutative colligations in the general (infinite-dimensional) case consists of the discrete part and the continuous part. As before we let X be a real smooth projective plane curve of degree n whose set of real points XR, # 0, and let Y1°2 - y2a1 + y be a determinantal representation of X that has sign e and corresponds to a point C in J(X). We start with the discrete part of the solution. Let ) (t) = (A('), a2'))(i = 1, ...) be an infinite sequence of non-real afire points on X that is bounded in C2 and all of
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whose accumulation points are in XR. As in (4.8), assume that
i9[( + F, ' =, (A(i) - T(i))](A('+1) - a(41)
> 0 (i = 01 ...) O[C + E =1(.\(i) - aii))l(o)E(A(i+1), V(;+1)) As in (4.10), we write down the system of recursive equations: 0,
($(1')a2 - . 2')01 ?(;+1)
2a4k') (k = 1, 2), = 7(i) + i(ai0('V')*a2 a2(b('Y')*a1),
-
7(1) = 7
for i = 1, .... It follows from Theorem 4.1 that this system is solvable (uniquely up to multiplication of O(i) by scalars of absolute value 1) and for each i Y1a2 - y20`1 + 7('} is
a determinantal representation of X that has sign e and corresponds to the point C + r,'='(A(i) - Ai)) in J(X). As in (4.12), we form infinite matrices: A(')
0
z¢(2)*ak(p(1)
.1
(2)
... ...
0
0
0
0
(k = 1, 2),
Ak= z j(`)*0k(2) ...
0(1)
...
!a(i)
... )
It turns out that A1i A2 are bounded linear operators in 12 and is a bounded linear mapping from 12 to C' (we write elements of 12 as infinite column vectors) if and only if ry = limi_., 7(i) exists. In this case C = (A1, A2,
12,,p, C", a1,
0`2, 7,')
(5.4)
is a regular commutative colligation. The joint spectrum of A1i A2 is {A(')}°O1.
(0) - )(;)) in Theorem 5.1 The limit y = lim;_ 7(') exists if and only if the series J(X) converges and O(( + E°°1(A(i) - T7))) # 0. In this case the determinantal representation y1a2 - y201 +7 of X has sign a and corresponds to the point (_ C + E 1(0) - a(;)) in J(X). In the special case 0`2 > 0, the conditions (5.1) reduce to X3'.42') > 0(i = 1,...) < 00. and the conditions of Theorem 5.1 are just We pass now to the continuous part of the solution to the inverse problem. Let c : [0,1] -i XR, be a function from some finite interval into the set of real affine points of X, such that c(t) = (c1(t), c2(t)), where c1(t), c2(t) are bounded almost everywhere continuous functions on [0, fl. We write down the following system of differential equations
361
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(Waksman [23], Liv"sic [12], Kravitsky [9]), which is the continuous analog of (4.10) and (5.2):
(cl(t)o2 - c2(t)(71 +7(t))O(t) = 0, (k = 1, 2), O(t)*oko(t) = Edyk(c(t)) w(c(t)) d-y
dt = i(ai (t)O(t)*o2 - 0'20(t)O(t)M0`1),
7(0) = 7
for 0 < t < 1. By a solution of this system we mean an absolutely continuous matrix function 7(t) on [0,1] and an almost everywhere continuous vector function ¢(t) on [0,1] such that (5.5) holds almost everywhere. w is a real differential on X, defined, analytic and non-zero in a neighbourhood of the set of left and right limit values of the function c : [0,1] --+ XR, whose signs on the different connected components of XR correspond to the real torus in J(X) to which the point c belongs; there is a version of the relation (4.6) for real points that shows that the required normalization of ¢(t) is always possible; see [18] for all these. A change of the differential w corresponds to a change of the parameter t.
Assume that the system (5.5) on the interval [0, 1] is solvable (uniquely almost
everywhere up to multiplication of 0(t) by a scalar function of absolute value 1). Then [23,12,9] for each t y10'2 - y2o1 + 7(t) is a determinantal representation of X. For f (t) E L2[0,1] define
(Akf)(t) = Ck(t)f(t) + i f t00(t)'okO(s)f (s)ds (k = 1, 2),
c f= f 1 fi(t) f (t)dt 0
A1, A2 are triangular integral operators on L2 [0,1] (continuous analogs of triangular ma-
trices) and r is a mapping from L2[0,1] to C. It turns out [23,12,9] that Al and A2 commute, and L C= (5.7) is a regular commutative colligation. The joint spectrum of A1, A2 is the set of left and right limit values of the function c : [0,1] --+ X.
Theorem 5.2 Let t
+E2
f0
W, C 9 WLs
ds
(5.8)
Wcs W e(a))
where w1i ... , w9 are the basis for holomorphic differentials on X that was chosen in the construction of the Jacobian variety. The system (5.5) is solvable on the interval [0,1] if and only if 9(C(t) # 0 for all t E [0,1]. In this case the determinantal representation Y10'2 - Y20'1 + 7(t) of X' has sign e and corresponds to the point ((t) in J(X).
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In the special case 0'2 > 0, the conditions of Theorem 5.1 are automatically satisfied, so the system (5.5) is always solvable; this was obtained by Livsic [12] for the case when the image of c consists of a single point. Theorem 5.2 gives not only an explicit condition for the solvability of the system of non-linear differential equations (5.5), it also shows that this system is linearized by passing from determinantal representations to the corresponding points in the Jacobian variety. The point ((t) given by (5.8) determines the equivalence class of the determinantal representation y1o2 - y20r1 + 7(t); one can go further and determine explicitly the representation y10r2 - Y20ri + 7(t) inside the equivalence class, i.e. integrate explicitly the system (5.5). We present the answer for the simplest case. Let X be the real smooth cubic (2.1) and let y102-y20r1+7 be the determinantal representation (2.3) of sign a corresponding to the point v in the period parallelogram (v 0 0, v + v - 0). Let c(t) = (C1, c2) for all t E [0,1], where (Cl, c2) is a real affine point on X corresponding to the point a in the period parallelogram under the parametrization (2.2). Assume for definiteness that e = 1 and sa = 0. As a real differential in (5.5) we may take w = -7 = -du (where u is the uniformization parameter (2.2)); note that as a basis for holomorphic differentials on X we take wl =f I v2 Let v(t) = v - it, and let et, dt, qt, pt be the numbers appearing in the determinant( representation (2.3) corresponding to the point v(t) (v(t) 0- 0). Then the solution of the system (5.5) is given by
pt+rt2(gt-lt)- a + z
7(t) _
-dt + rt(qt - lt) 2
qt- 2 - s rt =
(32)2(((v(t))
+
r2`
z dt-rt(gt-lt)+ a -nqt2et + ri rt
- ((v) - ip(a)t),st = -' 4ipt(a)t
-rt -1
(5.9)
Here ((u) is the Weierstrass (-function. If v E T1 (see Fig. 2.1), the system (5.5) is solvable on the interval [0, 1] if and only if l < i3v. If v E To, the system is solvable on any interval and the solution is quasiperiodic in the sense that Y10'2 - y20'1 + 7(t) and Y102 - y20r1 +'y(t + sr) are equivalent determinantal representations for any t (since v(t+2'-r) = v(t)). Of course, one can also write down explicitly, using Weierstrass functions, the vector function 0(t) and the commuting integral operators (5.6). We can solve now the inverse problem for regular commutative operator colligations in the general case by coupling (5.4) and (5.7). Let 0> = (A( `), ) (t))(i = 1, ... , N; N < oo) be a sequence of non-real affine points on X that is bounded in C2 and all of whose
accumulation points are in XR. Let c(t) = (cl(t),e2(t))(0 < t < 1;0 < I < oo) be real affine points on X, where cl(t), c2(t) are bounded almost everywhere continuous functions on [0, 1]; we order the connected components of XR,, choose a basepoint and an orientation on each one of them, and assume that c : [0,1] --+ Xg is continuous from the left everywhere, continuous at 0, and non-decreasing in the resulting order on XR. We call a('), c(t) the spectral data. Assume that the conditions (5.1) and the conditions of Theorems 5.1 - 5.2 are
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satisfied
ie[C + E'=,(A(i) - (i3)](\(i+1) - (i+1)) e
e(C + E;=1(A(i) - A(i))](0)E(0+1), 0+1))
>0 (z=0,...,N-1),
E(,\(') - A(')) converges , 9(C + E(P) - (`))) m,es N
t7
0,
i=1
i=1
C + E(\(`) - A(1)) + ei
It
c(s))
ds
# 0 (t E [0, 1])
(5.10)
es
i=1
a,cs
where w, w1i ... , wg are as before (if N < oo the second condition is not needed). Write down the system of recursive equations (5.2) followed by the system of differential equations (5.5) (A1(')a2
- As ')a1 +
0,
2a4') (k = 1, 2), = 7(') + i(010('Vi)r0'2 - 0'20('Y')`0`1), 7(1) = 7, i = 1, ... , N; 7(i+l)
(c1(t)0'z - c2(t)0`1 + 7)q5(t) = 0, Edyk(c(t)) (k
w(c(t)) d-y
dt
= 1,2),
= i(a1O(t)O(t)*a2 - 0`20(t)0(t)`al),
ry(0) = lim -YO), 0 < t < I
(5.11)
i- 00
(if N < oo, y(0) = 7(N+1)). The system of recursive equations is solvable by Theorem 4.1, lim4-.0 ry(') exists by Theorem 5.1, and the system of differential equations is solvable by Theorem 5.2.
Theorem 5.3 Let \(i)(i = 1, ... , N; N < oo), c(t) = (c1(t), c2(t))(0 < t < 1) be a spectral data satisfying (5.10), and let ry(i), V(i), ry(t), q5(t) be determined by (5.11). Then C = (A1, A2, H, 4,, C", al, 0`2, 7, 7)
(5.12)
is a regular commutative colligation, where H = 12 ® L2[0,1] and
(E -i
`fv
)00
(t)) _- ( i/(t)"akc(')vi + i Io 0(t)"ak/(s)f(s)ds + ck(t)f(t)) 4, (fi)) ,- O(')v, + 1' 0(t)f(t)dt, Ak
1
y = ry(1)
for v
1
(k =1, 2),
(5.13)
E 12, f(t) E L2[0,1] (if N < oo, replace 12 by CN and oo by N in the
above formulas). The joint spectrum of A1, A2 is {A(i)};_1 U {c(t)}tE(0,1, and the output
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364
determinantal representation y10`2 - Y20`1 + y of X has sign e and corresponds to the point t
in J(X), where N
E(P) - ,\(0) + ei J
w1 MW w(e(t))
dt
(5.14)
w ce w e t)
We call the solution (5.12) of the inverse problem the triangular model with discriminant curve X, input determinantal representation Y1012 - y2a1 +y and spectral data .1W, c(t). We can state now our main result.
Theorem 5.4 Let C = (Al, A2, H, C", a1, a2, y,y) be a regular commutative colligation with smooth discriminant curve X that has real points. Let S be the joint spectrum of Al, A2 (restricted to the principal subspace H of C in H). There exists a spectral data aW(i = 1,. .. , N; N < oo), c(t) = (cl(t), c2(t))(0 < t < 1) satisfying (5.10), such that S = {al')}N1 U {c(t)}tE[°,t) and C is unitarily equivalent (on its principal subspace H) to the triangular model with discriminant curve X, input determinantal representation Y1 2 - y20'1 + y and spectral data 0), c(t) (on its principal subspace). In the special case when one of the operators A1, A2 is dissipative, say a2 > 0, Theorem 5.4 has been obtained by Liv"sic [11] for dim H < oo, as we noted in the previous section, and by Waksman [23] for commuting Volterra operators (the joint spectrum S = (0, 0)) whose discriminant curve is a real smooth cubic. We can not prove Theorem 5.4 by imitating the proof of Theorem 4.3, since
we do not have, in the general case, enough direct information on common invariant subspaces of A1, A2. Therefore we shall adopt a function-theoretic approach. We shall associate to a regular commutative colligation its characteristic function. The coupling of colligations corresponds to the multiplication of characteristic functions, and the reduction of the colligation to the triangular model corresponds to the canonical factorization of its characteristic function. Since the characteristic function will turn out eventually to be a function on the discriminant curve, this will also tie the theory of commuting nonselfadjoint operators and the function theory on a real Riemann surface, much in the same way as the theory of a single nonselfadjoint (or nonunitary) operator is tied with the function theory on the upper half-plane (or on the unit disk) (see e.g. [14]).
6
Characteristic Functions and the Factorization Theorem
We first recall (see [10]) the basic definition of the characteristic function of an operator colligation.
Let C = (A1, A2i H, -11, E, al, a2, y, y) be a regular commutative colligation. The complete characteristic function of C is the operator function in E given by S(6, X2, Z) = I + i(S1a1 + e2a2).(41Ai + f2A; -
(6.1)
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where S1, 2;2, z E C. This function is a regular analytic function of b1, 1;2i z whenever z V spectrum (C1Ai + f2Ai). The following are the basic properties of the complete characteristic function.
Theorem 6.1 ([10]) Let a (finite-dimensional) space E and selfadjoint operators o'1, a2i y, y in E be given; assume that det(ei0`1 + e2a2) $ 0. Then the complete characteristic function S(6i ez, z) determines the corresponding regular commutative colligation up to the unitary equivalence on the principal subspace.
Theorem 6.2 ([10]) Let C = C' V C", where C',C",C are regular commutative colligations, and let S', S", S be the corresponding complete characteristic functions. Then S(61, 62, z) = S'(61, 62, z)S"(61, E2, Z)
-
For the one-point colligation (4.1) determined by a non-real affine point A _ (A1, A2) on the discriminant curve X S(C1, e2, z) = I + i(Sla1 + e2a2)
00"
e1A1+e23z-z
It follows from Theorem 6.2 and some limiting considerations that for the colligation (5.4) determined by an infinite sequence of non-real affine points A(') _ (al'), A2'))(i = 1, ...) on
X S(6 z, z) _
I + i(l;lal + l2o2)
0(i)
1 I
(6.3)
1') + 2A2') - z/ It can be also shown by standard techniques (see [1]) that for the colligation (5.7) deter'=1
mined by a function c : [0, 1] ---> XR. into the set of real affine points of X S(l;l, S2, Z) =
tt*
J exp (ziai + S2a2)6c1(t) (+)l2(c)2(t)
- z) dt
Let now X be a real smooth projective plane curve of degree n whose set of real points XR # 0, and let Y162 - Y261 +7, Y10`2 - y2a1 +,t' be two determinantal representations
of X. As in the previous sections we identify the space E in the colligation with C", so that the complete characteristic function is an n x n matrix function.
Theorem 6.3 Ann x n matrix function S(61, .z, z) is the complete characteristic function of a regular commutative colligation with discriminant curve X, input determinantal representation y1a2 - y2a1 + y and output determinantal representation y1a2 - y2a1 + y if and only if:
1) S(6i S2, z) has the form S(e1, e2, z) = I + i(Slal + S2a2)R(e1, S2, z)
(6.5)
where R(e1, e2, z) is holomorphic in the region K. = {(e1, e2, z) E C3: Izi > a(Ill I2 + 1e212)1'2} for
some a > 0, and R(tel,te2itz) = t-1R(e1,e2iz) for all t E C,t # 0.
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366
2) For any affine pointy= (Y1, Y2) on X, S(C1,1;2, Slyl +
maps L(y) = coker(y102 -
y20'1 +Y) into L(y) = coker (y10'2 - y201 + ry) and the restriction S(e1, S2, C1y1 + S2y2) I L(y) is independent of Sl, b2 ((Cl, S2, Slyl + S2y2) E Ka).
3) For any bl, e2 E R, S(el, S2, z) is a meromorphic function of z on the complement of the real axis and S(Sl, SZ)(60'1 + 5tt20'2)S(St, S2, z), S(Cl, y2,
z)(tt1b101
5101 + C20`2 (9z > 0),
+ 52a2)S(ttSl, S2, z)0 = 5101 + 52a2 (Osz = 0)
(6.6)
((6, 6, C1 Y1 + C2 Y2) E Ka).
The "only if" part of this Theorem, and the "if" part in the special case 02 > 0, have been obtained by Liv-sic [12]. It follows that if S(Gi e2, z) is the complete characteristic function of a regular
commutative colligation with discriminant curve X, input determinantal representation Y102 - y2a1 +Y and output determinantal representation y1a2 - y2a1 +7 ', we can define for each affine point y = (yl, y2) on X the mapping
S(y) = S(6, 6, 6Y1 + 6y2) I L(y) : L(y) - L(y)
(6.7)
We call the function S(y) of a point y on X the joint characteristic function of the colligation. It is a mapping of line bundles L, L on X, holomorphic outside the joint spectrum of the operators A;, A2 (restricted to the principal subspace of the colligation). Theorem 6.4 The joint characteristic function of a regular commutative colligation determines the complete characteristic function. In the special case a2 > 0 this has been obtained by Liv"sic [12]. Using (6.3)-(6.4) we see that Theorem 5.4 on the reduction to the triangular model is equivalent to the following: for every matrix function S2, z) satisfying the conditions of Theorem 6.3, there exists a spectral data A(`)(i = 1,... , N; N < oo), c(t) _ (cl(t), c2(t))(0 < t < 1) satisfying (5.10), such that y(l) = y and
t,
S(bl
tt b2 ,
+ i(5101 + 662)
z)
x
a
0
tt
b,A(') + ttb2A(')
i=1
exp
-z
(t)O(t)`
(i(iai + b202)`Slel(t + &2(t) - % dt) )
where y('), O('), y(t), 4(t) are determined by (5.11). Now, functions of several complex variables do not admit a good factorization theory. However, we see from Theorem 6.4 that the complete characteristic function reduces to the function on the one-dimensional complex manifold X. We shall therefore reduce (6.8) to the factorization theorem on a real Riemann surface. We first want to express the contractivity and isometricity properties (6.6) of the complete characteristic function in terms of the joint characteristic function. To this
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end we introduce a hermitian pairing between the fibers L(y(1)), L(y(2)) of the line bundle y(2) = L(y) = coker (y1v2 - Y20`1 + 'y) over non-conjugate affine points y(1) = (y(,l), Y(21))
(y(2), Y(22)) on X:
u(S1Q1 + S2 r2)y#
1
[u, vJ y(1),y(2) = 2 (1(TJ1
- y(2)) + S2(y21) - Y22))
(u E L(y(1)), v E L(y(2)); y(1)
y(2))
This is in fact independent (see (4.6)) of 6, 6 E R. In particular, taking y = y(1) = y(2) , we get an (indefinite) scalar product on the fiber L(y) over non-real affine points y on X. We also introduce a hermitian pairing between the fibers L(y), L(y) over conjugate affine points:
ayvM (u E L(y),v E L(-g)) (6.10) [u,v]yy = Zu((dyl + f This is again independent of 6, S2 E R, and we get in particular a scalar product on the
fiber L(y) over real affine points y on X (to get a value in (6.10) we have to choose, of course, a local parameter on X at the point y = (yl, y2)) Theorem 6.5 Let S((1i(2iz) be a matrix function satisfying the conditions 1)-2) of Theorem 6.3, and let the function S(y) be defined by (6.7). Then S((1i b2, z) satisfies (6.6) if and only if S(y) satisfies the following: for all affine points y, y(1), ... , y(N) on X in its region of analyticity (y(') # yW) G
([u(i)
u(i)] q,y(,))
=1,...,N
(u(`) E L(y(`)); i = 1, ... , N),
[uS(y), vS(y)]yy = [u, v]y-y (u E L(y), v E L(y))
(6.11)
Let now S(y) be the joint characteristic function of a regular commutative colligation with discriminant curve X, input determinantal representation Y1012 - Y2011 +y and output determinantal representation y10`2 - y201 +'Y, where the representations Y102 Y20'1 +Y, y1 0`2-y2a1+y have sign e (the input and the output determinantal representation
have always the same sign) and correspond, as in Theorem 2.1, to the points (, ( in J(X) (9(() # 0, 9(c) # 0, ( + ( = e, ( + ( = e). Since ( and ( are, up to a constant translation, the images of the line bundles L and L in the Jacobian variety under the Abel-Jacobi map µ, and S is a mapping of L to L, it follows that S can be identified, up to a constant factor of absolute value 1, with a (scalar) multivalued multiplicative function s(x) on X, with multipliers of absolute value 1 corresponding to the point (- (in J(X). More precisely, let A1, ... , A9) B1,. .., B9 be the chosen canonical integral homology basis on X, let Z be the g x g period matrix of J(X) (the period lattice A C C9 is spanned by the g vectors of the standard basis and the g columns of Z), and let (= b + Za, ( = b + Za, where a, b, a, b are vectors in R9 with entries a;, a;, b,, b, respectively; then the multipliers X, of s(x) over the basis cycle are given by Xs(Ai) = exp(-2iri(a1 - a,)) (i = 1, ... , g),
X3(Bi) = exp(27ri(b; - b,)) (i = 1,... , g)
(6.12)
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368
See e.g. [2] for more details. We call s(x) the normalized joint characteristic function of the colligation (since it arises from the joint characteristic function by choosing sections of L and L with certain normalized zeroes and poles). We have essentially seen in Theorem 4.1 that the pairing
(6.9) on the line bundle can be expressed analytically; the same is true of the pairing (6.10). We obtain thus from Theorem 6.5 a complete analytic description of normalized joint characteristic functions of regular commutative colligations.
Theorem 6.6 Let y1a2 - y20`1 + y be a determinantal representation of X that has sign e and corresponds to the point (in J(X), and let be another point in J(X), 6(() # 0, (+( _ e. A multivalued multiplicative function s(x) on X with multipliers of absolute value 1 corresponding to the point (- ( is the normalized joint characteristic function of a regular commutative colligation with discriminant curve X, input determinantal representation y1o2 - y2o1 + y and an output determinantal representation Y102 - Y201 + ry that has sign e and corresponds to the point ( in J(X) if and only if. 1) s(x) is holomorphic outside a compact subset of affine points of X. 2) s(x) is meromorphic on X \Xp,, and for all points x, x(1), ... , x(N) on X in its region of analyticity (x(') # x(A) 1
3(x(t)) S( x (.7))
i9[(](x(`)
T(7) e
8[(](0)E(x(t),x(i))
- T(T)
9[(](0)E(x(+),x(3))
s(x)s(-Z)
(6.13)
In the special case when one of the operators A1, A2 in the colligation is dissipative, say 0`2 > 0, the "weights" e t (o)E(xz) e[s)(o)E(xx) are positive on X+ and negative on X_ (see comments following Theorem 4.1), and it turns out that the matrix condition in (6.13) can be replaced by eJs(x)I < e (x E X+) (6.14) We conjecture that in general the matrix condition is equivalent to e
2 < e 2B[(](x - x) (x E X \ XR) 9[(](0)E(x, T) - 6[(](0)E(x, T)
( 6 . 15 )
Let now X be a compact real Riemann surface (i.e. a compact Riemann surface with an antiholomorphic involution x i-+ 7; for example, a real smooth projective plane curve). Let XR be the set of fixed points of the involution; assume XR, 0. Let (, be two points in J(X), O(() # 0, 9(c) 0, ( + e (the half-period e of e, ( + Theorem 2.1 is defined for every real Riemann surface). A multivalued multiplicative function s(x) on X with multipliers of absolute value 1 corresponding to the point ( - ( is called semicontractive, or, specifically, ((, (')-contractive, if it is meromorphic on X\XR, and for all points x, X(I),. .., x(N) on X\XR (x(') # x(.i)):
(sj x
s x(i)
s(x)s(Y) = 1
28[(](x(`) - x(1) 1 x(i))
< (_iO[(](x(') - x(i)) e[(](o)E(x('), x(is) (6.16)
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Theorem 6.6 states that normalized joint characteristic functions of regular commutative colligations with a smooth discriminant curve (that has real points) are precisely semicontractive functions on the discriminant curve (for sign e = 1) and their inverses (for sign e = -1). The factorization (6.8) of the complete characteristic function follows from the following factorization theorem for semicontractive functions on the real Riemann surface X.
Theorem 6.7 Let s(x) be a ((,()-contractive function on X. Then s(x)
(exp (7rim(')t(A(') +
)
+
t=1
x exp
- 7rk-1 E i=O
!m(t)iHm(t)
2
1
/
exp
(- 27r(a () -
(
)Yx
r
Ixk
nav(y) - 27ri J xR
w(y)
W (y)
+ i r dvlnE(x,y)dv(y) 1
E(x, (i))
l
Yxdv(y) (6.17)
I
JxR
,
w(y)
Here .\(')(i = 1, ... , N; N < oo) are the zeroes of s(x) on X \XR and v is a uniquely determined finite positive Borel measure on XR; wl, ... , w9 are the chosen basis for holomorphic differentials on X; w is a real differential on X, defined, analytic and non-zero in a neighbourhood of supp v C XR, whose signs on different connected components Xo,... , Xk_1 of XR correspond to the real torus in J(X) to which the points (, ( belong [18]; Z = (H, Y real) is the g x g period matrix of J(X); m(')(i = 1, ... , N), ni(i = 0, ... , k - 1) are integral vectors depending on the choice of lifting of the points \(t) and the components Xi respectively from J(X) = C9/A to C9. Furthermore, the following hold: sH+iY-1
iO[( + E)=,(A(i) - T(i))]('\(i+1) -
9[( +E =1(1i) -
o (i = o, ... , N - 1), aT)1(o)E(A(i+1),
00
0-0
E('\(')
converges , 6(( +
i=1
E("'(i)
i=1
- ('))) T 0 (if N =
oo),
v W(v)
N
+ E(A(i) - ,)) + i Je i=1
dv(y)
# 0 (for all Borel sets B C XR),
wy W(v)1
N
E(P) - a(i)) + i J i=1
W (Y)
xR
(:)
dv(y)
(6.18)
W.
When X is a real smooth projective plane curve (and s(x) is holomorphic outside a compact subset of affine points of X), the two factors in (6.16) are the normalized joint characteristic functions of the colligations (5.4) and (5.7) respectively (c : [0,1] --+ XR is the left-continuous non-decreasing function determined by v(B) = m(c 1(B)) for Borel sets B C XR, where m is the Lebesgue measure on [0,1], 1 = v(XR)). Decomposing the measure v into singular and absolutely continuous parts (with respect to the measures induced on XR by the usual Lebesgue measure through local coordinates), we obtain the
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factorization of a semicontractive function into a Blaschke product, a singular inner function and an outer function, generalizing the Riesz-Nevanlinna factorization for bounded analytic functions in the unit disk (see e.g. [7]). Our factorization is better compared though to Potapov factorization for J-contractive matrix functions (see [15]), since the weights e { 0 E x i e[t](o)E(zzl are not, in general, positive or negative everywhere. In the special case of (6.14), the Blaschke product - singular inner facter - outer factor decomposition was known ([22,5,6]), without, however, explicit formulas for the factors in terms of the prime form E(x, y). It is my pleasure to thank Prof. M.S.Livsic for many deep and interesting discussions.
References [1] Brodskii,M.S., Livsic,M.S.: Spectral analysis of nonselfadjoint operators and intermediate systems, AMS Transl. (2) 13, 265-346 (1960). [2] Fay,J.D.: Theta Functions on Riemann Surfaces, Springer-Verlag, Heidelberg (1973). [3] Griffiths,P., Harris,J.: Principles of Algebraic Geometry, Wiley, New York (1978). [4] Harte,R.E.: Spectral mapping theorems, Proc. Roy. Irish Acad. (A) 72, 89-107 (1972).
Invariant subspace theorems for finite Riemann surfaces, Canad. J. [5] Hasumi,M. Math. 18, 240-255 (1986). [6] Hasumi,M. : Hardy Classes on Infinitely Connected Riemann Surfaces, SpringerVerlag, Heidelberg (1983).
[7] Hoffman,K.: Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, NJ (1962). [8] Kravitsky,N.: Regular colligations for several commuting operators in Banach space, Int. Eq. Oper. Th. 6, 224-249 (1983). [9] Kravitsky,N.: On commuting integral operators, Topics in Operator Theory, Systems and Networks (Dym,H., Gohberg,I., Eds.), Birkhauser, Boston (1984). [10] Livsic,M.S., Jancevich,A.A.: Theory of Operator Colligations in Hilbert Space, Wiley, New York (1979).
[11] Livsic,M.S.: Cayley-Hamilton theorem, vector bundles and divisors of commuting operators, Int. Eq. Oper. Th. 6, 250-273 (1983). [12] Livsic,M.S.: Commuting nonselfadjoint operators and mappings of vector bundles on algebraic curves, Operator Theory and Systems (Bart,H., Gohberg,I., Kaashoek,M.A., Eds.), Birkhauser, Boston (1986). [13] Mumford,D.: Tata Lectures on Theta, Birkhauser, Boston (Vol. 1, 1983; Vol. 2, 1984). [14] Nikolskii,N.K. : Treatise on the Shift Operator, Springer-Verlag, Heidelberg (1986).
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[15] Potapov,V.P.: The mulptiplicative structure of J-contractive matrix functions, AMS Transl. (2) 15, 131-243 (1960). [16] Taylor,J.L.: A joint spectrum for several commuting operators, J. of Funct. Anal. 6, 172-191 (1970).
[17] Vinnikov,V.: Self-adjoint determinantal representations of real irreducible cubics, Operator Theory and Systems (Bart,H., Gohberg,I., Kaashoek,M.A., Eds.), Birkhauser, Boston (1986).
[18] Vinnikov,V.: Self-adjoint determinantal representions of real plane curves, preprint. [19] Vinnikov,V.: Triangular models for commuting nonselfadjoint operators, in preparation. [20] Vinnikov,V.
preparation.
:
Characteristic functions of commuting nonselfadjoint operators, in
[21] Vinnikov,V.: The factorization theorem on a compact real Riemann surface, in preparation. [22] Voichick,M., Zalcman,L. : Inner and outer functions on Riemann surfaces, Proc. Amer. Math. Soc. 16, 1200-1204 (1965).
[23] Waksman,L. : Harmonic analysis of multi-parameter semigroups of contractions, Commuting Nonselfadjoint Operators in Hilbert space (Livs'ic,M.S., Waksman,L.), Springer-Verlag, Heidelberg (1987).
DEPARTEMENT OF THEORETICAL MATHEMATICS, WEIZMANN INSTITUTE OF SCIENCE, REHOVOT 76100, ISRAEL
E-mail address: [email protected]
1980 Mathematics Subject Classification (1985 Revision). Primary 47A45, 30D50; Secondary 14H45, 14H40, 14K20, 14K25, 30F15.
Operator Theory: Advances and Applications, Vol. 59 © 1992 Birkhauser Verlag Basel
372
ALL (?) ABOUT QUASINORMAL OPERATORS
Pei Yuan Wul) Dedicated to the memory of Domingo A. Herrero (1941-1991)
A bounded linear operator T on a complex separable Hilbert space is quasinormal if T and T T commute. In this article, we survey all (?) the known results concerning this class of operators with more emphasis on recent progresses. We will consider their various representations, spectral property, multiplicity, characterizations among weighted shifts, Toeplitz operators and composition operators, invariant subspace structure, double commutant property, commutant lifting property, similarity, quasisimilarity and compact perturbation, and end with some speculations on possible directions for further research.
1. INTRODUCTION
The class of quasinormal operators was first introduced and studied by A. Brown [4] in 1953. From the definition, it is easily seen that this class contains normal
operators (TT = T T) and isometries (T T = I). On the other hand, it can be shown [36, Problem 195] that any quasinormal operator is subnormal, that is, it has a normal extension. Normal operators and isometrics are classical objects : Their properties have been fully explored and their structures well-understood. It has also been widely recognized that subnormality constitutes a deep and useful generalization of normality.
After two-decades' intensive study by various operator theorists, the theory of subnormal operators has matured to the extent that two monographs [17, 18] have appeared which are devoted to its codification. People may come to suspect whether the in-between quasinormal operators would be of any interest to merit a separate survey paper like this one. The structure of quasinormal operators is, as we shall see below,
')This research was partially supported by the National Science Council of the Republic of China.
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indeed very simple. They are certainly not in the same league as their big brothers : Their theory is not as basic as those of normal operators and isometries and also not as deep as subnormal ones. However, we will report in subsequent discussions some recent progresses in the theory of quasinormality which serve to justify the worthwhileness of
our effort. One recent result (on the similarity of two quasinormal operators) establishes a connection between the theories of quasinormal operators and nest algebras. Another one (on their quasisimilarity) uses a great deal of the analytic function theory. These clearly show that there are indeed many interesting questions which can be asked about this class of operators. It used to be the case that the study of quasinormal operators was pursued as a step toward a better understanding of the subnormal ones. The recent healthy developments indicate that quasinormal operators may have an independent identity and deserve to be studied for their own sake.
The interpretation of our title "ALL (?) ABOUT QUASINORMAL OPERATORS" follows the same spirit as that of Domingo Herrero's paper [391 : The
"ALL" is interpreted as "all the author knows about the subject", and the question mark "?" means that weever really know "all" about any given subject. The paper is organized as follows. We start in Section 2 with three representations of quasinormal operators. One of them is the canonical representation on which all the theory is built. Section 3 discusses the (essential) spectrum, various parts thereof, (essential) norm and multiplicity. Section 4 gives characterizations of quasinormality among several special classes of operators, namely, weighted shifts, Toeplitz operators and composition operators. Section 5 then treats various properties related to the invariant subspaces of an operator such as reflexivity, decomposability, (bi)quasitriangularity and cellular-indecomposability. The three operator algebras
{T}', {T}" and Alg T of a pure quasinormal operator T are described in Section 6. Then we proceed to consider properties relating a quasinormal operator to operators in its commutant. One such property concerns their lifting to its minimal normal extension. We also consider the quasinormal extension for subnormal operators as developed by Embry-Wardrop. Sections 7 and 8 are on the similarity and quasisimilarity of two quasinormal operators. Section 9 discusses the problems when two quasinormal operators are approximately equivlaent, compact perturbations and algebraically equivalent to each other. We conclude in Section 10 with some open problems which seem to be worthy of exploring.
This paper is an expanded version of the talk given in the WOTCA at Hokkaido University. We would like to thank Professor T. Ando, the organizer, for his
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invitation to present this talk and for his efforts in organizing the conference. 2. REPRESENTATIONS
We start with the canonical representation for quasinormal operators first obtained by A. Brown [4]. This representation is the foundation for all the subsequent developments of the theory. THEOREM 2.1. An operator T on Hilbert space H is quasinormal if and only if T is unitarily equivalent to an operator of the form
where N is normal and A is positive semidefinite. If A is chosen to be positive, then N and A are uniquely determined (up to unitary equivalence). Recall that A is positive semidefinite (resp. positive definite) if (Ax, x) > 0 (resp. (Ax, x) > 0) for any vector (resp. nonzero vector) x. In fact, in the preceding theorem N and A may be chosen to be the 1
restrictions of T and (T*T)2 to their respective reducing subspaces nP91 ker (Tn*Tn
TnTn*) and H e (ker T e ran
.
-
If A is the identity operator on a one-dimensional
space, then 0
A A0 0
reduces to the simple unilateral shift S. (Later on, we will also consider S as the operator of multiplication by z on the Hardy space H2 of the unit disc.) For convenience, we will denote
0
AO AO
by S ® A without giving a precise meaning to the tensor product of two operators. Note that S ® A is completely nonnormal, that is, there is no nontrivial reducing subspace on which it is normal. We will call the uniquely determined N and S 0 A the normal and pure parts of T, respectively. If T is an isometry, then these two parts coincide with the unitary operator and the unilateral shift in its Wold decomposition. In terms of this representation, it is easily seen that every quasinormal operator N ® (S 0 A) is subnormal with minimal normal extension
AO AO
NO
AO
where a box is drawn around the (0, 0) -entry of the matrix. Since A
0
S®A=
IO
I0
A A
is the (unique) polar decomposition of S 0 A (with the two factors having equal kernels), an easy argument yields the following characterization of quasinormality [36, Problem 137].
THEOREM 2.2. An operator with polar decomposition UP is quasinormal if and only if U and P commute. There are other representations for quasinormal operators. Since every
positive operator can be expressed as the direct sum of cyclic positive operators, this implies that every pure quasinormal operator is the direct sum of operators of the form S ® A, where A is cyclic and positive definite. (Recall that an operator T on H is cyclic
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if there is a vector x in H such that V{Tnx : n > 01 = H.) The second representation which we now present will be for this latter type of operators.
By the spectral theorem, any cyclic positive definite operator A is unitarily equivalent to the operator of multiplication by t on L2(µ), where p is some positive Borel measure on an interval [0, a] in R with p ({0}) = 0. Let v be the measure on C defined by dv(z) = dOdp(t), where z = teie, and let K = V zImzn : m, n> 0} in L2(v). Then, obviously, K is an invariant subspace for M, the operator of multiplication by z on L2(v). Finally, let TA = M I K. THEOREM 2.3. For any cyclic positive definite operator A, TA is a pure quasinormal operator. Conversely, any pure quasinormal operator S ® A with A cyclic is unitarily equivalent to TA.
This representation is obtained in [19, Theorem 2.4]. The appearance of the space K above is not too obtrusive if we compare it with the space in the statement of Proposition 3.3 below.
We conclude this section with the third representation. It applies to pure quasinormal operators S 0 A with A invertible. This is originally due to G. Keough and first appeared in [19, Theorem 2.8]. Let A be a positive invertible operator on H, and let H 2A be the class of sequences {xn}n=0 with xn in H satisfying 00
E lIAnxnIl2 < ao.
n=0 It is easy to verify that H 2A is a Hilbert space under the inner product 00
= E (Anxn, AnYn), n=0 where (,) inside the summation sign denotes the inner product in H. Let SA denote the ({xn}, {yn})
right shift on H 2A SA ({x0,
x1, ... }) _ {0, x0, xl, ... }.
THEOREM 2.4. For any positive invertible A, SA is a pure quasinormal
operator. Conversely, any pure quasinormal operator S ® A with A invertible is unitarily equivalent to SA.
It is clear that the unitary operator
U({xn}) = {Anxn}
from HA onto H ® H ® implements the unitary equivalence between SA and S ® A. As an application, we have THEOREM 2.5. If T = S ® A is a pure quasinormal operator on H and R is any cyclic operator on K with IIRII < IIAII, then there exists an operatorX : H -+ K with dense range such that XT = RX. The preceding theorem is proved in [19, Theorem 4.2] first for invertible A
and then for the general case. We remark that if T is a pure isometry then X can be chosen not only to have dense range but have zero kernel [51].
3. SPECTRUM AND MULTIPLICITY
For the spectrum of quasinormal operators, we may restrict ourselves to the pure ones since putting back the normal part does not cause much difficulty. THEOREM 3.1. Let T = S 0 A be a pure quasinormal operator. Then (1) ap(T)
(2) ap(T
_ {A : IAI < IIAII},
(3) o(T) = oap(T) _ {A :
IAI
(4) oap(T) = o,,(T) _ {A (5) ae(T) = are(T) = {A :
AI
IAI
<_ IIAII},
E o(A)), and <_ IIAIIe} U {A :
Here o,(.), op(. ), aap(. ), oe( ),
and
IAI
E o(A)}.
ore(') denote, respectively,
the spectrum, point spectrum, approximate point spectrum, essential spectrum, left essential spectrum and right essential spectrum of its argument. The spectrum and essential spectrum of S 0 A were first obtained in [55, Corollary 2] and the approximate
point spectrum in [57, Theorem 2.1]; other assertions either are obvious or can be derived from the results in [55]. Alternatively, a (S ® A) and aap (S ® A) can also be obtained as in [12, Lemma 2.2] via the more general result on the spectrum of tensor product of operators [5]. Note that an immediate consequence of the above is that IIS 0 All = IIS 0 AIle = IIAII
The multiplicity p(T) of an operator T on H is the minimal cardinality of vectors {xa : a E ul} in H satisfying
V{Tnxa:n>0,aEft}=H.
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The next proposition gives an expression of the multiplicity of a pure quasinormal operator S ® A in terms of A.
PROPOSITION 3.2. If T = S ® A is a pure quasinormal operator on , then p (T)= dim H. In particular, p(A) < µ(S®A). H(°') Indeed, if {xl, , xm} are vectors in H 00 with = V{Txj : n > 0,
H(w) = H ® H ®
1 <j< m}, then, for any y E K, y ®0 ®0 e . . . is in H(°D) = V{Tnxj} whence y is a linear combination of the first components of the xj's. Therefore dim H < m.
Conversely, if H is spanned by {yl,
, ym}, then, since A is invertible
, Anym} for any n > 0. We infer that
on H, H is also spanned by {Anyl,
H(°°)
_
V{Tnyn>0,1 <j<m}, where yj=yj®0®0 ,1 <j <m. Thusp(T) < rm. l Note that, in general, µ(A) and u(S 0 A) are not equal: If A = [ 2J, then
µ(A) = 1 but p(S 0 A) = 2. Nevertheless, as the following proposition shows, the multiplicity of A may also be expressed in terms of S 0 A. PROPOSITION 3.3. If T = S 0 A is a pure quasinormal operator on H,
then µ(A) equals the minimal cardinality of vectors {xa : a E Sl} in H satisfying VI ITI
mTnxa:m,n>O, aEfl}=H, where ITI _(T T)1 2.
The proof is the same as the one for [19, Proposition 2.3] which we omit.
Next we consider the multiplicity of the adjoint of a pure quasinormal operator. Since the adjoint of any pure isometry is cyclic [36, Problem 160], we may not
be too surprised to find out that the same is true for adjoints of pure quasinormal operators.
THEOREM 3.4. p(T ) = 1 for any pure quasinormal operator T. This is proved in [58, Corollary 3]. It follows from a more general result [58, Theorem 2] that any operator of the form
0T12T13. 0
0 T23
0
0
0
.. ..
with Tn n+1 having dense range for all n > 1 is cyclic.
4. SPECIAL CLASSES
In this section, we give characterizations of quasinormal operators among three special classes, namely, unilateral (bilateral) weighted shifts, Toeplitz operators and composition operators. An operator T on H is a unilateral (resp. bilateral) weighted shift if there are an orthonormal basis {en} and a sequence of bounded complex numbers {wn}, n =
(resp. n = 0, ±1, ±2,
0, 1, 2,
. ),such that Ten = wnen+l for all n. As usual, we
may assume that the weights wn are all nonnegative.
THEOREM 4.1. A unilateral (bilateral) weighted shift T is quasinormal if = and only if there is an integer n0 such that wn = wn0-1 = = 0 and wn 0
wn0+2
= ..
0+1
.
The proof follows by an easy computation with the defining property of quasinormality and can be found in [36, Problem 139]. Note that the only normal unilateral (bilateral) weighted shift(s) is (are) the one(s) with all the weights equal to zero (all the weights equal), and the subnormal shifts have also been characterized (cf. [17, Theorems III. 8. 16. and III. 8. 17]). We next consider Toeplitz operators. For 0 in LO, the Lebesgue space on the unit circle, the Toeplitz operator TO is the operator on the Hardy space H2 defined
by T f = P(of) for f E H2, where P is the orthogonal projection from L2 onto H2. 0
THEOREM 4.2. The Toeplitz operator TO is quasinormal if and only if one of the following holds : (1) 0 is a linear function of a real-valued function in LOD,
(2) 0 is a constant multiple of an inner function. This result is due to Amemiya, Ito and Wong [1]. Note that condition (1)
above completely characterizes normal Toeplitz operators and condition (2) yields multiples of unilateral shifts. Historically, this theorem answers positively for quasinormal operators a question of Halmos : Is every subnormal Toeplitz operator TO
either normal or analytic (that is, with an analytic symbol 0) ? Its eventual negative solution is obtained by Cowen and Long [21] (compare also [20] for a survey of this problem).
Let (X, fl, p) be a u-finite measure space and T : X -+ X a measurable
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transformation. The composition operator CT on L2(p) induced by T is, by definition, given by CTf = foT for f E L2 (p). It has long been known [44, p.39] that a necessary
and sufficient condition for CT to be bounded is that µoT
1
be absolutely continuous
with respect to p and h =_ d(µoT 1)/dµ be in L0D(p). When are CT and CT quasinormal? The next two theorems from [54] and [37] provide complete answers. THEOREM 4.3. (1) CT is quasinormal if and only if h = hoT a.e. [µ].
(2) If µ is a finite measure, then CT is quasinormal if and only if T is measure-preserving. In particular, it follows from (2) above that CT is quasinormal if and only
if CT is an isometry at least when p is finite.
THEOREM 4.4. CT is quasinormal if and only if
(1) for any A in S1, A f1 supp h is in the completion of the a-algebra T-1 (Q), and (2) h = hoT a.e. [µl on supp h. 5. INVARIANT SUBSPACES
The existence of nontrivial invariant subspaces for normal operators is an easy consequence of the spectral theorem. For subnormal operators, this is more difficult to prove ; a subtle analytic approach would be needed [7]. In this respect, as in all others, quasinormal operators are in-between. THEOREM 5.1. Any quasinormal operator T on a space of dimension greater than 1 has a nontrivial invariant subspace. Moreover, if T is not a multiple of the identity operator, then it has a nontrivial hyperinvariant subspace.
The proof for the first part which makes use of the spectral theorem, Fuglede's theorem and the existence of invariant subspace for the simple unilateral shift appears in [36, Problem 196]. Alternatively, it also follows from the second part. As for
the proof of the latter, if T is represented as N ® (S ® A) on H1 ® H2, then H1
ran ® ) is a nontrivial hyperinvariant subspace for T since 0 E op((S ® A)*) by Theorem 3.1 (2) and there is no nonzero operator X such that XN = (S ® A) X by [55, Theorem 2].
A much stronger notion than the mere existence of invariant subspaces for
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operators is that of reflexivity : T is reflexive if Alg Lat T, the algebra of operators leaving invariant every invariant subspace of T, equals Alg T, the weakly closed algebra generated by T and I. Since Alg T is obviously contained in Alg Lat T, the reflexivity of T means that it has so many invariant subspaces as to make Alg Lat T the smallest possible. This notion is first proposed by Sarason [49] who showed that every normal operator is reflexive. The reflexivity of isometries and quasinormal operators are proved by Deddens [24] and Wogen [59], respectively. THEOREM 5.2. Every quasinormal operator is reflexive. The proof of the reflexivity of subnormal operators [45] came shortly; it is much deeper.
A class of operators with a reasonably rich spectral theory is that of decomposable ones. An operator T on H is decomposable if for every finite open covering G1, , Gn of a(T) there exist spectral maximal subspaces Kl, , Kn of T
such that o(T I Kj) C Gj for all j and H =
(Recall that an invariant subspace K
of T is a spectral maximal subspace if it contains every invariant subspace L of T satisfying v(T I L) C o(T 1K).) This notion is first introduced by Foias [30]. It is not difficult to show that normal operators are decomposable. On the other hand, there are subnormal operators which are not decomposable (as for example the simple unilateral shift) and subnormal decomposable operators which are not normal [48, Corollary 1]. Can the latter operators be quasinormal? The next Theorem provides a negative answer as would be expected. It appeared in [8]. THEOREM 5.3. A quasinormal decomposable operator must be normal. Another property of operators which is closely related to the invariant
subspace problem and stirred up many research activities in the 1970s is that of quasitriangularity. According to one of its equivalent definitions, an operator T is quasitriangular if there exists an increasing sequence of finite-rank projections {Pn}
such that Pn approaches to I in the strong operator topology and APn - PnAPn approaches to 0 in norm [34]. If both T and T are quasitriangular, then T is called biquasitriangular. Again, it is easy to show that normal operators are biquasitriangular and the simple unilateral shift is not (cf. [34]). The next theorem characterizes (bi)quasitriangularity among pure quasinormal operators. THEOREM 5.4. A pure quasinormal operator S ® A is (bi)quasitriangular if and only if A satisfies a(A) = [0, IJAII]. This result appeared in [57, Corollary 2.2]; it is proved via the observation
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that a hyponormal operator is (bi)quasitriangular if and only if its spectrum and approximate point spectrum coincide [56, Theorem 3.1], and the descriptions of these spectra for pure quasinormal operators (Theorem 3.1). We conclude this section with the notion of cellular-indecomposability first proposed by Olin and Thomson [46]. An operator T is cellular-indecomposable if any two of its nonzero invariant subspaces have nonzero intersection. One such operator which comes to mind immediately is the simple unilateral shift [36, Corollary 2 to Problem 157]. It turns out that, among quasinormal operators, multiples of the simple unilateral shift are the only ones having this property. PROPOSITION 5.5. A quasinormal operator is cellular-indecomposable if and only if it is a multiple of the simple unilateral shift.
This is easy to prove if we note that for a pure quasinormal operator T = S ® A, every spectral subspace of T*T = A2 ® A2 ® . . . reduces T. 6. COMMUTANT
In this section, we first determine the three operator algebras associated
with a pure quasinormal operator T: {T}', the commutant, {T}", the double commutant, and Alg T, the weakly closed algebra generated by T and I. (Recall that {T}' = {X: XT = TX} and {T}" = {Y: YX = XY for any X in {T}'}.) The commutant is the easiest to determine (cf. [19, Lemma 3.1]). PROPOSITION 6.1. Let T = S ® A be a pure quasinormal operator on H(00 = H ® H ®
.
Then an operator D = [D..]00
on H(00) commutes with T if and
only if Dij = 0 for any j > i and ADij = Di+1 j+lA fori > j. As for {T}" and Alg T, their characterizations lie deeper. Recall that the simple unilateral shift S satisfies {S}' = {S}" = Alg S = {o(S) : 0 E HOD} (cf. [36, Problems 147 and 148]). That the commutant and double commutant cannot equal for general (higher-multiplicity) unilateral shifts is obvious. The next theorem says that the remaining equalities (with a slight modification) still hold for any pure quasinormal operator. THEOREM 6.2. For any pure quasinormal operator T = S ® A, the equalities {T}" = Alg T = {O(T) : 0 E HI} hold, where r = JITh1 and Hi denotes the Banach algebra of bounded analytic functions on {z E C : IzI < r}.
Thus, in particular, operators in {T}" = Alg T are of the form
a0I 0 alA a0I
0
a2A2 a1A a01
where the an's are the Fourier coefficients of a function O(z) = En0D0 anzn in H. These results were proved in [19]. For nonpure quasinormal operators, the double commutant property ({T}" = Alg T) does not hold in general. Actually, this is already the case for
normal operators; the bilateral shift U on L2 of the unit circle is such that {U}" = {-O(U) : -i E L00} and Alg U = {o(U) : 0 E H°°}. A complete characterization of quasinormal operators satisfying the double commutant property is given in [19, Theorem 4.10]. The conditions are too technical to be repeated here. We content ourselves with the following special case which was proved earlier in [52].
PROPOSITION 6.3. Any nonunitary isometry has the double commutant property.
We next consider the commutant lifting problem: If T is a quasinormal operator on H with minimal normal extension N, when is an operator in {T}' the restriction to H of some operator in {N}'? That this is not always the case can be seen from the following example.
r
Let T = S ® A, where A = I ), where B = 10 i]. Since AnBA n =
1
2]
,
` [0
and X = diag (B, ABA 1, A2BA 2, (1/2)n]
for n > 0, X is indeed a bounded
operator. That X belongs to {T}' follows from Proposition 6.1. A simple computation shows that if X can be lifted to an operator Y in the commutant of the minimal normal extension
N=
0
A0 AO
of T, then Y must be of the form diag ( , A2BA2, A1BA, B, ABA', . ). However, as
A2BA_2
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2n 1 A B A ' = 101,
for n > 0, this operator cannot be bounded. This shows that X cannot be lifted to {N}'. Note that in this example T is even a pure quasinormal operator with multiplicity 2. A complete characterization of operators in {T}' which can be lifted to {N}' is obtained by Yoshino [62, Theorem 4]. THEOREM 6.4. Let T be a quasinormal operator with minimal normal
extension N and polar decomposition T = UP. Then X E {T}' can be lifted to Y E {N}' if and only if X commutes with U and P. Moreover, if this is the case,then Y is unique and IIYII = 11X11-
In particular, if T is an isometry, then operators in {T}' can always be lifted [27, Corollary 5.1]. These results are subsumed under Bram's characterization of commutant lifting for subnormal operators [3, Theorem 7].
Another version of the lifting problem asks whether two commuting quasinormal operators have commuting (not necessarily minimal) normal extensions. An example of Lubin [43] provides a negative answer. Indeed, the two quasinormal operators Ti and T2 he constructed are such that both are unitarily equivalent to S ® 0, where 0 denotes the zero operator on an infinite-dimensional space, T1T2 = T2T1 = 0
and Ti + T2 is not hyponormal. Again, a complete characterization in terms of the polar decomposition is given in [62, Theorem 5]. THEOREM 6.5. Let Ti and T2 be commuting quasinormal operators with
polar decompositions T1 = U1P1 and T2 = U2P2. Then Ti and T2 have commuting normal extensions if and only if U1 and P1 both commute with U2 and P2.
In this connection, we digress to discuss another topic which may shed some light on the commutant lifting problem. As is well-known, every subnormal operator has a unique minimal normal extension [36, Problem 197]. That it also has a unique minimal quasinormal extension seems to be not so widely known. This fact is due to Embry-Wardrop [28, Theorems 2 and 3]. THEOREM 6.6. Let T be a subnormal operator with minimal normal extension N on H. If K = V{=0(N*N)lxi : xj E H, n> 0}, then N I K is a minimal
quasinormal extension of T and any minimal quasinormal extension of T is unitarily equivalent to N I K. Moreover, N is also the minimal normal extension of N I K.
Thus, in particular, the lifting of the commutant for subnormal operators
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can be accomplished in two stages: first lifting to the commutant of the minimal quasinormal extension and then the minimal normal extension. Studies of other properties of subnormal operators along this line seem promising but lacking.
A problem which might be of interest is to determine which subnormal operator has a pure quasinormal extension. As observed by Conway and Wogen [58, p.169], subnormal unilateral weighted shifts do have this property. We conclude this section with properties of a class of operators considered by Williams [57, Section 3]. A result which is of interest and not too difficult to prove is the following.
THEOREM 6.7. If T is a quasinormal operator, N is normal and TN = NT, then T + N is subnormal.
Starting from this, he went on to consider operators of the form T + N, where T is pure quasinormal and N is a normal operator commuting with T. It turns out that such operators have a fairly simple structure. If we express T as S ® A on H H ® ... and use Proposition 6.1, we can show that N must be of the form N0 ® N0 An easy consequence of this is
THEOREM 6.8. If T is a pure quasinormal operator, N 10 is normal and TN = NT, then T + N is not quasinormal. For other properties of such operators, the reader is referred to [57]. 7. SIMILARITY
In this section and the next two, we will consider how two quasinormal operators are related through similarity, quasisimilarity and compact perturbation. We start with similarity. For over a decade, the problem whether two similar quasinormal operators are actually unitarily equivalent remains open [41]. This is recently solved in the negative in [12]. In fact, a complete characterization is given for the similarity of two quasinormal operators. Note that the similarity of two normal operators or two isometrics implies their unitary equivalence (even the weaker quasisimilarity will do).
For normal operators, this is a consequence of the Fuglede-Putnam theorem [36, Corollary to Problem 192]; the case for isometrics is proved in [40, Theorem 3.1]. On the other hand, there are similar subnormal operators which are not unitarily equivalent [36, Problem 199]. Against this background, the result on quasinormal operators should have more than a passing interest.
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THEOREM 7.1. For j = 1, 2, let Ti = Ni ® (S ® A ) be a quasinormal i
operator, where Nj is normal and Aj is positive definite. Then T1 is similar to T2 if and
only if NI is unitarily equivalent to N2, a(A1) = o(A2) and dim ker (A1 - AI) = dim ker (A2 - Al) for any A in Q(A1).
Thus, in particular, similarity of quasinormal operators ignores the multiplicity of the operator Aj in the pure part except those of its eigenvalues. From this observation, examples of similar but not unitarily equivalent quasinormal operators can be easily constructed. One such pair is TI = S ® A and T2 = S ®(A ®A), where A is the operator of multiplication by t on L2[0,1].
As for the proof, we may first reduce our consideration to pure quasinormal operators by a result of Conway [16, Proposition 2.6]: Two subnormal operators are similar if and only if their normal parts are unitarily equivalent and their pure parts are similar. For the pure ones, the proof depends on a deep theorem in the nest algebra theory. Here is how it goes. Recall that a collection )/of (closed) subspace of a fixed Hilbert space H is a nest if (1) {0} and H belong to X, (2) any two subspaces M and N in A' are comparable,
that is, either M C N or N C M, and (3) the span and intersection of any family of subspaces in Y are still in X. For any nest X, there is associated a weakly closed algebra, Alg X, consisting of all operators leaving invariant every subspace in X; Alg X is called
the nest algebra of A The study of nest algebra is initiated by J.R.Ringrose in the 1960s. Since then, it has attracted many researchers. A certain maturity is finally reached in recent years. The monograph [23] has a comprehensive coverage of the subject. Before stating the Similarity Theorem which we are going to invoke, we need
some more terminology of the theory. A nest X is continuous if every element N in X equals its immediate predecessor N =_ V {N' E A N' N}. Two nests X and Jl on spaces J H1 and H2 are similar if there is an invertible operator X from H1 onto H2 such that XX = JL A major breakthrough in the development of the theory is the proof by Larson [42] that any two continuous nests are similar. This is generalized later by Davidson [22] to
the similarity of any two nests: X and X are similar if and only if there is an order preserving isomorphism 0 from Xonto Al such that for any subspaces N1 and N2 in
X with N1 C N2 the dimensions of N2 a N1 and 0(N2) a 0(N1) are equal. In particular,
this says that the similarity of nests depends on the order and the dimensions (of the
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atoms) of the involved nests but not on their multiplicity. (A multiplicity theory of
nests can be developed via the abelian von Neumann algebra generated by the orthogonal projections onto the subspaces in the nest.) This may explain why the Similarity Theorem has some bearing on our result. Its proof is quite intricate. Before embarking on the proof of our result, we need a link relating pure quasinormal operators to nest algebras so that the Similarity Theorem can be applied. For any positive definite operator A on H, there is associated a natural nest ,VA, the one
generated by all subspaces of the form EA([0,t])H, t > 0, where
denotes the
spectral measure of A. The result we need is due to Deddens [25]. It says that the nest algebra Alg"A consists exactly of operators T satisfying sup00 IIAnTAnII < oo. Now we are ready to sketch the proof of Theorem 7.1.
If Al and A2 are positive definite operators on HI and H2 satisfying o(Al) = a(A2) and dim ker (AI - AI) = dim ker (A2 - Al) for A in a(AI), then define the order-preserving isomorphism 0 from )'Al to AA by 2
0 (EA1 [0,A]HI) = EA2 [0,A]H2
if A E v(Al)
and
0 (EA1[0,A)HI) = EA2[0,A)H2 if A is an eigenvlaue of Al.
Our assumption guarantees that 0 is dimension-preserving. Thus it is implemented by an invertible operator X by the Similarity Theorem. Letting
A=
Al 0 0
A2
and Y =
LX 0
1]
nll
we have Y E Alg OVA. Therefore, Deddens' result implies that supn>0 IIAnYA
< ao
or, in other words, sup IIA2XA1nII < oo and sup IIAnX 1A2niI < ao. Thus Z = diag(X, A2XA1 1, A2XA12,
) is an invertible operator satisfying Z(S 0 AI) = (S 0 A2)Z.
This shows that S 0 Al and S 0 A2 are similar. The converse can be proved essentially by a reversal of the above arguments. 8. QUASISIMILARITY
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Two operators Ti and T2 are quasisimilar if there are operators X and Y
which are injective and have dense range such that XT1 = T2X and YT2 = T1Y. In this section, we will address the problem when two quasinormal operators are quasisimilar. As we will see, this problem is much more complicated than the similarity problem which we discussed in Section 7.
If two quasinormal operators are quasisimilar, then, necessarily, their spectra and essential spectra must be equal to each other. The former is true even for quasisimilar hyponormal operators (cf. [13]), and the latter for subnormal operators (cf. [55, 61]). However, things are not as smooth as we would like them to be. The pure parts of quasisimilar quasinormal operators may not be quasisimilar [55, Example 1] although their normal parts are still unitarily equivalent [16,Proposition 2.3]. Thus, in the case of quasisimilarity, we cannot just consider the pure ones but also have to worry about the "mixing effect" of the normal and pure parts.
A complete characterization of quasisimilar quasinormal operators is given in [12]. We start with the pure ones. THEOREM 8.1. Two pure quasinormal operators S ® A 1 and S ® A2 are quasisimilar if and only if the following conditions hold:
(1) m(A1) = m(A2) and dim ker (A1 - m(A1)I) = dim ker (A2 m(A2)I),
(2) IIA1Ile = IIA2IIe and dim ker (Al - AI) = dim ker (A2 - Al) for any A > IIAlIIe, and, in case there are only finitely many points in a(A1) n (IIAlIIe'
(3) dim ker (Al - IIAlIIe I) = dim ker (A2 - IIA2IIe I). Here m(Aj) = inf {A : A E o(Aj)}, j = 1, 2.
In particular, this theorem says that for quasisimilar pure quasinormal operators S ® A and S ® AT the part of the spectrum of Aj in (m(Aj), IIAjIIe) can be 1 quite arbitrary. This is the source of examples used to illustrate the nonpreserving of various parts of the spectrum under quasisimilarity (cf. [56, Examples 2.2 and 2.3] and [38, p.1445]). In particular, in view of Theorem 3.1, this is the case for the approximate point spectrum of quasinormal operators. Another consequence of Theorem 8.1 is that every pure quasinormal operator is quasisimilar to an S 0 A with A a diagonal positive definite operator.
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Note that condition (1) (resp. (2) together with (3)) is equivalent to the injective similarity (resp. dense similarity) of S 0 Al and S a AT (Two operators Ti and T2 are injectively (resp. densely) similar if there are operators X and Y which are injective (resp. have dense range) such that XT1 = T2X and YT2 = T1Y.)
The proof for the necessity of conditions (1), (2) and (3) is elementary; that for the sufficiency is more intricate. Here is a very brief sketch. First decompose Ai on Hi, j = 1, 2, into three parts : Aj = Bj ® Cj ® Dj so that Bj, Cj and Dj are acting on the spectral subspaces
{m(Aj)}Hj, EA,(m(Aj), IIAjIIe] Hj and EAl
(IIAjIIe, IIAiII]
EA]
Hj, respectively. Correspondingly, we have the decomposition
S®Aj=(SeBj)®(SoCj)®(SoDj), j=1,2. The proof is accomplished by showing that (a) (S 0 Bl) ® (S ® Cl) < S ® B2 and (b)
S 0 Dl < (S 0 C2) ® (S 0 D2). (Recall that, for any two operators Ti and T2, Ti < T2
means that there is an injective operator X with dense range such that XT1 = T2X.) By our assumption, (a) is the same as m(A1)(S 0 I) ® (S 0 Cl) < m(A1)(S 0 I). The operator S ® Cl can be further decomposed as S 0 C1 = En ® (S 0 En), where En acts on
the spectral subspace EA1 (an, an_i]H with ap = IIAIIle and the sequence {an} decreasing to m(A1). Using the observation that S 0 A < m(A)(S 0 I) for any invertible
A, we obtain S 0 C1 < En ® an(S 0 I).
Thus the proof of (a) reduces to showing
S ® (En ® (bnS)) < S, where bn = an/m(A1) > 1. This is established through modifying
the proof of a result of Sz.-Nagy and Foias [51] that aS(n) < S for any a, I al > 1, and n, 1 < n < oo. On the other hand, following our assumptions, (b) is the same as S 0 D2 < (S a C2) ® (S 0 D2). The proof, based on the fact that IIC211 < m(D2)1 is easier (cf. [12, Lemma 3.14 (a)]).
We next turn to the quasisimilarity of general quasinormal operators. The following theorem gives a complete characterization. THEOREM 8.2. For j = 1, 2, let Ti = N1 ® (S 0 A ) be a quasinormal i
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390
operator. Let a = min{m(A1), m(A2)1 and d = max{dim ker (A1 - al), dim ker (A2-
al)). Then Ti is quasisimilar to T2 if and only if N1 is unitarily equivalent to N21 S Al is densely similar to S ® A2 and one of the following holds:
(1) So A 1 is quasisimilar to S 0 A2; (2) d = 0 and o(Nl) has a limit point in the disc {z E C
zI
< a);
(3) d > 0 and the absolutely continuous unitary part of N1/a does not vanish;
(4) d > 0 and the completely nonunitary part ofNl/a is not of class CO.
Some explanations for the terminology used above are in order.
Any
normal operator M on H can be decomposed as M = M1 ® M2 ® M3, where Ml, M2 and M3 act on EM(D)H, EM(8D)H and EM(C\ D )H, respectively (D is the open unit disc on
the plane).
M1, being a completely nonunitary contraction, is called the completely
nonunitary part of M. The unitary M2 can be further decomposed as the direct sum of
an absolutely continuous unitary operator and a singular unitary operator. These are the parts referred to in conditions (3) and (4) in the above theorem. A completely nonunitary contraction T is of class C0 if q(T) = 0 for some 0 E H. (For properties of such operators, the reader is referred to [50].) The proof of the sufficiency of the conditions in Theorem 8.2 involves a great deal of function-theoretic arguments. For simplicity, we will present one typical example for each of the conditions (2), (3) and (4) followed by a one-sentence sketch of its proof which somehow gives the general flavor of the arguments. EXAMPLE 8.3. If N is the diagonal operator diag(dn) on 12, where {dn} is a sequence satisfying 0 < I do I < c < 1 for all n and converging to 0, then S ® N < N.
The operator X : H2 ®12
12 defined by
X(f ® {an}) = {cn(f(dn) + an exp(-l/ I do I ))} can be shown to be injective, with dense range and satisfying X(S (D N) = NX. EXAMPLE 8.4. If N is the operator of multiplication by eit on L2(E), where E is a Borel subset of the unit circle, then S ® N < N.
The operator X : H2 0 L2(E) -. L2(E) required is defined by
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X(f w g) = (f I E) + 0g,
where 0 is a function in L°D(E) such that 0 # 0 a.e. on E and JE log I o I =
-
EXAMPLE 8.5. If N is the diagonal operator diag(dn) on 12, where {dn}
is a sequence of points in the open unit disc accumulating only at the unit circle and satisfying En(1-I dnI) = oo, then S a N < N.
The proof for this case is the most difficult one. The operator X : H2 a 12 12 defined by 1
X(f a {an}) = {f(dn)(1-I do I2)2/n + anbnexp(-1/(1-I do I )2)}, where {bn} is a bounded sequence of positive numbers satisfying lim supn I B(dn) I /nbn >
1 for any Blaschke product B (the existence of {bn} is proved in [12, Lemma 4.8]), will meet all the requirements. The difficulty lies in showing the injectivity of X. 9. COMPACT PERTURBATION a
Two operators Ti and T2 are approximately equivalent (donoted by Ti T2) if there is a sequence of unitary operators {Un} such that IIUnT1Un - T211 -4 o; they a are approximately similar (donoted by Ti a T2) if there are invertible operators Xn such
that sup {IIXnII, IIXn'II} < oo and IIXn1TiXn - T2II -+ 0. Using Berg's perturbation
theorem [2], Gellar and Page [31] proved that two normal operators T1 and T2 are approximately equivalent if and only if a(T1) = a(T2) and dim ker (T1 - AI) = dim ker
(T2 - Al) for any isolated point A in a(T1). This is later extended to isometries by Halmos [35]:Two isometries Ti and T2 are approximately equivalent if and only if either
both are unitary and are approximately equivalent or their pure parts are unitarily The corresponding problem for quasinormal operators was considered by Hadwin in his 1975 Ph.D. dissertation [32]. Using the notion of operator-valued spectrum, he obtained necessary and sufficient conditions for two quasinormal operators to be approximately equivalent. Recently, this result is reproved by Chen [11, Theorem
equivalent.
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2.1] using more down-to-earth operator-theoretic techniques. THEOREM 9.1. For j = 1, 2, let Ti = N1 ® (S ® Aj) be a quasinormal operator. Then the following statements are equivalent: a
(1)T1NT2; a (2) T1 v T2; a
(3) Al N A2, o(Nl)\oap(S ®Al) = o(N2)\oap(S ® A2) and dim ker(Nl AI) = dim ker(N2 - Al) for any isolated point A in o(NI)\oap(S ® Al).
The basic tool for the proof is a theorem of Pearcy and Salinas [47, Theorem 1] that if N is a normal operator, T is hyponormal and o(N) C a (T), then N a is TNT. Note that approximately equivalent operators are compact perturbations a
of each other; this is because that if T1 "_ T2 then unitary operators Un may be chosen such that not only U n T 1 U n - T2 approach to zero in norm but are compact for all n (cf. [53]).
Thus the following definitions are indeed weaker: T1 and T2 are equivalent
modulo compact (resp. similar modulo compact) if there is a unitary U (resp. invertible k
*
X) such that U TIU - T2 (resp. X-1 T I X- T2) is compact. We denote this by T1 N T2 (resp. TI
k
T2). The classical Weyl-von Neumann-Berg theorem implies that for k
normal operators T1 and T2, both T1 "_ T2 and TI oe(T2).
k
T2 are equivalent to oe(TI) _
There is an analogous result for isometries [11, Proposition 2.8].
As for
quasinormal operators, a complete characterization for the pure ones is known, but not for the general case. The following two theorems appeared in [1l]. THEOREM 9.2. For j = 1, 2, let Ti = S 0 Aj be a pure quasinormal operator. Then the following statements are equivalent: k (1) T1 " T2; k
(2) Ti u T2; a (3) Al A2.
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THEOREM 9.3. For j = 1, 2, let Tj = Nj
(S 0 A
be a quasinormal
k
operator. IfTl ti T2, Then oe(A1)\{0} = oe(A2)\{0}. That the conclusion of the preceding theorem cannot be strengthened to oe(A,) = oe(A2) can be seen by letting T1 = En ® (N/n) and T2 = T1 ® (S 0 A), where k
, .)(that T1 N T2 follows
N is a normal operator with o(N) _ 1 and A = diag(1,
from the Brown-Douglas-Fillmore theory
[6]).
There are, of course, the usual
Fredholm conditions for two operators to be equivalent (similar) modulo compact. Thus a necessary and sufficient condition in order that two quasinormal operators Ti = N1 (S ® Aj), j = 1, 2, with at least one Aj compact be equivalent (similar) modulo compact
can be formulated. In particular, we obtain PROPOSITION 9.4. No pure quasinormal operator is similar modulo compact to a normal operator. This result is first noted in [57, p.313]. There is another notion which is weaker than approximate equivalence. Two operators Ti and T2 are algebraically equivalent if there is a *-isomorphism from *
*
*
C (T1) onto C (T2) which maps T1 to T2, where C (Tj), j = 1, 2, denotes the C -algebra generated by T and I. That this is indeed weaker is proved in [33, Corollary 3.7].
If the *-isomorphism above is required to preserve rank, then this yields
approximate equivalence. By the Gelfand theory, we easily obtain that two normal
operators are algebraically equivalent if and only if they have equal spectra.
A
necessary and sufficient condition for the algebraic equivalence of isometrics is obtained by Coburn [14]. The next theorem from [11, Theorem 3.6] treats the quasinormal case. THEOREM 9.5. Two quasinormal operators Ni 0 (S ® A1) and N2 ® (S ®
A2) are algebraically equivalent if and only if o(Al) = o(A2) and o(Ni)\oap(S ® Al) _ o(N2)\oap(S ® A2).
10. OPEN PROBLEMS
So, after all these discussions, what is the future in store for quasinormal operators? What are the research problems worthy of pursuing for them? One place to look for the answers is probably among isometries. There are problems which are solved
Wu
394
for this subclass but never considered for general quasinormal operators. Here we propose three such problems as starters. More of them are waiting to be discovered and solved if the theory is to reach a respectable level. Along the way, if some unexpected link is established with other parts of operator theory or even other areas of research in mathematics, then so much the better. Our first problem concerns the multiplicity. In Proposition 3.2, it was proved that the multiplicity of a pure quasinormal operator S 0 A equals the dimension of the space on which A acts. Will putting back the normal part still yield a simple formula for the multiplicity? For isometries, this is solved completely in [60].
The second one concerns the hyperinvariant subspaces of quasinormal operators. Their existence is guaranteed by Theorem 5.1. Is there a simple way to describe all of them? This problem does not seem to have been touched upon before even for pure ones. Playing around with some special case such as S 0 A with A = 0 b] , a > b > 0, may lead to some idea on what should be expected in general. This
was done recently by K.-Y. Chen. Further progress would be expected in the future. The case with isometrics is known (cf. [26]).
Finally, as discussed in Section 9, the problem when two quasinormal operators are compact perturbations of each other has not been completely solved yet. Bypassing it, we may ask the problem of trace-class perturbation, that is, when two quasinormal operators T1 and T2 are such that U T1U - T2 is of trace class for some unitary U. In this case, the answer does not seem to be known completely even for isometrics and normal operators (cf. [9, 10]). How about finite-rank perturbations or even rank-one perturbations? All these problems are crying out for answers. Hopefully, their solutions
will lead to a better understanding of the structure of the underrated quasinormal operators.
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Department of Mathematics National Chiao Tung University Hsinchu, Taiwan Republic of China E-mail address: PYWU©TWNCTU01. BITNET
MSC: Primary 47B20
399
WORKSHOP PROGRAM Tuesday, June 11, 1991 9:30
Welcome by T. Ando
9:35
Opening address by L Gohberg
9:50-10:40
C. R. Johnson Matrix completion problem
11:10-12:00
H. Langer Model and unitary equivalence of simple selfadjoint operators in Pontrjagin spaces
12:10-12:40
H. Bart Matricial coupling revisited
14:00-14:40
A. Dijksma Holomorphic operators between Krein spaces and the number of squares of associated kernels
14:50-15:30
A. Gheondea The negative signature of defect and lifting of operators in Krein spaces
16:00-16:30
H. J. Woerdeman Positive semidefinite, contractive, isometric and unitary completions of operator matrices
16:35-17:05
J. I. Fujii Operator mean and the relative operator entropy
17:15-17:45
V. Vinnikov Commuting nonselfadjoint operators and function theory on a real Riemann surface
17:50-18:20
E. Kamei An application of Furuta's inequality to Ando's theorem
Wednesday, June 12, 1991
9:00- 9:50
I. Gohberg Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts
10:00-10:40
M. A. Kaashoek Maximum entropy principles for band extensions and Szego limit theorems
400
11:10-12:00
H. Widom Asymptotic expansions and stationary phase for operators with nonsmoot symbol
12:10-12:40
A. C. M. Ran On the equation X + A*X -1 A = Q
14:00-14:40
K. Izuchi Interpolating sequences in the maximal ideal space of H-
14:50-15:30
D. Z. Arov (j, J)-inner matrix-functions and generalized betangent CaratheodoryNevanlinna-Pick-Krein problem
16:00-16:30
S. Takahashi Extended interpolation problem for bounded analytic functions
16:35-47:05
T. Okayasu The von Neumann inequality and dilation theorems for contractions
17:15-17:45
P. Y. Wu Similarity and quasisimilarity of quasinormal operators
17:50-18:20
T. Nakazi Hyponormal Toeplitz operators and extremal problems of Hardy spaces
Thursday, June 13, 1991
9:00- 9:50
A. A. Nudel'man Some generalizations of the classical interpolation problems
10:00-10:40
T. Furuta Applications of order preserving operator inequalities
11:10-12:00
J. Ball A survey of interpolation problems for rational matrix functions and connections with H°° control theory
13:00
Excursion
17:00
Barbecue party
Friday, June 14, 1991
900- 9:50 10:00-10:40
V. M. Adamjan Analytic structure of scattering matrices for big integral schemes
H. Dym On a new class of reproducing kernel spaces
401
11:10-12:00
P. A. Fuhrmann Model reduction and robust control via LQG balancing
12:10-12:40
D. Alpay Some reproducing kernel spaces of analytic functions, sesquilinear forms and a non-hermitian Schur algorithm
14:00-14:40
R. Mennicken Expansion of analytic functions in series of Floquet solutions of first order linear differential systems
14:50-15:30
E. R. Tsekanovskii Accretive extensions, Stieltjes operator functions and conservative systems
16:00-16:40
J. W. Helton A symbol manipulator for aiding with the algebra in linear system theory
16:50-17:30
L. A. Sakhnovich Interpolation problems, inverse spectral problems and nonlinear equations
17:40-18:10
F. Kubo Museum for Selberg inequality
18:10
Closing remarks by T. Ando and I. Gohberg
402
LIST OF PARTICIPANTS
Adamyan, Vadim M., Odessa University, Odessa, UKRAINE Alpay, Daniel, Weizmann Institute of Science, Rehovot, ISRAEL Ando, T., Hokkaido University, Sapporo, JAPAN
Arov, D. Z., Odessa State Pedagogical Institute, Odessa, UKRAINE Ball, Joseph A., Virginia Polytechnic Institute and State University, Blacksburg, U.S.A. Bart, H., Erasmus University, Rotterdam, THE NETHERLANDS Chew, T. S., National University of Singapore, SINGAPORE Dijksma, A., University of Groningen, Groningen, THE NETHERLANDS Dym, Harry, Weizmann Institute of Science, Rehovot, ISRAEL
Fuhrmann, Paul A., Ben Gurion University, Beer Sheva, ISRAEL Fujii, Jun Ichi, Osaka Kyoiku University, Kashiwara, JAPAN Fujii, Masatoshi, Osaka Kyoiku University, Osaka, JAPAN Furuta, Takayuki, Science University of Tokyo, Tokyo, JAPAN Gheondea, Aurelian, Mathematics Institute of Romanian Academy, Bucharest, ROMANIA Gohberg, Israel, Tel Aviv University, Ramat-Aviv, ISRAEL Hayashi, Mikihiro, Hokkaido University, Sapporo, JAPAN
Helton, J. William, University of California, La Jolla, U.S.A. Hiai, Fumio, Ibaraki University, Mito, JAPAN
Inoue, Junji, Hokkaido University, Sapporo, JAPAN Ishikawa, Hiroshi, Ryukyu University, Okinawa, JAPAN Ito, Takashi, Musashi Institute of Technology, Tokyo, JAPAN Izuchi, Keiji, Kanagawa University, Yokohama, JAPAN Izumino, Saichi, Toyama University, Toyama, JAPAN
Johnson, Charles R., College of William and Mary, Williamsburg, U.S.A. Kaashoek, M. A., Vrije Universiteit, Amsterdam, THE NETHERLANDS
Kamei, Eizaburo, Momodani Senior Highschool, Osaka, JAPAN Katsumata, Osamu, Hokkaido University, Sapporo, JAPAN Kishimoto, Akitaka, Hokkaido University, Sapporo, JAPAN
403
Kubo, Fumio, Toyama University, Toyama, JAPAN
Kubo, Kyoko, Toyama, JAPAN Langer, Heinz, University of Wien, Wien, AUSTRIA Mennicken, Reinhard, University of Regensburg, Regensburg, GERMANY Miyajima, Shizuo, Science University of Tokyo, Tokyo, JAPAN Nakamura, Yoshihiro, Hokkaido University, Sapporo, JAPAN Nakazi, Takahiko, Hokkaido University, Sapporo, JAPAN Nara, Chie, Musashi Institute of Technology, Tokyo, JAPAN Nishio, Katsuyoshi, Ibaraki University, Hitachi, JAPAN
Nudel'man, A. A., Odessa Civil Engineering Institute, Odessa, UKRAINE Okayasu, Takateru, Yamagata University, Yamagata, JAPAN Okubo, Kazuyoshi, Hokkaido University of Education, Sapporo, JAPAN Ota, Schoichi, Kyushu Institute of Design, Fukuoka, JAPAN Ran, A. C. M., Vrije University, Amsterdam, THE NETHERLANDS Saito, Isao, Science University of Tokyo, Tokyo, JAPAN
Sakhnovich, L. A., Odessa Electrical Engineering Institute of Communications, Odessa, UKRAINE Sawashima, Ikuko, Ochanomizu University, Tokyo, JAPAN Sayed, Ali H., Stanford University, Stanford, U.S.A. Takaguchi, Makoto, Hirosaki University, Hirosaki, JAPAN
Takahashi, Katsutoshi, Hokkaido University, Sapporo, JAPAN Takahashi, Sechiko, Nara Women's University, Nara, JAPAN Tsekanovskii, E. R., Donetsk State University, Donetsk, UKRAINE Vinnikov, Victor, Weizmann Institute of Science, Rehovot, ISRAEL Watanabe, Keiichi, Niigata University, Niigata, JAPAN Watatani, Yasuo, Hokkaido University, Sapporo, JAPAN
Widom, Harold, University of California, Santa Cruz, U.S.A. Woerdeman, Hugo J., College of William and Mary, Williamsburg, U.S.A. Wu, Pei Yuan, National Chiao Tung University, Hsinchu, REPUBLIC OF CHINA Yamamoto, Takanori, Hokkai-Gakuen University, Sapporo, JAPAN Yanagi, Kenjiro, Yamaguchi University, Yamaguchi, JAPAN
404
Titles previously published in the series
OPERATOR THEORY: ADVANCES AND APPLICATIONS
BIRKHAUSER VERLAG
1. H. Bart, I. Gohberg, M.A. Kaashoek: Minimal Factorization of Matrix and Operator Functions, 1979, (3-7643-1139-8) 2. C. Apostol, R.G. Douglas, B. Sz.-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Topics in Modern Operator Theroy, 1981, (3-7643-1244-0) 3. K. Clancey, I. Gohberg: Factorization of Matrix Functions and Singular Integral Operators, 1981, (3-7643-1297-1) 4. I. Gohberg (Ed.): Toeplitz Centennial, 1982, (3-7643-1333-1) S. H.G. Kaper, C.G. Lekkerkerker, J. Hejtmanek: Spectral Methods in Linear Transport Theory, 1982, (3-7643-1372-2) 6. C. Apostol, R.G. Douglas, B. Sz-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Invariant Subspaces and Other Topics, 1982, (3-7643-1360-9) 7. M.G. Krein: Topics in Differential and Integral Equations and Operator Theory, 1983, (3-7643-1517-2) 8. I. Gohberg, P. Lancaster, L. Rodman: Matrices and Indefinite Scalar Products, 1983, (3-7643-1527-X) 9. H. Baumgartel, M. Wollenberg: Mathematical Scattering Theory, 1983, (3-7643-1519-9) 10. D. Xia: Spectral Theory of Hyponormal Operators, 1983, (3-7643-1541-5) 11. C. Apostol, C.M. Pearcy, B. Sz.-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Dilation Theory, Toeplitz Operators and Other Topics, 1983, (3-7643-1516-4) 12. H. Dym, I. Gohberg (Eds.): Topics in Operator Theory Systems and Networks, 1984, (3-7643-1550-4) 13. G. Heinig, K. Rost: Algebraic Methods for Toeplitz-like Matrices and Operators, 1984, (3-7643-1643-8) 14. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, D.Voiculescu, Gr. Arsene (Eds.): Spectral Theory of Linear Operators and Related Topics, 1984, (3-7643-1642-X) 15. H. Baumgartel: Analytic Perturbation Theory for Matrices and Operators, 1984, (3-7643-1664-0) 16. H. Konig: Eigenvalue Distribution of Compact Operators, 1986, (3-7643-1755-8)
405
17. R.G. Douglas, C.M. Pearcy, B. Sz.-Nagy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Advances in Invariant Subspaces and Other Results of Operator Theory, 1986, (3-7643-1763-9) 18. I. Gohberg (Ed.): I. Schur Methods in Operator Theory and Signal Processing, 1986, (3-7643-1776-0) 19. H. Bart, I. Gohberg, M.A. Kaashoek (Eds.): Operator Theory and Systems, 1986, (3-7643-1783-3) 20. D. Amir: Isometric characterization of Inner Product Spaces, 1986, (3-7643-1774-4) 21. I. Gohberg, M.A. Kaashoek (Eds.): Constructive Methods of Wiener-Hopf Factorization, 1986,(3-7643-1826-0) 22. V.A. Marchenko: Sturm-Liouville Operators and Applications, 1986, (3-7643-1794-9) 23. W. Greenberg, C. van der Mee, V. Protopopescu: Boundary Value Problems in Abstract Kinetic Theory, 1987, (3-7643-1765-5) 24. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics, 1987, (3-7643-1843-0) 25. G.S. Litvinchuk, I.M. Spitkovskii: Factorization of Measurable Matrix Functions, 1987, (3-7643-1843-X) 26. N.Y. Krupnik: Banach Algebras with Symbol and Singular Integral Operators, 1987, (3-7643-1836-8) 27. A. Bultheel: Laurent Series and their Pade Approximation, 1987, (3-7643-1940-2) 28. H. Helson, C.M. Pearcy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Special Classes of Linear Operators and Other Topics, 1988, (3-7643-1970-4) 29. I. Gohberg (Ed.): Topics in Operator Theory and Interpolation, 1988, (3-7634-1960-7) 30. Yu.I. Lyubich: Introduction to the Theory of Banach Representations of Groups, 1988, (3-7643-2207-1) 31. E.M. Polishchuk: Continual Means and Boundary Value Problems in Function Spaces, 1988, (3-7643-2217-9) 32. I. Gohberg (Ed.): Topics in Operator Theory. Constantin Apostol Memorial Issue, 1988, (3-7643-2232-2) 33. I. Gohberg (Ed.): Topics in Interplation Theory of Rational Matrix-Valued Functions, 1988, (3-7643-2233-0) 34. I. Gohberg (Ed.): Orthogonal Matrix-Valued Polynomials and Applications, 1988, (3-7643-2242-X) 35. I. Gohberg, J.W. Helton, L. Rodman (Eds.): Contributions to Operator Theory and its Applications, 1988, (3-7643-2221-7) 36. G.R. Belitskii, Yu.I. Lyubich: Matrix Norms and their Applications, 1988, (3-7643-2220-9) 37. K. Schmiidgen: Unbounded Operator Algebras and Representation Theory, 1990, (3-7643-2321-3) 38. L. Rodman: An Introduction to Operator Polynomials, 1989, (3-7643-2324-8) 39. M. Martin, M. Putinar: Lectures on Hyponormal Operators, 1989, (3-7643-2329-9)
406
40. H. Dym, S. Goldberg, P. Lancaster, M.A. Kaashoek (Eds.): The Gohberg Anniversary Collection, Volume I, 1989, (3-7643-2307-8) 41. H. Dym, S. Goldberg, P. Lancaster, M.A. Kaashoek (Eds.): The Gohberg Anniversary Collection, Volume II, 1989, (3-7643-2308-6) 42. N.K. Nikolskii (Ed.): Toeplitz Operators and Spectral Function Theory, 1989, (3-7643-2344-2) 43. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, Gr. Arsene (Eds.): Linear Operators in Function Spaces, 1990, (3-7643-2343-4) 44. C. Foias, A. Frazho: The Commutant Lifting Approach to Interpolation Problems, 1990, (3-7643-2461-9) 45. J.A. Ball, I. Gohberg, L. Rodman: Interpolation of Rational Matrix Functions, 1990, (3-7643-2476-7) 46. P. Exner, H. Neidhardt (Eds.): Order, Disorder and Chaos in Quantum Systems, 1990, (3-7643-2492-9) 47. L Gohberg (Ed.): Extension and Interpolation of Linear Operators and Matrix Functions, 1990, (3-7643-2530-5) 48. L. de Branges, I. Gohberg, J. Rovnyak (Eds.): Topics in Operator Theory. Ernst D. Hellinger Memorial Volume, 1990, (3-7643-2532-1) 49. I. Gohberg, S. Goldberg, M.A. Kaashoek: Classes of Linear Operators, Volume I, 1990, (3-7643-2531-3) 50. H. Bart, I. Gohberg, M.A. Kaashoek (Eds.): Topics in Matrix and Operator Theory, 1991, (3-7643-2570-4) 51. W. Greenberg, J. Polewczak (Eds.): Modern Mathematical Methods in Transport Theory, 1991, (3-7643-2571-2) 52. S. Prbssdorf, B. Silbermann: Numerical Analysis for Integral and Related Operator Equations, 1991, (3-7643-2620-4) 53. I. Gohberg, N. Krupnik: One-Dimensional Linear Singular Integral Equations, Volume I, Introduction, 1991, (3-7643-2584-4) 54. I. Gohberg, N. Krupnik (Eds.): One-Dimensional Linear Singular Integral Equations, 1992, (3-7643-2796-0) 55. R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii: Measures of Noncompactness and Condensing Operators, 1992, (3-7643-2716-2) 56. I. Gohberg (Ed.): Time-Variant Systems and Interpolation, 1992, (3-7643-2738-3) 57. M. Demuth, B. Gramsch, B.W. Schulze (Eds.): Operator Calculus and Spectral Theory, 1992,(3-7643-2792-8) 58. I. Gohberg (Ed.): Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations, 1992, (ISBN 3-7643-2809-6)
