LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES' Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB, England The books in the series listed below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 4 5 8
11
12 13 16 17 18 20 21 23 24 25 26 27 29 30 31 32 34 35 36 37 38 39 40 41
42 43 44 45 46 47 48 49 50 51 52 54 55 56 57 58
59 60 61 62
63
Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS New developments in topology, G.SEGAL (ed) Symposium on complex analysis, J.CLUNIE & W.K.HAYMAN (eds) Combinatorics, T.P.McDONOUGH & V.C.MAVRON (eds) Topics in finite groups, T.M.GAGEN Differential germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P.ROURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups, C.KOSNIOWSKI (ed) Skew field constructions, P.M.COHN Pontryagin duality and the structure of LCA groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras,A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Representation theory of Lie groups, M.F. ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed) Affine sets and affine groups, D.G.NORTHCOTT Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A.BELLER, R.JENSEN & P.WELCH Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds) Finite geometries and designs,P.CAMERON, J.W.P.HIRSCHFELD & D.R.HUGHES (eds) Commutator calculus and groups of homotopy classes, H.J.BAUES Synthetic differential geometry, A.KOCK Combinatorics, H.N.V.TEMPERLEY (ed) Markov process and related problems of analysis, E.B.DYNKIN Ordered permutation groups, A.M.W.GLASS Journees arithmetiques, J.V.ARMITAGE (ed) Techniques of geometric topology, R.A.FENN Singularities of smooth functions and maps, J.A.MARTINET Applicable differential geometry, M.CRAMPIN & F.A.E.PIRANI Integrable systems, S.P.NOVIKOV et al The core model, A.DODD Economics for mathematicians, J.W.S.CASSELS Continuous semigroups in Banach algebras, A.M.SINCLAIR
64 65 66 67 68 69 70 71 72 73 74 75 76 77
78 79 80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
Basic concepts of enriched category theory, G.M.KELLY Several complex variables and complex manifolds I, M.J.FIELD Several complex variables and complex manifolds II, M.J.FIELD Classification problems in ergodic theory, W.PARRY & S.TUNCEL Complex algebraic surfaces, A.BEAUVILLE Representation theory, I.M.GELFAND et al. Stochastic differential equations on manifolds, K.D.ELWORTHY Groups - St Andrews 1981, C.M.CAMPBELL & E.F.ROBERTSON (eds) Commutative algebra: Durham 1981, R.Y.SHARP (ed) Riemann surfaces: a view towards several complex variables,A.T.HUCKLEBERRY Symmetric designs: an algebraic approach, E.S.LANDER New geometric splittings of classical knots, L.SIEBENMANN & F.BONAHON Spectral theory of linear differential operators and comparison algebras, H.O.CORDES Isolated singular points on complete intersections, E.J.N.LOOIJENGA A primer on Riemann surfaces, A.F.BEARDON Probability, statistics and analysis, J.F.C.KINGMAN & G.E.H.REUTER (eds) Introduction to the representation theory of compact and locally compact groups, A.ROBERT Skew fields, P.K.DRAXL Surveys in combinatorics, E.K.LLOYD (ed) Homogeneous structures on Riemannian manifolds, F.TRICERRI & L.VANHECKE Finite group algebras and their modules, P.LANDROCK Solitons, P.G.DRAZIN Topological topics, I.M.JAMES (ed) Surveys in set theory, A.R.D.MATHIAS (ed) FPF ring theory, C.FAITH & S.PAGE An F-space sampler, N.J.KALTON, N.T.PECK & J.W.ROBERTS Polytopes and symmetry, S.A.ROBERTSON Classgroups of group rings, M.J.TAYLOR Representation of rings over skew fields, A.H.SCHOFIELD Aspects of topology, I.M.JAMES & E.H.KRONHEIMER (eds) Representations of general linear groups, G.D.JAMES Low-dimensional topology 1982, R.A.FENN (ed) Diophantine equations over function fields, R.C.MASON Varieties of constructive mathematics, D.S.BRIDGES & F.RICHMAN Localization in Noetherian rings, A.V.JATEGAONKAR Methods of differential geometry in algebraic topology, M.KAROUBI & C.LERUSTE Stopping time techniques for analysts and probabilists, L.EGGHE Groups and geometry, ROGER C.LYNDON Topology of the automorphism group of a free group, S.M.GERSTEN Surveys in combinatorics 1985, I.ANDERSEN (ed) Elliptic structures on 3-manifolds, C.B.THOMAS A local spectral theory for closed operators, I.ERDELYI & WANG SHENGWANG Syzygies, E.G.EVANS & P.GRIFFITH Compactification of Siegel moduli schemes, C-L.CHAI Some topics in graph theory, H.P.YAP Diophantine analysis, J.LOXTON & A.VAN DER POORTEN (eds) An introduction to surreal numbers, H.GONSHOR Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN (ed) Low dimensional topology and Kleinian groups, D.B.A.EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D.REES Lectures on Bochner-Riesz means, K.M.DAVIS & Y-C.CHANG An introduction to independence for analysts, H.G.DALES & W.H.WOODIN Representations of algebras, P.J.WEBB (ed) Homotopy theory, E.REES & J.D.S.JONES (eds) Skew linear groups, M.SHIRVANI & B.WEHRFRITZ
London Mathematical Society Lecture Note Series. 76
Spectral Theory of Linear Differential Operators and Comparison Algebras
H. 0. CORDES University of California, Berkeley
CAMBRIDGE UNIVERSITY PRESS Cambridge London
New York
Melbourne
Sydney
New Rochelle
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521284431
© Cambridge University Press 1987
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1987 Re-issued in this digitally printed version 2007
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data Cordes, H. O. (Heinz Otto), 1925Spectral theory of linear differential operators and comparison algebras. (London Mathematical Society lecture note series ; 76) 1. Differential operators. 2. Linear operators. 1. Title. QA329.4.C67
1986
515.7'242
ISBN 978-0-521-28443-1 paperback
85-47935
P R E F A C E
The main purpose of this volume is to introduce the reader to the concept of comparison algebra, defined as a type of C*-algebra of singular integral operators, generally on a noncompact manifold, generated by an elliptic second order differential expression, and certain classes of multipliers and 'Riesz-operators' As for singular integral operators on In or on a compact manifold
the Fredholm properties of operators in such an algebra are governed by a symbol homomorphism. However, for noncompact manifolds the symbol is of special interest at infinity. In particular the structure of the symbol space over infinity is of interest, and the fact, that the symbol no longer needs to be complex-valued there.
The first attempts of the author to make a systematic presentation of this material happened at Berkeley (1966) and at
Lund (1970/71). Especially the second lecture exists in form of The cases of the Laplace compari(somewhat ragged) notes [CS] .
son algebra of Rn and the half-space were presented in [C1]
.
In the course of laying out theory of comparison algebras we had to develop in details spectral theory of differential operators, as well as many of the basic properties of elliptic second order differential operators. This was done in the first four chapters. Comparison algebras (in L2-spaces and L2-Sobolev spaces) Finally, in chapter X we are discussed in chapters V to IX .
recall the basic facts of theory of Fredholm operators, partly without proofs. The material has been with the author for more than 20 years and has been subject of innumerable discussions with students and
vi
associates. Accordingly it is almost impossible to recall in detail the origin of the various concepts introduced. Especially we are indebted to E. Herman, M. Breuer, E. Luft, M. Taylor, R. McOwen, A. Erkip, D.Williams, H. Sohrab, in chronological, not alphabetical order. We are indebted to S.H.Doong, S.Melo, R.Rainsberger, M.Arse-
novic for help with proof reading. This volume originally was planned under the title 'techniques of pseudodifferential operators', but then split into two parts, with the second yet to appear. We are grateful to the publisher, Cambridge University Press, for cooperation and patience in waiting for the manuscript.
Berkeley, September 1986
Heinz 0. Cordes
TABLE OF CONTENTS
Chapter 1. Abstract spectral theory in Hilbert spaces
1
1.1. Unbounded linear operators on Banach and Hilbert spaces 1.2. Self-adjoint extensions of hermitian operators
1 6
1.3. On the spectral theorem for self-adjoint operators.
12
1.4. Proof of the spectral theorem
17
1.5. A result on powers of positive operators
24
1.6. On HS-chains
29
35 Chapter 2. Spectral theory of differential operators 2.1. Linear differential operators on a subdomain of En 35 2.2. Generalized boundary problems;ordinary
differential expressions
41
2.3. Singular endpoints of a 2r-th order 49 Sturm-Liouville problem 2.4. The spectral theorem for a second order expression 53
Chapter 3. Second order elliptic expressions on manifolds
59
3.1. 2-nd order partial differential expressions on manifolds;Weyl's lemma; Dirichlet operator
60
3.2. Boundary regularity for the Dirichiet realization
67
3.3. Compactness of the resolvent of the Friedrichs extension
71
3.4. A Green's function for H and Hd, and a mean value inequality
78
3.5. Harnack inequality; Dirichlet problem; maximum principle
87
3.6. Change of dependent variable; normal forms; positivity of the Green's function Chapter 4. Essential self-adjointness 4.1.
of the Minimal Operator Essential self-adjointness of powers of H0
93
103
104
viii 4.2. 4.3. 4.4. 4.5.
Essential self-adjointness of H0 Proof of theorem 1.1
109
Proof of Frehse's theorem More criteria for essential self-adjointness
118
111
Chapter 5. C -Comparison algebras 5.1. Comparison operators and comparison algebras 5.2. Differential expressions of order < 2 5.3. Compactness criteria for commutators
123 125 127 132 137
5.4. Comparison algebras with compact commutators 5.5. A discussion of one-dimensional problems 5.6. An expansion for expressions within reach of an algebra C Chapter 6. Minimal comparison algebra and wave front space.
144 150
155
161
6.1. The local invariance of the minimal comparison algebra
162
6.2. The wave front space
169
6.3. Differential expressions within reach of the algebra J0 6.4. The Sobolev estimate for elliptic expressions expressions on a compact 0 Chapter 7. The secondary symbol space
175
181 186
7.1. The symbol space of a general comparison algebra 7.2. The space M\W , and some examples
189
7.3. Stronger conditions and more detail on ffi\W
200
7.4. More structure of 2s , and more on examples
.
193
210
Chapter 8. Comparison algebras with non-compact commutators 8.1. An algebra invariant under a discrete translation group 8.2. A C -algebra on a poly-cylinder. 8.3. Algebra surgery
8.4. Complete Riemannian manifolds with cylindrical ends Chapter 9. Hs-Algebras; higher order operators within reach 9.1. Higher order Sobolev spaces and Hs-comparison algebras
218
220 228 239
246
250
251
9.2. Closer analysis of some of the conditions (1.) and (m.) J
3
258
ix
9.3. Higher order differential expressions
within reach of C or C
s
262
9.4. Symbol calculus in Hs
267
9.5. Local properties of the Sobolev spaces Hs
272
9.6. Sobolev norms of integral order
279
9.7. Examples for higher order theory;
the secondary symbol
283
291 Chapter 10. Fredholm theory in comparison algebras 10.1. Fredholm theory in C C L(H) 292 10.2. Fredholm properties of operators within reach of C 296
10.3. Systems of operators, and operators acting on vector bundles
303
10.4. Discussion of algebras with a two-link ideal chain 309 10.5. Fredholm theory and comparison technique in Sobolev spaces Appendix A. Auxiliary results concerning functions
315
on manifolds Appendix B. Covariant derivatives and curvature
320
Appendix C. Summary of the conditions (xj) used
324
List of symbols used
327
References
328
Index
340
322
TO HILLGIA
CHAPTER 1. ABSTRACT SPECTRAL THEORY IN HILBERT SPACES.
In this chapter we give a short introduction into spectral theory of abstract unbounded operators of a Hilbert space. In sec. 1 we give a discussion of general facts on unbounded operators. In sec.2 we discuss the v.Neumann-Riesz theory of self-adjoint extension of hermitian operators. Sec.3 gives a general discussion of the abstract spectral theorem for unbounded self-adjoint operators. We discuss a proof of the spectral theorem in sec.4. Also, in sec.5 we discuss an extension of a result by Heinz and Loewner useful in the following. Finally an abstract result on Fredholm operators in a certain type of Frechet algebra related to a chain of Hilbert spaces generated by powers of a self-adjoint positive
operator is discussed in sec.6. The typical 'HS-chain' is a chain of L2-Sobolev spaces. The chapter is self-contained and elementary, and only requires some familiarity with general concepts of analysis and functional analysis of bounded linear operators. 1. Unbounded linear operators on Banach and Hilbert spaces.
The term "(unbounded) linear operator" (between Banach spaces X and Y) is commonly used to denote any linear map A:dom A -+
Y
from a dense linear subspace dom A of X to Y. The space dom A C X then is called the domain of A. Here we distinguish between a linear map X + Y and a linear operator: A linear map X -F Y by definition has its domain equal to X .
The term "unbounded linear operator" will be used with the meaning "not necessarily bounded linear operator", so that the bounded linear operators are special unbounded operators. A bounded linear operator, satisfying (1.1)
sup {UAuU/UuI
:
0
9 u E dom A} < - ,
I.1. Unbounded operators
is necessasily continuous, hence admits a unique extension to X in which dom A is dense. This is why we usually assume that a bounded linear operator also is a linear map. The class of all such unbounded linear operators between two given spaces X and Y will be denoted by P(X,V). In particular the class L(X,Y) of all continuous linear maps X -. Y then is a subset of P(X,Y) .
Since unbounded linear operators are not linear maps from X (but only from their individual domain to Y ) their sum and to V product needs the following special interpretation: For A,BEP(X,Y) ,
and C E P(W,X) we define the sum A+B E P(X,Y) and the product AC E P(W,Y) by setting (1.2) dom (A+B) = dom A n dom B
,
(A+B)u = Au+Bu for u E dom(A+B),
and
(1.3) dom AC = {uEdom C
:
CuEdom Al
(AC)u=A(Cu) for u E dom AC
,
where it is assumed that dom(A+B) and dom AC are dense in X (or else, we will say that dom(A+B) or dom AC is not defined ). Also we define cA = with the identity operator 1 E L(X,X) A linear operator A E P(X,V) is uniquely characterized by its graph, defined as the linear subspace .
(1.4)
graph A =
{(u;Au)
:
u e dom Al
of the cartesian product XxY = {(u;v)
:
uEX
,
v(=-Y}
, where (u;v)
denotes the ordered pair. Vice versa, if for any linear subspace T C XxY the set of all first components is dense in X , and if T does not contain elements of the form (O;u) other than (0;0), then a unique unbounded operator A E P(X,Y) is defined by setting (1.5) dom A = {uEX:(u;v)ET for some vEV}
and then we have graph A = T
, Au=v , for u E dom A ,
.
For two linear operators A,B E P(X,Y) we shall say that A extends B (or that B is a restriction of A ) if graph A D graph B. We then will write A D B (or B C A )
.
Notice that the cartesian product XxY of two Banach spaces is
a Banach space again, for example under the norm II (u; v) II = II uIl +II vII Therefore it is meaningful to speak of a closed subspace of XxY An unbounded operator A is defined to be closed if its graph is
2
I.1. Unbounded operators
a closed subspace of Xxy . The class of all closed operators in P(X, Y)
It is clear that a continuous linear
is denoted. by Q(X, Y) .
map A E L( X, Y) is closed, since (uk;Auk) ->(u;v) implies v=lim Auk
, hence (u;v) = (u;Au) E graph A.
= Au
An operator A E P(X,y) is called preclosed if the closure of graph A is a graph again. Then Ac with graph Ac= (graph A)closure is called the closure of A .
In the following we will be mainly interested in unbounded = P( H, H) , where p is an infinite dimensional separable Filbert space with inner product (u,v) and norm
linear operators A E P( H) Ilull
=
{(U, U) }1/2
.
In that case the graph space HxH becomes a Fil-
bert space again, under the inner product and norm
(1.6)
((u;w),(v;z)) _ (u,v) + (w,z)
, I1(u;w)II
=
(I1 ull2+I1w11
2)112
The following facts, regarding adjoint and closure all work with proper amendments. However for general Banach spaces X Y ,
,
we will get restricted to the case X=Y=H in all of the following. For an operator AEp(H) we will say that the (Hilbert space) ,
adjoint A E P(H) exists if the space TA = J(graph A) is a graph and " L" again. Here J:HxH -. HxH denotes the map (u;w) (w;-u) denotes the orthogonal complement in the graph space (with respect A" Then we define the adjoint to the inner product (1.6) E P(H) -
,
.
of A by setting
(1.7)
graph A
TA
Notice that A , if it exists, is closed, since all orthogonal complements are necessarily closed and since J is inverted by -J
It is clear that A C B implies B C A assuming that A and B exist. The proposition, below, translates the definition of the adjoint into a more transparent form. (The proof is left to the reader. )
Proposition 1.1. Assure that A E Q(H) exists, for some AE: P(H) Then dom A* consists precisely of all u E: H for which there exists
such that
an element v E
it
(1.8)
(u,Aw) = (v,w)
,
for all w E dom A.
Moreover the element v thus defined for each u E dom A determined, and we have A..u=v
is uniouely
.
to
Proposition 1.2. An operator A_E
P(H) admits an adjoint A
if and
3
I.l. Unbounded operators
only if it is preclosed. Moreover, then A C 'xx A and the closure A of A equals A
4
k
also admits an adjoint
,
Proof. Let first A E P(H) have an adjoint Ask E Q(H). Then let T =
(graph A)clos* contain the element (O;z) It follows that there exists a sequence wk E dom A with (wk;Awk) - (0;z) Substitute w = wk in (1.8), and conclude that (u,z) = 0 for all u E dom A .
.
since the inner products in (1.8) allow a passing to the limit.
Since dom A
is dense, it follows that z=0, so that no element of the form (O;z) is in T, except (0;0). Also, T J graph A, which implies that the set of first components of elements in T is dense. Therefore T indeed is a graph of some AcE P(H) and A is ,
preclosed.
Vice versa, let A be preclosed, and let again T be the closure of graph A
the set P of all first components of elements 1If(i.e., of all second components of (graph A) L
in TA=J(graph A)
)
is not dense in H then there exists 0 (z,v) = 0 for all (u;v)E(graph A)
9
z E H with ((0;z),(u;v))=
But this implies that (O;z) (graph A)clos =T. However, for preclosedness of A it is required that T does not contain such elements. Thus the set .
A)H
E (graph
must be dense in H. On the other hand if (O;v) E TA = graph A then (1.8) yields (v,w)=0 for all w E dom A , so that v=0 since V
,
dom A is dense. This shows that then indeed A is a well defined operator in P(H), hence in Q(H) q.e.d. All continuous linear maps in L(H) are closed,hence have an adjoint. Moreover L(H) is adjoint invariant, and, for an A E L(H), ,
the above adjoint coincides with the well known Hilbert space adjoint A of the bounded operator A Also, if A E P(H) is (pre-)closed, then yA+B is (pre-)closed, for every B E L(H), 0 # y E M. Then (YA+B)o=YAo+B, (yA+B) * A*+B in the sense of (1.2)
Our main interest, in the following, will focus on selfadjoint unbounded operators. Here an operator A is called selfadjoint if (A exists,and) A =A First of all a self-adjoint operator allows a spectral decomposition as a direct generaliza.
,
tion of the principal axis transformation of a symmetric matrix. Second we will learn about important classes of differential operators which are unbounded self-adjoint operators, and therefore allow such a spectral decomposition. Third, we will show how results on unbounded non-selfadjoint differential operators can be achieved by " comparing" them with certain standard self-adjoint
I.I. Unbounded operators differential operators.
Note that a bounded operator (i.e., a continuous linear map) A is self-adjoint if and only if it satisfies the relation (u,Av) _ (Au,v) for all u,v E dom A ,
(1,9)
(where dom A = H ).A general unbounded operator A satisfying (1.9) needs not to be self-adjoint, because (1.9) just implies that A D A , not that A = A . Such an operator is called hermitian If the closure of a hermitian operator A is self-adjoint then we speak of an essentially self-adjoint operator(i.e., A*=At ).
Note that a hermitian operator A indeed has an adjoint:If ukEdom A Uk+O , Auk--w, then we may substitute u=uk into (1.9), for fixed v, and pass to the limit, resulting in 0 = (w,v), for all v E dom A It follows that w=0
,
so that A is preclosed, and A
*
exists, by
prop. 1.2. Comparing (1.8) and (1.9), it then follows at once that A E P(H) is hermitian if and only if A C AF If A is (essentially) self-adjoint, and B bounded hermitian then yA+B is (essentially) self-adjoint for all 0#y E R
.
More generally, two operators A, B E P(H) will be said to be in adjoint relation if (1.10)
(u,Av) = (Bu,v), for all u E dom B
,
v E dom A
.
The above conclusion, showing that hermitian operators have
adjoints, can be repeated to prove that operators A, B in adjoint relation must have adjoints (both)
and that,moreover, A and B are *
in adjoint relation if and only if A C B
(or if and only if
BCA* ). One of the first major problems occurring in our discussion of differential operators, in later sections, will be the construction of all self-adjoint extensions of a given hermitian ope-
rator. It turns out that not all hermitian operators possess selfadjoint extensions. On the other hand, the problem of characterizing all self-adjoint extensions was solved by v. Neumann [vN11
and F. Riesz [Ri11. We will discuss the v. Neumann-Riesz theory in section 2, together with some other constructions of self-adjoint extensions.
5
1.2. Riesz-v.Neumann extension 2. Self-adjoint extensions of Hermitian operators.
In this section we discuss the v. Neumann-Riesz theory of self-adjoint extensions of hermitian operators.
It is at once clear that a hermitian operator A has a hermisY st
;:'e
sk
is
C A Since a tian closure Ac = A , because A C A implies A self-adjoint extension B = BC of A is necessarily closed, it also .
must be an extension of the closure A Therefore, in looking for self-adjoint extensions of a given hermitian operator A we may .
look for such extensions of the closure, and thus assume that A is closed, without loss of generality. Also if A C B = B then B = B C A extension B of A is a restriction of A ,
A C B = B
( 2 . 1 )
,
so that every self-adjoint as well: We have
,
C A
Proposition 2.1. A hermitian operator A satisfies the identity (2.2) l1(A-a)u02=11(A-Re A)u112+(Im A)2Hu112
,
for all uEdom A
,
AEC
Proof.We have II(A-a)u11 2=((A-u-iv)u, (A-u-iv)u)=((A-p)u,(A-u)u)
where we have written u = Re A + v2(u,u) - 2Re ((A-u)u,ivu) Here the last term vanishes, due to (Au,iu) + (iu,Au) v = Im A ,
.
i((Au,u)-(u,Au)) = 0 , using that A is hermitian, q.e.d. For a closed hermitian operator A it is implied by prop.2.1 Indeed, consider that im (A-A) is closed for every nonreal A E E a sequence uk E dom A such that (A-a)uk -> v It follows that .
.
(A-X)(uk-ul) } 0 (2.3)
,
as k, 1 1 °°
U(A-A)ull
>
But (2.2) implies the inequality
.
I Im AINO
,
u E dom A
.
Substituting u=uk-ul into (2.3) yields Iluk-ulll - 0, since Im A 90, u for some u E H by assumption. Hence uk and (uk;Auk) -. (u;v) ,
in HxH. But graph A is closed since A is closed. Thus it follows that (u;v) E graph A , or, u E dom A , v=Au .
In the following we first consider the special case A = ±i Since we have im (A±i) closed, by the above, we obtain a pair of
orthogonal direct decompositions
1 (2.4)
H = im(A±i) $ V.
,
D..
= (im(A±i))
The two spaces D+=D+(A) are called the defect spaces of the closed
6
1.2. Riesz-v.Neumann extension hermitian operator A indices of A
.
and their dimensions are called the defect
,
We write
def A = (dim(im(A+i))
(2.5)
i ,
dim(im(A-i))
L (v+
JA-) )
Note that the defect spaces D,. are just the eigenspaces of We have the adjoint operator A to the eigenvalues ±i :
D+ _ (im(Ati))1 = ker(A* i)
(2.6)
Indeed, f E D+ ,for example,amounts to 0 = (f,(A+i)u) for all u E dom A
We write this as (f,Au) = (if,u) concluding that f E dom A*
.
with (1.8) X =
±i
(2.7)
,
,
,
u E dom A A*f = if
,
and compare
As another consequence of prop.2.1 we note that (2.2) implies
for
,
=
(IIAuN2+llull2)1/2
n(Ati)ull =
hl(u;Au)H
u E dom A .
,
Note that graph A. as a closed subspace of HxH is a Hilbert space under the norm and inner product of HxH Moreover graph A is in linear 1-1-correspondence with dom A which may be used to trans.
fer that Hilbert space structure of graph A to dom A In other words, dom A is a Hilbert space under the (stronger) norm .
(2.8)
IIuIIA =
(Ilull2+llAull2)1/2
with inner product (2.9)
(u,v)A =
(u,v) + (Au,Av)
,
u,v E dom A
The latter is true for every closed operator B E 0(H)
,
not
only for hermitian operators. In particular we may apply it to the adjoint B=A* of our closed hermitian operator A , obtaining a cor* and inner product (u,v)
responding norm llull
A
,k
,
u,v E dom A
A
In fact, we then get Pull
Pull A ,for u E dom A C dom A* , and A
*
dom A appears as a closed subspace of dom A under graph norm. Note that (2.7) may be interpreted as follows: The two ope,
In fact these rators (A±i) are isometries dom A ; im(A±i) = D. im(A-i)-+im(A+i) isometries are 'onto'. Therefore V =(A+i)(A-i)71 .
:
defines an isometry between the two closed subspaces of H Proposition 2.2. For a closed hermitian operator A we have *
(2.10)
dom A
= dom A ® D +(A) ® D _(A)
7
1.2. Riesz-v.Neumann extension
8
as an orthogonal direct decomposition of the Hilbert space dom A* under its norm and inner product. Proof. We already noticed that dom A is a closed subspace of
* * dom A , under graph norm. The eigenspaces D+ of A are closed subspaces of H , as nulspaces of the closed operators A*'+i. Moreover,
on D+ we have
_ V Hu II ,
nu II *
so that D+ are also closed under -
A
graph norm. For f+ E D+ one confirms that (f+,f ) *= 0, using that - A *
A f+ _ -+if+. Also, for u E dom A
,
(Au,±if+) + (u,f+)
(u,f+)
A 0, so that the three spaces in the decomposition ±i((A±i)u,f+) = *
(2.10) are orthogonal. Suppose f E dom A
satisfies 0
=
(f,u) * A
=(A*f,A*u)+(f,u) for all u E dom A. Comparing this with (1.8) it
is found that A*f E dom At, hence f E dom (A*)2, and (A*)2f+f = One also may write this as (A*+i)(A*-i)f = (A*-i)(A*+i)f = 0 *
*
*
0.
*
In particular, f E dom (A +i)(A -i) = dom (A -i)(A +i), in the sense of (1.3). Now we write f = (A*+i)f/2i - (A*-i)f/2i = f++f noting that (A +i)f+ = 0. This proves that every f orthogonal to and thus is in D+®Ddom A , under graph inner product of A* completes the proof. ,
,
Corollary 2.3. A closed hermitian operator A is self-adjoint if and only if its defect indices vanish (i.e. def A = 0 ). Or, for both, "+" and equivalently, if and only if im (A±i) = H ,
Indeed, A = A
*
:e
implies dom A = dom A , so that (2.10) gives D+ = {0}, hence def A = 0. Vice versa, if def A = 0 , (2.10) gives *
*
*
dom A = dom A , hence A =A since A DA , q.e.d. We now state the v. Neumann-Riesz extension theorem. ,
Theorem 2.4. The closed hermitian extensions B of a given closed hermitian operator A are in 1-1-correspondence with the extensions W W- -> W+ of the isometry V = (A+i)(A-i)-1 im(A-i) + im(A+i) This as an isometry between the closed subspaces W. D im(A±i) :
:
.
correspondence is established by assigning to B D A the operator W = VB = (B+i)(B-i)-1 (which is an isometry extending V between the spaces 0+ = im(B±i) D im(A±i)). Vice versa, given an isometry one must observe that W , as W :W--t W+ extending the isometry V ,
an extension of V, is determined by its restriction W0=WIW°, where W0= W n(im(A-D) = W nD is a subspace of the defect space D and ,
where W0 is just any isometry W0, W0 = W+nD+C V+. Then we have the
1.2. Riesz-v.Neumann extension closed hermitian extension B given by dom B = dom A $
(2.11)
{W0-
E W0}
:
,
where the direct sum again is orthogonal in (.,.) * A
.
The proof is almost self-explanatory. It is clear from the above that W = (B+i)(B-i)-1 is an isometric extension of V
for
,
every closed hermitian extension B of A . Vice versa, that W0 determines W
,
as described, follows from the well known fact that
isometries preserve orthogonality. Then, of course, the operator B
,
if it exists,should satisfy
(2.12)
W(B-i)u = (B+i)u
u E dom B = dom A $ ZO
,
,
with a certain subspace Z0 C D+$D-, because dom A C dom B C dom A* and due to (2.10)
.
For u E Z0 let m = Bu-iu
X = Bu + iu = W0
,
It follows that u = (W04)-4))/2i, Bu = (WO4)+4))/2, in agreement with (2.11). Now one simply must verify that the operator B of (2.11)
is closed and hermitian. The closedness follows if we show that * ZO = 4 E WO } is a closed space, under graph norm of A {W0$-4
But we have (2.13)
11W0 _O*2 =
11W04)_$112
+
11W04)+4)112
=
211W04)112 +
2114)112
=
4114)112,
is, in fact an isometry (up to the factor 4). Since W is closed, Z0 also is closed. To verify that B of (2.11) is hermitian is only a calcuwhich shows that the map 4)-W04)-4), taking W0 onto Z0
,
lation; since we know that Bldom A = A is hermitian one must show that (AiYu,v)=(u,A*v) for all u,v E Z0, and for u E dom A, v E 20. Both follow trivially, q.e.d.
Theorem 2.4 has the following important consequence. Corollary 2.5. A closed hermitian operator A admits a self-adjoint extension if and only if def A = (v,v), with v=0,1,2,...,- arbi-
trarily given. In other words, we must have (2.14)
codim im(A+i) = codim im(A-i)
.
Then every self-adjoint extension B of A is obtained by picking an arbitrary isometry W0
:
D_
D+
between the two defect spaces
(2.6), and then defining B with (2.11)
,
and WO , with WO±=D±
The proof is evident.
Although the v.Neumann-Riesz theory completely clarifies
9
1.2. Riesz-v.Neumann extension
the problem of self-adjoint extensions, other criteria are useful, of course, because the construction of isometries between defect spaces is not always practical. In particular not every closed hermitian operator A satisfies the condition (2.14), so that a self-adjoint extension need not always to exist. There are two well known general criteria giving existence of self-adjoint extensions. Shortly, 'real' hermitian operators as well as 'semibounded' hermitian operators always have self-adjoint extensions. The concept of real operator refers to a given involution u
u of the Hilbert space H
.
In most applications we will have
H = L2(X,dp) with some measure space X and measure dp
,
and then
refer to the complex conjugation u(x) - u(x) of the complex-valued function u(x)EH=L2. However, one may think of an abstract space K and an involution map u -r u (U-)_ = u
,
with the properties
(c1u+c2v)
,
= clu +c2v
(2.15)
(u ,v
)
_
(v,u)
,
for u,v E H
,
cjE T
,
j=1,2.
Then a real operator is defined as an operator A E P(H) satisfying (2.16)
(dom A)
= dom A
and
(Au)
= Au , for all u E dom A
Now, if a closed hermitian operator A is real with respect to any such involution of H
,
then one confirms at once that
D+- _ fu-: u E D+} = D-
(2.17)
.
Indeed if f E D+, i.e., (f,(A+i)u)=0 for all u E dom A,then we get ,((A+i)u)_)=(f
0
= ((A+i)u,f)=(f
,'(A-i)u ), hence f E(im(A-i))
using (2.16). This conclusion may be reversed, so that (2.17) follows. Also it is clear that D+ and D+- have the same dimension. We have proven:
Proposition 2.6. A closed hermitian operator A which is real
with
respect to some involution of H has equal defect indices and hence admits a self-adjoint extension.
A hermitian operator A E P(H) is called semi-bounded below, if there exists a real constant c such that
10
1.2. Riesz-v.Neumann extension (2.18)
(Au,u)
> c(u,u)
for all u E dom A .
Similarly one speaks of semi-boundedness above if (2.18) holds
with ">" replaced by "<". In such case c is called an (upper or lower) (semi-) bound of A . Theorem 2.7. A semi-bounded hermitian operator (regardless whether above or below) admits a self-adjoint extension. We will prove thm.2.7 by constructing a distinguished self-adjoint extension B of a semi-bounded operator A , where B has the same semi-bounds as A. This operator B will be referred to as the Friedrichs extension of A (cf. K.O.Friedrichs [Fr2], and M.H.Stone [St11). In the construction of B we assume that A is semi-bounded below, and that c=1 is a lower bound. The general case may be re-
duced to this by considering the operator ±A + y with suitable real y and sign ± .
In the case of c=1 we conclude from (2.18) that (u,v)..=(Au,v)
=(u,Av) defines a positive definite inner product in the space dom A C H (lull-
=
.
One may complete dom A under the corresponding norm containing
((u,u)-)1'2, and obtain a new Hilbert space H
dom A as a dense subspace. It is easily confirmed that every class of equivalent Cauchy sequences under 11.11., is contained in a unique class of equivalent Cauchy sequences under 11.11 an imbedding of H"" into- H
,
.
This provides
so that we get dom A C H- C H
.
Now the Friedrichs extension B of A is defined as the restriction of A (2.19)
4e
to the space (dom A )nH-
dom B = (dom A )CH"
.
Bu = A u
,
In other words,
u E dom B
.
First of all, B still is hermitian: for u,vEdom B =H"'fldom A
there exist sequences uk
vk E dom A with uk'u
in H" by construction of H-. Then (u,Bv) = (u,A v) = lim (uk,A v) _ ,
,
= lim (Auk,v) = lim (uk,v)_ _ (u,v)-= lim (u,vk)-=
(Bu,v)
Also it follows that (2.20)
(u,Bu) = (u,u). >
(u,u)
,
for all u E dom B
,
so that B has the same lower bound 1 as A .
In order to show that B is self-adjoint refer to (1.8) and Now consider the linear functional 1(u) = (g,u), for g as, above and all let (f,Bu)=(g,u), for all uEdom B, and a given f,gEH
u E dom A . We get 11(u) l
<
II glI II ull
<
.
II gll II ull _ , using (2.18). Hence
11
1.2. Riesz-v.Neumann extension
12
1(u) is a bounded linear functional over a dense subspace of H-, and we may write 1(u) _ (g,u) _ (h,u)- , with some h E H-, by the Frechet-Riesz theorem (or Hahn-Banach-theorem). As above, we find that (g,u) _ (h,u)- _ (h,Au), for all u E dom A. Therefore it also
follows that h E dom A = dom B
and Bh = g.
,
*
*,
and that A h=g. Therefore, h E H-
om A
We get (f,Bu) _ (g,u) = (Bh,u) = (h,Bu)
,
or (f-h,Bu) = 0, for all u E dom B. Note that we also have shown,
with the above conclusion, that im B = H, because the construction of h works for every g E H. Thus it follows that f-h=0. Or f = h E dom B and Bf = Bh = g. This proves that the Friedrichs extension B is self-adjoint, and completes the proof of thm.2.7 Note that, in the special case of the lower bound 1 consi.
B-1
dered we find that the operator is bounded and hermitian, since we proved that im B = H , while (2.20) implies injectiveness u,v E dom B implies of B, so that B-1 exists. Also (Bu,v)=(u,Bv) f,g E H Furthermore, (f,B-lg) = (B-lf,g) ,
.
,
IIBuIl2= II(B-1)u+u112=11(B-1)u112+llu112+2((Bu,u)-(u,u)) >
(2.21) or, IlBul
>
B_1(=- L(H)
Pull ,
,
IIB-111
u E dom B , which yields < 1
PB-1fII
(1/2)(B-1-l)0
.
(2.22)
,
or
be the unique
=
bounded positive self-adjoint square root of B-1 B-1
f e H
The proposition, below, will be useful.
.
Proposition 2.8. Let C = B-1/2=
=
,
Ilull2
,
satisfying C2
Then C:H -> H" is an isometry between H and H"
H- = im C
,
and (lull
= PCuF`, for all u E H
.
We have C-1, with dom Cr -1H"' C H a self-adjoint operator in P(H),
with lower bound 1. Moreover, the restriction C-11dom A still is essentially self-adjoint.
The proof is left to the reader (cf.also sec.4). 3. On the spectral theorem for self-adjoint operators.
The resolvent of a closed operator A E Q(H) is commonly defined as the inverse R(a) _ (A-a)-1
similarly as for linear maps. More precisely, the resolvent set Rs(A) is defined as the set of all X E M such that (3.1)
im(A-A) = H
,
,
and P(A-A)ull > cllull
,
u E dom A ,
1.3. Spectral theorem, general discussion with a positive constant c
.
It is clear that (3.1) holds if and
only if the linear map A-A between the spaces dom A and H has an inverse (A-A)-1
H -> dom A C H which constitutes a bounded
:
operator of H, with 11(A-A)-lII
well defined in this sense,for all A E Rs(A) of A is defined as the complement of Rs(A)
The spectrum Sp(A)
.
Sp(A) _
:
cC
\ Rs(A)
It is easily seen that Rs(A) is an open set,so that Sp(A) must be closed: Note that (3.1), for a=a0, and IA-A01
11(A-A)u°I
>
II(A-A 0)uII_ja_a0j11u11
> (c/2)IIuII
,
u E dom A
so that the second condition holds for a neighbourhood of A0 Regarding the first condition we observe that (A-A)u = (1+(a0-A)R(A0))(A-A0)u
(3.3)
,
u E dom A
by a simple calculation. We know that im(A-A0) = H , and boundedness of R(a0). For small JA_a0) the first factor at right of (3.3) is of the form 1+ E takes H onto H A-A01
,
,
11E11
< 1 , hence is invertible in L(H)
,
and
Thus (3.3) shows that im(A-a)=H for all small
.
and Rs(A) is open.
Now we conclude from (3.3) that R(a) = R(A0)(l+(\0-A)R(A0))-1
(3.4)
,
JA-A01
< c
where only bounded operators in L(H) occur. It is evident that the right hand side of (3.4) provides a norm convergent power series expansion of the operator R(a) in powers of (A-A0) for A close to \0
.
,
Therefore it follows that R(A) is an analytic
function from the resolvent set Rs(A) to L(H)
.
Let us now return to self-adjoint operators. Theorem 3.1. For a self-adjoint AE Q(H) we have Rs(A) J cC\
g, i.e.,
Sp(A)C 1, and
(3.5)
IIR(A)fl
<
I
If in addition A>c (or A
IIR(A)II
< (dist(A,(c,oo)))-1
,
(or IIR(A)II< (dist(A,-oo,c)))-1)
A hermitian AC=P(H) is essentially self-adjoint if(f) for some im(A- 7) dense in H. If in addiA, Im A d 0, we have both im(A-A) tion A>c (or A
13
1.3. Spectral theorem, general discussion real (y
c) we have im(A-y) dense in H Proof. The estimate (3.1), for a non-real is an immediate consequence of prop.2.1, which also gives a value of the constant c .
).
in agreement with (3.5). On the other hand, for X = ±i the first condition (3.1) is a consequence of cor.2.3. In fact, it is noticed that the conclusions leading to cor.2.3 can be repeated for every pair of non-real complex conjugate numbers. Therefore we indeed get the first statement and (3.5). On the other hand, for a semi-bounded operator we must repeat the conclusions leading to prop.2.8., particularly (2.21). Details are left to the reader. The spectral theorem for unbounded self-adjoint operators provides a Fourier type integral representation of every self{E(X) a E 1} adjoint operator by a so-called spectral family :
In this section we only will give a general discussion of the spectral theorem, while the proof will be discussed in sec.4. A bounded hermitian operator A semi-bounded below by 0 will be called positive, and for two bounded hermitian operators A , B we will write A < B (or B > A) if B-A is positive. An orthogonal projection (here shortly 'projection') of the Hilbert space H is a bounded hermitian operator P satisfying P2 = P (i.e. an idempotent). One easily verifies Proposition 3.2. The orthogonal projections P of H correspond to the orthogonal direct decompositions of H :
(3.7)
= im P $ ker P
H
For u E H we get u = v + w
,
.
corresponding to (3.7) , where v = Pu
and Pw = 0. For any direct decomposition H = M ® N with M , N orthogonal the assignment u - v defines a bounded hermitian idempotent
operator P such that M = im P , N = ker P Pu=u in M Moreover, for two orthogonal projections P,Q we have P
.
and only if im P C im Q and then Q-P is the orthogonal projection onto the complement (im Q)n(im P)1 . In particular this holds ,
if and only if (3.8)
PQ = QP = P
.
Proof. For u E im P get u=Pv Pu=P2v=Pv=u Hence for w E ker P get (u,w) = (Pv,w) _ (v,Pw) = 0, so that im P and ker P are ortho,
.
gonal. Also, for z E H write z=Pz+(l-P)z, where clearly Pz E im P so that we get the P(1-P)z = (P-P2)z=0 , hence (1-P)z E ker P ,
14
1.3. Spectral theorem, general discussion
desired orthogonal direct decomposition. Vice versa we confirm easily that for an orthogonal decomposition H=M$N the map given is a hermitian idempotent. If P,Q are projections, and im PCim Q
,
then we get H=im P
1 ® (im Q)f(im P)
where P = 1,0,0
® ker Q
an orthogonal direct decomposition,
,
and Q = 1,1,0 in the three spaces. Hence Q-P= , which shows that Q-P projects onto the second space. Also
0,1,0
,
,
we then clearly get (3.8). Vice versa,for projections P,Q, let P
_ (u,(l-Q)u) < (u,(l-P)u) =
0
((1-Q)u,(l-Q)u)=(u,(1-Q)2u)
=
, hence Qu=u
,
and im P C im Q
Then, of course Q-P projects as described, and we have QP=PQ=P. Finally, if PQ=QP=P
,
one computes (Q-P) 2=Q2+P2-QP-PQ=Q-P
so that Q-P is a projection. If (Q-P)u=u then find Qu=Q(Q-P)u=u, Pu=O
,
and vice versa, so that Q-P projects onto (im Q)n(ker P),
as stated, q.e.d.
A spectral family is defined to be an increasing one parameter family {E(A) :AE g} of orthogonal projections which is left continuous, in strong operator convergence of L(H) and such that (3.9)
lime-SE(A) =
, limX-EM = 1
0
,
also in strong convergence. In particular, (3.10)
E(A) < E(p)
,
as l < u
>
and
lima}u-OE(A) = E(u)
(3.11)
It is clear from prop.3.2 that the projection operators E(A)
of a spectral family all commute, and that HA = im E(A) forms an We have increasing family of closed subspaces :
HA C Hu ,
(3.12)
as A < p
.
For similar reason it is found that, for a spectral family E(A),the two limits (3.13)
E(A+0) = limo}A'u>aE(p)
, E(A-0) =
in strong operator convergence of H similarly as for a real-valued function. In fact for p > A we have exist at each A E IR
15
1.3. Spectral theorem, general discussion
(3.14)
= (u,E(u)u) - (u,E(X)u)
II(E(p)-E(A))u112
uEH
,
,
hence the upper and lower limits will exist, just as they do for the monotone real-valued functions (u,E(A)u), u E H. Using (3.14) for an orthonormal basis of (the separable space) H one easily except concludes that E(a) is strongly continuous over all of I ,
perhaps at certain 'jumps', and that there are at most countably many jumps. Then the existence of the limits E(±-)=lim E(A) in strong convergence may be derived from the monotony of E(a) only, without the explicit requirements (3.9) .
For an interval A = (a',a"] of the real axis, left open and right closed, let EA=E(a")-E(a'). If we have a partition (3.15)
a'
=
A0
< Al <
.
..
< AN-1 < aN =
a"
,
of A into finitely many sub-intervals Aj = (aj_l,aj]
,
j=1,...,N,
then we get a corresponding orthogonal decomposition
im EA = s T=1 (im EA )
(3.16)
.
J
For arbitrary u E H we get (3.17)
IIEAu1l2
= N_1 IIEA ul12 J
in view of the orthogonality. One may use this fact to define the
J (A)dE(A), for a continuous function
Riemann-Stieltjes integral
A
(a), as the limit, in norm convergence, limS,0' Q , with Riemann sum S and maximum interval length S (with points a.E[aj-l,x.}), where Jj S =
(3.18)
d
Here the existence of the limit
= Max {(aj-aj_1) :j=l,...,N} for continuous functions
is
easily derived from the observation that, for the Riemann sum S' of any refinement of the partition (3.15),we have
(3.19)
II(S-S.')ull < /_60EAull
,
by the same orthogonality. (Since any pair of partitions, with sums S,S"
and maximal lengthes S,d" has a common refinement sum
,
S1, we get 11S-S1'11<,/_6+,/_6
, insuring convergence of the integral.)
Also, (3.20)
11
A
A
A
16
1.3. Spectral theorem, general discussion
with real-valued Riemann-Stieltjes integrals at right. For u E H and a function ¢ E CU) one then may look for existence of the improper Riemann-Stieltjes integral Iu =
It is
found that Iu exists if and only if (3.21)
111u112
=
(This follows by applying (3.20) to the difference IAu-IA,u , with two large intervals A
,
A'
,
and IAu = Jo $(a)dE(A)u .)
In the theorem, below, we will say that an unbounded operaif we have BA C AB , with tor A and a bounded operator B commute ,
products defined by (1.3)
.
One confirms easily that this holds
true if and only if B commutes with the resolvent R(X)=(A-a)-1 for every A E Rs(A) .
We now can state the spectral theorem.
Theorem 3.3. For every self-adjoint operator A E Q(H) there exists a unique spectral family {E(A)} , with E(a) commuting with every commuting with A bounded hermitian operator B such that ,
J-XdE(X)
A =
(3.22)
,
,
in the sense that (3.23)
dom A = {uEH
:
and (3.24)
Au =
J-AdE(X)u
,
with an improper Riemann-Stieltjes integral at right. formula (3.22) Vice versa, for every spectral family E(a) and E(A) commutes with every ,
defines an unbounded operator A
,
bounded hermitian operator, commuting with A . 4. Proof of the spectral theorem.
In this section we discuss a proof of thm.3.3. Note that there is a selection of proofs, mainly differing by the type of
17
1.4. Proof of the spectral theorem construction used for the family E(A)
.
Here we define E(A) as
the orthogonal projection onto the null space of the bounded operator SA = TA+(TA2)1/2
(4.1)
with the resolvent R(p) _
Tx = (R(A+i)+R(A-i))/2
,
(A-u)-1 of the self-adjoint operator A.
More precisely, since the above would give a right (not left) continuous spectral family, the above family of projections is called and then E(A) = lim u-,a,u<XPu = PX-0 For a similar proof cf. Nagy [NSz1], [RN]. PX
,
.
For a self-adjoint operator A the resolvent R(p) is defined (and holomorphic) in all of U\1 (thm.3.1), and we have (A-p)'
_
= A-1i , hence R(p)
Therefore the operator TA of (4.1) is bounded and self-adjoint, since we may write TA=(R(u)+R(u)")/2, with u=A+i
= R(u)
.
In fact we get IIT II<1 , using (3.5) . Accordingly we a and the binomial have TA2 bounded self-adjoint and O
,
series of prop.2.8
,
i.e., 1--0(1/2)(TA2
U _ (T2)1/2 =
(4.2)
- 1)0
j
J
defines a bounded self-adjoint operator Ux satisfying O
for x=1, as well known. We get -1<(Tx2-1)<0<1, hence IITx2-111<1
(using prop 4.3), so that the series (4.2) converges absolutely. Thus the Cauchy-type reordering of the product is correct, just as for a series of numbers, and we get
U
The coefficients of the series are real hence all terms are self-adjoint. Moreover, all terms but the (1+(TA2-1))
TA2
.
0-th power are <0
,
=
and one estimates -Ii < 1
,
so that UA=1+1i20.
It is clear that TA and Ux commute, by definition of Ux (We have TAUA = UATA). Accordingly, 0=(TA2-UA2)=(TA+UA)(TX-UA). For a uEim TA we get u=TAv=(TA+UA)v/2 + (TA-UA)v/2 = w+z , where w = (TA+UA)v/2 E ker (TA-UA) z = (TA-UA)v/2 E ker (TA+UA) using that TA and Ux are Also, (w,z) = 4-1(v,(TA2-UA2)v) = 0 ,
,
self-adjoint. This means that w and z are orthogonal. Since TA is self-adjoint, we have (4.3)
clo H = ker TA $ Um TA)s.
18
1.4. Proof of the spectral theorem which is an orthogonal direct decomosition. Also, ker TA = ker U If TAu=O, then UA2u = Tx2u = 0 hence IIUAuIi2 = (u,UA2u) = 0, so that ker TIC ker UA and by symmetry we get equality. It follows ,
,
that ker(TA+UA) n ker (TA-UA) = ker TA = ker UA (4.4)
Hx = ker(TA+UA)
,
Nx = ker TA
.
Define
y
,
KA= ker(TA-UA) n (Nt ),
and conclude from the above that
H = Hx ® KX
(4.5)
,
with an orthogonal direct decomposition. Also, (u,Txu) = -(u,Uxu) <0 on HA = ker(TA+UA), (4.6)
(u,Txu) = (u,UAu) >0 on K, C ker(TX-UA) Finally, in this chain of arguments, note that
2Tu = R(A+i)u + R(A-i)u = R(A+i)((A-A+i)+(A-A-i))R(A-i)u
= 2R(A+i)(A-A)R(A-i)u , uEH
,
implying that
TA = R(A-i) (A-A)R(A+i)
(4.7)
Combining this with (4.6) we get (4.8)
(R(A-i)u,(A-A)R (X-i)u) <
0
, as uEHA
,
>0
,
as uEKx
.
Furthermore we observe that R(v)R(p) = R(p)R(v), for every p,vET, which also implies that R(p) commutes with TA and U . We conclude we have from there that, for all p E ®\R ,
R(p)HA = dom A n Hx , R(p)Kx = dom A fl K .
(4.9)
Indeed, for uEHx we get R(p)uE dom A R(p)(TA+UA)u =
0
,
,
so that R(p)u E dom A n HA
and (TA+UA)R(p)u = .
For uE dom A fl HA
one may write u=R(p)v, v=(A-p)u, and then 0=(TA+UA)u=R(p)(TA+UX)v. conhence vEH, Since R(p) is 1-1, it follows that (TA+UA)v=O ,
,
firming the first equation (4.9). Similarly one finds that R(p)(ker(TA-UA)) = dom A n ker(TA-UA) . Also it is clear that R(p)(ker TA)= ker T, = ker(A-A) C dom A , using (4.7) and (4.8),
19
1.4. Proof of the spectral theorem so that also the second (4.9) follows. One then finds that dom A = (dom A n HA)
(dom A n KA)
in the sense that every uE dom A allows a unique sum decomposition u=v+w with v,w in the spaces at right. Such decomposition arises if (A-p)u is subject to the decomposition (4.5), and then R(p) is applied. Let R(4)X = R(u)IHA then it follows that ,
and im R(p)' is dense in HA , while R(p)A is 1-1, hence has an inverse in Q(HA) and thus finds that One gets R(u)A .
,
AX = (R(p)A)-1 + u defines an unbounded self-adjoint operator in Q(HA) with domain dom A n HA Moreover, AX is the restriction of A to that space, and is independent of p. Similarly for the restriction of A to KX. .
We summarize the above: Theorem 4.1.
We have
(4.10)
ker(A-A) C HX , ker(A-A) n K. _
{0}
and
(4.11) (u,Au)>A(u,u), uEdom AnHA, and (u,Au)
,
,
,
(4.12)
A:(dom A n H ) - HA , A:(dom A n KA) X
5
while dom A n H, , and dom A n KA are dense in HA and KA , respectively, and the restrictions of A to these spaces are unbounded self-adjoint operators of the Hilbert spaces HA and KA resp. Thm.4.1 already holds the principal information of the ,
spectral theorem, at least as far as existence of the spectral resolution is concerned. We complete this as follows. Corollary 4.2. For every self-adjoint operator B E L(H), commuting with every R(p), 4ET\l the restrictions BX=BIHA ,and BA=BIKA ,
are maps HA+HA and KA+KA , respectively In other words, every bounded self-adjoint operator B, com.
muting with the resolvent R(u)
is reduced by the direct decom-
,
position (4.5).
Indeed, such an operator B commutes with TA, hence with U,, As above we thus conclude that given as a power series in TA-1 .
20
1.4. Proof of the spectral theorem (TA+UA)Bu=B(TA+UA)u=0 B: KA+KA
,
as u E H ,
so that B: HX.HX . Similarly
.
It is convenient now to introduce the orthogonal projections PA by setting
P Au=u on H, , P Xu=0 on K,
(4.13)
.
Then, as a consequence of cor.4.2 we find that
(4.14)
PAPX, = PA,PX ,
for all A,
EA .
At
Indeed, the operator PA is bounded and self-adjoint, and commutes The latter is an immediate consequence of thm.4.1:
with R(4)
.
For u E HA we get R(p)u E HA and uEKA implies R(u)uE KA , hence PXR(p)u=R(p)u=R(p)Pu , uEHX , and (1-PX)R(p)u=R(p)u=R(p)(l-PX)u ,
uEKA
,
or also PR(p)u=R(p)PXu , for uEKX. Since H=HX®KX we get
PXR(p) = R(p)PX . Self-adjointness and boundedness of PA is a con-
sequence of the orthogonality of HA and KA Using cor.4.2 it thus follows that B is reduced by any of .
the other direct decompositions (4.5) (4.15)
.
Thus again, we get
PA,PXu=PAu=PAPX,u, uEHV (1-PA,)PXu=PA(1-PA,)u, uEKX,,
implying (4.14).
Next let A',AER
,
a'
.
For uEH., write
u = PAru = PA,PAu + PX,(l_PX)u = v+w .
(4.16)
The first term, at right, is EHA, and the second term wEHXtnKX using the commutativity (4.14)
.
Thus the second term satisfies
A(w,w) < (w,Aw) < A'(w,w)
(4.17)
.
since A'
,
HA, C HA ,
(4.18)
as At
.
Now we invoke prop.3.2, and conclude that JP X:AER}
,
is
a nondecreasing family; we have (3.10). As discussed in sec.3, this implies existence of the monotone limits (3.9) and (3.13)
and these limits must be orthogonal projections, since the functional equations P2=P and P = P are continuous. Thus we may set E(A) = PX-0
,
as postulated.
21
I.4. Proof of the spectral theorem
In particular E(--) and E(w) are the projections onto H_ _ nHx, and Hm (uHx), respectivey, with union and intersection over I
.
For uEH__ we must have (u,Au)
< A(u,u) for all AER , which is
only possible, if u=0. For uEK.= (H.)1 we must have (u,Au)>a(u,u) for all AER
implying u=0 again. This yields H
,
or, E(-=) = 1-E(w)
=
(K.)-L= 0
,
confirming (3.9). Similarly we conclude = 0 (3.11), by construction. Therefore E(A) indeed is a (left conti,
nuous) spectral family, while {PX} is right continuous. Next consider an interval A=(a',a"] a, with finite EA as in sec.3 observe that
and a partition (3.15). With EA
,
7
= EX (1-EX). Thus (4.10) implies
EA
J-1
J
J
(4.19)
HE
Aj
ull2
-< (E Aj u,AE Aj u) -< aJ IIE
In particular one gets (4.19) first for uE dom A n H a space dense in im EA
n H
= Hx J
J
n K
aj
aj_1
by a direct sum argument,
,
j-l
, uEH
Ajull2
as in the proof of thm.4.1. This implies boundedness of (the restriction to im EA of) A in im EA , using prop.4.3, below. J
J
Since that restriction is a closed operator, it follows , hence im EA
= im EA
dom A n im EA
7
J
C dom A J
From (4.19) we conclude that (4.20)
< (EAJu,(A-x.
0
3-1
)EAJu) < (Aj-aj_1)(u,EA7u)
,
uEH
Summing this over j one has (4.21)
with S and
0
,
< (EAu,(A-S)EAu) <
of (3.18)
,
for a
=
a.
(u,EAu) ,
,
and q(a) =
X
. Here we
j
used that (EA u,EA u)= (EA u,AEA u)=0 for j91 1
J
J
1
Now we use (4.21) and prop.4.3, below, and conclude that AEA = lA A dE(a)
(4.22)
with an operator norm convergent Riemann-Stieltjes integral.Then, AE u , which equals the for u E dom A we get Au = lim improper Riemann-Stieltjes integral of (3.22). Vice versa, if that ,
integral exists, or, equivalently, if
{-a2dIIE(A)u112
<
,
then
22
1.4. Proof of the spectral theorem E6u}u , AEAu
JIR
AdE(A)u
,
23
so that uE H, and (3.24) holds, by
closedness of the operator A.
This completes the proof of thm.3.3, except for uniqueness of E(a)
.
Now, if QX is another spectral family such that QX commutes with every self-adjoint BEL(H) commuting with A (i.e., with every R(u)), then we get (4.23)
QxPU = PuQA
,
using that Pu commutes with R(v) by construction. Then the argument leading to (4.14) may be repeated, showing that also ,
in QX, C HX C im QA , as A'< A < A"
(4.24)
Then, by monotony of QX we get (4.25)
QX-O < PX < QX+O
resulting in equality QX=PX=E(X) at all points of continuity of QX. Since both E(A) and QX are left continuous we get E(A)=Qx also at the 'jumps' of QX using the fact that im(QX+O-QX-O)=ker(A-X). ,
This completes the proof of thm.3.3.
Proposition 4.3. Assume that a hermitian operator T satisfies (4.26)
l(u,Tu)j
< c(u,u)
,
for all u E dom T
,
with some nonnegative constant c. Then T is bounded, and we have 11 T11
.
Proof. Apply (4.26) to u+v and u-v both, and subtract the two inequalities, for (4.27)
-2c(11u11 2+11v112)
< 4Re(u,Tv) <2c(Ilull2+Iivll2), u,v E dom T
Here we introduce u=Tv/c (if necessary, let u run through a sequence converging to Tv/c). It follows that (4.28)
(2/c)IITvll2 <
(1/c)IITvll2 + clivll2
which implies the statement.
, vE dom T
.
I.S. Powers of positive operators 5. A result on powers of positive operators. In the lemma,below, let H be a separable Hilbert space.
Lemma 5.1. Let B E L(H) be self-adjoint, and let A be an unbounded self-adjoint operator with domain dom A . Assume A positive, and B positive definite, so that B-1 exists as an unbounded selfadjoint operator, and the positive powers As Bs , s>O are well defined positive self-adjoint operators. If AB is bounded, then also ASBS is bounded, for every 0<s
,
< IIAB U5 .
IIASBS N
If AB is compact, then also ASBs is compact, for every O<s
condition that the unbounded operator product AB has dense domain and extends continuously to H Indeed, if dom A D in B , then .
the product AB has domain H , while we get (AB)* D BA , due to self-adjointness. Hence dom(AB) * is dense, which implies boundedby the closed graph theorem. If AB has dense domain ness of AB and a continuous extension to H , then for all uEH there ,
e Sc
exists uk E dom (AB) with uk-, u , ABuk+ v= (AB) bounded we also get Buk-> Bu , A(Buk) - v
.
u Since B is The selfadjoint opera.
tor is closed. Thus we conclude that Bu E dom A , hence dom A D in B Thus indeed all 3 above conditions are equivalent. .
Remark (c): Clearly boundedness of ASBs in lemma 5.1 again may be interpreted in each of the ways of rem.(b). Also, clearly, since compactness of AB implies boundedness of AB , all of remark (b) applies in the case of AB E K(H) , where K(H) denotes the (closed 2-sided) ideal of compact operators in L(H). The proof of the lemma will require some preparations.
Let BEL(H) be self-adjoint, positive definite. Let H0 be the completion of H under the norm 9up0 = {(u,u)0}"2 , with inner product (u,v)0 = (u,Bv) = (Bu,v)
,
u,v E H
.
From simple arguments it
24
I.S. Powers of positive operators follows that H may be regarded as a dense subspace of HO, Now let us assume that A with domain dom A is an unbounded closed operator of H which is hermitian with respect to the inner product (.,.) 0 That is, we have (u,Av)0 = (Au,v)0 ,for all u,v E dom A Also let the resolvent set Rs(A) of A contain at least one real point X0 .
Proposition 5.2. The linear map A:dom A } H0 defines an essenIf A0 denotes the closure of tially selfadjoint operator of HO .
A in H0, then the resolvent set Rs(A0) contains Rs(A). Moreover, (5.2)
II(A0-a)-1110 <
II(A-a)-111 for all A E Rs(A) (A-A0)-1
Proof. Let R = R(A0) _
E L(H)
.
.
We get ((A-X0)u,v)0 =
(u,(A-X0)v)0 for all u,v E dom A , which implies (f,Rg)0 = (Rf,g)0 for all f,g E H tian in H0
In other words, the operator R E L(H) is hermi-
.
.
Proposition 5.3. We have R bounded in HO
and,
,
<
IIR110
IIRII
Proof. With a spectral family E(a) of the operator B let P. 1 - E(E)
for e
,
of H the norms
S
Bull2
>
0
and let S. = im PE
,
11.11 and
11.11
0
.
In the closed subspace
are equivalent. We have
Ellull2<
IIBIIIlull2. The projection PE: H I SE is orthogonal with
respect to both inner products (.,.) and (.,.)0 . Define R. PEAIS£
=
.
_
(PEu,Rv)
_ (u,REv) Now it is easy to see that we must have IIREII0 ,
_
(u,(PRISE)v) (u,REv)0 = (P6u,Rv)0 = (u,Rv)0 = (Ru,v)0 = (REU,v)0
For u,v E SE we get (u,Rv)
<
IIRE11
Indeed, the
.
spectrum, and hence the spectral radius p of R. is independent of the norm under which SE is considered. For a bounded hermitian operator we get IIREII0
=
P
, while generally IIRE11 > P
Thus we
.
indeed have the above inequality. It follows that (5.3)
sup{j(u,Ru)I/(u,u)
0#uESE} <
IIP6RII
<
IIRR
.
is dense in But the space of all u with u = P E u for some E > 0 since B was assumed positive definite. The proof is complete. H ,
,
Continuing in the proof of prop.5.2 it now follows that the linear map A:dom A - H0 is an essentially self-adjoint operator, since
(A-A0)-1
is bounded and densely defined (cf. thm.3.l).
The closure of this operator (in H0) is self-adjoint. Its resoland is a normal operator vent exists for all non-real points A ,
wherever it exists. Also, at each point A of the resolvent set
25
I.S. Powers of positive operators
Rs(AO) we find that A-X must be invertible as a linear map, at (A0-a)-1
least, and the inverse is the restriction of to H trivially have Rs(A)\l C Rs(A0) since Rs(A0) contains E\1
.
We
,
For every real A E Rs(A) prop.5.2 applies as well as to A0 thus Rs(A) C Rs(A0) Hence we also get RS(A)r C Rs(A0) ,
,
as stated. Finally, for real A E Rs(A) prop.5.3 implies the inequality (5.2). For non-real A E Rs(A) we know that A E and that (A0-a)-1 is normal. If P and p0 denote the spectral radii of (A-A)-1 and (A0-X)-1 then again pO= II(A0-a)-111,
Rs(A0)
p < II(A-a)-111 and we must have p0 < p since p and PO are just the reciprocal minimum distances from A to Sp(A) and Sp(AO) , and ,
(Rs(A))comp J tes the proof of prop.5.2 as well. since Sp(A)
(Rs(A0))comp=
Sp(AO)
_
.
This comple-
Proposition 5.4. Under the assumptions of prop.5.2, if we have A E K(H) then also A0 E K(H0) .
Proof. The resolvent set Rs(A) of an operator A E K(H) is known 9e to be the complement in 1 = (E\{0} of an at most countable set of eigenvalues, each of finite algebraic multiplicity, and with only possible cluster point at zero. In particular, for each eigenvalue A
#
0
,
(5.4)
the spectral projection P
=
i/(2Tr)Jr R(u)de
,
R(u) = (A-0-1
a
with a sufficiently small positively oriented circle rA of center A not containing any other eigenvalue, is a bounded operator of ,
finite rank. The resolvent R(A) is a holomorphic map Rs(A)- L(H), and the integral (5.4) exists in norm convergence of L(H) Now we apply prop. 5.2 to conclude that Sp(A0) C Sp(A) so that Sp(A0)\{0} is a collection of some of the eigenvalues A (A0-u)-
of A For each such A we get R0(u) _ as the continuous in view of (5.2), and uniformly so extension of R(u) onto HO .
,
over r . It follows that (5.5)
R(u)du O
PO = i/(2,r)Jr u
Since PX is of finite is the continuous extension of PA to HO rank, this is true for PO as well It follows that all non-vani.
.
shing points of Sp(A0) of the self-adjoint operator AO are isolated eigenvalues of finite multiplicity. This proves that A0 is compact and our proof is complete. Proof of lemma 5.1. First we introduce the inner product (.,.) 0
26
1.5. Powers of positive operators and space H0 with B of lemma 5.1
Then if AB E L(H)
.
27
it may
,
be observed that (u,ABv) 0 = (u,BABv) _ (ABu,Bv) _ (ABu,v)0 for all u,v E H so that AB satisfies the conditions of prop.5.3. ,
,
It follows that IIA1/2B1/2112= IIB1/2ABl/211 have used the C -algebra property of L(H)
IIABIIO<
= ,
(i.e.
where we
IIABII
=
II X112
IIX"`XII
,
for X E L(H)), and the fact that B1/2 defines an isometry H - HO, for X E L(H0) so that 11X110 = IIB1/2XB-1/211 Accordingly, we have .
,
proven (5.1) for s = 1/2 Similarly, if AB E K(H) , we may apply prop.5.4 instead of prop.5.3 to conclude that B1/2AB1/2 .
E K(H)
Since Bl/2AB1/2 = that ASBS E K(H) for s=1/2
(Al/2B1/2(A1/2B1/2)
it then follows
.
In a similar way one now proves the lemma for all rationals of the form s = k/21 0 as follows. (We only indicate < k < 21 ,
,
how to proceed for s=1/4 and s=3/4 to make the method evident) For s=1/4 we may just repeat the above, now using (u,v)O = B1/2
as the operator A of prop.5.2 For s and Al/2 so if AB E L(H) we already know that Al/2B1/2 E L(H) = 3/4 B1/2A3/2B = (A1/2B1/2)'`AB E L(H) The operator C is that C = as easily checked. hermitian with respect to (u,v)0 = (B1/2u,v) (u,B1/2 v)
.
,
,
.
,
Hence it also is bounded in the corresponding norm. With the isometry B1/4.H-> O H we get that IIA3/4B3/4II2= IIB3/4A3/2B3/411 UC110 < UCH
<
IIA1/2B1/2IIIIABII
thus get (5.1) for s=3/4
.
=
Taking square roots, we If AB E K(H) we find that C E K(H) < IIABII3/2
.
so that prop.5.4 yields ASBs E K(H) for s=3/4 Note finally that boundedness of ASBS= E
.
is equivalent to s
the inequality (5.6)
IlAsull
< IE 1111B-SuII for all u E dom B-s
-
.
s
< 11E11IS is correct for all rationals r=p/q, then it must be true for all 0<s
If (5.6) with 11Es1I
q=21
,
0<s
,
.
B is bounded selfadjoint we have dom
B-s
= imBs decreasing, as s 1.
increases. Let s
.
be a sequence of rationals with denominators 2
2
J
with sji s, sj<s. Then u E dom s
C dom A J
s ,
B-s
B- sj
C dom As implies u E dom
s.
and B- Ju -* B-su , A Ju -- Asu
,
as j ;
.
Using this
in (5.6) we conclude that this inequality is correct for all 0
< s < 1 if it is true for s=l
.
Finally, regarding compactness of ASBS for general s we will have to discuss the function F(z)=AZBz of the complex argument z
.
1.5. Powers of positive operators For u
,
v E H define the function (z) _ (u,AZBzv)
28
, where it may
be noted that the general powers AZ and BZ are well defined also for complex z The expression (z) is meaningful for 0 .
<
assuming that AB is bounded, and using our knowledge on boundedness of ASBs for 0<s
,
.
z
,
z
we get A B v = Ait(AsBs)(Bity) well defined, since the operators Alt and Bit are unitary, of course. We get (5.7)
IIABII5IIuIIIIvII
,
0<s
,
--
Note that t(z) also is analytic in the open strip 0<s
if we
,
assume that, for some O2 tion 1 = X(X) + w(X) 0< X,w E C°°(R) X=l for )
.
,
,
,
we write, with some 0<6<Min{c,Rez}
,
,
A)AbX(A))u,(AZ-dBz-d)(BSv))
fl(z) _ (((log
+
(5.8) ((A6u),(AZ-dBZ-d)(Bdlog
((log A)w(A))u,(AZBZ)v) +
B)v)
.
All expressions at right are meaningful, in view of the selfadjointness of A and B using that B is bounded and positive definite, while A is possibly unbounded, but still positive. In parti,
cular, the function X(X)X6 log , extends bounded and continuously = O(TE) in [0,.) and adlog ) = 0(1) to [0,-) , while w(a)log ,
on the bounded set Sp B C [0,o) Note that the restriction u E dom AE can be removed. For a .
general u E H let the sequence ukE dom AE converge to u in H get k(z), with uk instead of u, analytic in 0<s
Wz)-Wz)I
=
I(u-uk,AZBZv)I
<
.
We
IIABIISIIvIIIIu-uk'
which implies uniform convergence Wz) -> (z) in the entire strip is bounded and analytic in 0< Re z <1 Accordingly for any arbitrary u,v E H For a given 0<s
(5.10)
n-1 Max{IO(C)I:Ic-sI=n}
<
Ilunllvil Max{1,IIABH)/n.
From this estimate we conclude that
(5.11) with
(u,(ASBS-AtBt)v)
_
sdTOu,v(T) = 0(Is-tIIIu1111viI)
t
0(I s-tI II ull 11 vll) < c(s,t) I s-tI II un 11 vll
,
the constant c(s,t)
I.S. Powers of positive operators independent of u,v
and bounded in compact subintervals of (0,1).
Note that (5.11) implies the proposition,below. Proposition S.S. Under the assumptions of lemma 5.1, if AB E L(H)
then the operator family F(s) = AS BS
,
continuous (even Lipschitz continuous) (5.12)
IIASBs - A t B t I
,
0
< s< 1
,
is norm
We have
.
< c(s,t)js-tl
Moreover, the corresponding is true for the more general family F(z)=AZBz 0 < Re z < 1 We then get (5.12) with s,t replaced ,
.
by z,z', with c(z,z') bounded in conpact subsets of the strip. We will use only (5.12), not its complex extension. Prop.5.5 also provides the final link in the proof of If AB E K(H) then certainly AB E L(H) so that prop.5.5 applies and ASBS is norm continuous in the open interlemma 5.1
.
,
Since AS BS is compact in the dense subset of all rationals of the form s=k/2 1 it must be compact for every s E val (0,1) (0,1)
.
since the compact ideal is closed in norm topology. of course, is not in K(H). Accordingly Notice that A0B0=I ,
,
it is clear that F(z) = AZBz cannot be norm continuous at z=0 6.
.
On HS-chains.
In this section we shall discuss an abstract result on certain families of Hilbert spaces similar to the so-called Sobolev spaces. For a concrete example cf. [C1],ch.III and IV, where corresponding facts are derived for the Sobolev spaces over Rn (we shall apply the abstract result in VI,3 and ch.IX below). Let H be a (separable) Hilbert space and let H be a selfadjoint operator of H with domain dom H
.
We assume H semi-bounded
below and that 1 be a lower semi-bound, i.e., H>l
Also we assume Note H properly unbounded, i.e., that its spectrum extend to +_ that the complex powers HS, sEU are well defined closed operators. .
.
For example we may use the spectral family E(X) of H and define (with As = es log A log A real-valued) ,
(6.1)
HS = I'AsdE(A), dom HS ={uEH:
Ix2Re(5)d11E(x12<.}
For a formal convenience, useful later on, we introduce the inverse positive square root A Hs=A-s/2 write .
=
H-1'2 , and henceforth will also
29
1.6. HS-chains
For every sEg let H
denote the Hilbert space obtained by
completing dom Hs/2 = dom A s C H under the norm
(6.2)
ilulls
=
IIA- sull .
Evidently we get Hs = dom AS C H for s>0 and the norm 11.115 is equivalent to the graph norm of the unbounded operator A-5. For ,
s<0 the norm 11.l1 s is weaker that the norm of H , whence Hs DH
For example one may consider Hs
,
tion of all 1-parameter families up, as p
,
and that
Ilull2
=
.
s <0, represented by the collecsuch that E(p)u, _
JX2Re(s)dlluX112
<.
.
The same represen-
tation of uEHs as a function of ), also holds true for general sEiR. In this way one finds that Hs is naturally imbedded in Ht as s>t, so that {Hs:sE1R} is a decreasing family of spaces. We still introduce the locally convex spaces (6.3)
H_ = n{HS:sER}
,
H__ = U{Hs:sElk}
,
where the union carries the inductive limit topology, while H carries the Frechet topology induced by all the norms 11.0s sE1. A collection of spaces Hs, -=<s<0, as constructed above will :
be referred to as an HS-chain. It is natural to extend a projection E(a) of the spectral
family to an operator E(X)-_: H- -, H_ by defining E(X)__u=uA for u={up>1}, as X>1 , and E(X) u=0 for X
Proposition 6.1. The space H its topology. Proof. (6.4)
For u E H5
,
Qu-uX112 =
is dense in every Hs,
IsL<=
,
under
s<0, we clearly have u, = E(a)_.u E H. J_p2Re sdllE(p)u112 + 0
,
and
as Ai.
For general uEH- we must have uEHs for some 5E1. Then uA+u in Hs as above, implying u1+u in H q.e.d. ,
We now define the order classes O(m).
Definition 6.2. For mE1 we denote by O(m) the class of all linear operators A E L(H operator AEP(H H
)
such that, for every sEl the induced unbounded ) with dense domain H CH extends to a (conti-
s s s-m nuous) operator AEL(Hs,Hs_m) (from Hs to Hs-m) It is clear that an operator AEO(m) also admits a continuous
extension to H__ . In particular we must have Asu=Atu for u E
30
1.6. HS-chains HsnHt=Hr
, r=Max(s,t)
Clearly every 0(m) is a linear space, and
.
AB E 0(m+m') for all AEO(m)
(6.5)
31
,
BEO(m')
.
Accordingly 0(0) and 0(o) = U0(m) are algebras and 0(-oo)=ff0(m)
is an ideal of both algebras.
In particular we have A-m E 0(m) (i.e., more precisely, the restrictions A-mIH0, are contained in 0(m) , for every mEt ). MoreA-C
over the operators (A-C)
found that
are in 0(Re ) for every GC . Also it is
s constitutes an isometry Hs-*Hs-
, E=Re C , for every real s and complex 0 . Also E(a) (or rather E(a)IH.) is an
and E(A). above is its canonical extension. One may introduce a locally convex topology on 0(m), men, using the collection of all operator norms operator of 0(-W)
(6.6)
RAN
,
HA
s,s-m=
II
s
s-m=
s,
sup{UA uU s
s-m
:
uEH s,
OUR
<1)
s
-
,
5E1
.
Theorem 6.3. The topology of (6.6) is a Frechet topology, and 0(m) is a Frechet space. In particular we have (6.7)
RAN
<
r,r-m -
UAU(t-r)/(t-s)
s,s-m
IIAII(r-s)/(t-s)
s
t,t-m
< r < t
Proof. It is sufficient to discuss the last statement. Moreover, we may consider the case m=0 only, since RANpip-m=11AmAnpip= HAmAIIP.
This is a matter of Calderon interpolation (a straight application of the Phragmen Lindeloef principle, cf.[Sel]): For fixed s
(6.8)
,
z E T
.
It was seen before that As is well defined for real and complex s. We have 2d/dz(As(z)w)=-(t-s)As(z)log H w, with complex differention, for every wEHw. Thus f(z) is analytic in T. Write f=( Ps
Am-s(z)
,AA
m+s(z
q) a P= A-mu s 1 =A-mv s Ps4 EH ms
m=max{IsI
> tI),
to see that f is bounded. For Re z=0, z=iy, get s(z)=s+iy(t-s), hence, If(z)I=I(Aiy(t-s)
(6.9)
u,A-sAAsAiy(t-s)v)I
using that Alp is a unitary operator of HO = H and that IIAUs=UA-sAAs110
(6.10)
.
If(z)I
,
for all real p
Similarly, for Re z = 1,
< IIAUtHlui01IvI10
.
1.6. HS-chains
From the Phragmen-Lindeloef principle we thus conclude that (6.11)
lf((r-s)/(t-s))
IIuIIDIIv11IIAllst-r)/(t-s)IIAlltr-s)/(t-s)
<
Also note that f((r-s)/(t-s)) _ (u,A-rAArv)
.
Setting u=A-rAArv
and using that (6.12)
IIAIIr
=
IIA-rAArll
= sup {IIA-rAAruq: uEH
we conclude (6.7), q.e.d. An operator AEO(m) is said to have order m .
Hull 0<1}
,
Since A is
determined by any one of its extensions As we often will refer to an operator A:H -*H and say that AEO() (or that A is of r r-m order m) whenever AIH, EO(n) Next let us discuss adjoints of operators in L(H ). For .
s
A E L(Hs) the standard interpretation of adjoint would be that of Hilbert space adjoint, an operator of L(H ) again. However, s
for the present purposes it is more practical to regard H_s as using the 'pairing' the adjoint space of Hs ,
(6.13)
{u,v} - (u,v)
Here we write As =
(Asu,A_sv)
, where u E Hs
(A-5)S for the isometry
,
v E H_s
Hs}HO=H mentionned
above and (.,.) denotes the inner product of H=H0. A continuous linear functional 1: Hs-}1 may be uniquely written in the form as an immediate 1(u) = (f,u) = (A_sf,Asu) , with some f E H_s consequence of the Frechet-Riesz theorem. One then may regard ,
the adjoint Banach space Hs identified with the space H_s. Accordingly for A E L(H ) we define the adjoint operator s
A
as an operator in L(H_s)
(6.14)
(u,Av) = (A*u,v)
,
defined by the relation
,
u E H-s
,
v E Hs
,
with the pairing (.,.) of (6.13). Similarly, for AE L(H5,Ht) there exists a unique operator A*EL(H_t,H_s) such that (6.14) holds for uEH_t, v E Hs
.
Proposition 6.4. For an operator AEO(m) there exists a unique 1:
operator A EO(m) such that (6.15)
(As )*
_
(A*) m-s
exists and define contiProof: For AEO(m) all the adjoints (As) Moreover, these maps coincide for s and t, nuous maps H -H m-s -s .
32
1.6. HS-chains on uE H
m-s
nH
m-t
,
due to
, vEH
(As u-At u,v) _ (u,(A5-At)v)=0
(6.16)
since H0 is dense in every H
It follows that (A )*IHt = A* s takes H_ to itself and is independent of s . Moreover, A* admits A*s continuous extensions defined by (6.15), and hence has all
r
.
et
norms IIAIIs m finite. This implies that A EL(H.) , q.e.d.
Now we come to the main result of this section, a result on Fredholm operators in 0(m). Recall that an operator AEL(Hs'Ht) is called Fredholm if
we have a(A)=dim ker A -, 6(A)=dim(Ht/im A)<-. A Fredholm operator always has closed image, and it follows that 6(A.)= dim ker A*. Here we may interpret A as the above adjoint operator A :H-t+H-s. The Fredholm index of A is defined as ind A = a(A)-S(A). It is known that Fredholm property and index are invariant under small bounded and arbitrary compact perturbations. For a more detailed discussion cf.X,l. For fully detailed proofs cf.[C1],AI. Definition 6.5. An operator AEO(0) is said to be K-invariant under H-conjugation if we have AS AS A-1 - AO = K(s) E K(H)
(6.17)
with the above isometry As:Hs+H0
,
,
s E R
defined as As=(A-s)s
Theorem 6.6. Let AEO(0) be K-invariant under H-conjugation, and let A0 be Fredholm. Then all operators As5 sER, are Fredholm, and independent of s Moreover, we even have we have ind A = const. .
s
*
ker As and ker(A ) s independent of s
ces of H-
,
and both spaces are subspa-
.
Proof. Since As and its inverse are isometries it is trivial that As is Fredholm if and only if ASAA-1= AO+K(s) is Fredholm. The latter is correct, since A0 is Fredholm, and K(s) is compact. Similarly we get ind A = ind A A A-1 = ind(A +K(s)) = ind A s
s s s
0
0
In particular ind As is independent of s On the other hand (6.18)
ind As = dim ker(A5) = dim ker As - dim ker(A )-s
using (6.15). Here both terms at right cannot increase, as s increases: Indeed we must have ker As D ker At as s
33
1.6. HS-chains
as s
,
so that S(As) = dim ker(A )-s is nondecreasing.
It follows that both dimensions at right of (6.18) also must be constant. Theorefore the two null spaces do not depend on s and must be contained in H.
.
This completes the proof.
Remark 6.7. Let P and Q denote the orthogonal projections in H=HO
onto ker AO and ker A
respectively. Recalling that
0,
Qu = I'Pj4j,u) , uEH
(6.19)
with orthonormal bases
,
} of the finite dimensional
¢.} and
spaces ker A0 and ker A 0 respectively, it is clear at once that P,QE 0(-o) , using that cj, 0j E H_ , and using the pairing (6.13).
Later on we shall construct a distinguished Fredholm inverse B0 of A0, such that B0A0 = 1-P , A0B0 = 1-Q It follows thus that such operator B0 is an inverse even mod operators H_.-H. of finite .
rank (called a Green inverse, later on). Problem 6.8. Let F- denote the class of all operators T E 0(-o) of the form T = If j)(gj
(6.20)
,
fj, gj E H_
,
i.e.,
Tu = If.(gj,u) , uE H-. , with (.,.) of (6.13)
.
gj and a finite sum. Show that the closure of F. in the Frechet topology of 0(0) coincides with the class of
with given fj
,
,
all AE 0(0) such that ASE K(HS) for all sE Ik Hint: Repeat the proof of [C1],p.144, lemma 3.2 in abstract form. ,
.
34
CHAPTER 2. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS.
In the present chapter we will apply some of the abstract principles developed in ch.l to the case of a differential operator. Here a differential operator appears as a realization of a differential expression aa(x)D°
L =
(0.1)
,
xEQ
where H is an open subset of Rn , and multi-index notation is used (cf.sec.l). The coefficients as will be complex-valued functions over 2 Later on we also will consider matrix-valued .
coefficients, or differential expressions on manifolds, acting on crosssections of vector bundles (cf. X,3). The term realization is discussed in sec.l, below.
All of our discussion will be restricted to hypo-elliptic expressions only. In fact, our main effort will be devoted to elliptic expresions, and even only ordinary differential expressions. In sec.l we discuss the v.Neumann-Riesz extension theory for the minimal operator of a hypo-elliptic self-adjoint expression. In sec.2 we discuss generalized boundary conditions, in the regular and singular case. In sec's 3 and 4 we focus on the spectral theorem of an essentially self-adjoint realization of an ordinary differential expression. Specifically a detailed proof is discussed in the case of a second order expression. 1. Linear differential operators on a subdomain of
In
In the present section we will be concerned with differential operators, as a special class of unbounded linear operators, regarding their spectral theory. It is convenient to use multiindex notation of In aj
= 8/2x
= ax J
,
J
,
Dj
as follows = -iaj
>
D =
(D1,.... Dn)
'
a = (°l....,an)
II.1. Differential operators
a Da = D1a1D2a2...D
generals
,
36
a n
8a = 81a1...an n
,
a
xa = x1a1...xn n
for x = (x1,...,xn)
,
,
etc.
For multi-indices a,6 it also will be convenient to abbreviate lal
1
= al + ... + an
,
a!
= al!...an!
(6)
I
z
(61)(62)...
=
.
Suppose we are given a differential expression aa(x)Da as E C(C) a _
L =
,
adjoint L* is defined, also a differential expression,by (Lu,v) = (u,L*v)
(1.2)
u,v E CO(C)
,
with the inner product and norm of the Hilbert space H = L2(C) (u,v) = J,,u(x)v(x)dx
(1.3)
,
=
(lull
Iul2dx)1/2
{J Q
By a partial integration one finds that, for all u,v E CO (Lu,v) =
dx
J
C
(1.4) =
la
laI
a
a
()D uu v =J S2 dx laI
, w =
la
since all boundary terms cancel, for such functions. A differential expression is determined by the values Mu, for all u E CO since they determine all aa Thus M is determined by all values of (u,Mv), for u,v E CO, since C 0 is dense in H.Accordingly we have
(1.5)
Lt
Daaa
=
laT
Using Leibnitz'formula (1.6)
P(D)(au) = J(D6a)(P(S)(D)u)/R!
,
PM(o
_
(3Yp)(
)
6
*
one may write L
in the form (1.1) again, with new coefficients
as E CO. Note that as = as for IaI=N
,
so that the principal part
LN (i.e,the sum of highest order terms) of L conjugating the coefficients of LN A:X--Y
is obtained by
Generally, a differential operator will be a (linear) map between spaces of functions or distributions X and Y such ,
II.1. Differential operators
,
generals
for all u E X , with some differential expression L. Normally, unless stated otherwise, we shall assume the expression that Au=Lu
,
L to have C'-coefficients, and that C'(D)C X C D'(D) , with the space DI(D) of all distributions over H.
Conventionally spectral theory is considered only for operators defined by hypo-elliptic differential expressions. An expression L (of the form (1.1)) is called hypo-elliptic if the differential equation Lu=f admits only infinitely differentiable solutions, whenever f is C'
,
for all open subsets Sl' C Q.
(Speaking precisely, every distribution u E D'(H') solving Lu = f in H' necessarily is C°°(H'), and a classical solution of Lu =
f.
In [C3] we will discuss criteria for hypo-ellipticity of a differential expression (cf. also [Ho1] [Ho2])
.
Here we note that
an elliptic differential expression necessarily is hypo-elliptic. A differential expression L of the form (1.1) is called elliptic
if, for all x E H
,
and all 0 # E E In
,
aa(x)Ea #
0
(1.7)
ja =N
Particularly an ordinary differential expression (1.8)
L =
a.()D1 0<
,
a < x <
with x=(x1) E il, and 0 = (a,8) C 1, an open interval, is elliptic if and only if aN(x) #
0
,
for all x E (a,g).
We shall require hypo-ellipticity only in the slightly weakened form of lemma 1.2, below. Detailed proofs of such weaker
regularity property, in the cases of an ordinary expression L, or a second order strongly elliptic expression are discussed in 11,2, and III,1.
Presently we shall study unbounded operators of the Hilbert space H above, in the sense of I,1, which are differential operators. For a given differential expression L one first introduces the minimal operator L0 and the maximal operator L1 defi,
ned by setting Lju = Lu for u E dom L. (1.10)
dom LO = C'(Q) C H
,
,
j=0,1
,
where
dom L1 = {u E C°°(H)
Clearly dom L0 and dom L1 are dense in H
,
:
u, Lu E H}
so that L0 and L1 are
unbounded linear operators in the sense of I,1. We have L0 C L1.
An operator A E P(H) with L0 C A C L1 will be called a realization of the expression L. (In later chapters we will use the term
37
II.1. Differential operators
,
generals
'realization' somewhat more generally for an operator AEP(H) with C A C L1* . Actually this notation is in general use for unbounded linear operators A of more general Hilbert or Banach
spaces in a similar relation to the (closures of) the minimal and maximal operator. For the present chapter we appoint that L0 C A C L1 is required for a realization.) For a realization A we must have LOAD ADD L1 D (L)0 implying that dom Ax D dom (L") 0 is dense in H. Thus A is preclo-
sed. Note that realizations normally are not closed. Apart from realizations also the (restrictions of) their closures and (Hilbert pace) adjoints are differential operators in the distribution sense, as easily verified. ,t
A differential expression L is called self-adjoint if L =L. For a self-adjoint expression L the minimal operator clearly is hermitian, but the maximal operator in general is not hermitian. For spectral theory of differential operators one will be interested in the self-adjoint extensions of L0 As a restriction of L1 .
a realization cannot be self-adjoint, but at most will be essentially self-adjoint. In view of the Riesz-v.Neumann extension theory of 1,2 the question arises whether all self-adjoint extensions of L0 are closures of essentially self-adjoint realizations of the given expression L.
As a general assumption let us assume that all expressions L-A, for A E M are hypo-elliptic. Then L will be called strongly hypo-elliptic. If L is not of order 0 and is either elliptic or
of constant coefficients and hypo-elliptic, then it is strongly hypo-ellitic. Similarly for the general formally hypo-elliptic expressions fitting into the definitions of [C3],III, (having a local parametrix of negative order). All this is discussed in
detail in [ C3]
.
Lemma 1.1. Let the expression L be self-adjoint and strongly hypoelliptic. Then the defect spaces D+= (im(L0±i))L are contained in dom L1 and are just the eigenspaces of L1 to the eigenvalues ±i respectively. Moreover, every eigenvector of L0 also is an eigenvector of Ll.
Proof. It suffices to prove the last statement.Let L0u = Au, then (1.11)
(u,(L-A)v) =
0
,
for all v E CA(D) = dom L0
Since L is self-adjoint and u E H C D'(D)
,
.
(1.11) implies that
38
II.1. Differential operators , generals (L-A)u=0
,
xES2
,
in the distribution sense. Thus u E C°'(c), since
by assumption L-A is hypo-elliptic and the function 0 is C°°(D). Then Lu=au in the sense of standard partial derivatives. Since uEH we get Lu=au E H, hence u E dom Ll which completes the proof. The lemma, below, is a straight extension of lemma 1.1. Lemma 1.2. Let L be hypo-elliptic, not necessarily self-adjoint. Suppose for some f E H and g E C°'(c)nH we have (f,L*u) _ (g,u)
(1.12)
for all u E C0(D)
,
Then we have f E C°°(D), and, moreover, f E dom L1
.
,
Lf=g
,
x E n
with standard partial derivatives, possibly after changing f on a set of measure zero. Again the proof is immediate if we realize that (1.12) for an L2-function f implies that f solves Lf=g in the distribution sense. A function f E H satisfying (1.12) (for some gEH) is called a weak L2-solution of the differential equation Lf=g. Evidently such f is just a distribution solution which also belongs to H Lemma 1.2, for the special case of a second order elliptic
expression, is generally known as Weyls Lemma. It was proven by H.Weyl for the potential equation [We2) We will refer to the .
property of an expression to have at least every weak L2-solution
in C°° (for gEC°°) as (the property) of Weyl's lemma. This property is weaker than hypo-ellipticity, but it will be sufficient for all of our purposes. It will be more convenient to use since we do not have to refer to distribution calculus. For the remainder of sec.l let us assume that he expression **
L is self-adjoint. For the closure L0 of the minimal operator (a closed hermitian operator) we now have the well known direct decom
position (1.13), below. Moreover, it is orthogonal with respect to (u,v)+(L*u,L*v) (cf.I,(2.9)). the "graph inner product" (u,v)L 0
**
dom LO
(1.13)
=
dom LO
®D+ ® D_
Let us introduce the amended (slightly larger) minimal operator LOS
LO as the restriction of to dom.,LO = C°° n dom LO. As a restriction of the closure Lp* of L0 ,L0 can be reached from L0 by
That is, each u Edom L0 is the limit of a sequence un closing E C' such that also Lun 3 LOu. We get (LOu,v)=(LO*u,v)=(u,LOv) .
=(u,Lv)=(Lu,v) for u E dom LO
,
vECO(D), using (1.4). Since C-(D)
39
II.l. Differential operators
,
generals
40
is dense-in H, we get L0u = Lu for all u E dom L0. In other words, LO is a realization of the expression L. ,
Notice that also L1 is the restriction of L0 to by the same conclusion: We get dom L1 C C- n dom LO
L0, ,
using (1.4).
Also, L0f=g, for some f E C', amounts to (1.14), below, using (1.4) again. (1.14)
(g,u)
_
(f,L0u) = (Lf,u)
This implies L0f = Lf E H C-ndom LO
.
,
for all u E C'O.
,
so that f E dom L
, hence dom L1 D
Thus L1 indeed is the stated restriction.
Theorem 1.3. For a self-adjoint, strongly hypo-elliptic expression L we have the algebraic decomposition (1.15)
dom L1 = dom L-0
+
D+
D_
+
where every pair of terms at right intersects at 0 only. Every self-adjoint extension A of L0 (or L"'0) is the closure of a unique essentially self-adjoint realization B determined as the
restriction of A to dom A n C'(D) = dom B. Moreover, B is determined as a restriction of L1 to the space (1.16)
dom B =
dom L0 +
{
(u-Wu)
:
u E D- }
,
D+ is an isometry onto D+ , with D+ Hilbert spaces, where W: Das closed subspaces of H. Also B is uniquely determined by W, and (1.17)
Bu = Lu on dom L0
,
B(u-Wu) = -i(u+Wu)
,
u E D_
The decomposition (1.16) of dom B is orthogonal, with respect to the inner product (u,v)L=(u,v)+(Lu,Lv), defined for u,v E dom L1. The proof of (1.15) is a consequence of Lemma 1.1, and the facts on L0 and L1 derived. The remainder of the theorem then follows from the v.Neumann-Riesz theorem (cf. I, thm.2.4 ).
Theorem 1.3 amounts to a strong simplification of spectral theory, for hypo-elliptic expressions, because all discussions may be restricted to C"-functions, and no generalized derivatives, Sobolev spaces, etc., have to be considered. This is under the assumption that simple criteria for hypo-ellipticity can be derived, which in turn do not use generalized derivatives. Note, that the concept of distribution may be avoided, above, because all conclusions already follow, if the statement of Weyl's Lemma
II.1. Differential operators
generals
,
is known to be true.
Let us still mention the Carleman alternatives. Carleman [Call, in essence, arrived at the conclusions of thm.1.3 in the case of a Schroedinger operator L = H = -A+q 'potential' q(x)
where def L0= 0
.
,
,
for Q=fin
, with a
He distinguishes between the definite case and L0 possesses precisely one self-adjoint
extension, and the indefinite case, where def L0 = (v,v) Similarly, in the case of an ordinary differential expression, H.Weyl speaks of the limit point case, and the limit v>0
.
circle case, respectively, with a notation referring to his special construction. 2.Generalized boundary problems;ordinary differential expressions. In section 1 we derived a direct decomposition (1.15) of the domain of the maximal operator, assuming that the expression (1.1) is strongly hypo-elliptic and self-adjoint. In particular we concluded that all self-adjoint extensions of the (hermitian) minimal
operator L0 are given as closures of certain restrictions of the maximal operator L1, called e.s.a. realizations of L. In turn the e.s.a. realizations are characterized by an isometry W:D- -+ D+ between the defect spaces D+
,
similar as in the v. Neumann-Riesz
theorem (I, thm.2.4).
For a given e.s.a.-realization A of the expression L the condition
uEdom A ', imposed on a function uEdom L1, amounts to a generalized boundary condition, because if u e dom L1 satisfies '
u E dom A , then so does u + v = w , for all v E C0
,
due to C0
C dom A. In other words, the condition u e dom A is not influenced by the behaviour of u away from the boundary, it depends only on the properties of u in some (arbitrarily small) neighbourhood of the boundary of the domain D. In this sense we associate to every e.s.a.-realization of L a (generalized) boundary problem
One will be tempted to ask for classes of e.s.a.-realizations with boundary conditions of the conventional type. For an investigation of this kind involving partial differential operators, and more generally, dissipative and accretive boundary conditions cf. [ CFO]
.
In this and the following section we will get restricted to n=1, i.e.
,
to the case of an ordinary differential operator.
Spectral theory of "ODE's" was completed to a high degree of per-
41
II.2. Boundary problems
,
ODE
fection, starting with the work of H.Weyl [We1]. From the large number of contributions to this subject we mention the work of
Hilb [ H11] , K.Kodaira [ 1 < 0 1 ] E.C.Titchmarsh [Ti1]
,
M.G.Krein [ Kr1] , N.Levinson [Lei] ,
,M.A.Naimark [Ne1], E.A.Coddington [Cd1],and
others (cf. also the monographs [ Ne2] , [Oh] ,
[ Bz1] , [ RS] , [ CdLi] )
We write L =
(2.1)
J jN=0
a.(x)3
3
with extended real numbers a<S
.
a < x <
= d/dx
For simplicity assume that a. E
C°°((a,s)) are real-valued. The adjoint of L is given by (2.2)
L
3=0(-1)030a.(x)
_
_ (-1)NaN3N+ terms of lower order
Assume L*=L, and L elliptic (i.e., aN#0). Particularly aN=(-l)NaN follows from (2.2), so that N must be even. Also assume NiO, so that N=2,4,6,...
.
We give a proof of Weyl's lemma (i.e., Lemma 1.2), for this case. Lemma 2.1. If (Let-X)f = g, for some g E HnC°°(S2), then fEdom L1.
Proof. We require a fundamental solution of L, that is, in distribution terms, a solution e(x,t) of the differential equation (2.3)
(Lx-a)e(x,t) = 6(x-t)
with the'Delta-function'
6
.
a < x,t <
,
S
,
In simple terms this is a CN-2-func-
tion in the square of (2.3) being CN in each of the two triangular regions a < x < t < g
and a
Moreover, at x=t we require the'jump condition' (2.4)
3N-le(x+O,x)
-
3N-le(x-O,x) = 1/aN(x)
,
a < x <
It then is easily verified that (2.3) holds.
Note that the general existence and uniqueness theorem for ODE-s, and the known results on differential dependence of solutions on initial data at once supply the existence of (an infinite family of) fundamental solutions: One may prescribe arbitrary initial data 3Xe(t±O,t), subject to (2.4) and 3Xe(t+0,t)=3Xe(t-0,t), O<j
unique function e(x,t) satisfying the above requirements. If, in addition, we require the data in C°° then also e will be C°° in the
42
11.2. Boundary problems
,
ODE
43
two triangles. The continuity of the derivatives then also gives (2.5)
aN-le(x,x+0)
aN-le(x,x-0)
= 1/aN()
-
,
a < x < 6
.
In the proof of Lemma 2.1 we may assume that a=0, due to g E C, or,
N L>2. Suppose we have LQf = g
,
(f,Lu) _ (g,u)
(2.6)
,
,
for all u E CO
Let us focus on some point x0 E C _ (a,s)
with X=l near x0, and ECOU) , X20 in R
We can find x E C0(C)
.
=
,
0
,
IxI?1,
and then let E(x) = Vx/E)/E . The function (2.7)
u(x) = u5(x) = X(x)J
then will be in CO(C)
Yx0-y)e(x,y)dy
,
0
< E
and may be substituted into (2.6).
Indeed, we have v(x) = Ja
E(x0-y)e(x,y)dy
=
Ja + JS , with both
integrands being Cam, so that v E Cam, hence u E C
.
Moreover (2.3)
suggests that (2.8)
Lv(x)
=JOE(x0-y)6(x-y)dy
as also may be easily verified by a calculation, using (2.4). Moreover, for sufficiently small E ,
(2.9)
Lu(x) = Yx0-x) +
Y={L x(X(x)e(x,y))}f
with the 'function part' {d}f of the distribution d (i.e., {d}f omits the delta-function term arising from differentiating at the discontinuity x=y). Clearly the integrand vanishes identiusing that Y = (Lxe(x,y))f=(6(x-y))f = cally for small 1x-x01 (due to E=1). Moreover, we have lx0-yI small in supp E(x0-y), ,
0
so that the possible singularity of Y(x-y) is outside supp E unless Ix0-xl is small, in which case E=0 anyhow. Thus, for small for all x,yEC Substituting into (2.6) and interchanging integrals we get
E>0 we have the integrand C
J E(x0-y)dyJX(x)g(x)e(x,y)dx
=
Jf(x)0E(x0-x)dx
(2.10)
Clearly (2.10) remains valid if x 0 is replaced by x
1
close to x0,
11.2. Boundary problems
where the functions X and e
->
0
c
,
> 0
,
ODE
may be kept the same. In the limit
, one obtains the equation
f(xl) =/j e(x,xl)X(x)g(x)dx -J. dxf(x)y(x,xl)
(2.11)
for almost all x1 E D , with xl close enough to x0 so that X(x)=l still holds near xl . Evaluation of these limits requires ,
a common technique known as regularization (or mollification)
(cf. [Cl1,I). Note that the right hand side is C', as a function of xl since the first term looks like v of (2.7) , while the kernel y
of the second term is C , as seen above. Thus by changing f on a null set, keeping it within the equivalence class, representing the given element of H , we find that fEC' , near x0 Since the construction is valid for every x0 E S2 the proof is complete. .
,
The above proof is valid without the assumption of selfadjointness for L or reality of coefficients, as long as L is elliptic, as easily checked. Also, along similar arguments, one easily confirms the following. Corollary 2.2. The space dom L0 consists precisely of all u E CN-1(0) with absolutely continuous N-1-st derivative such that u(N is defined as derivative L u E H , where the N-th derivative almost everywhere of u(N-1) . Moreover, for fE dom L* Lgf=g we ,
equal to 1 near x 1 E 0 , and the C'-function y(x,y)=Lx(X(x)e(x,y)), (ignoring still have (2.11) valid, for any C0'(c2)-functions X(x)
,
the delta-function arising from the discontinuity of the (N-1)-st derivative of e(x,y) at x=y). Indeed, we have proven (2.11) for all fE dom L0 with L0f=g, where gEC°°nH, but notice that exactly the same derivation works
for fE dom L0, g=L0f, regardless whether or not g E C'. By differentiating (2.11) it then follows at once that such f must be and that its (N-1)-st derivative must be still absolutely continuous. Also the differentiation implies that L"f=g E H CN-1(Q)
,
Vice versa, if fECN-1(0) has its (N-1)-st derivative absolutely continuous, and if g=L*f , with f(N) existing only almost everywhere, is in H, then (2.6) follows by partial integration, so that f E dom L0 q.e.d. and L0f=g ,
,
Note that, after Lemma 2.1, we now have complete control of sec.1 , without having to rely on the hypo-ellipticity of L
(which in general is discussed in [C3). ).
44
11.2. Boundary problems
,
ODE
If an endpoint a or a is finite and has the property that all coefficients aj may be extended up to it, as Co-functions, keeping aN90 also at that endpoint, then we shall speak of a ,
regular endpoint. Otherwise the endpoint is called singular. Lemma 2.3. If a is a regular endpoint, then dom L C CN-1([a,g)). 1 Similarly for a regular endpoint S If both endpoints are regular .
then dom L1 C CN-1([a,s])
.
Proof. If a is a regular endpoint, then all functions of the
defect spaces D± are C`,ta,s)), as solutions of the differential equation (L±i)u=0. Indeed, it is well known that a solution of a linear differential equation with C--coefficients stays C- in the entire interval where the coefficients are defined, as long
as aN#0, also at the boundary (c.f. [CdL1] for example). In view of the decomposition (1.13) it thus is sufficient to show that dom L-0 C CN-1([a,s)) .
We again use a fundamental solution e(x,y). Since the solutions of Lw=O extend smoothly into the left endpoint the same is true for the fundamental solution e(x,y) we constructed. In fact, we may assume e(x,y) defined for all x,yE [a',s)
a slightly larger interval. Let first fE CD
,
, with a'
g=Lf
,
Then (2.11)
.
follows even for the more general case of a cut-off function X E and is valid for xl near a The , X=l near a =0 near a
C-(1)
,
,
.
reason for this extension is that (2.6) now holds not only for u with compact support, but for general u with LuEH , hence u of the form (2.7), with our new function X define y with this new X near x=$
,
.
Of course we also must
and then get yE C'([a',B)x[(X',S)), y=0
.
The same formula then holds for general f E dom L0, because Lfj}g and setting a sequence fj E C' may be choosen with will give that f=f. in (2.11), and passing to the limit j near its left endformula. Given the new representation of f ,
,
point, one may differentiate N-1 times to confirm that f E
CN-1
near a. Similarly for the other endpoint. Q.e.d.
The extended formula (2.11) of the proof of lemma 2.3 also gives the following.
Corollary 2.4. If a is regular then (2.12) dom LO ={uEdom L1: u=u'=...=u(N-1)=0 at a, u satisfies BS}, where 8 S denotes a certain generalized boundary condition at s.
45
11.2. Boundary problems
ODE
,
46
A corresponding formula holds, if S is regular. If both, a and are regular, then dom L""0 consists precisely of all uE dom L1 $ with derivatives up to order N-1 including vanishing at a and S ,
.
Proof. The statement may be expressed as follows: (i) a function u E dom L0 must have all derivatives at a equal to zero, up to order N-1 Moreover (ii) if v e dom L1 has that property near a .
and vanishes near $ , then we get v E dom LO To prove this we apply our modified (2.11) (of the proof of lemma 2.3). For .
u E dom LO , w=L0u we get
u(a)=limaau(x)=JQe(y,a)X(y)w(y)dy-J.y(y,a)u(y)dy
(2.13)
Any such u is the limit of a sequence ujECo for which the right hand side of (2.13) vanishes. Hence we also get the left hand side zero. Similarly we may differentiate up to N-1-times, for u(1)(x)
(2.14)
=
K1(y,x)w(y)dy -J0X1(y,x)u(y)dy
,
1
with kernels K1, Al given as the 1-th x-derivatives of the kernels in (2.13). These kernels still are L2 in the integration variable as x is near a. y, and as L2-functions vary continuously with x Therefore we also get u(1)(a)=O 1=1,...,N-l. This proves (i) ,
,
To verify (ii), let v E C' be 0 near S , and let 0 at a. (That is, ) Then we just show that v E dom L"'0, by constructing a
=limxyav(N-1)(x)
v=v'=...=v(N-1)=
=0
.
sequence vj E CD with vj + v
,
Lvj -r Lv
.
Let w E C-U)
,
w =
0
j=1,2,..., near a = 1 near R. Then vj(x) = v(x)w(a+j(x-a)) has the desired properties, as a calculation shows. Similarly ,
,
if s is regular or if both a and S are regular. Q.E.D. Remark 2.5. Note that, for an elliptic ordinary differential expression L a characterization of dom LO is possible along the lines of cor.2.4: If the endpoint a is regular, then dom L0 is given as the set of (2.12), with dom L1 replaced by dom L Similar for a regular endpoint B or regular a and S Let us first assume that both endpoints a and B of a given .
,
expression L are regular. We introduce the sesqui-linear form (2.15)
Q(u,v) _ (u,Lv) - (Lu,v)
, u,v E dom L1
.
Proposition 2.6. For uEdom L1 let us introduce the column vectors
11.2. Boundary problems us"'=
(u(a),...,u(N-1)(a))
ua"=
(2.16)
,
,
ODE
47
(u(g),...,u(N-1)(6)) ,
(well defined in view of lemma 2.3), and the (2N-column-) vector
_ (ua",u
u'"
(2.17)
)
.
There exists a nonsingular skew-symmetric 2Nx2N-matrix
(2.18)
with NxN-matrices Q `,
Q
the inner product of T2N
real coefficients such that, with
a
or or TN , respectively,
Q(u,v)=(u-,Q-v"')=(us ,Q6-v)-(ua"',Qa-va-), u,vEdom L1.
(2.19)
Proof. Letting a < a' < g'
<
13
,
we get
Q(u,v) = lima',R'-a,8 Ja,(av - L*uv)dx
(2.20)
Using a partial integration as in (1.4) we find the right hand integral equal to a sum of boundary expressions of the form (2.19) with a,13 replaced by a',S'. The matrices Qa
, Qare composed
from derivatives of the coefficients a.. Only products u(j)v(1) with j+lN-l. Moreover, for j+l=N-1 the coefficient is either
or ±aN(13 ). In the
limit a'ia, 6'-6 we get (2.19). the matrix coefficients remain the same, except that the values of a(0) are taken at a,1 now. Hence Qa and Q8
are nonsingular as triangle matrices with non-vanishing
diagonal terms, referring to the 'anti-diagonal' j+l=N-1. Finally, (Q-u-,v-) =
(2.21)
v ,Q"u
= -((u,Lv)-(Lu,v)) = -(u-,Q"v")
,
confirming that Q"' is skew-symmetric. (All 2N-vectors u",v" are assumed, as u,v vary through dom L1.)
A linear subspace W of T2N will be called a maximal null space of the matrix Q"' if (i) (u-,Q-u-)=0 for all u- E W, and (ii)
every subspace V with W C V C T2N and with property (i) coincides with W.
Theorem 2.7. If both endpoints of an even order expression L with real coefficients are regular, then the class of all e.s.a.-reali-
11.2. Boundary problems
,
ODE
zations A of L is in 1-1-correspondence with the class of all maximal null spaces W of the form Q-, as follows. For a maximal null space W an e.s.a.-realization is defined by (2.22)
dom A = {u E dom L: u- E W} , Au = Lu for u E dom A
.
Vice versa, for an e.s.a-realization A the space W is defined by (2.23)
W =
{u-: there exists u E dom A with boundary values u-}.
A proof of Theorem 2.7 will be discussed in section 3. Here let us get some insight into the nature of a maximal null space W. The non-singular skew-symmetric matrix Q- has pure imaginary eigenvalues. Since Q- is real there are precisely N eigenvalues with positive and negative imaginary part each, and the corresponding eigenvectors span spaces W+ with dim W+ = N We have the g2N=W+ orthogonal direct decomposition We get positive $ W_ definite sesqui-linear forms F+ defined by .
.
(2.24)
F+(u`,v-) = +iQ-(u-,v-)
, u-,v- E W+
That is, the matrix Q- is reduced by the above direct decomposition, and, up to a factor ±i, F+ are the forms induced in W+. Then the maximal null spaces of Q are given by (2.25)
W = {u-+W-u-: u E W-1
where W- is any isometry W
,
- W+, the latter two spaces considered
under norm and inner product given by the forms (2.24).
Note that Q also is reduced by the direct decomposition 2N = gN $ gN, as manifested by (2.18). Some (but not all of its) maximal null spaces are of the form (2.26)
W = (u-=(u a-,u): ua- E Wa
with maximal null spaces Wa
,
,
u
E Ws }
I
WS of Q_ and QS', resp. The corresWe get
ponding boundary condition then will be called separated (2.27)
dom A = {u E don L1: ua-E Wa, uS-E WS }
.
Note that (2.25) implies that all maximal null spaces 0 of Q have dimension N, and all maximal null spaces of Qa, QS have dimension N/2 (an integer). As easily seen, any N-dimensional null space of Q is maximal. Similarly for
Q,_'
Q
$_*
48
11.2. Boundary problems
ODE
,
Theorem 2.7 then simply is a consequence of the fact that, for regular endpoints a,$, every e.s.a.-realization has defectindex (N,N), and the e.s.a.-realizations are characterized as the hermitian extensions A , L0 C A C L1 , with dim (dom A/dom L0)=N. This will be discussed in sec.3.
3. Singular endpoints of a 2r-th order Sturm-Liouville problem. Let us next discuss the defect index of LO
,
under the
assumptions of section 2.
Lemma 3.1. Under the general assumptions of section 2 we have def LO = def L0'= (v,v)
(3.1)
0
,
< v < N.
Moreover, if one endpoint is regular, we get r=N/2 < v < N both endpoints are regular we have v = N
.
If
.
Proof. Recall that the functions of D+ are Cm-solutions of Lu=±iu in 0
.
It is well known that there are precisely N linearly
independent such solutions of either equation. Clearly D+ will
consist of precisely the squared integrable solutions, hence must have dimension between 0 and N. Both dimensions are equal since D+ = D , using that L has real coefficients. If both endpoints are regular then the solutions of Lu=±iu as already observed. Hence all solutions are
are C'({a,s])
,
in H, and v = N follows.
From (1.13) we get dim(dom L1/dom L0-)=2v. On the other hand, if one endpoint, for example the point a, is regular, then we have C'(La,s)) C dom L1 , while Corollary 2.4 implies that all functions of dom L0-are CN-1([a,s)) and have their derivatives It follows that any two functions u,v up to order N-1 zero at a .
E dom L0` with different boundary values ua-9 va-are incongruent q.e.d. modulo dom LO-5 so that 2v =dim(dom L1/dom li0) > dim UN=N Now we discuss the proof of Theorem 2.7. First of all, for LO-C A C L1, the space Ui={u-:uE dom A) any hermitian extension A ,
,
and a null space of the form Q, by is a linear subspace of M2N definition of the hermitian operator. From (1.15) and lemma 3.1 It follows from cor.2.4 it follows that dim (dom A/dom L0-) = N ,
.
that uEdom A is in dom L0- if and only if u-= 0. Accordingly, for an e.s.a.-realization, we must have dim M = dim(dom A/dom L0-)=N. so that Ul is a maximal null space. Therefore (2.23) indeed defines
49
11.3. Singular Sturm-Liouville problems
50
a maximal null space for every e.s.a.-realization. Vice versa, for a maximal null space (d, let a linear operator A be defined as restriction of L1 with domain (2.22). Then A is hermitian, in view of (2.21), since GJ is a null space of Q"
.
Also, dim (dom A/dom L0")=dim (1 =N, so that A must be essentially self-adjoint, again by lemma 3.1 and (1.15), q.e.d.
Let us consider the boundary form Q of (2.15), in case where endpoint a is regular, while s may be singular. From (2.20) we get (3.2)
Q(u,v) _ -(ua",Qa"ua") + limy,}s-0 (usr",QS,"us,")
where the limit at right exists, although it no longer be expressed in the form of (2.19). Define (3.3)
Qa(u,v) = lims,-s(us,",QS,vs,") = Q(u,v) + (uaQava") Note that QS(u,v) = 0 if either u or v are in LS = dom L0
{u E dom L1: u=0 near
.
Similar as in the proof of Lemma 3.1
we get dim (dom L11L S) = 2v-N = 2(v-r) = 2vs
,
r = N/2, 0 < vS
= dom L1/LS as to the (abstract) boundary Clearly QS induces a form space at the right boundary point 5 we get Qs(u,v)=(us,QS"vs") , with a certain skew-hermiover M One may refer to p(
.
and the cosets uvs" E NS of u,v. tian operator QS of pas With this new meaning of us",vs",QS", (3.2) assumes the form (2.10), and the construction of e.s.a.-realizations may be carried ,
M2N
and out similar as in section 2, with replaced by ®N® Ms= N Qa"®QS"' One obtains the same 1-1-correspondence between e.s.a. ,
Q"
realizations and maximal null spaces of Q" as stated for the regular case in Theorem 2.6, with the only difference, that a maximal null space now has dimension r+vs < N. Also the construction of maximal null spaces from isometries remains unchanged from (2.25). We may distinguish separated boundary conditions, etc. It is easy to construct explicit e.s.a.-boundary conditions by first choosing a basis in MS
,
a matrix for Q" and Qand then constructing
an explicit maximal null space.
In case of two singular endpoints it is convenient to first consider the expression L in the two subintervals (a,y] and [y,s), Let the restricted expressions be with some point y , a < y < a .
denoted by La and LS respectively. Ls as above, for La and Ls Proposition 3.2. We have, with La ,
,
respectively,
II.3. Singular Sturm-Liouville problems d = dom L1 /dom L- = dom La/L a ® don LS/LS = Pa
(3.4)
.
V
Moreover, the defect index V = v+ = v- of LO now is given by V = va + vs , with va and vs of La and LS the expressions with ,
one regular endpoint. The simple proof is left to the reader.
Clearly the projections u ->ua- and u -u- may be defined, just as above. Then, with two abstract boundary spaces M a and MS the construction of e.s.a.-realizations from maximal null spaces assumes the old form. We now will state the spectral theorem for singular (even order) Sturm-Liouville problems. (Note that, conventionally, the eigenvalue problem Au = Au, for an e.s.a.-realization A of a selfadjoint elliptic ordinary differential expression of even order, and with real coefficients is called a Sturm-Liouville problem. The special point will be that, for an ordinary expression do*
the spectral family of A can be linked to the finite dimensional space of solutions of the differential equation Lu=au L
,
For the remainder of this section we fix a fundamental set u1(x;A)
(3.5)
,
...
,
uN(x;A)
(i.e.,a basis of the solution space) for the differential equation by imposing the condition
Lu = Au
33ul(Y,a) = 6j+l,l
(3.6)
,
j+l,l = 1,....N
where y is any point, a < y < S, kept fixed for all the discussion
From standard results on ODE's we conclude that uj E Cfor fixed A E ® , and that 31u1(x;A) is an entire function of A , for
every fixed x E (a,B) and every j,l. For the following the conditions (3.6) are not essential, but it is important that the u. form a fundamental set with the above smoothness and analyticity. Let o(A)=((a.l ))be a matrix-valued function, j,1=1,...,N' real -valued, and a(A) The coefficients are assumed for A E I is symmetric, for every A. We shall call a non-decreasing if a(A) .
_< a(U), as A < U, in the matrix sense (i.e., a. = a(U) - a(A) satisfies jaijl 7cl > 0, for all E E ®N ). It is clear that the
diagonal elements ajj are real-valued, non-decreasing functions. Moreover,
51
11.3. Singular Sturm-Liouville problems (3.7)
10
X14
1/2 -< (a 4jj *'All)
i/2(a
4jj
+ a
411
)
follows, so that all elements of a are of bounded variation, in any compact subinterval of I . An inner product and a norm on MN-valued C--functions may be defined by setting (3.8)
(u,v)a =
J
u.(A)v1()da.1(a)
IN
J,1=1
J
,
[lull
=
(u,u)al/2,
J
with an improper Riemann-Stieltjes integral, at right. The linear space Ha of all Co-function with finite norm may be completed to a Hilbert space Ha It is known that a Riesz-Fischer theorem holds, .
so that H = L2(1,da) with a corresponding Lebesgue-Stieltjes integral. We shall not use this fact, however. In all of the following we assume a continuous from the left. Theorem 3.3. (Spectral theorem for an e.s.a.-realization A )
For every e.s.a.-realization A of L there exists a non-decreasing called the spectral denmatrix-valued function p(a), - < A < sity matrix, such that f defined by ,
1s
(3.9)
j uj(x;A)f(x)dx
$. U) =
j=1,...,N, f E
,
a
and (3.10)
f
f(x) =
,
defined by
J+
-
E Hp j,1=1
give a pair of mutually inverse unitary operators H H Hp, defined by continuous extension of the maps (3.9) and (3.10). Moreover, if U: H -* Hp denotes the operator (3.9), then UAU
operator of multiplication by A in Hp (3.11)
UAU
.
E U(dom A)
(A) =
coincides with the
That is, ,
A E I
After choosing the fundamental set fu j} the spectral density
matrix is uniquely determined as a left continuous function, up to an additive constant matrix, by the realization A .
A proof of theorem 3.3 will be discussed in section 4, (but only for the simpler case of N=2
,
in the interval
[0,oo), with
0 a regular endpoint, and for the Dirichlet boundary condition
at x=0). Our proof is that of M.G.Krein, c.f. the book of Neumark [Ne2]. An extension of the proof to general N would meet only minor technical obstacles.
52
11.4. 2-nd order spectral theory 4. The spectral theorem for a second order expression.
Let us consider a second order self-adjoint expression L in the interval [0,-), (i.e., with a regular endpoint at 0). From the self-adjointness we conclude easily that (4.1)
L = -d/dx p(x) d/dx + q(x)
,
x E [0,o)
,
with real-valued p, q (-= C'00,-)), and p(x)# 0 Usually one assumes p>0. Expressions of this form are known as (second order) .
Sturm-Liouville expressions. From section 3 it follows that all realizations with sepa-
rated boundary conditions have domains of the form (4.2)
{u E dom L1
:
u0 E WO
, u: E W.
with certain maximal nulspaces WO We get u0 = (u(0),u'(0)), W. while W. C M = dom L1/L_. We have dim WO = 1 = N/2, so that the ,
.
condition 'u0 E R0 may be expressed as a conventional boundary jaj2+ lb12 = 1 ), condition (with certain constants a,b '
,
au(0) + bu'(0) =
(4.3)
0
A calculation shows that the real boundary conditions, i.e., the conditions (4.3) with real a,b , precisely characterize all maximal nul spaces at 0. On the other hand, at the singular endpoint W
the space W. has dimension v_ , where v. + 1 is the number of linearly independent squared integrable solutions of the differen,
tial equation Lu = iu (or Lu = -iu). By lemma 3.1 and proposition 3.2 the defect index of L** must be v,+l > N/2 =1, so that there are always either one or two linearly independent squared integraeach. In the first case dim Wm= dim M_ ble solutions of Lu=±iu ,
=0, so that u- E W. holds for all u E dom L1. That is, we need no boundary condition at - in this case.(H.Weyl calls this the limit point case.) In the other case (the limit circle caee,after Weyl),
we get dim W. = v_ =
1
, dim M
= 2v_ =
,
2
so that 'u_ E W_
'
To express this condiamounts to one boundary condition at tion explicitly requires a basis in the two-dimensional space M. .
We refer to Neumark [Ne2] for details As a special case of the spectral theorem (thm.3.3) we for.
mulate and prove theorem 4.1, below, which refers to a second order Sturm-Liouville expression in [0,oo), under a Dirichlet-boun-
53
11.4. 2-nd order spectral theory
dary condition at 0, i.e., the condition (4.3) with a=l, bzO (4.4)
dom A = {u E dom L1
u(0) =
:
0
Or,
E
u
,
.
54
where the last condition is void if we have the limit point case. For convenience we introduce the operator L2 (4.5)
{u E C0'([0,oo))
dom L2 =
:
,
u(0) =
L2 C A C L1 , with 0
A E B 0 < x < : be the unique realvalued solution of the differential equation Lu = Au satisfying
Theorem 4.1. Let u(x;A)
,
,
,
,
u(0;a) =
(4.6)
0
,
u'(O,A) = 1/p(0)
.
There exists a non-decreasing real-valued function p(A), -:
(A) =
(4.7)
Ju(x;X)f(x)dx
,
f E dom L2
0
and
E C0(1)
f(x) =
(4.8)
by define a pair of mutually inverse unitary operators H H HP continuous extension in H and H . Moreover, if U denotes the ope,
P
rator of (4.7), then we have UAU
(4.9)
(A)
=
A
(A)
,
for all
E U(dom A)
The function p(a) is uniquely determined as a left continuous function with p(O) =
0
,
with the above properties.
Our proof of theorem 4.1 is based on the abstract spectral theorem for unbounded self-adjoint operators (i.e., I, thm.3.3 ), and some other preparations are required. The result was first who did not use the spectral proven by H. Weyl [We11 in 1910 theorem for his proof. (This was, in fact, earlier than the abstract theorem. ) The use of the spectral theorem in the proof ,
became customary with the extension of the result to general order (i.e., thm.3.3). For f E (4.10)
I+ _ [0,:), we introduce the entire function D(A) _ D(A;f) =
l-u(x;a)f(x)dx 0
Proposition 4.2. We have
11.4. 2-nd order spectral theory (4.11)
= X'(X;f)
VA;Af) _ V(A;Lf)
,
f E
dom L
2
Moreover, if (D(A;f) = 0 for some AEI, then there exists w E dom L2 such that
f = (L2-a)w
.
Proof. (4.11) is a matter of partial integration, using that f=0 for large x, and that u and f both satisfy the Dirichlet condition at 0. Introducing z(x) as the unique solution of Lz=Xz satisfying z=l z'=0 at 0 , we define
x u(y)f(y)dy 0 x Note that the 'Wronskian expression' W = p(uz'-zu') is constant (its derivative vanishes), so that Wsl , using the values of u and z at 0. Thus one confirms by explicit differentiation that (L-X)w=f. For large x the first integral in (4.12) vanishes while the second equals '(X;f) (which also vanishes,by assumption),using that f E dom L2 vanishes for large x. Hence w = 0 for large x. q.e.d. since u(0)=0 . Hence w E dom L2 Also w(0) = 0 It is easy to find some p E CQU+) such that D(0;p)#0. Then VO) = VX;ip) can have at most countably many zeros al, A2' ..., and lim lajl = m, if the sequence is infinite. Applying proposition 4.2 for a=a1 we get w=i1 E dom L2 such that _ (L-al)ipl (4.12)
w(x) = u(x)
z(y)f(y)dy - z(x)
,
,
_ '(A;Lw)-'(a;w) = (X-X1)4(a,i1). Accorexcept that Al has its has the same roots as
and (4.11) yields (D(A;ip)
dingly
multiplicity reduced by 1 (or no longer is a zero). It is clear that we may iterate this proceedure to arrive at the lemma, below. Lemma 4.3. For any compact interval [a,b] C R there exists a 0 for all a E [a,b] function p E C°([ 0,w)) such that f(A;i) ,
Lemma 4.4. Let E(A) be the left-continuous spectral family of the and let self-adjoint operator B = A Let y be as in Lemma 4.3 ,
.
A = [a,b). For any interval A'=[u1,42) C A
,
EA,=E(u2)-E(pl),
and any f E C'(]R+) we have
dE(A)V
EA,f =
(4.13)
Proof. It suffices to show, that, for v E (4.14)
(EA,f,v)
=
1A' 1 (A)d(E(a)V,v)
,
y(7,)=
since the existence of the integral (4.13), in norm convergence as a Riemann-Stieltjes integral, is well known. Note that (4.14)
55
11.4. 2-nd order spectral theory amounts to K(u) =
, with
0
u
(4.15)
K(u) =
(d(E(X)f,v) - y(X)d(E(A)ip,v))
u E A
,
a
We shall prove that the complex-valued function K is differentiable and that its derivative vanishes. We have K(u)=(h(u),v), u
h(u) _
(dE(X)f - y(X)dE(X)p)
, and want to show that
a
(4.16)
h(u+6) - h(p) = o(6)
,
as 6 - 0
.
For a moment let 6 > 0 , and let E = E(p+6) - E(p) . Write u+6
(y(a)-y(u))dE(a)i
h(u+6) - h(u) = E(f-Y(u)$) -
(4.17)
u
u+6
In the second expression, called J2, we may write
J
u+0 u+6
instead of
1
,
since the integrand vanishes at p
.
This gives
u
u+6 (4.18)IIJ2n2= u+0 y
For the first expression, J1, in (4.17), let w = f - Y(V)* Note that (4.19)
Vu;w) J ()(u;f) - Y(u)(D(u;ip)
by definition of y
=
0
,
Thus, by proposition 4.2 again, write w =
.
(A-u)w, with some w E dom L2. If u is an eigenvalue of A then one may add a, multiple of the corresponding eigenfunction u(x;u) as to obtain a revised w which is orthogonal to u(x,u) is in dom A . It follows that
,
and still
u+6 (4.20)
11J112
= IIE(A-1)0
2
(X-u)2dfE(A)wI12 = 0(62)
=
u+0
Similarly for 6 < 0
,
q.e.d.
Lemma 4.5. We may write (4.13) in the form (4.21)
EA,f = Jo,D(A;f) dE(X)g
,
g = Jo
Here the function g E H is independent of the specific choice of subject to the conditions of lemma 4.3. * ,
Proof. It is clear that we may write (4.13) in the form of
56
11.4. 2-nd order spectral theory (4.21). To show that g is independent of , let i"' be any function with the properties of 'P . Applying lemma 4.4 with f = '_ EA,*' = J6,0(a;*`)/D(X;'P)dE(a)' , hence g" =
we get
dE(A)V`/D(a;V')
JA
This completes the proof.
We will write g = gO
,
since this element g E H
is determined by the choice of the interval A (4.22)
gp, = EA,gA
Introduce a function
g
I + H
:
(4.23) g(a) = lime-++0,u+a+0
,
as A' C A
.
evidently
One finds that
.
by setting as A>0
g[e,u)
,
=-g[a,0)
as 1 < 0.
Then one confirms that gO, = g(u2) - g(ul). For f E CO we thus get b (4.24)
(1) dg(a) , (X) = (D(a;f)
(E(b) - E(a))f = a
b ? 0, with 'P as in (4.10). Also (4.23) implies
for all a<0
,
(4.25)
IHEOf112
f
=
l¢(X)12dP(X)
J
11fll2
Ilg(A)112
we confirm ?arseval's relation
Letting a,b tend to (4.26)
P(a) =
,
=
1'P(X)I2dp(1)
J
,
f E
and it is found that
f f =
(4.27)
'P(a)dg(A)
f E
,
with a strongly convergent improper Riemann-Stieltjes integral. From (4.26) it is clear that the linear map f + 'P defines an isometry in the dense subspace CO(l+) of H , which may be conVice versa, let 'P E C0(I) tinuously extended to a map H - H .
P
be given, then we may define f by (4.27), and will find that fEH. Then a sequence f. E C'(I+) may be constructed with f. + f in H. Let $j be given by (4.7) with f = fj It follows that for f and 'P .
.
J
J
.
We know that (4.26) holds
.
(4.28)
f-fj =
J
('P-'j)dg(1)
=
Of - fjll2
Taking norms we get (4.29)
J Il2dp()
This proves that the above isometry maps onto HP
,
since C0(1+)
57
11.4. 2-nd order spectral theory is dense in H P
Next we prove the inversion formula (4.8). From (4.25) we get (4.30)
(f,g) _
with , defined by (4.7)
,
f,g E H
with f and g
Choose f E H
,
g = X,
the characteristic function of some interval A = [t,t+d] C 1+
Calculate that (X) =
Therefore (4.30) yields
t+d
(4.31)
t+d
f(x)dx
=
t
We assume
E C U +) 0
,
j $Q)( )
-
u(x;X)dx)dp(A)
t
and then may interchange the integrals at
right, then divide by S and let d
->
0. In the limit we get (4.8).
Finally the uniqueness of p is evident, because
JI
412da for a pair of left-continuous
functions p
,
a
,
for all 4 E
This completes the proof of thm. 4.1.
JI
2dp =
nondecreasing real-valued implies that p-a = const.
58
CHAPTER 3. SECOND ORDER ELLIPTIC EXPRESSIONS ON MANIFOLDS.
In the present chapter we turn to a discussion of a variety of properties of second order self-adjoint strongly elliptic differential expressions with real coefficients defined on a C--manifold of dimension n. In local coordinates such an expression will be of the form (0.1)
H =
-K-1a
xj
with function K,q>O
,
Khjk3 k + q x
,
and a positive definite tensor ((hjk))
It then is self-adjoint with respect to the inner product (u,v) =
Juvdp , with the measure du=Kdx
.
First we again discuss Weyl's lemma for expresssions of this type, and a boundary extension of Weyl's lemma as well, similar to that discussed for ordinary expressions in 11,2,3. Instead of the fundamental solution used in 11,2 we here use a E.E.Lewy type parametrix of the form (1.9), below. With this preparation we can in case of give a full discussion of the Dirichlet realization ,
a regular boundary, as an extension of the minimal operator with closure coinciding with the Friedrichs extension.
Next we will require criteria for the Friedrichs extension, hence the closure of the Dirichlet realization, to be of compact resolvent. Such criteria are developed in sec.2.
It is well known that differential expressions of the form (0.1) possess a variety of important special properties, such as the maximum principle, and the so-called Harnack inequality. These facts will be required in the sequel, and will be discussed in detail in sec.4. Also, in sec.3, we will discuss a detailed proof of the existence of a (positive) Green's function for the Friedrichs extension, again as required in the following. The majority of the results mentioned are local results. However, it is important to see them in this global surrounding,
111.0. Introduction
where they will have to be applied. Finally we will be concerned with two normal forms for expressions of the form (0.1). Both may be achieved by a transformation of dependent variable, but the second only if the minimal operator of the expression H is positive definite. This is discussed in sec.6.
1. 2nd order PDE on manifolds; Weyls lemma; Dirichlet operator.
Let 0 denote a Co-manifold without boundary, of dimension n > 1, not necessarily compact, but paracompact (and connected). In fact, for convenience we assume existence of a countable atlas. On 0 we assume given a positive Co-measure du locally of the ,
form du = Kdx , with K being C' in the local coordiates x. By H we denote the Hilbert space L2(O,dp), with inner product and norm ,
, NO = (u,u)1/2
(u,v) = J2uv du
(1.1)
u,v E H
,
Also we assume given on 0 a second order formally selfadjoint strongly elliptic partial differential expression hikK2 k + q
H = -K-18
(1.2)
x
x]
with Co-coefficients. In particular hjk denotes a symmetric positive definite contravariant tensor with real Cm-coefficients: hjk =
hkj
=
-k0
>.0
,
,
as E 9
0
,
and q denotes a scalar real-valued C -function defined over 0 In the next sections we use the summation convention to always sum from 1 to n over a pair of an upper and a lower index, in a tensor, denoted by the same symbol. Also we will write local not coordinates with superscript indices x=(x1,x2,...,xn) E In ,
subscripts, as for a subdomain 0 C In , according to convention. As in II,1 the expression H induces minimal and maximal differential operators of H domain we get (1.3)
(u,H0v) = J0(hJku
v Ix7
For the minimal operator H0 with
.
Ik + quv)dp = H(u,v)
lx
,
u,v E dom H0
Clearly H0 is hermitian and real, in the sense of I,prop. hjk are real. Accordingly there exist self-ad2.6, since q and joint extensions of H0
.
60
III.l. PDE on manifolds
In much of the following we will be interested in the case where the sesqui-linear form H(u,v) of (1.3) is an inner product, In fact if we assume q > 1 in G , then we get called (u,v)1 .
(1.4)
>
Ilulll
Hull
,
for all u E dom HO
with the norm Ilulll=((u,u)1)1/2
,
,
since the tensor hjk is positive
definite. Condition (1.4) will be crucial in ch.5f, although the condition q > 1 will be weakened later on (cf. prop.5.2).
For an expression H with (1.4) the Friedrichs extension of H0 is well defined as restriction of H* to (dom H*) fl H1, with the completion H1 of dom HO = C3(Q) under the norm Ilulll (cf. the proof of I, thm.2.7 ). In fact, the norm Ilulll of (1.4) precisely coincides with the norm null
of that proof, and the completion H- defi-
ned there will be denoted by H1 here. We know that H1 is naturally imbedded in H In ch.5 we will assume that (1.4) holds, and then denote the .
Friedrichs extension by H again. This will be a self-adjoint operator,satisfying H > 1 We then shall speak of a comparison ope.
rator H
The space H1 will be called the first Sobolev space of the operator H Clearly H-1 E L(H) has a unique self-adjoint .
.
yk=0(-1)k(lk2)(1-H-1)k
positive square root A = H-1/2 = < A < 1
0
(1.5)
, and
We have (cf. I, prop.2.8)
.
H1 = dom A-1
,
(lull
=
IlAupl
,
u E H
,
Hvfl1 =
RA-1vll, v E H1.
That is, A is an isometric isomorphism between the Hilbert spaces Generally the restriction of the self-adjoint operator A-1 to dom H0 still is essentially self-adjoint, and A-1dom H is H and H1
.
dense in H
,
(while the corresponding is not generally true for
H dom H0 = im H0 ). Note that H1 may be identified as the space of all functions u E H such that a sequence um E dom H0 exists which is Cauchy in the sense of H1 and converges to u in H It follows that there exists a unique covariant tensor (called the gradient of u , and .
_
(u
,
even though the components are not proper derivatives)
Ix j)
which is p-measurable and satisfies
S2
h0k(u Ixj - um Ixj )(u Ixk - um Ixk )du
->
0
, m
61
III.1. PDE on manifolds
It is clear that u
.
is the local distribution derivative of u E
Ix]
L2(O,dp) C D'(O) while the above entitles us to speak of the strong L2-derivative. One may introduce L2-norm and inner product of gradients just as for covariant tensors in general by setting (1.6)
hlku
(pu,pv)
Ix3
v
Ixlc
dp
=
Ilpull
,
(pu,pu)1/2
Returning to the general case, where (1.4) needs not to be satisfied, we observe that the expression H is elliptic, since we assumed the tensor
hjk
positive definite
(1.7)
0
,
:
We have
for all E
0
at each point x E 0 It was mentioned in II,1 that ellipticity implies hypo-ellipticity, but this will be discussed in general .
only in FC31. Since we need (strong) hypo-ellipticity (or at least Weyl's lemma) in the following we shall offer a short independent proof of Weyl's lemma here, which is quite similar to the proof of II, lemma 2.2, in the case of an ODE .
It is clear why a proposition like Weyl's lemma is desirable: Thm.l.3 of ch.II carries over literally, with the same proofs, to the present case of an elliptic operator on a manifold. We express this in thm.l.l, below, the proof of which will be left to the reader, (except for our proof of Weyl's lemma, i.e., thm.1.2. ) Theorem 1..1. The defect spaces 4-(H ) are subspaces of ao 4:'a **0 H1= H0 (i.e.,"weak" _ HnC (Q) . Moreover, we have H0= Hl e
,
"strong"), and the direct decomposition (1.8)
dom H1 = ((dom H** )nC'(0)) $ D+(H0) ® D-(H0) ,
which is orthogonal with respect to the inner product of graph H0. The self-adjoint extensions A of H0 precisely are given as the closures of the 'e.s.a.-realizations' A- obtained from an arbitrary isometry W: + D , using formulas II,(1.16) and II,(1.17) Theorem 1.2. If (HO-A)f=g, for any fE dom H0 and gE HnC(2), then we have f E C°10) and Hf-Xf = g , hence f E dom H1 ,
.
,
The proof depends on use of an (E.E.Levy-type) local parametrix of the form as n > 2 e(x,y) _ (p(x,x-y))2-n
,
(1.9)
= log p(x,x-y)
,
as n=2
62
III.l. PDE on manifolds
with p=p(x,z) _ (hjk(x)zizk)1/2 of generality.
(We may assume A=0, without loss
) Note that e(x,y) is defined only
locally, in local coordinates, for x, y E WC 0 , with a chart 0'. We note the proposition, below, which also will be useful later on.
Proposition 1.3. For a function m E
let
v(x) = Je(x,y)$(y)dy
(1.10)
Then we have v E C-(Q') , and (1.11)
J.,
Hv(x) =
y(x,y)$(y)dy
with a positive C'(Q')-function c(x)
,
,
xEQ'
and with a function
y(x,y) of the form (1.12)
y(x,y) = y0(x,Y,x-Y) + 2 i(y3(x,y,x-Y)) x
Here the functions y!(x,y,z) j=0,1,.... n, are C,(Q'xO'xjjn*) In and homogeneous of degree 2-n in the variable z , as n>2 ,
.
the case n=2 we have (1.13)
yi(x,y,z) = h1(x,y,z)log p(x,x-y)
,
and homogeneous of degree 0.
where the h0 are
Proof. We only consider the case n>2
.
The other case n=2 may be
treated similarly. Observe that the function e(x,y) may be written in the general form (1.14)
e(x,y) = f(x,y,x-y)
,
f(x,y,z) E C,(O'xQ'x]Rn*)
where f is homogeneous in the variable z (1.15)
.
,
(Presently we have
=(hjk(x)zjzk)1-n/2
f(x,y,z)
,
independent of y and homogeneous in z of degree 2-n. For any function g(x,y) = f(x,y,x-y) of the form (1.14) we have (1.16)
(axi + a j)g(x,y) = fj(x,y,x-Y)
n
y
where fj again has all the properties of (1.14) homogeneity degree as f
.
, with the same
This follows, because a function a(x-y)
63
III.1. PDE on manifolds is constant in the directions x. yj hence has (2
.+8
x
Now a single derivative 2
.)a=0.
y
may be applied under the inte-
gral sign of (1.10), because the differentiated integrand still is integrable. Applying (1.16), and a partial integration, we get (y)dy +
vlx.(x) =
(1.17)
where gj(x,y) = fj(x,y,x-y) has exactly the properties (1.14) again, (with homogeneity degree 2-n). In particular the partial integration first may be performed on the integral over Ix-yI>e where boundary terms may be explicitly evaluated. These boundary terms tend to zero, as 6+0 , so that no special terms at x=y from the partial integration appear in (1.17).
Accordingly the process may be iterated, it follows that indeed all partial derivatives v(a)(x) exists (1.18)
v(a)(x)
=
for x E Q'
,
, and
I
a
(1.19) H=Hw+Aw, Hw=-hjk(w)8
8
xi x
k,
Aw=a.k(x,w)B
2 k+bj(x)2
x x
x
+c(x)
where wEO' is arbitrary, fixed, and a0k(x,w)=hOk(w)-hOk(x), it follows that (1.20)
with h0
x .(y)dy ,
h0 again of the form (1.14) (degree 2-n)
.
In (1.20) the first integral, at right, called Il, contains the constant coefficient operator Hx (where x is only a parameter) and the function e(x,y), depending on x-y only, as long as x is kept fixed. One may make the linear substitution n=M(y-x) in the integral Il, with M = ((hjk(x)))1/2, which brings it onto the form (1.21)
I1 = -det((1
Jk(x)))-1/2JInI2
nAPdn
,
with the Euclidean Laplace operator A , and
(1.22) JIn 12-nAW(n)dn=lime+0
I
Here,
=(2-n)limE+O
ITIL using partial integration, and that
(n)Inll-ndSn
In =e An(InI2-n)=0
. Also that
64
III.1. PDE on manifolds 2-n
lim
where dSn denotes the surface element
JInI
In =e
on the sphere InI=e, and 2v=-2 p, p=InI, the radial derivative.
From (1.22) we conclude that cn = (n-2)wn-1
I1 =
(1.23)
with the area wn-1 of the sphere InI=l in
In .
The above holds
for n>2. In case of n=2 one finds that cn = c2 = 27. Since the determinant is C' and >0 this term accounts for the first term in (1.11).
On the other hand another partial integration in the third term of (1.20) will remove the derivative from and give an ,
(x,y)¢(y) instead, confirming (1.12). (In view of
integrand h3 Iv'
(1.16) we again may write
h
l
+k with k(x,y,x-y) (degree
-hlYi=
2-n), etc.) This completes the proof of prop.1.3.
Now the proof of thm.1.2, in essence, is a repetition of the argument used in the proof of II, 1.2.1. Using our parametrix e(x,y) of (1.9) in place of the fundamental solution e(x,y), we derive a local integral equation of the form 11,(2.11) for any pair of functions f,gEH satisfying II,(2.6), with L=H, and with the present Hilbert space H=L2(c)
.
Then, if gEC-, the right hand
side is continuous, using the nature of the kernels. Hence f(x), after correction on a null set, also is continuous. Knowing that f is continuous, one concludes that the right hand side even is C1
,
so that fEC1, etc. Continuing on, one concludes that fEC
Let us get such formula in detail. Let u(x) =
(1.24)
with a local cut-off function X(x)=l near some x0EO', XEC0(Q') c>0 , where ¢ECD(In). and a "regularizing kernel" Assume X=1 in 1"C2'
, 0 open, and let x`EO" , and e so small that
supp 4e(x--.)C 0'. Then we may apply (1.11) for Hu(x) = c(x)me(x--x) +
(1.25)
with c(x) of (1.11), and 6(x,y) of the form (1.12) again. In fact, 6 is a sum of X(x)y(x,y) and terms of the form a(x)e(x,y), (*x,y), with a,b3 E C' (c2') , where a and b3 vanish out-
b3(x)e Ix
65
III.1. PDE on manifolds side supp x
.
All these terms may .be written in the form (1.12).
Substituting u and Hu into the (present) equation II,(2.6), and passing to the limit a+0, we get (1.26)
K(x")C(x")f(x") + JQ,S(x,x")f(x)dux
Jduxg(x)X(x)e(x,x-),
=
valid for x"E2". This relation is of the general type of II,(2.11) However, the kernel 6 requires a somewhat refined proceedure. If g E C_ , then the right hand side is a C--function h(x"), by prop.1.3. The integral at left is at least Hoelder continuous, since we find that the function x" + S(.,x") , with values in H is Hoelder continuous: We get 2/lx"-x"Ie = 0(1)
(1.27)
,
L
for a suitable e>0 and x", x" E Q" , by a calculation. Hence (1.26) shows that f is Hoelder-continuous as well. To show that f is even C1 we write (1.26) as (1.28)
(KCf)(x) = h(x) - JS(y,x)f(y)dvy
and investigate the function (1.29)
e(x) = JS(y,x)f(y)dpy
The difference quotient ve = (9(x+e)-e(x))/e may be written as (1.30)
ye = Jvd(y)f(y)duy
Jvd(y)(f(y)-f(x))duy
=
where My) _ (S(y,x+c )-S(y,x))/e (1.31)
Jvd(y)duY
=
+
f(x)Jvd(y)duy
Now we use (1.12) for
.
My .
J(V6 0K-V6JK ly]
The right hand side limit exists under the integral sign, and
represents a Cm-function of x, since again the derivatives may be taken over to the function K , using (1.16) The other integral in (1.30) has a limit as well, since f is Hoelder conti.
nuous, so that the limit of the integrand is L1 (1.32)
8 0(x) = k(x)f(x) + JSI
.
Thus we get
(y,x)(f(y)-f(x))dpy
,
xj
confirming that fEC'
.
In particular, k(x) is a Cm-function.
Knowing that f is Cl we may integrate by parts in (1.32) (1.33)
(KCf)Ixi = hlx7 - kf +
J
S°(Y,x)flx.(y)dy
,
, for
66
III.1. PDE on manifolds with a function d° of the general for (1.12) again.
It is clear now that this may be iterated, proving that Kcf , hence f is C_ , q.e.d.
2. Boundary regularity, for the Dirichlet realization.
It will be of interest to note the following variant of thm.1.2, applying in the case where g is only continuous, not necessarily smooth.
Corollary 2.1. If g of thm.1.2 is only continuous, then we still get f E C1(0) , and its first derivatives are Hoelder continuous.
Moreover, if g is Hoelder continuous, then also fEC2(0) follows, and f is a classical solution of (H-X)f=g
.
Indeed, it is clear that (1.26) is valid for general f,gEH satisfying (HD-X)f=g . If g is only continuous then the right hand side of (1.26), called V(x) still is C1(0) and p ,
,
IxJ
satisfies a Hoelder condition, by the above arguments, using an estimate like (1.27) for the first derivatives of e(x,x-)
.
Moreover, if g is Hoelder continuous, then the conclusion leading to (1.32) may be applied to integrals representing i
1.
.
It fol-
1 exist and are conti-
lows then that also the derivatives lxj
x
nuous. This shows that, in the first case, f will be C1
,
and
that (1.33) still holds. Also the first derivatives of f then still will be Hoelder continuous. For Hoelder continuous g thus we may iterate once more, getting that also fEC2 , q.e.d. As in 11.2 we next try some boundary application of our Let us investigate the Friedrichs extension in the special case where we have H of the minimal operator H0 only regular boundary points. Similarly as in the 1-dimensional case parametrix e(x,y)
.
of 11,2 it may occur that the manifold 0 is a subdomain with smooth boundary 30 of another manifold R^
,
and that the triple
{0,du,H} extends to a triple {0^,dp^,H^} on 0^ satisfying our general assumptions. (That is, 30 is an (n-l)-dimensional Co-submanifold of 0^
,
and we have du = dp^I0 , H = H-JQ .) In this
case we will say that 0 has the regular boundary 30
.
Moreover,
if even OU30 is a compact subset of Q^ then we will speak of an expression H (or a triple) with regular boundary, or we will ,
67
111.2. Boundary regularity
say that all boundary points of c are regular. In the case of a triple with regular boundary one defines the Dirichlet operator Hd J HO by setting (2.1)
dom Hd= {uEC'(SZUM): u=O on 8O}, Hdu=Hu , for uEdom Hd
Specifically the condition "u=0 at x(=-22" is referred to as the
Dirichlet (boundary) condition. Theorem 2.2. The Dirichlet operator Hd
,
for an expression,H with
regular boundary, is an e.s.a.-realization of H , and its closure ** H = Hd is the Friedrichs extension of the minimal operator HO Proof. First we note that Hd indeed is a restriction of the FrieFor clearly we have HdCHd* drichs extension H of HO (i.e., .
Hd is hermitian), since a partial integration confirms that (2.2)
(u,Hdv) _ (u,v)1 = (Hdu,v)
,
for all u,v E dom Hd
(The boundary terms vanish, since u and v vanish at 2D .) Also, HOCHd , trivially, hence Hd*CHO* so that we get HdCH O* For any u E dom Hd it is easy to construct a sequence ujE dom HO with nu-u101-0 , as j- , since only the first derivatives of u must be modified, while the 1-st Sobolev norm 11.111 contains only first derivatives as well. In details, for a boundary chart c', ,
thought of as an open set in &+ _
{xn>0} ,
X(xn)EC,(J)
let
,
X?0
X=0 near xn=0 , X=l for xn>1 , and let X5(xn) = X(Jxn) uj(x) = u(x)Xj(xn)
.
Then u-uj = u(1-X].) has support in O<xn<1/j.
If u has compact support in c' same is true for IIO(u-uj)U
, then
as 54
.
The
, whenever u=0 at xn= 0, since we get
(u-uj)Ixk=(1-Xj)ulxk as k
where vj=uXl n has support in O<xn<1/j, and is bounded in j and x, X due to u(x)Xjlxn = ju(x)X'(jxn) , where u = 0(xn)=0(1/j) in supp X'(1xn) . Accordingly we conclude that I1u-uj1114-0
,
since
the coefficients of H are bounded at the regular boundary. For a general u E dom Hd one uses a partition of unity, corresponding to a finite atlas of the compact manifold with boundary QU30
,
for a similar result. Therefore indeed we get dom HdCHln(dom Hg), the domain of the Friedrichs extension, and the above also shows that Hdu=Hu for uEdom H .
Our theorem is proven if we show that Hd is essentially
68
111.2. Boundary regularity
69
self-adjoint. For then it is clear that its (self-adjoint) closure **
Hd
,
as a restriction of the Friedrichs extension, must coincide
with the Friedrichs extension. Since we know that Hd>1
it suffices to show that im Hd is
,
dense. Thus, let fEfl satisfy (2.3)
(f,Hu) = 0 for all u E dom Hd
By thm.1.2 we then get fEC_(O)
.
.
We want to show, that, in addi-
tion we get fEC°O()U90), and f=0 on M. If this can be established dfp2=11f11i=(f,Hdf)=0, hence f=0, then (2.2) for u=v=f implies that so that indeed im Hd is dense.
Thus we again focus on a boundary chart n'
as above. After
another change of coordinates one may assume that the co-normal direction at xn=0 councides with the x n_direction. In other words, this means that we have hnl(x)= 0 , j=1,...,n-1, as xn=0 Let us again write the expression H in the form H = hjk2
(2.4)
+ c
3 k + b32
x3 x
x3 for simplicity, and let O" denote the 'doubled' chart 0' given by (2.5)
o" = 0' U O2,
,
O'
x=(xl,...,xn-l,-xn)EO'}
_
.
In 0" we define the doubled measure dp"=K"dx , with the even extension K" of the function K to O" (i.e., K"(x)=K(x)
,
xE92" ).
We drop the notation K" , du" in favour of the old notation K, du. Let an extension of H be defined by extending hnn , hjk , bi , c for j,k
We claim that then (2.3)
.
with integrals over 0" still holds for f an all such u
.
Indeed,
one may decompose (2.6)
u=ue+uo , ue(x)=(u(x)+u(x))/2
,
u0(x)=(u(x)-u(x))/2
.
Note that the extended H maps even functions into even functions and odd functions into odd functions. Since (the extended) f is
odd, we get (f,Hue)0 = J0" dpTHue =
0
,
as an integral over an
odd function with symmetric range. On the other hand, (f,Hu0)0 = 2(f,Huo)0, = 0 as well, since (the restriction to 0' of) u0
111.2. Boundary regularity
70
is a function in dom Hd (A continuous odd function vanishes at xn=0 ). Accordingly, the extended function f satisfies a relation of the form II,(2.6) with the extended expression H
.
Note that uEC0'(0") is not required, as long as u0 is conti-
nuous at zero, and u0In' is
Thus it appears that the proof of thm.1.2 will give fEC on the part of an bordering the chart 0' Since f was defined as odd function, it then follows that f=O on an , or, fe dom Hd .
.
However, it should be noticed that the assumptions of thm. 1.2 are not satisfied, since the coefficients of the extended H and K ) are not C_ accross xn =
0
.
cients hjk are continuous and C'(01) easy to make at least hnn E C1(H") y - _ (y 1
,
Instead all leading coeffi, since hJn=O at xn=0. It is by the change of variable
,...,n-11 x y _ (x,...,n-1 )
)
= x"
(2.7)
yn = xn(1 + xn.hnn(x-,0)/(4hnn
Ix
n(x-,0)))
,
which takes H' onto a similar region in y-space, and makes the normal derivative of hnn zero at xn=0 The lower order coefficients'c and b. , j
7
bn has a jump accross xn = 0 Furthermore we have g = A = 0, in the notation of thm.1.2. Going over the proof of prop.l.3 we notice that v(x) of .
(1.10) no longer is C-(0"), but will be C-(0') again, and continuous accross xn=O into (2.3)
.
Therefore we again may substitute u of (1.24)
We still get (1.25), as long as x- is not on an and then (1.26) follows as well, with ,
for sufficiently small c so that we have g = 0
,
,
f(x) = jnit y(x,y,x-y)f(y)dy , for xn90 , xE 0-
(2.8)
with a sufficiently small neighbourhood 2- of a boundary point x0. Here y(x,y,z) is continuous and of compact support in , and homogeneous of degree 2-n in z , and its restrictions to xn>O O"XQ"Xln*
and to xn<0 are C_ . In this way we conclude that indeed f is
continuous, and has continuous one-sided derivatives of all orders at xn=0
.
Thus indeed f E dom Hd , and the proof is complete.
Note that a small modification of the above proof gives the following. Corollary 2.3. Under the assumptions of thm.2.2, if fECW(0U3c2)
,
111.2. Boundary regularity relative to the imbedding 1-O^ for some XEU
,
and u E dom H
,
and if (H-X)u = (Hd -a)u = f Hd then we have uE dom Hd
= dom
,
and (Hd-A)u=f
Indeed, under these assumptions the interior regularity of u is a consequence of thm.1.2, while, at the boundary, the argument around (2.3) may be repeated, giving that also u E C_ at a boundary point. Thus cor.2.3 follows. 3. Compactness of the resolvent of the Friedrichs extension.
This section discusses various compactness criteria, for the resolvent of a comparison operator, as well as for the inverse with a square root A=H-1"2 and also, more generally, for aA function a(x)EC_(O) Its main applications will be found in V,3, ,
.
and V,4,6 , where compactness of commutators, and certain asymptotic expansions are derived and applied. On the other hand, compactness of the resolvent (H-a)-1 = R(a) will be essential in the discussions of the present chapter, below, involving the Dirichlet operator Hd
.
In each case we consider the (self-adjoint) Friedrichs for an expression of the form (1.2). Note that compactness of the resolvent R(a) is an extension of the minimal operator H0
,
immediate consequence of the compactness of A
, since we get R(a) _ (H-a)-1 = A2(1-XA2)-1, for points AERs(H), where
E L(H)
(1-XA2)-1
Since H=Hd for a manifold Q with regular boundary, and 0U30 compact, we get compactness of the resolvent of the Dirichlet .
operator in cor.3.9.
The requirements on a for compactness of aA will be diffeFor a clear formulation rent near the various 'infinities of 0' .
of these facts we introduce the concept of subextending triple, below.
For technical reasons, for a while, we consider differential expressions of the (formally) more general form L = q - K a
(3.1)
Khjk3 x
xi
ks
,
du = K dx
and with all other coefficients q with a C-(c2)-function $ > 0 hjk as in (1.2). Such an expression may be brought into the K , ,
form (1.2) (3.2)
:
One has
L = -K-1a xi
K82h3k3 k + q + q0 x
,
q0 = - a (h0kKs
)
1xi
ix
k
71
111.3. Compactness of the resolvent Indeed, we get (u,Lv) u v E dom HO , (u,qv) + (4(Bu),V(Bv)) where we write V(Bu) = $Vu + u48 , and apply a partial integration ,
,
to the term (BVB,uVv+vVu) = (BOB,V(uv)), from the second inner product, to free the term (uv) from its gradient. For more detail, cf. the derivation of formula (6.6), below. If we assume that (u,Lu) > (u,u)
(3.3)
u E
,
then the facts discussed for H in section 1 apply to L as well.
We get a well defined Friedrichs extension L of the minimal operator LO , and its inverse positive square root L-112 is a bounded The space L-1/2C0(St) is dense in H self-adjoint operator of H and L112 is essentially self-adjoint in dom HO .
Let us adopt the convention to write
(3.4)
limx
y(x) = Y0
(in S2
,
)
,
where y(x) is a function over 0 , (with values in a space X) , and if for every neighbourhood N of YO a compact set K C S2 YO E X We can be found such that y(x) E N whenever x E 0 is outside K Then condiare not excluding the case of a compact manifold ) ,
.
.
tion (3.4) is void: It is true for every function y(x) , limit y0° and space X whatsoever. In particular condition (3.5), below, is generally true whenever S2 is compact.
Theorem 3.1. Let the expression (3.1) satisfy (3.3) and
limxq(x) = - ,
(3.5) Then L 112
(in S2
)
.
is a compact operator of H = L2(c,dp)
The proof is rather technical,and will be prepared by a series of propositions. Let w,X E C'(l) be such that (3.6)
0<w,X
,
t < 0 , w(t)=0
,
t>l.
For 0 < A < - define the two functions (3.7)
XA(x) = w(q(x)-A)
,
uA(x)
= X(q(x)-A)
It follows from (3.5) that aA has compact support.In fact, we have (3.8)
q < A+ 1
in supp aA
q > A 9
in supp
uA
72
111.3. Compactness of the resolvent
Proposition 3.2. For every A > 0 there exists a constant cA with
(3.9)
(u,Lu) > 7 IIUAull2 + IIV(AABu)112 - cAIIXABull
u E C0(52)
,
Proof. We get (u,Lu) = J9
(Bu)
J9>A 11A2(hjk(Bu)
+
gIuI2)dp
Ik +
IXJ
x
(Bu)Ik + gIul2)dp x
Ix]
The second term at right is bounded below by AUPAuO2 first term we use (3.2), with B replaced by XA
,
.
For the
to obtain, with
a=XA , u=VA ,
JaA2h3k(Bu)
J82
kdu = Uy(AA6u)112+
(Bu) Ix7
K(Khjka
Ix
klul2dp. IxJ
Ix
Here we estimate the last term at right by cA'
Hull
llABull
cA'
,
= supxES2 IS(Khjkx
Note that Hull
=
II(A2+P2)uU <
IIX2ull
+ 1u2ull
(u,Lu) > AUP U112 + 114(aABu)U2 +
<
Ilaull
Ix
+ Uuull,so that
J(ga2+62K(KhjkX
A
kl
)
Ixi
k)IuI2du
)
Ixi
ix
with the last term estimated below by -cA"IIX uI12 -°A1llasull2 _ CA'llxBullUuuU, cA" =sup{-q/B2:xEsupp AA
supp XA}
cA'=
Using that
Zlluull2+ 1/(2c)UXBuU2
HABuflluuf<
e=A/cA , with cA = cA+ cA + cA2/(2A)
.
we get (3.9),setting
Q.E.D.
In the following let {S2i ) be a locally finite atlas of S2
, and
let w3.2 be a corresponding partition of unity, such that supp w, C S2.
,
wi E C-(Q)
,
wj >
0
,
w!2 = 1
,
on all of S2
.
We assume
without loss of generality that each S23 . is a subset of a larger
chart in which it has compact closure.
Proposition 3.3. For every A > 0 there exists a constant dA with (3.10)
(u,Lu) >
211uAull2+ Z V (XABwiu)U2- dA IIXABul2, u E C,(0).
73
111.3. Compactness of the resolvent
74
Proof. In view of (3.9) it suffices to show that (3.11)
11V(AARu)n2 > lj114(XARwju)H2 - cAIAARull2
Write IIV(XASu)112
(3.1) -> (3.2)
,
u E C0(S2)
= IjHwjo(AASu)112 , and apply the conversion
for S = wj
under the gradient.Since
to bring wj
,
aA has compact support only finitely many of the summands do not vanish,and the corresponding functions q0 of (3.2) are all bounded. Hence (3.11) follows, q.e.d. Q={x(=-]kn:O<xj<1,j=1,....
For a moment consider the cube
with the Riemmannian metric of In
,
Ipvl2 =
(i.e.,
and the functions sinafrx = Rn sin 7rajxj 1
,
a E Zn
n)
,
j=llaxjv12 ), As a consequence
of Parsevals relation for the Fourier sine and Fourier cosine series we have (3.12)
JQIvj2dx = flval2
,
JQIOv,2dx = j(a,a)jva12
with the sums taken over all multi-indices a with aj
,
>
vEC0,(S2), 1
,
and
(a,a) = Zjaj2 , and with the Fourier coefficients
va =
(3.13)
2n1Q
v(x) sin anx dx
Proposition 3.4. For every C > 0 there exists an integer N = N(C), such that v E C'(Q)
va = JQsinarx v(x)dx = 0 for jaj
,
JQIVv12
(3.14)
dx > C JQIv12 dx
.
Proof. It is clear that (3.14) is an immediate consequence of (3.12)
, setting N(C) = Max {Ial (a,a) < C ) Returning to our general case we next prove the following. .
:
Proposition 3.5. For every B > 0 there exists finitely many functions zj E C_ (S2)
,
j
= 1,...,M , M = M(B)
u E C0'(S2)
,
(zj,u) =
0
(u,Lu) > Bllull 2
,
such that
1,...,M
111.3. Compactness of the resolvent
75
Proof. Apply proposition 3.3 , with A = 2B , and assume without loss of generality that each of the compact sets supp wj relates to a subset of the cube Q under the coordinates of the corresponding chart Q.
.
In that compact set we also get a positive bound
below for the positive definite matrix function ((h3k(x))). Thus AO(AAwju)112
f
> (B + dA)IIAAw.u112
AAwj u sinalix dx =
0
,
I a I
< N.
.
Since only finitely many indices j are involved (i.e., those with supp wj n supp aA 1 0 (3.18) translates into (3.15), and (3.16) follows by combining (3.10) and (3.17), q.e.d. ,
Proof of theorem 3.1. Setting u =
L-1/2w
in (3.15)
,
(3.16) we get
(3.19) IIL-1/2 w112
Since L1/2C.(N) is dense in H, the estimate (3.19) follows for all 1,...,M . Since B may be choosen w E H orthogonal to L-1/2z. L-1/2, by Rellich's arbitrarily large this implies compactness of criterion ([C1], A1,4) IIL-1/2ITBI
(3.20)
where
:
<
We get B-1/2
,
TB = (L-1/2{zl,...,zM})1
TB has finite codimension. This completes the proof.
Corollary 3.6. Under the assumptions of theorem 3.1, let " C 0 Then the statement of the theorem holds for be a subdomain of the restriction of the differential expression L to Q" as well If L" is the Friedrichs extension of the minimal operator generaL"-1/2 E K(H") Moreover, the then ted by L in H" = L2(Q",du) .
:
.
,
same still holds for any expression (3.21)
Kh"3k3
L" = q"-(S/K)2 xj
x
k
of the form (3.1) with C--coefficients, defined only in SZ" , and satisfying (3.22)
Proof. For u E
q"> q
,
((h" 3k)) > ((h 3k))
setting (.,.)Q" =
1
in c"
.
..du
,
we get
111.3. Compactness of the resolvent (u,L-u),,- > (u,Lu) > B(u,u) = B(u,u)0-
(3.23)
,
whenever (zj,u) = 0 , as a consequence of prop.3.5. Here we were extending u to 0 by setting it zero outside QNotice that the .
conditions (zj,u)=0 may be written as (z-i,u),- = 0 , with z-i _ ziIsi` E HIn other words, we conclude that prop.3.5 is valid also for L- in the subdomain Sl- C n, except that now the functions .
no longer are C'(c-), but only in C'(Q)1H`
z`3 .
.
This weakened
condition still allows its use for the argument of the proof of thm.3.1 , because we still get (z-j,L--1/2w) _ (L--1/2z-j,w) q.e.d. E Hwhere L--1/2z-.
,
We now return to comparison triples. Let triples {c,du,H}, and {c^,dp^,H^} be given, both satisfying the basic assumptions of sec.l. We write {2,du,H}
(or {0^,du^,H^} (c> {Q,dp,H}
)
if
(i) 2 is an (open) subdomain of Q^
,
(ii) du = dµ^Ic
(3.24)
(iii) ((hjk(x))) > ((h^Ok(x))), q(x) > q^(x), for all x E Q.
In this case we will say that the triple {Q^,du^,H^} sub-extends the triple {P,dp,H}
.
The main point for introducing this more complicated notion is the fact that many such singular problems display a very different behaviour near a point x0 E 2Q C 2^ than at infinity (of 0^
)
.
q -
For example, as shown by cor.3.6, we do not have to require L-1/2 near such a point x0 to get compactness of This will still become more evident in V,3, where we inves,
tigate compactness of commutators of the generators of a comparison algebra C Here it may become important that a proper change of depen.
dent and independent variable may be necessary, before Q^ can be defined (cf. the examples in V,4 and V,5). Theorem 3.7. For a comparison triple {c1,du,H} let there exist a
triple {0^,dp^,H^} (c> {c,dp,H} satisfying (3.25)
q^(x) > 0 for all x e 2^\K
where K C Y^ is compact
.
76
111.3. Compactness of the resolvent i.e., with
Then, for every function a E C(Q) with a = a2 = 0(q^)
(3.26)
the operator aA =
,
and
aH-1/2
0
,
(in 2-
is compact in H = L2(Q,dp).
Proof. Assume first a has a real extension 0
.
a^-1H'a"-1
Consider L^=
and set , and note that L^+ Y0 , with
a sufficiently large positive constant 0 satisfies the assumptions of thm.3.1. Also the expression L+y satisfies the assump0 (L+yo)-1/2, (L+y0)1 are compact, and L+y0 tions of cor.3.6. Hence (therefore also L) has discrete spectrum. But A-1 = aH-1a is well defined, hence 0 cannot be eigenvalue of L, and L-1 also must
be compact. But L-1= (aA)(aA)*, so that aA is compact, since an operator A E L(H) is compact if and only if AAA is compact. For a general aEC(2) we get UaAuU
, where b = cJQ is
the restriction to 2 of O
A.1, with
.
just done above. The proof is complete.
Corollary 3.8. Under the assumptions of thm.3.7 we also have JaICAC E K(H)
,
for all O<e
.
This corollary is a straight application of I,lemma 5.1. Corollary 3.9. The operators Ae, O<e
(H-a)-lis compact for all XERs(H)=T\Z, where Z=Sp(H) consists of countably many eigenvalues of finite multiplicity without cluster. The same holds if {Q,dp,H1
Proof. Under these assumptions the conditions of thm.3.7 hold for a=1 , evidently, q.e.d. Lemma 3.10. Given two bounded operators A , B E L(H)
.
If A is
compact and B satisfies the inequality (3.27)
IIBuf
< UAuU
for all u E H
,
then B also is compact.
Proof. By Rellich's criterion (cf.[C1],App.AI,4) there exists a closed subspace S£ C H with IIAISEU < e
(3.27) implies that
,
for every e>0
.
But then
77
111.3. Compactness of the resolvent (3.28)
IlBull
Or, IBIS EI
< e
,
<
IlAull
78
< e Iluh for all u E SE
so that B must be compact, q.e.d.
4. A Greens function for H and Hd, and a mean-value inequality. For later application it will be useful to obtain a 'clean' fundamental solution for our expression H , and, moreover, a Greens function for our Friedrichs extension H
as well as for
Dirichlet operator Hd of sec.2. First we work locally, in a given fixed compact subset
Bint= ,
B C 0' of a chart S1', with nonvoid connected
B.
Let n0 denote the (positive) distance (in In ) from B to W. It is practical to consider the local expression H of the form (1.2) redefined outside S in such a way that it equals 1-t in n,compl , while still satisfying all other conditions of H . For example one may choose a cutoff function X E 0<X<1 X=l near B , and then define the measure and expression dp'=K'dx , H'
(4.1)
=
q'-K-12 xj
K'h'jk8 k x
e
over In , where (4.2)
K'=XK+(l-X)
, q'=Xq+(1-X)
,
h'Jk
Xhjk+(1-X)Sjk =
For a while (until thm.4.4) we will entirely focus on this expression over In . Hence we return to our old notation, and will assume that H and du coincide with 1-0 and dx outside some compact subset B of In The 'parametrix function' e(x,y) of (1.9) again will play a central part. Note that it now is defined for x,yEin. Moreover, it is clear that e(x,y) _ (x-y12-n (or = log Ix-yl , for n=2) In fact, we again will depart from (1.26), which as xEScompl..
will have to be iterated. Let us reinterpret these facts as follows.
Proposition 4.1. For x,y E In define the function (4.3)
h=det ((hjk)),
with the constant cn of (1.23), and with X(x)?0 denoting a cutoff function, XEC0(IxI<2), X=l at IxI
.
There
111.4. Greens function exist functions e3E C'(Inxinxin*)
,
j=O,...,n , with e3(x,y,z)
homogeneous of degree 2-n in z and support contained in the set {(x,y)Einxin:1x-y1<2}xin* such that, for every solution ,
u of Hu=v , with u,vEC_(1n), we have the relation (4.4)
u(x) -
J
dye(x,y)u(y) =
J
g(x,y)v(y)by
,
x E In
with
O(x,Y) = 00(x,Yex-Y) - (03(x,y,x-Y))
(4.5)
.
IY3
Here the functions e3 depend only on the choice of X but not on the functions u, v. and the coefficients of H in In ,
For the proof one must look at the derivation of (1.26) again, and just confirm that the functions O3 are as stated. A pair of Co-functions u,v satisfying Hu=v may be put into the places of f anf g of (1.26), of course, once we interpret II,(2.6)
with 2=St' =,n
.
Proposition 4.2. Let 0 and G denote the integral operators on In, with kernel e(x,y) and g(x,y) , respectively so that (4.4) assumes the form u-Ou=Gv. Then the iterates Gk=OkG , k=l,...,n , and ,
0n+1,
all are well defined integral operators over In with kernels On+l(x,y), respectively. In particular, 0n+l' gn-l'
gk(x,y) and
gn are C(inxin), while F,k EC(inx]kn\{x=y}), for k=O,l,...,n-2.
Moreover, we have gk = 0((x-y)-n+k+2) is 0((x-y)-n)
,
for all n>0
with kernel gly.(x,y)
,
.
, as O
If ri denotes the integral operator
then the kernels of 0
,
j,k=1,...,n,
are C(inxin\{x=y}) and 0((x-y)-n+k+l), as kO, as k=n-1, and are C(inxin), for k=n. All the above kernels have support in {lx-yl<2(n+l)}
In particular we may ite-
.
rate (4.4) n-times for (4.6), below, where again uEC-(in), v=Hu: (4.6)
u(x) _ (0n+lu)(x) + Ik=O(Gkv)(x)
,
for all xE8
The proof is an immediate consequence of prop.4.6, below. continuous over Proposition 4.3. Given two kernels kj(x,y) gnxRn\{x=y} , and with support contained in lx-yl
,
_V. (4.7)
kj = O(lx-yl
3)
,
Kju(x) = Jkj(x,y)u(y)dy
,
uEC0(in)
79
111.4. Greens function Then the operator K3=K1K2 is an integral operator of the same form -v
again, with kernel k3 = O(Ix-yl
3)
, whenever v3=-n+v1+v2 > 0
while the kernel k3 is continuous at x=y if v3 < 0 is O(Ix-yj-n)
,
for every n>0
,
and k3 still
if v3=0
Proof. If v3<0 then the integral k3(x,y) = Jk1(x,z)k2(z,y)dz
(4.8)
exists for all x,y including x=y . It is easily seen that the product ax (z) = k1(x,z)k2(z,y) defines a continuous family of ,
functions in L1(Rn)
nuous in x and y
.
, depending on x,y , hence k3 also is contiIf v3>0 the same still is true for x#y , but
no longer at x=y
.
On the other hand, as v3>0 one may estimate nlx-zl-vljz-yl-v2dz
k3(x,y) < c
(4.9)
.
f
R
The right hand side integral I(x,y) exists, since the integrand is O(Izl-vl-v2) for large IzI, with vl+v2>n , while the local singularities at x and y have order -v1.>-n . Moreover, (4.10)
a=Ix-yl
I(x,y) =
e1=(1,0,...,0),
,
as shown by a substitution of integration variable involving a suitable translation and rotation. Then another substitution n-v -v 1 z = aw shows that I(x,y) = a 2I(0,eI) , where I(0,e1) is a -v
constant. This confirms the estimate k3 = O(jx-yj3) . Finally, if v3=0 we still get the estimate (4.9) with vl replaced by v1+n, for sufficiently small n>0 (or, similarly, for v2, if more conveq.e.d. nient). Then we get (4.10) with 0=v3 replaced by v3+n=n>0 Note that prop.4.2 suggests that the function ,
(4.11)
g'(x,y) = lk=0 gk(x,Y) = g(x,y) +
Ik=1
gk(x,y)
is an improved parametrix of the expression H , because the corresponding integral operator G- satisfies (4.12)
u - 0n+lu = G`Hu
which seems better than (4.4) continuous at x=y
.
, for all u Ei C'(Rn) ,
,
since the kernel 8n+l of 0n+1 is
In order to find a fundamental solution of H
we seek to add another term f- to g-, such that g '=g-+f- satisfies
80
111.4. Greens function (4.13)
u
G°Hu , for all u E
=
Subtracting (4.12) from (4.13) we find that f` must satisfy (4.14) On+lu=JOn+1(x,y)u(y)dy=(G°-G-)Hu=Jf-(x,y)Hu(y)dy, uECQ(In).
Let us require (4.14) not only for uECO , but even for all uE dom H (the Friedrichs extension of H0)
. Also introduce the
functions (of y) bx(y)=On+1(x,y)/K(y) , and ax(y)=f'(x,y)/K(y). Observe that bx E C0(gn) C H = L2(gn) , and require that ax E H
Under these conditions (4.14) takes the form (bx,u) = (ax,Hu)
(4.15)
,
for all u E dom H
,
so that ax E dom H and Hax = bx follows, since the Friedrichs extension is self-adjoint. In particular, of course, we also get H*ax=bx
,
so that cor.2.1 applies, since bx E CO(In). Accordingly
ax at least is C1
,
and has its first derivatives Hoelder continH-1 E L(H) , and bx E C(In,H) as x varies,
uous. Moreover, since
,
we also get ax E C(ln,H) = g-+f-
Now let us carry this back to the function g°
Introduce the function cx(y) = g4(x,y)/K(y) = g-(x,y)/K(y)+ax(y) Clearly we still get cx E C(1n,H). Combining (4.12) and (4.15) get (4.16)
u(x) = (cx,Hu)
,
for all u E dom H and all x E
Applying (4.16) for u E C_(]n\{x})
,
In
for fixed x , one
concludes that cx is a solution of H0 xcx=0, with the minimal of the expression H in the punctured In Thus operator H O'x thm.1.2 implies that we have
.
cxEC-(I
\{x})
,
and Hcx=O , as y#x
with H acting on the y-variable.
and inteFinally let us multiply (4.16) by v(x) E CD(Ln) gn Note that the integration may be interchanover grate dux ged with the inner product, since the family {cx:xEIn} is C(In,H). ,
,
.
It follows that (v,u) _ (Jv(x)cxdux,Hu)
(4.17)
,
for all vEC_
,
uE dom H
Accordingly, (4.18)
w=Jv(x)cxdpx E dom H , and Hw = v
,
for all v E
But the Friedrichs extension H is self-adjoint and >1 , and
81
111.4. Greens function
has a well defined bounded hermitian inverse H-1 to (4.18) we conclude that
H-I
E L(H)
.
Applying
H-1v(y)=Jcx(y)v(x)dp =Jg (x,y)(K(x)/K(y))v(x)dx, vEC-(In).
(4.19)
Then the hermitian symmetry of H-1 implies that (4.20)
g (x,y)K(x) = g°(y,x)K(y) , x,yE In , x#y
We have proven the following result in the special case of a triple {In,dp,H) , coinciding with the Laplace triple outside a compact set. In case of a general manifold 0 we relate the
integral operator to the global measure du=Kdx again. Also we denote the Green's kernel by g(x,y), not g°(x,y) ignoring the ,
earlier meaning of g(x,y)
.
Theorem 4.4. For any triple {c2,du,H} satisfying (1.4) the Frie-
drichs extension has the form of an integral operator H-1u(x)
(4.21)
=
Jg(x,y)u(y)duy
,
u E H = L2(,,n)
with the 'Green's function' g(x,y) (E C-(HxQ\{x=y}) and with g(x,.), g(.,x) E C(H,H). Also, g(x,y) = g(y,x), x,y E H, and, in
local coordinates, on a chart n', g has an expansion of the form g(x,y) =
(4.22)
R7)e(x,y) + Ek=1gk(x,y)
,
x,yEH',
where e(x,y) denotes the parametrix of (1.9) and X(x) denotes an even cutoff function (X(-x)=X(x) X?0). X(=-C0 , X=1 near 0 ,
,
,
Also, gk(x,y) = 0((x-y)k-n+2), k=l,...,n-3, for all n>0 gn-1 E C(H'xH') , while gk , k=1,...,n-2, have Furthermore, the first support contained in a ball Ix-yl
gn-2(x,y)=0((x-y)-n),
,
.
,
,
lyj
Finally, for any x0EH , and 0((x-y)-n) for every n>0 for k=n-1 l-X1EC0_(H) , the funca cutoff function X1EC'(H) 1=0 near x0 .
,
tion *(y) = X1(y)g(x0,y) is contained in C.fldom H
For a proof of thm.4.4 in the general case observe that the function g(x,y) is uniquely determined by the relation u(x) = Jg(x,y)(Hu)(y)dpy
(4.23)
For xEH'
,
a chart of H
,
,
for all u E dom H
let 8 and H' be as initially in this
82
111.4. Greens function
83
section. Let g'(x,y) be the Greens function of (the Friedrichs extension of) H' in In
,
and let the cutoff function X have
support in 0' and be = 1 in B
Then we also get
.
u(x) = Jg'(x,y)(H(Xu))(y)duy
(4.24)
, uE C'(Q)
Subtracting (4.23) and (4.24) we get
(4.25)J(g(x,y)-X(y)g'(x,y))(Hu)(y)duy=J([X,H]g'(x,y))u(y)duy,uEC-0.
In particular we note that the commutator [H,X] acts on the y-variable, and is a first order expression with support in 1'\B so that the function [X,H]g'(x,y) = qx(y) is C-(c') tion of y only)
.
,
(as a func-
,
Setting px(y) = g(x,y) - X(y)g'(x,y)
,
(4.25)
assumes the form (px,Hu) _ (qx,u)
(4.26)
,
for all u E C0'(S2)
We even request that (4.26) holds for all u E dom H
.
This deter-
mines a Cm-function p E dom H such that Hpx=qx in the sense of x ordinary differentiation. Then we define g(x,y)=px(y)+X(y)g'(x,y).
At least for all uE dom H with HuEC' we also get uEC_(G) , and the right hand side of (4.25) may be written as (g'(x,.)[H,X]u) so that (4.23) follows.
For the last statement observe that (i,Hv) _ (gx0,H(X1v)) + ([H,X1]gx0,v)
(4.27)
,
vGCCOfldom H
E HnC"
with gx (y) = g(x0,y) , where z = [H,X1]gx
,
,
since the
0
0
commutator [H,X1] vanishes near x0
.
Note that the first term at
right of (4.27) equals GH(X1v)(x0) = 0. Hence the selfadjointness and essential self-adjointness of its restriction to as stated. Therefore the proof C'fldom H implies that Edom HnC' of H
,
of thm.4.4 is complete.
Next we will investigate the Greens function g(x,y) near its singularity. Given some fixed x0Ec let us ask for the surface (4.28)
g(x0,y)=6(x0)n2-n},
Cx001={yEc:
(or g=dlog n, as n=2)
with 6 = 6(x0) = cnVET-x T , as in (4.22) , where the constant n 0
,
111.4. Greens function
is assumed sufficiently large. More precisely, if g in (4.28) is replaced by g0 , then the surface (4.28) will be an ellipsoid, in close local coordinates. For g one expects a smooth surface C x0,n
to that ellipsoid, and with normal close to the normal of the ellipsoid, as n is small. We leave open the possibility that other points y exist in the set (4.28). These will be ignored, i.e., the statement "y sufficiently close to x0 should be added in (4.28). Let us assume without loss of generality that, at the point x0, we have hjk(x0)=6jk=0, as jtl, =1, as j=1, and that x0=0 so that the equation of (4.28) assumes the form if
g(x0,y)=dn2-n
,
(4.29)
IYI2-n + Y(Y) = .2-n
where we have (4.30)
0(ly15/2-n)
Y(y) =
o(IY13/2-n)
, VY(Y) =
.
For n=2 the first term in (4.29) must be replaced by -loglyl, and the second term is continuous with derivatives satisfying (4.30). We again focus on the case n>2, with n=2 to be treated similarly. It follows that for small IyI the first term at left of equation (4.29) is large as compared to the second term. Along each ray ty0 , with 1y01=l , 0
n,
+YO-VY(tyo)
as t is small. In particular, also in the expression for atgt,y0) 0(t3/2-n),
the first term dominates for small t, since y00y(ty0) = near t=0 , by (4.30). Let (4.32)
a0(t)=sup{Tn-5/2IY(TYO)l,
Tn-3/21yoVy(Ty0)l/(n-2):
and define a+(t) = t-1/2+a0(t). Then t1/2a+(t)-*l t5/2-na
,
< (t,y0) < t5/2-nX+(t)
(t)
T
as t-0, and ,
(4.33) (n-2)t3/2-na+(t)
(n-2)t3/2-nX-(t)
<
.
Choose a t0>0 such that I1-t1/2a+(t)I<1/2
,
for 0
Then the function 0 decreases with t, for every fixed y0 . Hence there will be precisely one root tE(O,t ) of (4.31) for each y 0
whenever n2-n>(3/2)t n, i.e.,
O
Moreover, then
84
111.4. Greens function (4.33) implies that
(t,y0) = n2-n < (3/2)t2-n
(1/2)T2-n<
(1/2)1/(n-2)n
.
Or,
(3/2)1/(n-2)n
< t <
(4.34)
.
This defines a surface Cn: t=tTI (y0), contained in the spherical shell (4.34)
.
Combining (4.34) with the second (4.33) we get
(4.35)
2/9(n-2)nl-n
.
6(n-2)nl-n
<
This implies on CT, , so that Cn is smooth. Also we get Cn inside C,, n as n
the function (y) are by (4.30), and (4.34), since the first term is constant along the spheres jyl = const. By (4.35) cnl-n
the radial derivative is bounded below by . Accordingly the unit normal of C (which is parallel to the gradient) has its ran
dial component of the form nr(y) _ ±1 + O(lyll/2)
(4.36)
,
as lyl-.0
Now we consider the integral (4.37)
In = JW (g(x0,y)-Sn2-n)Hu(y)duy TI
for small n , and some Co-function u , where Wn denotes the interior of C . With a cutoff function V=l near x0 =0 near CT, xOen ,
write u=ui+u(l-$)=ul+u2, and, correspondingly, In=ll,n+I2,n Clearly we get Il,n = u(x
)
Sn2-nJqu du
-
.
On the other hand,
1
0 g(x0,y)-6n2-n
setting w(y) =
,
I
(4.38)
2
JHwu2du
=
Jwqu du - Jdya
=
2,n
for a moment,
+
yj
(Khjku
2lyk
)w
JC dSyK(y)hjk(y)vk(y)w lju2(y) n y
with the surface element dS and the exterior normal components V.. Noting that Hwu2 = -Sn2-nqu2 clude that 0
(4.39)
,
and assuming that Hu=O , we con-
= In = u(x0) + JC dS p n u -6n2-nJW qudu n
n
with
p (y)=K(y)hJk(y)g n
(x0,y)vk(y) IYj
.
,
85
111.4. Greens function
86
Write the surface integral in (4.39) as an integral over the Sn-l unit sphere Sn-1={ly0I=1} , with the surface element dSl of Iyln-1/nr(y) noting that dS/dS1 = , with nr of (4.36) We get .
(4.40) u(x0)=6n2-nJWnqudu-JSn-lItn(y)In-lpn(tn(y))u(tn(y))/v rdS1
Izln-lpn(z)/nr(z) Here we introduce the function an(z) = , and observe that a(z) is bounded and bounded below by positive conLet us multiply (4.40) by nn-1 stants independent of n and z and integrate this do from 0 to .
n0-(2/3)1/(n-2)t0
,
:
(4.41)
n0u(x0) = P -
J
S
n-1dS1. Or. nn-ldn(au)(tn(y0)) y
Ino
with P= S 0
n W qudu T)
In the inner integral of (4.41) make the substitution t = tn(y0) of integration variable. The derivative tn= 8ntn(Y0)
may be obtained by differentiating (4.31) for n (4.42)
.
It follows that
tn((2-n)t1 n + y04y(ty0)) _ (2-n)nl n I
Clearly this implies that to is bounded and bounded below by positive constants independent of t,n, and u . Therefore the integral in (4.41) may be written in the form (4.43)
dys(y)u(y)
fW
,
cl<$(y)
rb
with constants cl, C1 independent of n and u
On the other hand
.
the integral P above may be estimated by (4.44)
O(n2JW
dyIu(y)I rb
We have proven the following result.
Theorem 4.5. Let the 'local balls' B(x0,n) be defined by (4.45)
B(x0,n) = {xEO
:
d(x,x0)
,
where d(x,y) denotes the geodesic distance from x to y, with respect to ds2=hjkdxJdxk. For every compact subset K of 0 there exist positive constants e0,c,C, which generally may depend on the triple {c2,du,H}, and on K, but do not depend on u, such that, for
every nonnegative solution of the differential equation Hu=O
every e>O with e<e0 we have the inequality
,
and
111.4. Greens function ce
(4.46)
nJB(x,e/2)udu
< u(x) < Ce-nJB(x,2e)udu
,
x E K
87
.
Indeed, our construction gives such inequality with the two 'balls' of integration replaced by the set Wn above. Moreover, it follows from (4.34) that B(x,e/2) C Wn,x C B(x,2e) smaller co
,
for suitably
, which implies (4.46) . Also, it is evident from our
construction that each point x0En has a neighbourhood Nx
such 0
that the constants c,C,e0 may be choosen independent of x, as long as xEN x
.
In particular, our construction was carried out in a
0
special coordinate system, for each fixed x. Hence we must involve a coordinate transform to establish the special coordinates, but the coefficients involved then are locally continuous in x, and all conditions are locally uniform. Since a compact set K may be , we thus can covered by finitely many such neighbourhoods Nx j
choose our constants to be valid in K,q.e.d.
5. Harnack inequality; Dirichlet problem; Maximum principle. In all of this section let H denote a second order strongly elliptic differential expression of the form (1.2), with real coefficients, and q(x) of any sign, not necessarily satisfying (1.4). Actually, formal self-adjointness is not required for thm.
5.1 nor thm.5.8. In fact, both results are known to be true under still more general assumptions not important for us (cf. [GT1])
.
In preparation for some results of sec.ll we next will discuss a proof of the Harnack inequality, using the inequality of thm.4.5.
Theorem 5.1. (Harnack's inequality). For every compact subset KC(2 there exists a constant c=cK such that, for every nonnegative solution (5.1)
of Hu=O we have sup{u: xEK} < c inf{u: xEK}
.
Remark. It is known that the constant c may be choosen to depend only on the dimension n, the ellipticity constant (i.e. the local lower bounds of
in a given fixed countable
atlas), on local bounds for all coefficients, and the choice of 0 and K (cf. Gilbarg-Trudinger [GT1]). We will not require (nor
III.5. Harnack inequality prove) this much stronger result here, however. Proof. First assume that K=B(x0,e) , as in (4.45) for some x0ES and that B(x0,16e)C92 as well, where 8e<eo , with e0 as in thm.4.5.
A solution u of Hu=O assumes its maximum and its minimum at xM and xm E K , respectively. Then we use (4.46) for u(xM) < Ce nJB(xM,2e)udu
(5.2)
<
Cc-nJB(xm,4e)udu<8nC/c u(xm)
using that d(xm,xM) < 2e , hence B(xM,2e)CB(xm,4e) . Accordingly (5.1) follows with the constant cK given by 8nC/c, where c and C are the constants of thm.4.5.
Now consider a general compact subset KCQ
.
It is no loss
of generality to assume that Kint is connected, and (Kint)clos=K since a larger K with these properties always can be found. The constant cK will also be valid for the given K, since the sup increases and the inf decreases, as K is enlarged.
Again the solution u assumes its maximum at xM and its minimum at xm, and these points may be connected by a path r within Kint j=1,...,N, may be Then a finite sequence of points xjEr .
,
found such that xO=xM , xN=xm , and that the intersections B(xj,e)fB(xj+1,e) , j=1,...,N-1 all are nonvoid, while e is choo-
sen to satisfy the above conditions. Accordingly, letting B B(xj,e) , we conclude that (5.3)
u(xM) <supB u
0
0
r
with the constant y=8nC/c of (5.2). This completes the proof. Proposition 5.2. Let 0' C 0 be a subdomain (an open connected subset) of 0 . If u e C'(2') solves Hu = Au , x e 0', with A E I and u is nonnegative in 0' then it must be either positive in all of 0' or identically zero in 2'
.
Proof. Follows from Harnack's inequality.
Next we will make some comments about the Dirichlet problem of the differential expression H in a subdomain 2' C Q with regular boundary, in the sense of sec.2. That is, we assume Q'U3Q' compact , and 8Q' a finite union of (connected) compact n-l-dimenNote We also allow the case of 32=O sional submanifolds of Q that the assumptions of cor.3.9 are satisfied, so that the Friedrichs extension H, (i.e., the closure of the the Dirichlet .
.
operator Hd) has compact resolvent. It follows that the spectrum is discrete. There exists an orthonormal basis of H=Ha*
88
111.5. Harnack inequality {,j
of H'=L2(SZ') consisting of eigenvectors of H
such that H*
=
(5.4)
A0 < Al < ....
X.
.
, with corresponding eigenvalues ,
aj i - , as j + .
.
Proposition 5.3. All eigenvectors are in C'(S2'U3S2')
.
If AEM
A#A1, 1=0,1,2,..., and if 3S2'9 0 , then the Dirichiet problem
(5.5)
u E C_ (S2' Ua c2')
, Hu = Au , x E SZ' U8 S2'
, u l 9 S2' = u 0
is uniquely solvable for all u0 E C0-(252')
Proposition 5.4. For the smallest eigenvalue A0 of Hd we have (5.6)
12/]lull'2
\0 =Min {J(u) :u E H1 \ {0} }, where J(u) =
Here Ilull 1 and Hull' denote the norms IluO1 and (lull
,
Pull
for n'
.
Proofs. Since H is of compact resolvent we find that A is not in the spectrum of H , so that the resolvent (H-A)-1 exists, as a bounded operator of H'. Hence the equation (H-A)u = f is uniquely solvable, with u E dom H, for all fEH'. Again, if f E CW(SZ'U8S2'), In partithen we conclude that uEC(S2'U3S2') , using cor.2.2 .
cular this also implies that the eigenvectors are For a given u0 E C'(8S2'), if 8S2' is nonvoid, it is easy to
construct v E C'(c') with vlac2' = u0
Just extend u0 a little
.
distance inside of D', carrying it to zero,using a partition, etc. then the problem (5.5) is equivalent to (H-A)w = - (H-A)v = f
(5.7)
,
where w = u - v , evidently. Since (Hd-a)w=f is uniquely solvable for f E C'(S2) proposition 5.3 follows.
For prop.5.4 we first notice that, for u E C'(') r
(5.8)
(u,u)1 = (u,HOu)'=(u,Hu)'= j=0Xlaj12
aj =(,Pj,u),
Taking closure in H1 one confirms equality of the first and fourth We know that H1 C H' . Hence term in (5.8) for general u E H1 r
the functional J(u) of (5.6) is well defined over H1 , and
(5.9)
Eajla.I2/1lajl2
J(u) =
> A0 = J(p0)
This completes the proof of prop.5.4. I
Proposition 5.5. Any u E H1 \
{0} which minimizes the functional
89
111.5. Harnack inequality
J(u) is in dom Hd ,and is an eigen function of Hd to the minimal eigenvalue a0
.
Proof. For a given function u with the assumed properties and any w E dom H
let 0(t) = J(u+tw)
ble function,since u+tw E Hl \
This is a well defined differentia{0} for small t
Using the fact
that u minimizes J we conclude that (5.10)
dJ/dt(0) = 2 Re ((u,w)/(u,u)' - (u,w)'/(u,u)'2) =
Or, replacing w by we
u
with suitable real u
,
0
it is found that
'Re' in the last equation (5.10) may be omitted. It follows that I
(5.11)
(u,w)1 =(u,Hw)' = A(u,w)'
A = J(u) ,for all w e dom H.
This implies that A is an eigenvalue of H hence of Hd ,by prop.5.3. It is clear,from prop.5.4 that A is the smallest eigenvalue of Hd
,
q.e.d.
Proposition 5.6. If u is a real-valued eigenfunction to the smallest eigenvalue a0 of Hd ,then either u - 0 or u 9 0 in all of 0'. Proof. In view of prop.5.2 it is sufficient to show that u cannot assume both positive and negative values. If the latter were true then the function v(x) = Max {u(x),0) vanishes identically in some open subset of 0' but does not vanish on all of 0' We claim that v is in Hl ,and that it minimizes J(u) This,if proven, implies that v E dom Hd ,and Lv = a0v a contradiction because v then will have to vanish identically,in view of prop.5.2. We require the following special case of Sard's theorem
(cf.
[ Mll] , [ GPl]
.
)
Proposition 5.7. For any real-valued C'(0)-function f the set of all critical values (i.e. values n E I such that n=f(x) at some xEO with Of(x)=0) has Lebesgue measure 0 on the real line I .
Using prop.5.7 we conclude the existence of a decreasing sequence {ej} ,ej + 0
such that the level sets u(x) = ej do not contain a critical point for each j Indeed the nulset of all .
critical values of u(x) cannot contain any interval (O,e) , e>0 It follows from the implicit function theorem that the sets = {x E 0': u(x) = e
ail;
are finite unions of smooth compact
J
n-l-dimensional submanifolds of ill. In particular, ;Q
naO'=o, J
since u=09ej on aO'
.
(In case of infinitely many components there
90
III.5. Harnack inequality would have to be a limit point in the interior of the compact set O'U8O' , which would make E. a critical value.) It follows 1
J
1
that the set &IE U8S2E , with S2E J
J
is a finite dis-
_ J
joint union of compact manifolds with boundary. Accordingly we may integrate by parts in these sets to obtain JonhJku (5.12)
u IxJ
kdu=-JOn(u-e1)K-1(KhJku
,
Or, setting S2+
kdu
)I
IxJ
I x
x
I S2"
=
S2
E1
, we get
= US2E 1
(5.13)
(h Jku
(v,v)1 =
k + gIul2)du = X0(v,v)' <
u
Jsl+ I
xJ
Ix
r
This indeed shows that v E H1 minimizes J ,q.e.d.
Finally, in this section we formulate and prove a maximum principle for second order elliptic expressions, needed below.
Again we will not require the strongest known results (cf.GT11). Theorem 5.8. (Maximum principle) Assume a differential expression of the form (1.2), with q(x) > 0
.
Let u E C2(O) be a solution of
the differential inequality
(5.14)
Hu < 0
, x E S2
Then, if u assumes a positive maximum at some point of 0, or if q(x)=0 , we have u = const.
Proof. Assume that uM=u(x0)>u(x) for all xEO, and some x0ES2. Let um >
0
.
Since we have q>0 , and since Vu = 0 for x=x0
,
it
then follows from (5.14) that (5.15)
hjkulxjxk > 0 at x = x0
.
At a maximum we also have the 'Hessian matrix' ((u
k)) = M IxJx
negative semi-definite,as well known. On the other hand, it is no loss of generality to assume that hJk(x0) = Sjk , by virtue of a suitable linear coordinate transform. Then (5.15) amounts to the condition that the trace of the matrix M is nonnegative. One thus concludes that the Hessian matrix must be zero at x0
.
A contradiction results if we even have Auu>O at x0 , since then it is implied that the trace of a negative semi-definite matrix is positive.
91
111.5. Harnack inequality
A proof of the maximum principle results by proving that a suitable modification of u indeed results in a function w with Hw>O at its maximum. The construction, below, has been adopted from Protter-Weinberger [PW1] p61f. ,
Suppose we have u(xl)
by a path r , and let x2 be the first point, coming from xl
,
where u=uM. Choose a chart for x2, and let B be a ball with center x2 , and radius d, in that chart. On the part between x1 and x2 of r choose a point x3 with distance from x2 less than d/2. Let B1 be the largest open ball with center at x3 such that u
.
Clearly B1 has radius
chart. Also there must be a point E at the boundary of B1 where u=uM . Finally let K be a ball inside of B1 tangent to B1 at and with smaller radius, and let y be its center. As a result of this construction we have a ball K (of radius r and center y) such that u=uM at precisely one point while u
of 3K,
Also KU3K is contained in a ball
contained in a chart (Fig.5.1). Let 0
the ball around C with radius r1. Since uM>O we can choose ri so small that u>O in K1 , so that APu>0 in Ki. The boundary 3K1 is the union of a closed arc C1' in and an open arc the set KU3K ,
Cl" outside K
Note that max{u:xEC1'} = uM-C
Fig.5.1
Now introduce the auxiliary function z(x) =
(5.16)
e-a x-y
2
- e-ar 2
with a>O to be determined. Confirm that (5.17)
A z = e aI x-Y12{4a2hjk(x-y)i (x-Y) k 2a(h» +bJ(xj-yj ))}, 11
where, for a moment we set 0
= hJk3
u
coefficients hjk
3 k+bj2
xi x
x
, and where the
By choosing a large bi all are taken at x we can arrange for L z>O in KUBKi, using ellipticity of H ar )- 1 Finally define w = u + ez , with Confirm ,
.
0<e<0(1-e-
.
92
222.5. Harnack inequality that w has the following properties. 2 e(1-e-ar
) () w0 in K1. Thus we get the contradiction mentioned above, and the theorem is established. .
6. Change of dependent variable; Normal forms; Positivity of g. Let us return to the configuration of section 1, of a triple {Q,du,H} of a measure du = Kdx , and an elliptic self-adjoint
The presence of expression H of the form (1.2) on a manifold 0 hjk the tensor on 2 suggests introduction of the Riemannian metric .
(6.1)
on (6.2)
ds2 = hjkdxJdxk
,
((hjk))-l
((hjk)) =
. Then one also has the corresponding surface measure
dS = 1 dx
, h = det ((hjk))
(det ((hjk)) )-l
and the 'Beltrami-Laplace expression' 0
(6.3)
la X]
Yi hJk2 k x
The choice which is self-adjoint with respect to the measure dS dp = dS , for a general Riemannian space 0 with of H = - 0 + 1 .
,
positive definite metric (6.1) will give a configuration meeting (1.4). Then the Friedrichs extension H will be called the Laplace comparison operator. Two triples {l1,du,H} will be called equivalent if they can
be transformed into each other by a "transformation of dependent variable". To explain this in more detail, if we set u=yu-, v=yv-, with a positive CO(Q)-function y then the inner product (1.1) assumes the form (u,v) = L u'v-y2du = (u-,v-)-. Also, the equation
Hu=v may be written as y-1Hyu" = v- . Accordingly we introduce (6.4)
dp-
= y2du
,
H- = y-1Hy
,
as a new measure and second order differential operator, and the triple (2,du-,H-) will be called equivalent to (c2,du,H). (We then
93
111.6. Normal forms; positivity of g write (H,dp,H)-(Q,dp",H-).
94
It is clear then that the linear map
)
u' i yu- defines an isometry L2M,du-)-L2(c2,dp) . Moreover, we get (6.5) (u,v)1= (u,Hv) _ (u-,y 2(y-1Hy)v-) = JHu-H-v-dp- _ (u`,v-)1,
for u,v E
C0-
(0)
, where (.,.)1 is the inner product (1.3) for H-.
Since (A-liCO)** = A-1
,
H`-112
and the corresponding for A` =
is
true, (6.5) implies that uyu- also defines an isometry H1+H1. Furthermore H-=y-1Hy , of course, is a second order diffe-
rential operator with principal part -hi13
1
a
xi
. We get (H-u,v)"
x
for u,v E CO(Q) (Y-1HYu,Y2v) = (Y2u,Y-1Hyv) = (u,H-v)` so that H- is formally self-adjoint,with respect to the new inner ,
product. In fact, H- again is of the form (1.2), with (6.6)
K- = Y 2 K
,
q- = Hy/y = q - (yK)-1(Khjky
)
Ix3
.
Ixk
To verify this we write, for u,v E CO =(yu,Hyv) = J(hjk(YU)
(u,H'- v)` =(Y2u,y-1Hyv)
=JhjkuIxjvIxkY2dp
+
+ gy2uv)dp
(Yv) Ixi
x Ix
J(hjkYIxjYIxk + gy2)uvdu + Jhjkyy
kdp
Ixj(uv) I
x
A partial integration transforms the last integral into -
JK-1(yKhJky
kuv du . Accordingly, by a calculation,
)
Jxj j
(6.7)
x
(u,H`v)` = J(h3kulxjvlxk + q`uv)dp-
, u,v e CO(Q)
K- of (6.6) Comparing this and dp- = K-dx with (1.3) one concludes that H` = y-1Hy indeed must have the
with q- of (6.6)
,
,
desired form.
It is natural to use the equivalence defined for introducing a suitable normal form for comparison triples. We shall find two specific normal forms convenient.
Proposition 6.1. Every comparison triple is equivalent to a unique triple of the form (6.8)
(n ,dS,-A + q")
with surface measure dS and Laplace operator A of the given metric ds2
, and with a C-(n)-function q"
,
explicitly given by
111.6. Normal forms; positivity of g (6.9)
Proof. Note that y
{K2/h}-1/4
= Hy/y
q
,
Y =
as defined in (6.9) indeed is a Cam-function
,
on 0 because the transformation properties of the measure du and hjk the metric imply coordinate independence of q" and y of (6.9), as easily checked. Then the proposition follows from the general facts about the transformation u = yu- derived above, noting that 2
K_ = Y K = F ,q.e.d.
A triple (6.8) will be said to be in Sturm-Liouville normal form.
We note that the condition q>l imposed on comparison triples It will guarantee that H is semiso far has only one purpose :
bounded below, and, moreover, that H > 1
, implying (1.4). On the
other hand, prop. 6.2, below, shows that for a triple with H>l always another normal form can be introduced -i.e. an equivalent In fact, if R is nontriple - such that q" is a constant >1 .
compact, we always can arrange for q-=l
In the following we often no longer require q > 1, but only that (1.4) holds. Proposition 6.2. Let (cl,du,H) satisfy all assumptions of early sec.
1, especially (1.4), i.e.,(6.10), below, but not necessarily q>l. (u,Hu) > (u,u)
(6.10)
, u E C_(Q)
Then, if 0 is noncompact, there exists a positive C-(0)-function y, solving the differential equation HY = y in all of 0 .
Using y in a transformation of dependent variable u = yu-,
as described, one arrives at an equivalent triple of the form {0,du-,l-&u-} with
-= K--18
(6.11) u
K-h3k xl
k
, du = K-dx
2
,
K- = Y K
x
in local coordinates.
On the other hand, if 0 is compact then the equation Hu=au,
for some real constant A > 1 admits a positive solution y which may be used as above to obtain an equivalent triple with q-=A > 1.
This proposition is an immediate consequence of the facts derived on transformations of dependent variable,as soon as we establish existence of a positive solution y of the equation HY=y.
Such a solution will be constructed below. In fact it will become
95
111.6. Normal forms; positivity of g
evident that the condition H > 1 is necessary and sufficient for existence of a positive solution. The result, below is required for prop.6.2,and perhaps of independent interest. It was shown by the author for an ODE and a compact interval in [CEdI, p.65 (in 1957) (also [CS],IV,1.9.),
and for elliptic operators of varying generality by Allegretto [Al1l, [Al21, [Al31, Piepenbrink [Ppl], [Pp2], Moss-Piepenbrink
[MP1l, Agmon [Agl]. We are indebted to M.Meier for the suggestion to use the Harnack principle, which independently also was used by Agmon (Agl] for the same result. ,
Theorem 6.3. Given a positive measure dp and differential expression H as in (1.2) on a manifold Q, with real symmetric positive definite hjk and real-valued q , all C` , but not necessarily q > 1 ,
.
(a) If D is noncompact, and the expression H-1 is positive, i.e., (6.12)
j(hJku lx
Iu lx k
+ (q-1)juj2)dp
> 0
,
for all u E Cp(D)
then there exists a positive solution u = y E C-(D) of the partial differential equation Hu = u .
(b) If D is compact then the same is true under the additional assumption that (6.12a)
inf {(Hu,u)/(u,u)
:
u E C0-(S2)
,
u 4 0) = 1
(c) Vice versa, regardless whether 0 is compact or not, if a posi-
tive C'-solution of Hu = u , defined over all of 0 , can be found then H-1 is positive (i.e. (6.12) holds. If 0 is compact then also (6.12a) follows. ,
Remark. Notice that a positive solution y of Hy = y may be used for a transformation of dependent variable, as in section 1, to take the triple (0t,dp,H) to an equivalent one {St,dp-,H-), with q=Hy/y=1, using (6.6). Since the new triple satisfies q"'=1>l, we trivially get (1.4), coinciding with (6.12). Moreover, if 0 is compact, then a positive solution of Hy = y trivially represents an eigenfunction of H to the eigenvalue 1. Since we know that (6.12) holds we then conclude that (6.12b)
1=(Hy,y)/(y,y) < (Hu,u)/(u,u), for all u E C0_(S2)=C"(0).
This evidently implies (6.12a) . Accordingly we already have pro-
96
111.6. Normal forms; positivity of g ven (c), using the above transformation of dependent variable.
Proposition 6.4. On a manifold n satisfying our general assumptions there exists a C'(O)-function 4) taking positive values on 0 such that limx_,°°4)(x)
_
°D
,
in the sense of sec.3 (or app.A).
Proof. If Q is compact then the function 0 = 1 will satisfy all with
assumptions. If Q is non-compact then set 4)(x) _
the partition of unity of app.A.
Q.E.D.
Proposition 6.5. If n is noncompact then there exists an infinite sequence
521C 522C ... C 52
, U SZj
=
52
,
52j
0
,
#
0
,
(6.13)
0i = U1 nil , S2j1r1 njm =
19m
,
where each 0j1 is a nonvoid open subdomain of n with compact closure and nonvoid smooth non-self-intersecting boundary (i.e., 8S2 j1
is an n-l-dimensional submanifold of Q ). Every sum in (6.13) is finite,and every inclusion in (6.13) is proper. Moreover, each nilclos is a proper subset of some unique SZj+l,m Proof. Using proposition 5.7 on the function 4) of proposition 6.4 we conclude the existence of a strictly increasing sequence nj + 1 < nl < n2 < ... < nj < ... such that the "level set" ,
an3 = {x E 0 = 0
:
4)(x)
= nil does not contain any point x with Of(x)
Indeed the set of all critical values cannot contain any
interval (n,°°) (6.14)
so that a sequence {nj) must exist. Then define SZj
=
{x E 0
:
4)(x)
< nj}
,
so that an. is the set defined above, containing no critical points. By the implicit function theorem we conclude that an
indeed is a compact n-l-dimensional manifold,not necessarily conis compact, by con-
nected. Each SZj is nonvoid and each Q.U3S2
struction of 4) Thus each connected component is the interior of a manifold with compact closure and smooth (compact boundary). There can be at most finitely many components. (Otherwise a subsequence of components must converge to a point x0E852., near which 8n. while it is supis homeomorphic to a connected open piece posed to contain an infinity of nonconnected boundaries of components.
) Clearly each component 0 j1 has a nonvoid boundary, since
97
111.6. Normal forms; positivity of g we assume 2 connected. Clearly we get 37
C
0j+l
, which implies
the remaining statements, q.e.d.
Let us observe next,that prop.5.6 establishes theorem 6.3 for the case of a compact manifold Q
.
Proposition 6.6. Let 2 be noncompact,and let
be defined as in 2j
(6.13) Then if X3 is the smallest eigenvalue of Hd in Qj the sequence {a0} is strictly decreasing and 0 lim 0X > 1
Remark: The operator Hd for Qj is defined as the direct sum of the corresponding operators for the components Qjl .
Proof. Notice that the minimal eigenvalue v of Hd is minimal eigen value for at least one of the connected components Pjl. But Sljl C S1 j+1'm ,
for some m , by prop.6.5, and this is a proper inclusion.
If i * 0 is a corresponding (positve) eigen function then define (6.15)
w = i in S1 jl
,
=
0 elsewehere in 0j+l,m
Then the conclusion of prop.5.5 may be repeated to show that w E H1 - for 0j+l, and J(u) = v, so that the minimal eigenvalue m, v' of Hd in Qj+1 cannot be larger than v If v=v' then w will have to be an eigen function of Hd in Qj+l,m which is impossible since w as defined in (6.15) vanishes in some open set, by prop.5.2, thus will have to vanish identically. This .
amounts to a contradiction unless v' > v Thus we indeed get a strictly decreasing sequence,as j increases.Evidently there is .
the lower bound 1 as well ,q.e.d.
For any j and some wjE C'M
,wj > 0 we now solve the
Dirichlet problem (6.16)
u E C-(Q j)
, uI92j = wj
, Hu = u in 2.U32j
.
This is possible,since 1 is less than the smallest eigenvalue of Hd for 0j (due to prop.6.6 and prop.5.3). The solution will be called u. Proposition 6.7. The function uj is positive in all of QjUaRj. This proposition follows exactly as prop.5.6
.
Proof of theorem 6.3. To complete the proof in the noncompact case let us now consider the sequence {vj = ajuj :j=1,2,...} the positive reals aj chosen to normalize vj according to (6.17)
sup {vj(x)
:
x E Q1 1 = 1
,
j
= 1,2,...
with
98
111.6. Normal forms; positivity of g Using the Harnack inequality (i.e., thm.5.1) we get sup {vj(x)
x E 03} < C inf {vj(x)
:
x E S23
:
}
(6.18)
< C inf {vj(x)
x E 01} < C sup {vj(x):x E 01) = C
:
where the constant C is independent of j
= 4,5,...
j
,
.
Now (6.18) at once implies a corresponding estimate (6.19)
J dp(glvjl2+hkly.
< C'
,
C' independent of j
Indeed, with a cutoff function X E C'(923)
,
X=l in 02
2
(6.20)
1 k vj lx lx
_ (Xv,A v)0
(Xv,(q-l)v)0 3
=-Jhkl(Xv) 3
,
kv lx
we get 1dp
lx
where we set v=vj, for a moment. In the right hand side write (Xv)lxk = Xvlxk + vXlxk , and, for the integral with the second
term, integrate by parts again, using that vvlx1 = (v2)Ix1/2
.
It follows that J52 2
(6.21)
hkly
lxk vI xldp <
JSt Xhkly 3
Ixkvl xl dp
Max{Iq-1IX+IOpXI/2:xE03}
=-(Xv,(q-1)v)0 3-(1/2)(A11 X,v2)
Clearly this proves (6.19). In turn, (6.19) implies existence of a subsequence {v.
}
k
converging in L2(S21) to a positive solution v of Hv=v. Indeed, with a new cutoff function X1ECO(S22)
,
X1=l in 0
1
we conclude
that wj= X2vj E Hl(S22), and that the norm I1wjII1 (with respect to
02) is uniformly bounded in j By cor.3.9 the imbedding H1-H (for 02) is compact. Thus wj must have a convergent subse.
quence in H(Q2)
.
Then the restrictions to S21 define a convergent . Again we get (H-1)v. =0 for x with
subsequence of vj in H(S21)
k
X1=1, i.e.,in a fixed neighbourhood 0" of SZl
.
It follows that
for all tECD(0"), i.e. v is a
(v,(H-l)4)H1V= lim(vj k
weak, hence a strong nonnegative solution of Hv=v Also Harnacks inequality gives v.>l/C in S21, since sup v. = 1. Accordingly v=0
-
is impossible.
A similar argument may be carried out for each 0k , instead of 921
Using the Cantor diagonal scheme one arrives at a sub-
99
111.6. Normal forms; positivity of g sequence of vj converging in every nj
.
The limit v cannot vanish
identically anywhere and will be the desired positive solution of Hv = v
.
This completes the proof of thm.6.3.
Finally, in this section we will show that (1.4) also implies positivity of the Greens function of the Friedrichs extension H .
Theorem 6.8. Assume that condition (1.4) holds. then the Greens of the minimal function g(x,y) of the Friedrichs extension H ,
operator H0
,
as constructed in thm.4.4, is positive for all
, x9y
x,yESI
Proof. First we note that it is sufficient to consider the case of q=const.>1, by virtue of thm.6.3, since we have (1.4). Indeed, the Greens function of the transformed operator y-1Hy=His given by (6.22)
g-(x,y) _ (y(y)/y(x))g(x,y)
,
which is positive if and only if g is positive. Notice that we certainly have g(x,y)>0 as x and y are sufficiently close together, by virtue of the expansion (4.22) Indeed, the first term in (4.22) is positive and is large in com.
parison to the other terms. Accordingly the positivity of g follows from the maximum principle (thm.5.8), whenever 2 is compact.
If G is noncompact, then we apply prop.6.5, constructing an increasing sequence Q . with smooth boundary, and let H
the Dirichlet operator of D.
d,j
be
Let gj(x,y) , x,yEGj be the Greens
.
function of Hd,j. From thm.4.4 we know that X(.)gj(x0,.)Edom Hd,j, as x0EDj, so that cor.2.2 gives gj(x0,.) EC'(PjUe3j\{x0)) , and gj(x0,y)=0 as yEaDj . Accordingly we again may use the maximum and x#y , and principle to conclude that gj(x,y)>0 as x,yENj ,
then Harnack's inequality to show that even gj>0
.
Now we conclude the same for g by showing a type of weak convergence gj-*g . Define the sequence of operators Gj`= GjXG
, with 7
of Qj , where it is understood
the characteristic function XN 3
that the function Gj-u , defined in G. is to be extended zero out-
100
111.6. Normal forms; positivity of g
101
Let vj=Gj-u-Gu. Note that vjE Hl, and that even Ilvjhl
side S1 j,
with c independent of j (6.23)
.
Indeed, we get GuEdom H , hence
= (Gu,HGu) = (Gu,u) =
IMGull2
Similarly one confirms that, with uj= ujStj
(6.24)
IlAull2
<
IluH2
,
IIGuH2 = IlGu112 < Hujll2 <
Ilull2
so that Hvll1<211uhl=c Next we note that, for uEHOC" , we have Hvj= HG.-u-HGu = as x E Stj
0
and vjE C'(O,) as well. This implies that
,
(6.25)
0
as
,
hence we also get limj.
(6.26)
0
for all 4E H1
,
.
(This follows first for all 4EC0 and then for H1, since Ilvjlll was seen bounded, and C- is dense in Hl ,
Proposition 6.9. We have
G(u - Gu
(6.27)
in weak operator convergence of L(H,H1 )
.
Moreover, we have strong operator convergence (in L(H,L2(K)) ) for each compact of the restrictions u + (G. u)IK (to (Gu)IK ) ,
subset K C 0
The first part of prop.6.9 was derived above. The second part follows since the restriction map Hl -> L2(K) is compact, by cor.3.9
,
so that a weakly convergent sequence is turned
into a strongly convergent sequence,q.e.d.
We will apply only the weak convergence (6.27): Suppose , Then the same holds for for some x0#y0 of x0 and y0 We xENx , yEN , with neighbourhoods Nx0 N
we have g(x0,y0)<0
.
.
,
0
y0
y0
may assume that N x0rN y0 (G4)(y)<0
,
as yE NY
=
0
.
) >0 , we conclude For any SEC-(N 0 x0
, and we get "<" whenever $ is not =
0
0
Accordingly we have (m,Gu) <0 whenever uECO(N y) , u>0, and both 0
u are not =
0
.
On the other hand it is clear that ($,G. u)>0
Since the weak convergence in H1 implies weak convergence in H,
111.6. Normal forms; positivity of g we get a contradiction. Accordingly we must have g(x,y)>0 , and This completes the proof Harnack's inequality then implies g>0 .
of thm.6.8.
102
CHAPTER 4. ESSENTIAL SELF-ADJOINTNESS OF THE MINIMAL OPERATOR.
In this section we will discuss a variety of criteria, all establishing essential self-adjointness of the minimal operator for an expression H of the form III,(0.1), or one of its powers. In other words, we seek to establish the definite case, HO
,
in the language of Carleman. This is a classical subject, with a large list of contributions, too numerous to be discussed in detail (cf. the bibliography in [F1] ). Our selection, below, is guided by the requirements in chapter's 5f. In particular we only consider the case of an expression H bounded below. All proofs offered are elementary. However we will require the statement of Weyl's lemma for expressions H as well as for their powers Hm , m=1,2..... . While the first was discussed in III,1, the corresponding property for powers can be easily derived with the same proof, using the explicit Green's function we have established for the operators H
.
Clearly Hm has the m-th
iterate of the integral kernel g(x,y) as its Green's function, and one easily analyzes the character of its singularity. Then the proof of III,thm.1.2 may be repeated.
Actually, the latter argument is not required if distribution calculus is used: It is a trivial fact that products of of hypo-elliptic operators are hypo-elliptic again. The proof of III,thm.1.2 is easily rewritten to show that the expression H there is hypo-elliptic. Hence Hm , for m = 1,2,3,... all ,
all must be hypo-elliptic. Hence Weyls Lemma holds for Hm The only objection to this argument is that distribution calculus must be used, which we have not introduced here (cf.[C1]).
Also, since the result is local, one may use a result in Rn, for an expression H equal to 1-L outside a compact set. Such a result is easily derived, even for general elliptic expressions of arbitrary order, using a Green inverse of the form introduced
IV.l. Summary of results
in [C1],IV,3. This indicates a third alternative for proving Weyl's lemma for Hm . Accordingly, and since the result is gene-
rally well known and contained in standard texts on elliptic partial differential equations, we will not discuss a detailed proof. 1. Essential self-adjointness of powers of H0. In this section we turn to the question about criteria to insure Carleman's definite case for an expression H of the form III,(1.2). We will not only look for H itself, but also for its powers which will be needed in chapter 5. It was shown by Gaffney [Gfl], and Roelcke [Rol] that the Laplace operator A on a complete Riemannian manifold always is in the definite case. This was generalized to all powers Am, m=1,2,..., by the author [CWS], together with a more general result on expressions H of the form (1.2) (cf. thm.l.l, below). Chernoff [Ch] gave a very elegant proof for the powers Am on a complete manifold, using
as an essential ingredient that the first order hyperbolic system 8u/3t = (d+6)u , with the operators d and S acting on the differential form u , has finite propagation speed, so that a solution of the Cauchy problem with CD-initial-values will be C' for all t. We shall discuss Chernoff's proof in [C3], in connection with theory of hyperbolic equations. On the other hand, for the special case of S2=Yn, dp=dx, m=1,
there is a multitude of results. We mainly will be interested in the case of a semi-bounded HO , and, moreover, q0. The most general set of results perhaps are those of J.Frehse [F1], some of which we will discuss in sec.4 (cf. thm.1.9). Notice that the case of n=1, under these assumptions, does not require any additional assumptions (cf.thm.1.10). For many other contributions to this subject we refer to the bibliography in [F1], c.f. also
Devinatz [ Dv1]
.
As in III,l we work with the expression (1.1)
H =-AU + q
,
Au = 1/K3 jKhJk3 k x
,
dp = Kdx
x
Theorem 1.1. Let us assume that there exists a non-negative Lipschitz continuous function a
defined over Q
the following conditions (i) a(x0) = 0 for some given fixed x0 E S2
and satisfying
104
IV.1. Summary of results
105
(ii) limx a(x)=- , (i.e., for N>0 there exists a compact set KN such that a(x) > N as x E n\KN); IVal2 < q x E 0 \ K , with some compact set K C Q (iv) x E 0 , with constants n,C > 0 lVol < Cena Then there exists a nonincreasing sequence nm , m=1,2,... ,
(iii)
,
,
such that (i),(ii),(iii),and (iv)TI implies essential self-adjoint m ness of H0m = (Hm) 0 (i.e., the definite case for the expression
Hm.) Moreover, nl may be choosen as any number with 0 < nl < 1 Remark 1.2. Notice that the conditions (i), (ii), (iii), (iv) of thm.l.l are equivalent to the following alternate conditions: There exists a positive Lipschitz-continuous function T(x)
xE0
, such that (i') T(x0)=l for some fixed x0E 0 (ii') limx
T(x) =
0
;
(iii')
VT(x) is bounded on 0
(iv')n
q(x) > n-2IVT(x)12/T2(x) some compact set K
for all x E CZ\K , with
,
.
Indeed, (i), (ii), (iii), (iv)T1 translate into (1'), (ii'),
(iii'), (iv')n, if we set T(x)=e-na(x), as shown by a calculation.
Remark 1.3. Notice that the conditions of the thm. imply semiboundedness of H0 , because we will have q > c0 in the compact set K
, and hence q bounded below in all of 0
> 0 outside K
,
since (iii) implies q
.
Our proof of thm.l.l is elementary, but quite technical. It will be discussed in the following sections. Presently let us discuss the theorem and some of its consequences. Proposition 1.4. For a semi-bounded H0
,
if
(H9+c0) > 1
,
and
(H0+c0)m is e.s.a., for some real c0, then (H0+c)k is e.s.a., for all c E I , and k=l,...,m . Proof. Let C'= (H0+c0)m , B'=H0+c0, for a moment, and let C be the unique self-adjoint extension of C'. Clearly B'>1 implies Hull2
< (u,B'u) < llulIfB'ull, or,
IIB'ull
>
Hull, u EC-, so that 0u112< IIB'u112
(u,B'2u) < (B'u,B'2u) =(u,B'3u)
,
so that C'> 1. It follows that C must be the Friedrichs extension of C', so we get C > 1 as well. As in the proof of I, prop.2.7 we C-1 find that is bounded, and 0
IV.l. Summary of results
since it is hermitian, and one easily concludes in Bm=H . Also, BmDB,m=C', so that C=Bm Now for f E H there exists fj=C'uj= .
B,(B,m-1u j) -* f, with u.E CO = dom B'. Note that vj=B'm-lu. E CO,
and v.- v, due to lB'z >Ilzll
and that B'o = B B'k(B'm-k u.)
,
.
It follows that im B' is dense in If,
so that B' is e.s.a. Similarly we may write f.
above,and conclude that B'k is e.s.a.
Now we are finished if we also can derive that (B'+n)k is e.s.a., for every real n , because clearly H0+c = B'+n , n=c-n For this we note that, for a given n there exist positive constants X. yl, y2 such that y1IIB'ku]2< II(B1+n)kull2+allull2< y2flB'ku12, for uEdom (BI )k
(1.2)
Indeed, the second inequality is evident, and the first follows from B'i < CB, k+ cE
(1.3)
,
j=l,...,k-1
,
valid for every e >
0 , and sufficiently large derived. Now (1.2) at once implies that (B'+n)k
c6>0 as easily =(B+n) , which ,
is self-adjoint, q.e.d.
Theorem 1.5. Let 2 be the interior of a complete Riemannian manifold Q0 with compact boundary 30
,
and assume that
q(x) > y/(distance(x,3c2))2
(1.4)
,
x E cA K
There exists a nondecreasing sequence ym , m=1,2,.... such that for y > ym , and H = -A+q , we have H0m = (Hm) 0 essentially self,
adjoint. Here condition (1.4) will be considered void (i.e., always satisfied) if the boundary 83 is void. Also, A is the Laplace operator of the Riemannian metric of 0 Proof. Let p(x) be the Riemannian distance from x to 83 and let K(x) be the distance from any given fixed point x0 E Q Then .
,
.
T(x) =Max { e-nK(x)
,
P(x)}
where we set p = - if 8G = 0 , so that then T = K , may be seen to satisfy all conditions ('). Indeed, the first three conditions are trivially satisfied, using that VK and Vp are bounded, and that p and K tend to - , of the same order, as x in view of the completeness of Q0 (cf. [KG1]). on 0 ,
In view of remark 1.3 and prop.l.4 we may consider H+c0,
106
IV.1. Summary of results for large co
instead of H Note that x = e-hk T = p near - (of 00 ). We get ,
.
(1.6) q+c0> VK near - of NO , and q+c0>
107
near 3C , and
y/p2>p-2Vp2/p2
near 3C
,
for sufficiently large y, depending on n. Thus the theorem follows Corollary 1.6. Under the assumptions of thm.1.5, if we have
limx;3 (q(x)(dist(x,aC))2 = + , then all powers H0m are e.s.a.
Corollary 1.7. For a complete Riemannian manifold without boundary all powers of the Laplace operator have their minimal operator essentially self-adjoint. The proofs are evident, in view of thm.1.5 and prop.l.4.
We shall discuss Chernoff's proof of cor.1.7 in [C3], since it depends on the basic properties of the solution operator of the hyperbolic equation Btu = Du to preserve the compact support of its initial values, as t propagates (i.e.,finite propagation speed). We note that his proof also applies to the case of ,
the Laplacian acting on differential forms, which we do not consider here. The theorem,below,is am immediate consequence of thm.1.5, and III,thm.6.3, regarding a normal form of H>l. It appears as an interesting generalization of a conjecture by F.Rellich [Re l1 for Schroedinger type operators in Rn, first proven by Wienholtz [Wil] (cf. also [Chi, sec.4).
Theorem 1.8. Let C be a complete Riemannian manifold (with countable atlas), and let a differential expressions H of the form hjk (1.1) be given, with the metric tensor of the space, but a general positive measure dp . Then, if the minimal operator H0 is semi-bounded below, all its powers H0m are essentially selfadjoint in the Hilbert space H = L2(C,dp).
Indeed, we conclude from III, prop.6.2, that the dependent variable may be changed such that q=y = constant. In view of I. prop.2.4 we may consider H+c0 instead of H and thus may assume that y=l . Then thm.1.5 may be applied with v(x)=dist(x,x0)
again, with a fixed point xO
Thus thm.1.8 follows. Finally let us mention some results of J.Frehse [F11 concerning the first power of a second order operator of the form (1.1) .
IV.l. Summary of results
0 = In. Actually we are stating weaker results, since we will get restricted to C--coefficients, as we always do here. Also, in on
[F1] one finds two more applications of refined nonlinear techniques to problems of essential self-adjointness of the minimal operator which we do not include. On In we of course can get restricted to a fixed global coordinate system x = (x1,...,xn) (written with subscripts again) Then, with fixed coordinates, let A(x) and a(x) be the largest, and the smallest eigenvalue of the nxn-matrix ((hjk)) J,k=l,...,n ' Let Br (x) be the closed ball with center x and radius r and let Br = Br(0) Also we get restricted to the case Kel ,
.
.
Theorem 1.9. Given the differential expression H = -Xax hJkax
( 1 . 8 )
+ q
, x E In
,
dp = dx
.
k
J
where hjk q E C,(ln), and the matrix ((hjk(x))) is real,symmetric and positive definite. Then each of the following conditions ,
is sufficient for essential self-adjointness of the minimal operator HO
.
1) q bounded, below, and (1.9)
sup{A(x):xEB2R\BR}/inf{X(x):xEB2R\Br} = O(R2+1) 2)
,
as R-+°
q bounded, below, and
(1.10) sup{A(x):xEBr(y)}/inf{A(x):xEBr(y)} = 0(1), with some r>0, The proof will be discussed in sec.4.
In the case of n=1, and a q bounded below it turns out that (1.9) and (1.10) are superfluous:
Theorem 1.10. For the singular Sturm-Liouville expression
(1.11)
H = -axp(x)ax + q(x)
, -°°<x<°°
, q(x)>0
,
in H=L2(1) the minimal operator H0 is essentially self-adjoint. Proof. We may assume that q > 0 , since H0 may be replaced by
H0+c. Since H is hypo-elliptic, it suffices to show that Hu=O uEC0'(I)f1H implies u=O
sequences
.
Now, since uEL2(1) there must exist
such that u(Ej )-+0
, u(Oj )-+0
, as j-+co.
However, since q>0 we have the maximum principle (III, thm.5.8) valid, so that Max{lu(x)I: C <x-"J.) + 0 , i.e., uSO q.e.d.
j-
,
108
IV.2. Limit. point case for H
2. Essential self-adjointness of HO.
It is practical to discuss the proof of thm.1.1 for m=1 first. This will be a matter of an established elementary technique, applied to manifolds by W. Roelcke. The lemma, below, is crucial.
Let G,du, H=-Am+q be as in (1.1). For any compact set ECG we define the 'local inner products' (u,v)E = JEu(x)v(x)du
(2.1)
(Vu,Vv)E = JEh3ku
,
IxjuIkdu
x Lemma 2.1. Let a(x) be a nonnegative Lipschitz continuous function over 0, satisfying o(x0) = R>0 let the set ER =
{xES2
:
for some given x0. For some
0
a(x)
R (2.2)
((Auu,v)E +(Vu,Vv) E )dp
J
0
p
J: IVoIIVuIIvIdu R
p
Proof. For given fixed e,p > 0
0 < p < R introduce the
,
continuous, piecewise linear function n = np'E on with This function is Lipschitz n(t)=0 on [p,°) , n(t)=l on (--,p-e] continuous, hence the composition fi(x) = n(a(x)) defines a Lipand we get V = n'Va , almost schitz continuous function over S2 .
,
everywhere. For u,v E CO(Q) we get Cv absolutely continuous, and is locally bounded. Thus Green's formula may be applied for (DUu vC + hjku
0 = JE
(2.3)
Ix.(cv)
R
I
x
k)du
It follows that
(2.4)
(Auu v + hjku
JE
JE
In'IIValIVulIvldu R
R
Note that (2.4) holds for all p
,
0 < p < R , and fixed e >
We may integrate (2.4) dp from 0 to R
,
0
observing that the only
term at right depending on p is the term n'
=
dnpeE/dt(a(x))
Hence we get R
(2.5)
J
0
As e
-*
x
x
R
converges to the characteristic function
, boundedly, and almost everywhere, while JRIn'Idp converges
of Ep
to 1
0 the function
v +h3kul jvI k) <JE duIVoIIVulIvl dpIn'I
,
as easily calculated. Therefore (2.2) follows, q.e.d.
109
IV.2. Limit point case for H
We now can prove thm.1.1 for m= 1 , as follows. Suppose a function a exists with the properties of the thm. since a-as x-, all the sets ER of lemma 2.1, formed with this function a, are compact. Now the operator H0 is >1
,
(if necessary we may
replace H by H+c0, using rem.1.3). Thus it is sufficient to prove that im H0 is dense (I, prop.2.7). Or, we may show that (f,H0u)=0 for all u E dom H0= C0 implies u=0. In view of Weyls lemma (i.e., III, prop.l.2) any such f must be C'(O), and then must solve Hf=O. Thus we are finished if we can prove the lemma, below. Lemma 2.2. Under the assumptions of thm.l.l
,
for m=1
,
if
f E H12W(O) solves Hf=O , then we get f=0 Proof. Define the functions $(p) , I(p) , A(p) , by setting .
VR) =
(2.6)
Note that
JRIIflValdg dp , *(R) = 0 p
JRIVfII2 dp, 0
j are integrals of nondecreasing functions over
,
[0,") each. They are continuous and have a left and right first derivativative at each point. Moreover, the first derivative exists at all points except at the (at most countably many) jumps of the integrand in (2.6). Moreover, 4 and L are convex. The same is true for A . We apply lemma 2.1, now valid for all R > 0 , and
use that Hf=O, i.e., Apf = of , to get R (2.7) 0
dp(IIV-qfl2 +HVf12 p p
< [IValfllE [Vf6E R
,
0 < R <
R
With condition (iii) of thm.l.1 and the above functions we have (2.8)
A(R)
< (4'(R+0)tp'(R+0))1/2 < A'(R+0)/2
,
0 C R < - .
If A does not vanish identiClearly cally then we get A > 0 , as p > pl , for some p1 > 0 A' is absolutely continuous. (2.8) may be integrated in [p1,")
Notice that A , A ' are > 0 , by ( 2 . 6 )
.
.
for A(p) > c e2p (2.9)
,
c= A(p1)e-2p' . Using (iv)
of thm.1.1 we get
Vol (< ena < CenP < C(A/c)n/2 = c1An/2, as x E Ep, P > P1. Now we go back to (2.6), and conclude that
< c2angfll2 =
c3an
c4/X-ran/2
(2.8), for A < (2.10)
,
A'
>
since IIfU .
<
.
This may be taken into
Or
c5A2-n
,
as p > pl
with a positive constant c5 . Now it is well known (and easily
110
IV.3. Proof of theorem 1.1
derived) that the differential inequality (2.10) cannot have a positive solution in an interval of infinite length, whenever n<1. Therefore, for n < 1 we get a contradiction, and aa0
,
i.e. f=0
follows, q.e.d.
3. Proof of theorem 1.1.
We now will discuss the proof of thm.l.l for general m=2,
The basic tool will be lemma 2.1 again, which will yield a system of differential inequalities instead of the single inequa3,...
lity (2.8). It then is a rather complicated matter to show that the system obtained admits only the identically zero solution. We start by arguing as before that the proof is accomplished if we can show that the only solution f E Hf1C'(Q) of the differen-
tial equation Hmf = 0 is the function f a 0 (Here we first have added a constant to q to insure that q > 1, and H m > 1, which may 0
be done, in view of prop.l.4). This works just like in sec.2 and will not be discussed again. For an f with above properties define g.
=
Hm-jf, j=0,±l,±2,
Clearly gj = 0 for j<0 , and we interpret Hk as a differential operator applying to a C"-function, for j=0,...,m . The negative powers Hm-j , j > m , will be interpreted as the (inverse) powers
of the Friedrichs extension H , a self-adjoint operator (and the closure of HO
,
since our assumptions (and n < 1) already give
the e.s.a.-property for H0 itself. It follows that gjE H
,
for all j E Z
.
Using Weyl's lemma
a finite number of times one also concludes that all gj are CW(SZ),
and we generally have H1gj = gj-1 , with derivatives in the ordinary sense.
Similarly as in sec.2. we now define the functions (3.1)
(R)
=
1RII,Vo,gjflrdr
, i.(R) =
JRIIVg.II 0
0
where we use the abbreviation 1.112
=
dr, j=0,±1,..., (cf.(2.1)).
II.12
r
r
Clearly 4j, ,j have similar properties as , $ of (2.6). Specifically they are absolutely continuous and the left and right deritives exist at each point and coincide, except possibly at countably many jumps. The first derivatives exist except perhaps at the jumps. In the estimates below we write '(R) for the right and derivative of 4 , i.e., q'(R) =
111
IV.3. Proof of theorem 1.1
112
Clearly than j4jt,V,j, j' are nondecreasing and nonnegative, and j , V,j are convex. We have the fundamental similarly for VP'(R)
.
R
theorem of calculus valid in the form (R)-4(S)
J
'(p)dp with
S
a well existing Riemann integral. For j
identically. Moreover, using (iii) of thm.l.l we get
(3.2)
1IVoIgj1I2+UVgj112
< II/gj112+11Vgj112= (gj,Hgj) =(gj,gj_1) < °°
,
where the partial integration without boundary terms is justified by the essential self-adjointness of H0, already proven (One may gj, Hg Hgj carry it out for a sequence g in Cp with
g
Notice that (3.2) yields boundedness of the derivatives 4j', i,j for all j
.
Proposition 3.1. The functions 0j
j satisfy the system (3.3)
,
of differential inequalities, where the series at right represent finite sums, since 0j Vj vanish for j<0 and where the argument ,
at right is R+0
,
not R (just as in (2.8)).
,
0 j+Vj < (0'IVj)l/2 +
erlr
(3.3)
j < C j
= 1,2,....,
0
< r < .o
.
Proof. This follows by repeated application of lemma 2.1 to the inequalities +Ipj
(3.4)
<
(UVgj112 + IIVgjU2)dr
0
R
j < enR
IIgjII
dr
0
which follow from (iii) and (iv)n of thm.1.5. For example, the first inequality (3.4) implies R +V'
<
0
R
(gj,(q-Au)g.)rdr +
j 0
((gj,ougj)r+ (Vgj,Vgj)r)dr
1R
(gj'gj-1)rdr + IIIVaIgjIIRUVgjUR
< 0 =
R
((q-Au)gj+l,gj_1)rdr + 4jV,j 0
.
IV.3. Proof of theorem 1.1
The first integral may be "integrated by parts" again, using lemma 2.1, for
jj +
_
((gj+l,(q-DU)gj-1)rdr 0
-
(R
((Vgj+l'Vgj-1)r+
J
('ugj+l,gj_1)r)dr
0
R +
<
m'
j'7j
+
((Vgj+1'Vgj-1)z,+ (gj+l,Apgj_1)r)dr
JO
Zj
+lvj-1 +
/V_-,.+, l;
j-1 + 0 ((q-Ap)gj+2'gj-2)rdr
It is clear now that this process may be iterated until an index j-l<0 is reached, at which time the last term vanishes. This results in the first formula (3.3). For the second inequality one starts with R 0
R
2 11gj11rdr =
0
((q-ou)gj+l,gj)rdr
and again uses lemma 2.1 to increase the index of the first factor, and decrease the index of the second factor, until again the inner product vanishes. Then the second (3.4) implies (3.3), completing the proof of prop.3.1. Only the first 2m inequalities (3.3) will be used, but we leave the system infinite for technical reasons.
We now reduce the system (3.3) as follows. First introduce new variables by setting
, IT 0j- Ij
aj = q.+ 1Pj
(3.5)
Using that
Zm1pk
$km +
<
by Schwarz' inequality, we get
kX
X2 j'2-uj'2
.
1
(3.6)
+11=0 j+1+lj-1-1
2(X.+
PI +.-1+1X1
uj)
Note that Xj
,
uj satisfy the following conditions.
(A): Both aj
,
pi are absolutely continuous and their right
derivatives aj', uj' exist everywhere. The functions XI are convex and Xj, X it are nondecreasing and nonnegative. The aj' are bounded
the uj' of bounded variation. Furthermore, Xj(0) = pj(0) = 0, and (3.7)
1 uj
< aj
,
I uj'I
< aj
as 0 < r < W
.
113
IV.3. Proof of theorem 1.1
Proposition 3.2. The inequalities (3.6) and conditions (A) imply < 3V7 X13/211(,7e nr/2x
(3.8) A
j
+
J+l
2rajaj+l+laj-1-1)
for j=1,2....., and 0
may have X.'=O only in some initial interval [O,p], but then will get aj'>0 in (p,co), (unless aj=0 for all r). We derive the j-th estimate (3.8) from the j-th pair of estimates (3.6). Note that k9j only in form of their all j-th estimates involve the Ak ,
derivatives. We thus may translate left by p and replace Ak by changing the j-th pair (3.6). If the j-th Ak-ak(p) , without (3.8) can be derived for these new A. then the same follows for J
the old aj
i
Thus, we may assume that X .>O for r>0
.
From (3.6) we obtain
/---(3.9)
a
+ 1l A
2' vlx
+l+laT 1
We plan to integrate (3.9). Note that (3.10)
JRajdr
2
JR(aj/aj)2d(aj)
=
with well existing Stieltjes integrals. In (3.10) we may let e-}0, R
A. exists, and the integral at right (called IE) increa-
since 0
ses, as a decreases. If Ie: [0,1]
.
then a/A
,
J
, hence X'/A?< C in 7
7
Integrate this to get 1/aj(6) - 1/aj(r) < C(r-6)
,
0<6
since aj(0)=0. Thus a contradiction results. Not only must the integral IO exists but also It follows that we must have A./A *0 , as r-'0 As 6-'0 the left hand side goes to -
,
.
R
A.I(2aj) <
(3.11)
aj(r)dr 0
Here we substitute aj from (3.6), and use the inequality R
Cdr < ab , as a,bEC1([0,R]) , a,a',b,b'>O
(3.12) 0
From Schwarz' inequality we get R
V777 dr < (Ja'dr Jb'dr)1/2<((a(R)-a(0))(b(R)-b(0))1/2
(3.13) 0
114
IV.3. Proof of theorem 1.1 confirming (3.12).] Thus we get
X?/(2X )
(3.14)
J
J
-'f <
J
J
1
J
a.J+1+1a.J_1_1
Here we substitute (X +uj)/2 from the second (3.6), for r
j1( cenraj47j+l+1X3-1
(3.15)
)j+l+laj-1-1
+
A /siimilar proceedure is now applied again to (3.15). Multi-
ply by X. 0
and integrate from 0 to R
,
as r-0
,
,
Clearly (3.11) implies
.
so that the integral at left exists. We
apply a partial integration as before:
As E->0
,
=XjXj_3/2IR_ JRa3d(A' 3/2
JR(X,)-3/2d(
(3.16) 3 JRX
the last integral, JE, is monotone, hence either conver-
ges or diverges to -- , as e->0. The latter proves impossible, because then a3/kr3/2 = (X2/X.)3/2 , while we showed above J
J
J
that it tends to 0 as c-0
.
J
It follows that
, -3/2(R)
(3.17)
R
aj dr
<
3XJX j
0
Combining (3.17) and the integrated (3.14) we g e IXX'-3/2(R)
<
1{TenR/2 JR
jaj aj+ly
--
tt +laj 1 dr
( 3 . 1 8 )
R + J 0
j+l+1X. 1-1
dr}
Using (3.12) twice, the first integral at right of (3.18) is estimated (3.19)
<
(X2/2J aj+l+laj-l-ldr)1/2
)1/4
<
Similarly the second integral is estimated by (3.20)
< (Rajaj+l+lxj-1-1)1/2
using the monotony of ak Now we may substitute (3.19) and (3.20) into (3.18) to obtain the desired estimate (3.8), q.e.d. For the derivation of thm.l.l from prop.3.2 we introduce the following two order relations, defined for the set of nonnegative functions a, b, defined over [0,-) We write a
a(r) < cb(r) , with a constant c, and we write a
< c(log(2+K(r))sb(r) , where K(r)=h= a.(r), with above functions 1 J
115
IV.3. Proof of theorem 1.1 aj Clearly both relations are transitive and reflexive, and are compatible with addition and multiplication of functions, and with taking positive powers. Moreover, a
Notice that K,K' are >0 and nondecreasing, and I
j=l,...,m, X
X
.
J
-
X
,
,
,
a
J
,
as j>m
J
Proposition 3.3. If K is not -0 , then we have eer 0 Proof. We sum over j in the first estimate (3.6), replacing u. J
by 0 and using (3.21). Thus it follows that K
+ y
,
for every y>0
lity ca < at
,
.
This amounts to the differential inequa-
for a=K=y an some e>0
.
Suppose K is not -0 , then
K(p)>0 for some p>0. For small y>0 we get a(r)>0 as r>p. Then we get e>a'/a=(log a). This may be integrated for or, K(r)>ceer , as r>p, with c >0, implying the statement. ,
a(r)>a(p)ee(r-p)
Corollary 3.4. We have either K-0 or r
.
For the reminder of this section we assume that K is not -0. This will lead us into a contradiction, thus completing the proof of thm.l.l. Define the functions am+1=0, aj - lk=jak
m.
Proposition 3.5. We have X.
K?1/2{K5/12a1/12
+
K1/3a 1/6 + K5/12 + K1/3
(3.22)
+ K5/12+n/6ea1/12 + K5/12+n/6e} J+l
Proof. User prop's 3.2 and 3.3. Note that F
(3.23)
Ti
(Xk-iaj+k-1)1/2
Taking the cube root we thus get (3.22) , also using prop.3.4. The proof is complete. Proposition 3.6. If n
116
IV.3. Proof of theorem 1.1
117
j=1,2,...,m, there exist finitely many positive numbers a3,k, Sj,k Yi,k, dj,k satisfying ,,3'k + Bj,k < 1, yj,k+d0'k < 1, such that a.
(3.24)
JK,aj,kKsJ,k
JK'YjekKdjek
'
k
k
Proof. We use induction and first note that am+l
= 0 am = Am so that (3.22) give the assertion for j=m. Assume it true for a certain j<m . Then substitute aj from (3.24) into (3.22) for j-1. It is clear then that ,
J{K5/12+gj,k/12 K,1/2+aj'k/12}
K5/12aj/12
(3.25)
'
which again is a sum of terms of the form in view of 5/12 +1/2 + (Sj'k+aJk)/12 = 11/12 + ($J,k+aJ'k)/12 < 1. Similarly for the other terms in (3.22)
This shows that a.-1 is dominated in +a. , of course, q.e.d. the described way, and we have a.-1 =a.-1 Note that K = a1 now is dominated by a sum of the form .
R
(3.24). Also we get K(R) <
K'dr < RK'(R)
,
by monotony of K
0
Thus we get K < rK', and cor.3.4 implies K
,
for j=1. It fol-
lows that
K
(3.26)
,
for some 0<E
.
Finally we observe that (3.26) implies K
(3.27)
because we have (log(2+K))s
(3.28)
,
for every c>0
as r, in view of prop.3.4. Now it is well know that the differential inequality (3.27)
since we know that K-
,
(or in details (3.29)
K < c K'Y
with constants c, y >0
,
,
y < 1 , does not have a nonnegative solu-
tion, defined in [0,-), except K(r)=0
.
Indeed, write (3.29)
in the form (3.30)
-(l/KT)'
=
TK'/KT+1 > c-
,
t = 1/Y - 1 > 0
,
c_>O
.
IV.4. Frehse's theorem
where r must be choosen large enough so that K90
.
Integrating
(3.30) we get
K-T(P) > K-T(P) - K_T(r) > c`(r-P)
(3.31)
where the right hand side tends to - , as r-
,
as r>P
,
, while the left
hand side remains constant. This results in a contradiction so that we must have K = 0 for all R. This, in turn, implies that f =
0
.
This completes the proof of thm.l.l. 4. Proof of Frehse's theorem.
In this section we finally want to discuss a proof of thm. 1.9. As in the discussion of thm.l.l we only must show that every C'-solution of the differential equation Hu=O in H=L2(In) vanishes identically, under the assumptions of the thm., such as (1.9). Here we may assume that us is real-valued, since the coefficients of H are real-valued. Let us assume q?0 again, without loss of generality.
In the following we consider open balls Br with center x0 kept fixed for a while, with radius r. For uEC(1n) solving Hu=O, and EEC_(Br) we have (4.1)
fB
0
r
In (4.1) we introduce $ = X2IulPsgn(u), with sgn u = ±1 and 0, as u>0 , u<0 , and u=0, respectively, and with a suitable Lipschitz
continuous cut-off function XEC0(Br) , to be chosen later on. Clearly such is not C_ but still Lipschitz-continuous. Nevertheless the relation remains valid, by an approximation argument, ,
(For example one may introduce a mollified me as long as p?l as in the proof of II,lemma.2.1, and then pass to the limit $E* , .
E-0 .) From (4.1) we conclude that (4.2)
JB hOkuli (X2IulPsgn u)Ikdx < 0 r
since the second term is nonnegative.
Let us write the integrand J in (4.2) in the form (4.3)
J = PX2IuIP-lhjkulJulk + 2XIuIP(sgn u)hJkulHXlk
.
118
IV.4. Frehse's theorem In (4.3) introduce u1jXIuI(p-1)/2= (
2/(p+l) (CIA-nj)
4.4) Xlul(p+l)/2sgn
C =
u
,
nj
= XIj IuI(p+l)/2sgn u
A calculation yields (4.5)
J = 4(p+l)-2(phJkCI.clk- hOknj nk -(p-l)hJkCIJnk)
Integrating (4.5) and applying (4.2) we thus get 0 > JBrh3k'lj'lkdx - p(JBrhOknjnkdx
(4.6)
Or, with A+=A+(r)= sup{A(x):xEBr}, a-=a-(r)= inf{X(x):xEBr}, with the largest and smallest eigenvalue A(x) and ),(x) of the matrix ((hjk(x))), respectively, JB dxIVCI2 < pA+/a-JB dxIVXI2Iulp+l
(4.7)
r
r
We now require a special case of the Sobolev estimate, as stated in prop.4.1, below. Proposition 4.1. For every Lipschitz continuous uEC0(Rn)
, and
every p satisfying 1
(4.8)
with the Lp(.n)-norms (
(4.9)
llullp
=
IlVullp
,
IVuI=(flu1J)1/2 , integrals over
{Jlulpdp}l/p
,
IlVullp =
n)
{JIVulp}1/p
The proof of prop. 4.1. will be discussed at the end of this section.
Let us assume n>2 , so that the assumptions of prop.4.1 hold for p=2. (The case of n=2 will be discussed later on.) Then we may combine (4.8), for u=r (4.10)
IlXIuI(p+l)/2b2n/(n-2)
< Cl
,
and (4.7) to obtain IIIVXIIul(p+l)/2fl2
with (4.11)
C1 =
2n-1/2(n-l)/(n-2)(A+/a-)1/2
Finally we replace p by p-l in (4.10), assuming p>2, and
119
IV.4. Frehse's theorem
take the 2/p-th power, and arrange for La-norms of u (4.12)
&X2/Pupgn/(n-2) <
(PC2)1/PIIIVX,2/Pulip
,
:
for all 2
The above is correct for every X meeting the above requirements, and it should be noted that the constant C1 depends on the choice of X only insofar as A+ and a- might change, according to the support of X . Also C1 is independent of p. Next let us choose a sequence of cutoff functions Xj, j=1,2, such that XjE CO(BR), and that X3.>0, and VXj= Xj-1VXj for all j. We explicitly choose Xj(x)=Pj(Ix-x0I/R) with tj=(2j)-+1/2, ...
,
(4.13)
,
.(r) =1 in rtj, =(r-tj)/(tj+l-tj) else
Then clearly Xj is Lipschitz continuous, and we get (4.14)
IVXjI
< 2j(j+l)/R = dj-l
In (4.12) let p=pj=2TJ , with T=n/(n-2), and let X = Xj+l' Using (4.14) and VXjXj-1=VXj, it follows that 1/g
2/p.
(4.15)
IIXj+1
Jullp
21 p
JIIXj
<
J-lullp
j=0,1, ...
,
J+1
J
This relation may be iterated for 1/pkjullp
1/pJullp .
IIXj+1
(4.16)
< IIk=O(dkpkC2)
J+1
0
Notice that the product in (4.16) is bounded; we get II(6kpkC2)1/Pk<e2Elog6K/Pk((T)l/p0)Ek/Tk((pOC2)1/po)ET-k
(4.17)
with infinite series, at right, taken from 0 to . All 3 series CT-k/2 , for large k converge, since T>l , and (log 6k)/pk < Accordingly we may replace the product in (4.16) by the constant C2 at right of (4.17), and then pass to the limit j;co Note that .
2/pJ l
(4.18)
HullL_(B
R/2
Therefore, expressing the (4.19)
lim
) =
J
R/2
pX J
uh J
pi.
at left by the sup, we get
sup {Iu(x)I: xEBR/2} < C2RUHL2(B R)
In particular we used that p0=2. The dependence of the constant C on R, A+ and a- is easily evaluated: Only the first infinite series in (4.17) involves the radius R. Using (4.14) we get (4.20)
21(log 6k)/pk=IT-k log 2k(k+l) -IT-k log R =C3-n/21og R
,
120
IV.4. Frehse's theorem with C3 independent of R
.
Similarly only the third term at right
of (4.17) involves the quotient A+/a-=a. With IT-k=(1-(n-2)/n)-1 =n/2, as used above, and with p0=2, the term becomes Cln/2= C4Qn/4
with C4 independent of a
.
In conclusion we get
C2(R,A+,a-) = C5 R-n/2(A+(R)/a-(R))n/4
(4.21)
with C5 depending on n only.
The above iteration technique was first applied by J.Moser [Msl), cf. also [GT1].
We have proven the following result in the case n>2. Proposition 4.2. If n>2, and q>O , then every C(I)-solution u
of Hu=O satisfies the estimate (4.18) with C2 of the form (4.19) where C5 depends only on n , while the center and radius of the ball BR are completely arbitrary. Exactly the same proof works in the case of n=2, with the following modification. Hoelder's inequality implies that (4.22)
Ilullp
< V(2-p)/2pnull2
, with V = 1B dx = nR2 R
and then apply prop.4.1 with p = r
Here we set (4.23)
(tR2)(2-p)/2pllVCll2
11CI2p/(2-p) <
,
p
1
,
,
for
< 2
Now we combine (4.23) with (4.7), replacing p with p-1: (4.24)
IIXIuIP/2112p/(2-p)
with cp=c YT (2Y)-1 of R,a,p (4.25)
.
,
<
cpR(2-p)/peal/2fIVXIIuIP/2112
Y=y(p)=p/(2-p)
,
c' a constant independent
Or,
IIX2/pupYP < (cpRl/Y(po)1/2)2/PIIIVXI2/PuIIp
,
P>2
.
-k
Notice that 0Y =p/2(p-1). Thus, for pj =2.Y3 the iteration lea-
ding to (4.16) yields (4.16) again, but now with the constant (4.26) c" depends on n only, and 6=6(p)=p/(p-1) > 2=n
,
6(p)-2,
It is clear that this is somewhat weaker than required in thm.1.9, at least under estimate (1.9), but will be enough for as p;2
.
121
IV.4. Frehse's theorem
an estimate (1.9) with '0(1+R2)' replaced by '0(1+R2-E)', e>0
.
However, for n=2, a simpler proceedure, not using the above technique, still will give the same result as for general n, as will not be discussed here. Proof of theorem 1.9. In view of thm.1.10 we may assume that n>2
,
so that prop.4.2 holds. Let us first focus on assumption
(1.10). For R=r/2 , with the r>0 of (1.10) we conclude that
sup{1u(x)I:ixl=n} < KJ n_r
(4.27)
since uEL2(Rn) , where K denotes a constant independent of n
.
Indeed, for each x0 with lx0H=n we get (4.19) with the ball BR around x0 and a constant independent of x0 and n However the .
maximum principle (III, thm.5.8) holds, since q>0
.
Therefore
un=Max{Ju(x)I:IxI
,
for large n, may be covered by balls Bn/2
with center on 3B 3n/2 and contained in the spherical shell B2n\Bn The corresponding concentric balls Bn/4 still cover 9B , and the constants n-n/2 (A+/a_)n/4 are uniformly bounded, 3q/2 in view of (1.9). Accordingly, again using the maximum principle, ,
.
we get (4.28)
Ju(x)J
<
c1n<JxJ<2n lu(x)I2dx + 0
This again implies u=0
,
as n,-
.
, hence H0 again is essentially self-ad-
joint, and theorem 1.9 is established. Finally, in this section, we discuss a proof of prop.4.1. (cf.[GT1]). We shall require Hoelder's inequality for n functions: (4.29)
Julu2...ukdx
valid if pl, ... (4.30)
< flk 111u1Ip
,
u E L p3(,n)
, pk E R satisfy the relation
1/pl + ... + 1/pk = 1
Also we use that, for k nonnegative numbers al, ..., ak the
'geometric mean' (H%-1a mean k-1yj=1ak
.
is never larger than the arithmetic
The latter is well known. Also, (4.29) is
easily derived from the well known case of k=2, by induction.
122
IV.4. Frehse's theorem For uEC0(&n) write X.
(4.31)
Iu(x)I
J Iluljldxj <
luljIdxj
Taking the product of these n inequalities, and the power 1/(n-1),
we get Iu(x)In/(n-1)
(4.32)
<
=1(Jluljldxj)l(n-1)
H
The inequality (4.32) may be integrated with respect to each xj, j=1,2,...,n. After each integration Hoelder's inequality is applied, with k=n-1 , P1-P2=. For example, the first factor, at right of (4.32) is independent of xi. Hence there are pn-1=n-1.
n-1 unknown functions under the integral Jdxl. We get
Jdx1Iu(x)ln/(n-1)
(4.33)
<
1Jdxldxjlul. }1/(n-1)
{Jdxllulll fl
For each subsequent integration one of the n factors at right is a constant, and (4.29) may be applied with n-1 factors. We get (4.34)
Ilulll
< (III=llluljll)1/n < n-lylluljlll < n-1/2UIVul1
Notice that (4.34) amounts to (4.8) for p=1. In particular the inequality may be obtained for Lipschitz continuous u E C0(1n) by an approximation argument.(For example one may use a mollifier as in the proof of III, thm 1.2).
The general inequality (4.8) then follows by applying We get (4.34) to the function v=lulY , with y = p(n-1)/(n-p) Substituting this into (4.34) and using IVIujYI = yIujY-ljVuI .
.
Hoelder's inequality we conclude (4.8) for general p. 5. More criteria for essential
self-adjointness.
In this section we discuss an extension of thm.l.5, useful in ch.8. Let us assume that the expression H
,
defined over S2
has an essentially self-adjoint minimal operator HO be an open subdomain of St
.
,
Let U C ci
with compact smooth boundary 3U, where
3U is assumed to bean n-l-dimensional submanifold of S2 . Assume the potential q of H bounded below on U , and let q'EC'(U) denote a nonnegative potential satisfying (1.4) in some neighbourhood of 3U , with a suitable constant y . Assume that q'=O outside another
123
IV.5. More essential self-adjointness
neighbourhood of 3U
.
Theorem 5.1. Under the above assumptions the minimal operator H'0 of the expression H'=H+q' , in the Hilbert space L2(U,dp), is essentially self-adjoint, whenever y>yl , where yl is a positive constant. Proof. One may repeat the proof of thm.1.5 in the following amenand define ded form. Let T(x)=dist(x,3U) ,
(5.1)
O(x) = n-llog(E/T)
for suitable constants E,n>0
as T(X)<E
,
.
,
=
0
,
as T(X)>E
,
Then use this redefined function
o(x) as in the proofs of lemma 2.1 and lemma 2.2, to get essential self-adjointness of H' .
In particular the 'spheres' ER again are defined by the inequality o(x)
for a suitable cutoff function, zero near 3U =1 outside a neighbourhood of 3U in UU3U . Since H0 (in 0) is essentially self-ad,
such that joint it follows that there exists a sequence zkEC'(U) k-* , in L2(O,dp) Using this, with a suitable zk-wf , ,
.
limit k- in (2.3), (2.4), (2.5)
,
one shows that the 'partial
integration' , taking JH'rfd14 into J(,Vf,2+qIfI2)dp still will
give no boundary terms at infinity of U , although there will be such terms at the boundary of ER , of course. In other words, one finds that inequality (2.7), for q+q' instead of q , remains intact. Therefore the remainder of the
proof in sec.2 may be repeated, and one finds that f=0 proves thm.5.1.
.
This
Remark 5.2. It appears that a corresponding statement, regarding essential selfadjointness of powers H' 0m , can be proven by repeating the chain of arguments in sec.3, with similar amendments. However, we have not checked this in detail.
124
CHAPTER 5. C -COMPARISON ALGEBRAS.
In this chapter we open the discussion of one of our main objectives: Comparison algebras are certain C -algebras of linear operators on an L2-space. In essence their operators may be explicitly represented either as Cauchy-type singular integral operators or else as pseudo-differential operators of order 0. However, we never use (or even discuss) such explicit representations. Except for a short excursion into Fredholm results for differential operators on compact manifolds in VI,3 and VI,4, in ch's V,VI and VII we will entirely focus on the discussion of the structure of such C*-algebras. The reader should keep in mind, however, that the results obtained will be used later on (ch.X) to discuss the properties of differential operators. For certain differential tial expressions (said to be 'within reach' of a comparison algebra C ) a realization will be defined (cf. def.6.2 below) In ch.X we will focus on the spectral (and Fredholm) properties of such realizations.
While we have discussed abstract spectral theory in ch.I, we refer to [C1],app.A2, regarding an abstract theory of Fredholm operators. Also we will use some results of abstract spectral theory not discussed in detail in earlier chapters, such as the discussion of the essential spectrum of a differential operator. (Note that at least a survey of some such abstract results is made in ch.X.)
We work in the Hilbert space H= L2(O,dp) 0
,
on a Cm-manifold
, with a positive Co-measure du, as introduced in ch.3. On Q we
also have a comparison operator H defined, where H is the (selfadjoint) Friedrichs extension of the minimal operator of a differential expression H, as in III,(0.1) In [Cl] we considered the .
special noncompact manifold In . There we introduced a class of
C*-algebras generated by multiplication operators and convolu-
V.0. Introduction
tion operators. Especially the singular integral operators A=(l-A)-1/2 , were used as such convolution generators. S. = DjA ,
In the present approach, on a general compact or non-compact manifold 0, we replace the generators Sj by operators of the form DH-1/2 DA = , with certain first order differential operators D defined on Sl
,
and with A = H-1/2 , with the above self-adjoint
positive definite operator H
Accordingly, a comparison algebra C is a C-algebra generated as norm closure from generators
(0.1)
a , DA
,
aE
A#
, DE
with certain classes A#
, D# , of bounded C--functions and first order linear partial differential expressions (folpde's), respec-
tively.
We work with general assumptions on A# , D which always imply that CDK(H) , with the ideal K(H) of all compact operators on H The commutator ideal E of C may or may not be K(H) as well. At any rate, for all examples considered there is a chain .
C D E D K(H)
(0.2)
,
where the second inclusion may be proper or improper, but such that both CIE, and E/K(H) are isometrically isomorphic to an algebra of continuous functions. Then the Fredholm property of an operator AEC is governed by one or two symbol functions, arising naturally from the above ideal chain. For more details and simple explicit examples we then refer to [C1], where both kinds of ideal chains are obtained.
It should be noted that algebras with two-link ideal chains also have been studied elsewhere (cf. Dynin [Dyl], Upmeyer, [Upl]) although, as it seems, not in connection with noncompact manifolds In applications it is found that the two-inversion symbol calculus has its disadvantages, insofar as the second symbol usually is available only after constructing an explicit E-inverse, not available from the first symbol alone. In some later examples we have remedied this by extending the second symbol to the entire algebra C Then usually the second symbol takes values in some algebra of singular integral operators over a lower dimensonal .
space. This will be discussed in CCMe1I, [CPo]
.
In the present chapter we only discuss some basic facts.
In particular it is proven that every comparison algebra C con-
126
V.1. Comparison algebras
tains the compact ideal, and that K(H) even is contained in the commutator ideal E of C (lemma 1.1). We explicitly discuss L2continuity of DA , and AMA , for certain first and second order expressions D and M, and derive explicit formulas for commutators (sec.2). In sec.3 we derive sufficient conditions for compactness
of commutators in a comparison algebra. In sec.4 and 5 we discuss a variety of examples. Specifically sec.5 considers the one-dimensional case. Finally, in sec.6, we prepare the later application of our theory by introducing the concept of an expression within reach of a comparison algebra. Specifically, a Taylor-type expansion (with integral remainder) is discussed for "H-compatible expressions", having all H-commutators within reach. Such expansions prove useful as an efficient substitute for the calculus of pseudodifferential operators, which we do not assume here. 1. Comparison operators and comparison algebras. Let ci denote a paracompact Co-manifold without boundary,
with a countable atlas, as in III,1, and again consider a measure dp=Kdx, and a second order elliptic expression H = -K-12 x3
hjkK3 k + q x
which is self-adjoint with respect to the inner product of the Hilbert space H = L2(D,du) As in III,1 we assume that H has real .
C°°-coefficients, that ((hjk))>O (1.2)
(u,Hu) > (u,u)
,
and that HO > 1
,
,
i.e.
for all u E CD(D)
Then we introduce the first Sobolev norm and inner product by (1.3)
IxjvIk + quv)dp = (u,v)1 , u,v E dom H
(u,H0v) = JD(hJku
x
The Friedrichs extension H is a well defined self-adjoint Clearly H-1 E L(H) has a selfrealization of H We get H > 1 .
.
adjoint positive square root A = and 0
.
H-1/2
=
1k_0(_1)k(1k2)(1-H-1)k
A is an isometric isomorphism between H and H1. We get
(1.4) H1 = dom A-1
,
(lull
=
IlAulll
,
u E H
,
Ovlll =
IIA-lvll
,
The restriction A-'Idom H0 is essentially self-adjoint, and
v E Hl.
127
V.1. Comparison algebras
A-ldom HO is dense in H , while the corresponding is not always true for H dom HO = im HO. For details cf.III,l. In fact, we already discussed this extensively. The operator H0 will have to be essentially self-adjoint for this. We also note the possibility of transforming the dependent variable,to get from the triple {c,du,H} to an 'equivalent triple' {c,du-,H"} with possibly better properties,such as a normal form etc. (cf.III,6). for H ,
If 1,dp,and H are given as described we shall speak of the comparison triple {S2,du,H}, and of the comparison operator H on n. Also H1 will be called the corresponding first Sobolev space. Let a comparison triple be given, and let
(1.5)
D = bl2
xj
+p
denote a first order differential expression on S2 , where (bi) and
p denote a contra-variant tensor field and a complex-valued function on 0 , both assumed Then the linear operator DA:CD()*
C
.
L2loc(St) is well defined, since the square root A of H-1 maps onto the first Sobolev space Hl (cf.III,(1.6)). In fact, assuming that IIDIIH
=
- ,
II(IbI2 +IpI2/q)1/2p
L_(Q) <
(with IbI={hjkFibk}1/2
)
(1.7)
< IIDIIH(u,Hu) _
IIDuf2
provided that q > 0 on 2
(1.8)
IIDAvu
,
it follows that
Using (1.4) < IIDIIHIIvII
PDIIHIIuII
,
,vE
it then follows that
im(A-lIC0-(S2))
where the space im(A-ljC-(c)) is dense in H. Accordingly, by continuous extension of D0A , we get a bounded operator A E L(H) and it is clear that A = DA , with A as a map H - H1 , and a differential operator D:H1 } H , defined with the strong L2-gradient of III,(1.6)
.
The adjoint operator A* of A E L(H) is in L(H) again, of course, and we have *
(1.9)
A u = AD*u , u E
C0'(52)
,
D* _ -K-18 jK50 + x
In chapter X we will study the Fredholm properties of
certain C-subalgebras * of the algebra L(H), called H-comparison
128
V.1. Comparison algebras
algebras. Here by an H-comparison algebra we mean a C*-algebra C C L(H) generated as operator norm closure of the smallest algebra CO containing all multiplication operators u -* au , and all operators A , A* of the above form A = DA , where a E A# D E V#. Here A# , D# denote a class of bounded continuous complex-valued func,
tions, and a class of first order linear partial differential expressions (folpde's) on Q. of the form (1.5), with IDBH < - . We will write C = C(A#,D#) , CO = CO(A#,D#) omitting the arguments if no confusion can arise. We require the following general assumptions: ,
Condition (a0): A# contains C'(c1), and its complex conjugates.
Condition (a1): A# contains the constant functions Condition (d0): D# contains its complex conjugates
j+p ,
T7=5j2
x
and contains all C'-folpde's with compact support. Here (a0) and (d0) are always imposed, and (a1) is required
unless explicitly stated otherwise. Especially there are two exceptions to this last rule: The minimal algebra JO defined ,
below, and the class A# 0 of VIII,3 normally will not contain all constant functions. In these cases we have the following condition instead. Condition (al'): For every D E D# there exists an a E A# such that Va has compact support, and aD = Da = D For every comparison triple we have a minimal comparison algebra, denoted by J , obtained by choosing AT as the class of functions a=a0+ c
,
of folpdes D = b3a
and D# as the class a0 E C with b3E C' , p E C'(D) The C -algebra ,
j+p
.
x
(not necessarily with unit) generated by norm closing the algebra generated by a and DA a and C'-folpde's for all ,
with compact support, will be denoted by JO. Clearly J is obtained from J0 by adjoining the identity 1. We shall see in VI,l that 1 and J0 have compact commutators. J0 is unital if and only if D is compact. Both J and J0 will be called minimal comparison algebras. Lemma 1.1. The minimal algebra JO , and hence every comparison algebra C , always contains the entire ideal K(H) of compact ope-
rators. Moreover, K(H) even is contained in the commutator ideal E = E(C) of any comparison algebra C , and we have E(J0)=K(H) In lemma 1.1. the commutator ideal E(C) of C denotes the smallest closed 2-sided ideal containing all the commutators of
129
V.1. Comparison algebras
operators in C
130
.
Proof. As in [C1],IV,thm.l.l, we use the fact that an irreducible C*-subalgebra A of L(H) containing at least one nontrivial compact operator, must contain the entire compact ideal (cf.Dixmier ,
[Dxl], cor.4.1.10). Here 'irreducible' means that the space Au is dense in H for every 0 # u E H ,
.
It is easy to construct nontrivial compact operators in a comparison algebra. For example, for any function a E C'(Q)
not vanishing identically we have aA=aH-1"2EK(H), by III,thm.3.7, below, while trivially aA 9 [a,bA]
,
0
.
In fact, also the commutator
for suitable a,b E C'(Q) is compact and #
sufficient to show that Cu is dense in H
,
0
.
Hence it is
for every 0 # u E H
,
(Clearly then the construction of [C1],IV,lemma 3.3 may be repeated to show that K C JO
,
and, starting with the given compact
that K C E holds as well. Indeed, one first will commutator # 0 construct an operator of rank 1 in C or E , of the form )( , ,,p 9 0 with ,1p c H Then the operator ,
,
.
comes arbitrarily close to every dyad f)(g
,
f,gE H
,
as A,BEL(H).
Accordingly C and E contain all such dyads, and hence K(H) closure of the class of operators of finite rank.) Note that the operator
H-1
,
the
may be written as an integral
operator (1.10)
H-Iu(x)
=
JQG(x,x')u(x')dp
where the 'Green's function' G(x,x') is positive on all of Q Indeed, from III,thm.6.2 we know that H - H-= 1 - Ap_ , after a .
suitable change of dependent variable. Such an operator satisfies
the conditions of the maximum principle, which insures that the Green's function is positive, since G solves HG(.,x')=0 , and is clx-x'12-n (or c loglx-x'j ) near its singularity x=x' , and is limit of the sequence of Green's functions Gk of the Dirichlet problem on an exhausting sequence Qk of subdomains of n with smooth boundary
Clearly the Gk are positive, since their boun-
dary values are.Hence G is positive. For details cf. III, thm.6.8. For any OiuEH we have 0 # H-ncu continuous for a suitable C0(c1)-function c, by the Sobolev imbedding lemma, (VI,1.4.2), or since the iterated Green's function has the corresponding smoothe-
ning property). Hence we may choose aECO(1)CA# such that aH-ncu = aA2ncu does not change sign, and is not always zero. Assume it >0, without loss of generality. Then w=A2aA2ncu is positive on 0,
V.1. Comparison algebras
131
and every v E C' can be written as v = bA2aA2ncu , where b = v/w E C0(O)
.
Choose bkE
+ 0 , then
with Ilb-bkll
let Ak = bkA2aA2nc and note that IIv - Akull } 0 Furthermore, if 1 , kE C'(O) , k(x) } 1 for all x E 0 0 < k then A A2n hence (A4.)2, A2, (A'. )2n in strong operator convergence. .
,
,
,
-
Thus we also get IIv - Bkull - 0
,
for Bk =
with a suitable sequence {jk} of integers
k
k
)2nc E J0,
jk This shows q.e.d. that indeed J0u is dense in H For a comparison algebra C we focus on the commutator ideal E , i.e., the closed 2-sided ideal generated by the commutators. Note that C normally has a unit (the function al). The quotient ,
.
,
C/E, as a unital C*-algebra, is isometrically isomorphic to a function algebra C(N), with a certain compact space N, called the symbol space, by the Gelfand-Naimark theorem. The homomorphism a:C; C/E;C(N) assigns to AEC a function aA' called symbol of A. In sec. 5 we will prove that N always contains a subset W homeomorphic to the bundle of unit spheres in the cotangent space T'O of 0 . We call W the wave front space (of the algebra C). The continuous function aA:M - U corresponding to the coset mod E of an operator A E C coincides with the principal symbol of D on W if A=DA , with a differential operator D In the simplest cases we will get E=K(H) equal to the ideal .
of compact operators on H We already mentionned that this is true for the minimal algebras J and J0 The space W mentionned above coincides with the maximal ideal space of J0/K On the other hand the set N \ W can be described as a certain set located "over - of the manifold 0 ". Then an operator A E C will be Fredholm if and only if aA # 0 on all of N However, in some more general cases we have a proper inclusion E D K(H) Then .
.
.
.
.
we shall tend to obtain a similar "function structure" for the quotient E/K. The Fredholm theory then will be somewhat more complicated. The techniques to be described in the following can be extended to other types of differential operators as comparison operators, not necessarily of second order, or elliptic, or selfadjoint. However, the fact that so many properties of these 'Schroedinger operators' are known makes the use of them particularly attractive. In particular the e.s.a.-properties discussed in ch.5 will be important, in this connection.
V.I. Comparison algebras
For many parts of this chapter it will be crucial to have Carleman's definite case of II,1 , for the operator Ho in the Hilbert space H In other words, no boundary conditions at infin,
.
ity will be required
;
the Friedrichs extension is the only self-
adjoint extension of HO , and it coincides with HO , the closure of Ho .
We shall use this condition in weaker or stronger form, as follows:
Condition (w) Condition (s)
:
:
H0 is essentially self-adjoint.
For all m = 1,2,.., the minimal operator (Hm)0=H0m of the m-th power Hm is essentially self-adjoint.
Condition (sk): (where k = 1,2,.... o)
(Hk)0 = H0k is essentially
:
self-adjoint.
for 1<1
It follows easily that (sk) implies (sl
ingly (s1) equals (w) and (s,) equals (s) (cf.IV, prop.l.4). Various results asserting the validity of (w) or (s) have been discussed in ch.4. In particular (s) holds whenever the manifold 0 is complete under the metric (1.7). On the other hand we never will have (w) nor (s) if 0 is the interior of a manifold with (nonvoid) boundary onto which the triple may be extended. 2. Differential expressions of order not larger than two. For a comparison triple {O,dp,H}
as in section 1, let us
,
consider more general differential expressions on the manifold 0 We speak of a differential expression L of order N on 0 if a linear operator L:C'(O) ; C_(O) is defined such that locally(within a chart with coordinates x) Lu is given by a differential expression of the form II,(1.1). Using the measure du = Kdx a general (<) second order differential operator M on 0 always may be written in the form (2.1)
M = -1 3 K
Kmjk 2 xj
xk
+ bj3
+ P xj
with contravariant C--tensor fields m5k degree, and with a C--function p over 0
,
.
b3
of second and first
,
This follows because for
an N-th order differential operator on 0 the local principal part defines a contravariant tensor of N indices. For N = 2 , and any -K-1 tensor field a jk one finds that Kajk 8 k= A defines a glo3 xi
x
132
V.2. 2-nd order differential expressions
133
bal second order differential operator with principal part -ajk. Indeed,for u,v with support in a chart one gets (u,Av) _ kdV which is coordinate invariant, by the well known x transformation properties of tensors. For a general second order operator M one may subtract this operator, formed with its principal part ajk = mjk to obtain an operator of order one or zero. But every first order global differential operator is of the form (1.5). Therefore (2.1) gives the general 2-nd order operator. For additional symmetry and later convenience let us replace (2.1) by fa3ku
(x7vI
M = - Km3k2
(2.2)
K xj
+ b38 xk
Kc3 + p
K-13
-
xj
xj
which is only formally more general, of course.
Now the comparison operator H , or rather its inverse square root A of (1.4) may be used to convert a general first or second order operator into an L2-bounded operator. For D of the form (1.5) and M of the form (2.1) we formally consider the operators A = DA
(2.3)
B = AMA
,
.
It is not hard to set up conditions insuring that A and B of (2.3) In section 1 we already found that
define bounded operators of H (2.4)
HAD
where A:H + Hl
,
=
.
IIDAII
<
IIDIIH
and D:H1 + H , with the strong L2-gradient in D
,
and with UDIIH of (1.6). Similarly one may look at (2.5) (u,Mv) = f(mJku
k+ubiv
v Ix
j
Ix
+cjuv Ixj
+puv)dp, u,v E C'(SZ) Ixj
by a partial integration. For the integrand m of (2.5) one gets (2.6)
Im(u,v)I2 < ci h(u)h(v)
, h(u) = hjku
u Ix
Ix
k
+ glul2
assuming finiteness of (2.7) , with the supremum over all x E SZ (2.7)cl= sup{(hijhklmikmjl+h]kFibk/Iql+hjkc3ck/IqI+IPI2/IgI2)1/2
by a calculation. From (2.5),(2.6),(2.7), and (1.4) we get (2.8)
I(u,Mv)I
< c1Pull 111vI11 = c11IA-luUUA-1vU
I(u,Bv)I
< c1Uuffvf
which yields (2.9)
,
u,v E A-Idom HO
,
V.2. 2-nd order differential expressions
Recall that A-1dom H0 is dense in H (2.10)
BBull
< IIMIH
(lull
so that (2.9) implies
,
IIMIIH = c1 (with Cl of (2.7))
,
,
for all u of a dense subspace. It follows that B admits a continuous extension to H , with the operator bound of (2.10). Finally it may be observed that we may improve on the estimates (2.4) and (2.10) by replacing IID0H and IIMIIH with
(2.11)
inf{IID"IIH-}
, int{IIM-11 H-.} ,
where the infimum is taken over all operators y-1Dy =D"', y-1My y>0 , representing D = M` , y-1Hy = Hy E C_(S2) M and H after a transformation of dependent variable, preserving the con,
,
dition q >
0
,
In fact, only one such norm needs to be finite, in
order to get A or B in L(H)
.
In the next few sections we mainly shall focus on first order expresssions D with finite fDIIH, so that A=DA is bounded in H. Our aim will be the investigation of algebras generated by classes of operators DA together with classes of multiplication operators Au(x) = a(M)u(x) = a(x)u(x). Of crucial interest will be the commutators of two generating operators [A,B] with A and B of either form DA , or a(M) In that respect we look at commutators of the .
for a = a(M), and first
forms [ a,D] , [ D, F] , [ H,a] , and [ H,D] , order expressions
D = b33. x + p
(2.12)
,
F = claxj + r
One calculates that (2.13)
[D,F]
= (bjcklx. - cjbklxjxk + (b1rlxj - c3plx.)
which is another first order expression.
For the commutator [H,D] it is useful to introduce covariant derivatives with respect to the Riemannian connection of the rhetric tensor hjk . With the Christoffel symbols (cf.app.B) (2.14)
rji k
=
hil(h .1 Ix
k + hkl
Ix
j - hjk
Ix
1
write (2.15)
bi
biij =
Proposition 2.1. We have
lx.
- bkrk.
,
bjIk
=
bjllhlk
134
V.2. 2-nd order differential expressions (2.16)
-K axjK(b3lk + bklJ)8xk _ D
[H,D]
+ D
b3q
p
I
xj
where we define Da = hJka
(2.17)
Ix
and where p (2.18)
j2 k x
a
,
E C1(S2)
is defined by Kax.Ks3
D*
+ p =
_S3 ax3
+ p
,
= P -..(Kb0) Ixj
p
The proof is a somewhat lengthy calculation. For D0=b3a x3 =
All
K-18 jKhJk8 k write x
[q,DO]
[H,D]
_ -[A ,DO]-[&u, p]+[q,D0], where
x
_ -b3q
j
For u,v E C'(Q) we get
.
Ix
-(u,[Au,p]v) _ (Vu,V(pv)) - (V(pu),Vv) = (Vu,(Vp)v)
-(u,(Vp).Vv) _ (DPu,v) - (u,Dpv) with Da as in (2.17). Accordingly, (2.19)
-[A
p]
Similarly
=
DP* - Dp
,
-(u,[AuDOjv) _ (Vu,VDOv) - (V(DO u),Vv)
using that D0
-bO - c
c = K-1(Kb3)
,
j
Therefore we get
IX
-(u,[DU,DO]v) iv
(hikb3(u
Ix
Ix
(Vu,VDOv) + (V150u,Vv) + (V(cu),Vv)
+ hlku
k Ix3
+ (Vu,cVv) =
[(h'jbk Ix3
+
ib3 Ix
hjkbi Ix3
-
+
kv Ix
hikb3
Ix3
k)dp
v
iu Ix
Ix 3
lx
+ (u,Dcv) hik Ixj
b3)u
v du + (u,D v) Ix1 Ixk c
Now we use (2.15) to express the derivatives b3
k by covariant Ix
derivatives. In particular we may differentiate for x1 and solve for him 1 to obtain Ix
(2.20)
him Ix1 = _hkmrikl - hijrm3l
h13hkj
= 6k
135
V.2. 2-nd order differential expressions using (2.14)
. With (2.20) we get h]Jbk
hjkbi
+
h17bk
- hik lxJ
lxJ
(2.21)
+ hJkb1 l7
=
bj 1X3
bilk + bkll
l7
Accordingly, (2.22)
-K-13 iK(bilk + bkli)8 k + Dc
-
x
x
and formula (2.16) follows. Q.e.d.
Since we introduced covariant derivatives for the commutator [H,D] we may as well use them in (2.13). This gives the formula [D,F]
(2.23)
_
(bJcklj
-
clbklj)3 k + (b7rlj - c7Plj) x
,
noting that first covariant derivatives of scalars are ordinary derivatives. The symmetry r k=tkjwas also used for (2.23). A simple calculation also gives *
[D,a] = b7alj
(2.24)
with D
a
,
[H,a] = Da
- Da
of (2.17).
Let us note, finally, that (2.23) is valid with any convenient affine connection, while (2.16) only holds for the Riemannian connection of the metric tensor hik It is easy now to set up conditions for boundedness in H of the operators (2.25)
[D,a]
,
[H,a]A
,
[D,FIA
,
and AIH,D]A
,
by just arranging for finiteness of the corresponding norms (1.6) or (2.10). Proposition 2.2. Let the tensors bJ and cJ be bounded , together with their first (Riemannian) co-variant derivatives. Assume that Vp* are p and r are and that Vp , Vr , and that Vq = 0(q) . Also let a , Va , and Aua be 0 ,
Then all operators (2.25) are bounded in H bounded The proof is evident, after the above calculations. .
The result will have to be refined, before it becomes useful. This we shall attempt in the sections, below.
136
V.3. Compactness of Commutators
137
3. Compactness criteria for commutators.
Throughout this section we shall assume a triple {Q,du,H} given with H = q - Au of (1.1) satisfying (1.2) , and such that condition (w) holds: the minimal operator H0 is essentially self-
adjoint (which implies that im H0 = H<(52)) is dense in H ). For most results we also will refer to a sub-extending {S2",du",H"} (c> {c,du,H} (which may be {52,du,H} itself) , and assume here in addihjk h"jk tion to III,(3.24) that the tensors and coincide on S2 , and that q" >,l outside a compact set K C 52" (we may achieve K=0 by (1.2) and III,prop.6.2, changing variablEs).In particular the norm of a tensor is unambiguously defined, since the two metric hjk and h"jk coincide. Generally (w) will be required for tensors {O,du,H} , but not for {52",du",H"}
.
In view of IV,thm.1.5 it is
naturalt expect q to tend to - near 30 C 0" whenever (w) holds. Accordingly thm.3.1, lemma 3.2 and lemma 3.3 often will have the best use for 0 = 02" , although they are formulated for general O C 0"
(Indeed the condition Va=o0(ge/2) is weaker than Va"=
oQ"(gne/2) near a point x0 E 30
, whenever q -* - as x -r x0 , since
a" and q" must stay continuous near such a point.
)
On the other
hand, thm.3.4 and the lemmata following it strongly depend on condition (3.13), below, which cannot be satisfied for q near a point x0 E 30 C 0" unless q stays bounded there (which is impossible if (w) is to hold). (This reflection assumes 0 to look like ) Note that near such
a manifold with smooth boundary near x0
.
point x0 the conditions (3.16) still allow some degeneration of the coefficients bi , p of a differential expression D , but weaker than near infinity (of 0")
.
The sequence of results, below, discusses conditions on the generating sets A# and D# implying compactness of commutators in the comparison algebra C(A#,D#). Thm.3.l deals with commutators of the form [a,DA], while thm.3.4 looks at [DA,FA], a E A#, D,F E D#.
We use the nctation oO(.)
,
etc., of app.A.
Theorem 3.1. Let aEC_(O) satisfy the condition Va=o0"(q"£/2)(in 01)
for some 0<s
,
and let the first order expression D (over 0),
of the form (2.2) satisfy b = 0(1) (3.1)
[a,DA] E K(H)
,
p =
Then we have
.
More precisely, the unbounded operator [a,DA] = aDA - DAa is
V.3. Compactness of Commutators
densely defined, and extends continuously to a compact operator. Proof. For u E im HO C CD(0) write [a,DA]u = -b3a .Au+D[a,A]u Ix
Here the first term is of the form Cu , with C E K(H), due to our
conditions on a and b (we get b1al j=on^(ne/)
0)),
x
by III,thm.3.7. For the second term we will need the two lemmata, below.
Lemma 3.2. Let 0 < c < 1 be given, and let the function a E CW(R) satisfy Va = o.^((q^)E/2)
(3.2)
Then the commutator [a,A2] = aA2 - A2a, defined on im HO = HC-(Q), extends to an operator in K(H) . Moreover, there exists compact operators C. E K(H), j=1,2, such that AC2A2-E
[a,A2] = A2-eC1A +
(3.3)
Proof. Use App.A, L.A.l to construct 0 < y E C'(0^), y=o,^(q^E/2), Va = 0(y) and define 6 = yl/E , so that Va = 0(Se), 6=oQ^(/ SEAS i.e., E K(H), by III,cor.3.8. For u E im H0 we have u, A2u ,
E CO
aA2u
.
-
Therefore, using (2.26), we get A2au
=
A2HaA2u
= AU-ED-A)
A2aHA2u
-
=
A2[H,a]A2u
=
A2Da*A2u - A2DaA2u
SEA')A2-eu - A2-E(AEse)(S-EDaA)Au
Since SEAS E K(H)
,
and a EDa has bounded coefficients, so that
(2.6) implies 6-ED aA E L(H) , we conclude that (3.3) holds.
It is trivial then,that also [a,A2] E K(H), q.e.d. Lemma 3.3. Let 0 < s < 2, and let the constants E,r,p,t,T satisfy (3.4)
0
< e,r,t < 1
,
0
< p,T < 2-s
,
r+p
,
t+T < l+s-c
.
Then, if we have (3.2) with c of (3.4) for some a E C-(0) , we get (3.5)
[a,A5]
= aAs - Asa = APCIAr + AtC2AT
where Cl. C2 are operators in K(H) Proof. The self-adjoint operator A2 has its spectrum on [0,1], and .
its resolvent R(a)
_
(A2 -
X)-1
is a holomorphic function from
138
V.3. Compactness of Commutators [0,1] to L(H) . Accordingly
U \
(3.6)
,
As = i/2n Jr R(A) As/2dX
with As/2 = es/2(loglal + i arg A)
s
,
,
vely oriented circle r = {la - 11 = 1
0
larg al
and the positi-
< n
}.
The integral in (3.6) exists as an improper Riemann integral in norm convergence of L(H) Its integrand is continuous, except s/2 = O(IAI-l+s/2) as A E r . We get at A = 0, where we have R(A) .
[a,R(A)]
=
R(A)(A2-A)aR(A) - R(A)a(A2-A)R(X)
=
R(A)[A2,a]R(X),
as follows for u E im(H0-X) = (H-X)C-, which is also dense in H,
for all A in the resolvent set, since H0 is essentially self-adjoint (cf.I,thm.3.1). Then the same follows for uEH, by continuous
extension.Next we apply (3.3) with c as in (3.2), and e,r,t,p,T satisfying (3.4) (3.7)
It follows that
.
As/2[a,R(A)]
= APC3(A)Ar + AtC4(A)AT
I
with C3(A) = _As/2A2-E-PR(A)C1A1-rR(A) (3.8)
C4(A) = -as/2A1-tR(A)C2A2 E-TR(A) Clearly C3 and C4 are compact for all A E r (3.9)
AsR(A) = 0(lal-l+s/2)
,
0
< s < 2
Also we have
.
,
A E r
from which we get IIC411=0(lal-1+(l+s-e-t-T)/2).
fC3H=O(lal-1+(l+s-e-r-P)/2)
(3.10)
,
It follows that the integrals (3.11) exists as improper Riemann integrals in L(H) and that ,
(3.11)
J
Cj(A)da = Cj E K(H)
,
j
= 3,4
with the ideal K(H) of compact operators of H. Thus (3.5) follows if we substitute (3.6) into the commutator, and use the above.
This completes the proof of lemma 3.3. Also we note that lemma 3.3 ,for s=1, p=t=1, r=T=O, and any e
D[a,A]u = DA(C1+C2)u = Cu
,
C E K(H)
,
139
V.3. Compactness of Commutators which completes the proof of thm.3.1 as well. Let us emphasize that the function a in thm.3.1 does not have to be bounded. Only the derivative is restricted by (3.2). This will be useful in thm.3.4 below for commutators of the form [qe
/2
,A]
, using the restriction (3.13)
On the other hand in
.
sec.4 we will discuss a class A# of function generators giving compact commutators, using thm.3.1.
While we note that no restrictions at all on the comparison operator H were needed for the commutator of thm.3.1, if one wants compactness of commutators of two generators DA of the second kind then it will be practical to require the condition Vq" = on"(q")
(3.13)
on the 'potential' q"
.
.
No further condition on q" will be impo-
sed, and none on q as well, if {O,du,H} _
{St",du",H"}
.
On the
other hand, if D is a proper subdomain of Sl", then at the boundary an of 0 the function q must be singular, in order to get (w)
,
while q" is smooth there. (Otherwise the Dirichlet and Neumann operators are different self-adjoint extensions of H0 ; hence H0 For that case we will cannot be essentially self-adjoint.) need a condition on the expression b3g1j, included in (3.16). x
Note that (3.13) holds trivially for the Laplace operator.
The comparison operator 1-A not only has q=l
with the function =
,
but also C=l,
h = det((hjk))-l. Note that
E C_(0),
due to the invariance properties of K and T . Similarly we define For an expression D of the form for the triple {Q",du",H"} (2.2) the adjoint D is given by (2.20). If we want D and D to satisfy the same type of estimates we must look at the term .
K-1(Kb3)Ixj
(3.14)
=
(-1(b0C)lj
,
with Riemannian covariant derivatives. We will require that the coefficientsbi p of D satisfy the conditions ,
(3.15)
(b3) = 0(1)
for D of (2.2)
,
,
p =
C-1(b3C)lj
= 0(1q) (in 0)
just to insure boundedness of D and D
finiteness of c1 in (2.6) for D and D* equals the 'divergence' b3lj
.
.
,
i.e.,
For H=1-A the term (3.14)
Note that a change of dependent
140
V.3. Compactness of Commutators variable can make either q=const or C=l, as was seen in 111,6.
For a result on compactness of commutators between two generators DA D E D# we also introduce the following conditions: ,
b01k
+ bk13
=
b3q
Vp =
for some 0 < c < 1,
= .
lx
Theorem 3.4. Let the comparison operator H" satisfy (3.13), and let the expression D of (2.2) satisfy (3.15) and (3.16). Then the * operators DA , D A E L(H) (by (2.6),(3.15)) satisfy (3.17)
(DA)* - D*A E K(H)
, DA - (AD)** E K(H)
where in particular AD, as an unbounded operator with domain H1 admits a continuous extension (AD)** . Moreover, for D and F of (2.12) both satisfying (3.15) and (3.16), we get [DA,FA] E K(H)
(3.18)
.
The proof again requires some preparations. Then the Lemma 3.5. Let D satisfy (3.16) for some 0<e<1 commutator DA2 - A2D (defined in extends to a bounded operator in H , which is compact. Moreover, we get .
AC2A1-e
(3.19)
[D,A2]
= Al-cCIA =
,
C.
= Cu(e) E K(H)
,
j= 1,2.
Proof. Again we get (3.20)
DA2u
-
A2Du = A2[H,D]A2u
u
E im H0 = HCO ,
where we may express [H,D] by (2.16). Accordingly, we must discuss the operators A2XA2 with X being one of the four terms of (2.16). First consider the term A2M0A2 , with (3.21)
MO=-K-13
k, mOk=b31k+bk10=9mik,
K M
xi
x
where B>0 denotes a C'(D^)-function with S=q^E/2, xEc^\K. Write (3.22)
Ml=-K-1a
M0= 8M1x
k
Kmlka k , 8 = log a
,
x
x
and observe that (3.13) implies (3.23) ck= mikO1j = e/2 mikq^lj/q^ (for xED^\K) = o,.(1) (on D).
141
V.3. Compactness of Commutators
For the second term in (3.22) we get =A2-ECA
,
A2-E(AEB)c33
x AA
of the proper form for (3.19). For the first term apply , p = t = 0 , r = T = 1 , for a = B , noting
lemma 3.3 with s = 1
that VB = BV6 =o,,,.(q"E/2), i.e., (3.2) holds. We get [A,B]=CA,
with CEK(H) , hence A2M1A2 = Al-e(AEB)(AM1A)A + AC(AM1A)A
,
where the second term is of the form ACA , by (2.10) The first E Indeed, we get A B E L(H), by (3.2) term is of the form Al-ECA .
.
and I,lemma 5.1, and AM1A E K(H) can be proven as follows: For 6 > 0 write mil = wmil+ Xmil , where w E C'(O') is chosen such that the tensor Xmil, with X=1-w, has norm <6. Correspondingly write M1= M2+M3, and note that IIAM3A<6. On the other hand, M2= wM1+ D2 , where the differential expression D2 is of the form yD4, with D 4 A bounded and y E C;(O')
It follows that AM2A = CAM1A + w(AM1A)+AD2A = C+wAM1A. Accordingly, (AEB)(AM1A)=E6+AEBw(AM1A)+C, .
where E6= (AEB)(AM3A) has norm = 0(6)
, while the other two terms
are compact. Since 6 may be chosen arbitrarily, we get (AEB)(AM1A) E K(H) , using that K(H) is norm closed. Thus we have shown that A2M0A2= Al-ECA is in the proper form for (3.19). A2bOgI.A2=J. Using (3.16) we Next let us focus on the term may write J = A(AB)y(BA)A, where 0< B E C'(O") B = is constructed according to lemma A.1 such that y=O(1) This ,
.
shows that J=ACA
, with compact C , using III, thm.3.7
Finally, for the two terms A2D
A2 and A2DP A2, it is suffi-
p
cient to look at J'= A2DpA2 only, for symmetry reasons. With B as
in the discussion of A2M0A2 and, similarly, y=q1/2 in O\K, O
, where D3 has coefficients o,,.(l) , by
the third formula (3.16). Then, J'= A2ByD3A2 = ABAyD3A2+ ACAyD3A2, =Al-E(AEB)(AyDaA)A + Al-E(AEB)C(AyD3A)A = Al-ECA , using the same techniques over again. In particular we used lemma 3.3 on B again. Note finally that we also may push the factor B to the right BYD3 = yD3B -c/2 y(D36)B, with the function 6= log q" again. Thus :
we also get J'= ACA1-e. This completes the proof of lemma 3.5.
Lemma 3.6. Under the assumptions of lemma 3.5 we have (3.24)
[DA,As]
= ArCAP
,
0 < s < 2
for every pair r,p of reals satisfying
,
C = Cr02sE K(H)
142
V.3. Compactness of Commutators (3.25) r+p+c
Proof. We substitute (3.6) into the commutator of (3.24) for [DA,As]
= i/2Tr
Xs/2R()[A2,D]AR(a) dA
)
r
Now (3.19) may be used in the integrand I(X) for I(X) _ X, /2 A1-ER(A)C1 A2R(a) = ArC3(a)AO , with ,
C3(X) _ Xs/2A1-r-ER(A)C1R(a)A2-p
=
Xs/2A1-rR(a)C2R(a)A2-p-E
Using (3.25) and the appropriate (3.19) we get C3(a) E K(H) I(X) = O(Iail+(1+s-r-p-E)/2) on r Thus (3.24) follows as The proof is complete. earlier (3.5) .
.
Lemma 3.7. For some 0<E<1, let the two differential expressions D,F , as in (2.12), satisfy the conditions (3.26)
bJcklj -
cObkWOO-
(qE/2)
, bOrlj -
c5plj-oQ^(q(1+E)/2),
Then the operator A[D,F]A is bounded in H, and may be written as (3.27)
A[D,F]A = Al-EC1
with a compact operator C1
.
Proof. Use lemma A.1 to construct a function 0<6 E C°°(Y^) with 6 =
such that $-1[D,F]A is bounded, using (2.23), (2.6) A1-E(AES)(S-1[D,F]A) =Al-EC1, with
and (3.26). Therefore A[D,F]A = C
1
E K(H), using III, cor.3.8.
Proof of theorem 3.4. For D satisfying (3.15),(3.16) we apply lemma 3.6 with s=p=l , r=0
,
= DA2 - ADA = CA
for [DA,A]
implies DAu = ADu + Cu , u E H1= im A
In particular AD
.
This
.
with
, must have a continuous extension (AD) E L(H) and DA = (AD)**+ C follows. Taking adjoints we get (DA)* = DA + C so that (3.17) follows. For (3.18) use (3.27) for E=l and the ** FAu + Cu = Cu + ADFAu = Cu + AFDAu second (3.17) for DDFAu = (AD) for all u c HC' , with C, + C'u = C"u + (AF)DAu = FADAu + C"'u C', C", C"' E K(H) , completing the proof of the theorem. domain H
,
,
143
V.4. Compact commutator algebras
4. Comparison algebras with compact commutators. In this section we will discuss some concrete cases of comparison algebras with compact commutators. Since all the discussion in sec.3 is based on the validity of condition (w) we first will engage one of the results of ch.4. First assume that the assumptions of IV, thm.1.5 hold for m=l. In details, let O^ be a smooth complete Riemannian manifold without boundary, with a positive Cm-measure dm^=K^dx and let the CW-function q^ be > 1 outside some compact set K , and satisfy condition (3.13) , i.e. (with the limit in c^), q^ k/q^2=lima (log q^) k=0. h^3k(log q^) (4.1) ,
Ixj
IxJ
Ix
Ix
We assume that O^ has a countable locally finite atlas. Our manifold 0 is a subdomain of 0^ , and its boundary 30 is assumed to be a smooth compact n-l-dimensional submanifold of 0^ On 0 we have the restricted measure du = du^IO = Kdx , and the restriction to 0 of the (contravariant) metric tensor is called hjk A comparison operator H is introduced by .
(4.2)
H = -K-12 X]
Khjk3 k + q x
with a real-valued Cm-function q>q^. Then condition (w) for the triple {0,dp,H} results if we require condition IV, (1.4), for q (4.3)
q(x) > yl/dist(x,30)2
,
for all x E Q\K'
,
with a suitable compact set K'CO, where yl is a certain positive constant defined in IV,thm.l.5. Now the results of sec.3 suggest introduction of the following generating sets A# and D# A# is defined as the class of all bounded functions a E c (That is, C,(c2), such that Va = o0(ge 2) , with some 0 < e < 1 .
.
the C-()-function q e/2hOka
a Ixj
k
is bounded over O\K , and has
Ix
) limit zero, as x- (in 0) D# is defined as the set of all first order partial differential expressions D=b03 j+p with Cam-coefficients, defined over .
x
satisfying (3.15),(3.16) with some 0<e
,
.
That is, in detail,
144
V.4. Compact commutator algebras h.
(4.4)
j/i
jbk , p/Vq , K-1(Kb3) Ix
are bounded over S2
q"-6(h..,h
» kk
(4.5)
and also, the 4 C-(S2)-functions (with C
,
,(bjlk+bklj)(bj'lk'+bk1Ij1)
,
bi(log q) Ixj
4 1q" ehikPlx-Plxk also are bounded over 0 (or at least as x-
(in 12")
R\K ), and have limit zero
.
It is trivial that A# and D# satisfy the general assumptions (a0), (a1), (d0) of sec.l. In addition D# contains its formal Hilbert space adjoints. Now thm.4.1, below, is an immediate consequence of thm.3.1, thm.3.4, and IV,thm.1.5. (We use thm.3.1 for S2=S2", but thm.4.4 with S2" as above.)
Theorem 4.1. Under the above assumptions the comparison algebra C
C
,D#) of section 1, formed with the above classes A
c
contains the class K(H) C is generated by a E A#
,
,
D
c c and has compact commutators. Moreover, DA D E D# alone, i.e., the generators ,
may be omitted.
(DA)
Let us shortly focus on some special cases onto which thm.
4.1 applies. In particular, for technical reasons, one often looks at the algebras of smaller generating sets. The above sets A# D# may be regarded as the largest practical such sets, allowing for compact commutators.
Example (A): Q=In, dp=dx, H=1-A,
j2, (i.e., the Euclidean x
Laplace comparison triple of In)
.
This case has been studied
We have condition (s), since In extensively in [Cl], ch.III,IV is a complete space (IV, cor.1.7). Since q(x)El is a constant, we .
also get q=q"=qE=q^eEl , so that the number e becomes redundant. AO consists of all bounded Co-functions with gradient vanishing at infinity. The largest comparison algebra of [C1],IV,l r'V
,
called
there, uses as D# the set of all folpde's with coefficients
in A
A calculation confirms that the class D# properly con-
tains D#
.
In [C1] we obtain precise knowledge about the 'symbol space' (i.e.,the maximal ideal space of
(/K(H) ).
145
V.4. Compact commutator algebras
Problem 4.2. Use the techniques of [C1] to characterize the symbol space of C(A#,Do) Example (B): The Laplace comparison algebra of the Euclidean Rn proves particularly simple because this space O=Rn is a locally compact abelian group (under translation). In [C1] we essentially use the Fourier transform for most of the proofs. Of course, the Fourier transform appears as the Plancherel transform of the translation group, which explains the simplicity of the case. Other cases of manifolds which are translation groups, are Mjk just as simple. For example, let p = = RjxTk, j+k=n, with the circle T = 1/(2rZ) For the simplest case of j=k=l, n=2 we obtain .
the 2-dimensional straight circular cylinder.
One may introduce global coordinates x=(xl,...,xn) , appointing that functions on M3k must be 2r-periodic in the last k variables. then a comparison triple again is given by the same dp=dx, H=1-A as under (A) ,
.
In [CS],V,2 we investigate the comparison algebra C(A#,D#), where D# again is a proper subset of D (defined as under (A)). Clearly our present triple satisfies (s), as the Laplace comparison triple of a complete Riemannian manifold, by IV,cor.1.7 There is an important difference between the symbol spaces for R2 (as under (A)) and the cylinder M1'1 , which will have to be discussed later on (cf.VII ,2, VIII,2.) Example (C): 0=Rn, du=dx, with the expression (where a,6 E R, a>O) .
(4.6)
H = (x) 2a - div( x) 2Sgrad = (x) 2a
(x) =(l+Ixl2)1/2.
x
j(x)
2R3
j x
We have hjk = (x) 286jk , hence hjk= (x) -2a6jk
Proposition 4.3. The manifold O=Rn , with metric tensor hjk =
(x)-26
(4.7)
1=
6 is complete if and only if S<1 jk Proof. For any smooth curve {x(t):tEJ) we obtain as arc length .
J((x(t))-2SIx,(t)I2)l/2dt J
=
J(x)-slx*(t)Idt J
.
For a moment define the real-valued function E(t)=Ix(t)I over the interval J We have .
Ix'(t)I
(4.8)
Accordingly, if J=(p,p') (4.9)
EV(PP
, we get
<
JJ(
<11
146
V.4. Compact commutator algebras This proves that the geodesic distance between two point x bounded below by the integral
147
, x- is
p=lxl, p_=Ix-J. This P
distance tends to -
as x-->- if and only if
J_(C)-Bd 0
This happens precisely for B<1 . Also (4.8) implies that ix.(t)l whenever the curve runs in a radial direction.
iE-(t)i _
O
Accordingly, if 0>1
,
:
Considering the expression (4.6) we thus have cdn.(s) whenby IV, cor.1.7. On the other hand, for a>1 we will
ever 6<1
,
change the dependent variable by setting
with X>O to
be chosen. This transformation yields a new expression H"' with
potential given by III,(6.6). That is, with B"=B-l
,
q` = q - (x) Adiv ((x) 2 B grad (x)) (4.10)
x) 2a
+ X(n+2B"-X)(x)2B -
Then the second term Let us first consider the case B>a+l at right is leading, as lxi is large. Its coefficient becomes .
largest if we choose X=$-+n/2 = B-l+n/2
.
Now, although we no
longer have a complete Riemannian manifold we still may apply IV, thm.l.l.
The proper choice of the function a will be a(x)=T log(x) with suitable T>0 . We then get (4.11) hjko
T2lxl2(x)28--2<
k=
a
q-, as T=("-E+n/2), any c>O,
x Ix
(at least outside a large compact set, which is sufficient). We trivially have a(0)=0 , and (ii) of thm.l.l holds, since T>0 Checking on (iv)n , we find that (4.12)
e- naiVal
whenever B""
,
=
E+n/2)) = 0(1)
0((x)B--n(B
for sufficiently small E>0
.
In other
words, we must require that (4.13)
n > B"'/(B"+n/2)
.
Notice that the right hand side of (4.13) is <1 while that expression tends to zero, as B'"+ 0
,
for all B"">0
and $->0
.
,
Thus we
V.4. Compact commutator algebras
always can satisfy (iv)n for n
in (4.11) we may set T=((6"-e+n/2)2+1)1/2 can allow any choice of T (4.14)
.
Similarly, for a>$- we
.
Correspondingly (4.13) improves to
n>S-/(1+(0-+n/2)2)l/2, as a=8
n>O
,
,
as a>S`
We have proven the result,below.
Theorem 4.4. For a comparison operator of the form (4.6) we always have cdn.(w) satisfied, regardless of the choice of a and 8. Moreover, we have cdn.(s) for all S<1 , regardless of the choice of a If 0
to +1 , we will have cdn.'s(sk) for larger and larger k
, such that for every integer k0>0 there exists Sk >1 such that (sk )
0
0
holds for every S<Sk 0
For a concrete example we now set q"=1, and then, with A#, D# as above, introduce the classes A#=A# , and
(4.15)
D#={DEV#: (x)-Sb7EA#, P=O((x)a'), p
j=o((x)-a+a)) 1x
The symbol spaces for some such algebras have indeed been . We shall discuss this in VII,thm.4.8.
worked out by Sohrab [S3]
Example (D): This example, and example (E), below, only point to special cases of thm.4.1 where the various conditions can be translated into a simpler (or different) form. None of the algebras have been investigated in generality, regarding their symbol or symbol space. Let 52 be an open subdomain of In with compact connected
smooth boundary 30 , where 2S2 is an n-l-dimensional smooth submanifold of In Let H = q - A , with the Euclidean Laplace operator A , above. Here the real-valued Co-potential q>l must satisfy (4.3), above, with the Euclidean distance. Set S2" = I dV = du" .
,
= dx, H^ = 1-A, to get all conditions satisfied, including (4.1). In other words, we have 0 either the interior or the exterior of a compact subdomain of In Now the class A# consists of all bounded C'(c2)-functions a c such that, for suitable e , 0<e
.
lalxj/qe/21
148
V.4. Compact commutator algebras
149
a suitable compact subset K6 of 0 , for every 6>0 Also the class D# consists of all (globally defined) D=b33 .
xj
+ p , with b3 , j=l,..,n
,
p E C'(0) and with the functions
b3, p/v, bjI j// bounded over 0
j=l,...,n
,
,
and with
x b3
k Ixk +b
,
b3q
,
p
Ixj/,q
,
(bl
Ixj/q
Ixj
l)
Ixj/V-q
,
all, = 0(1),
I x
for j,k=l,...,n, in the sense described for A# As a consequence of thm.4.1 we then get a comparison algebra with compact commutators, containing K(N) such an algebra seems to be unexplored.
.
The symbol space of
Example (E): Let 0 be the interior of a complete Riemannian It is known that then manifold QU3M with compact boundary 30 0U3M may be imbedded, in the sense required for thm.5.1, into a .
manifold Q^ which is a doubling of 0 (i.e., in two copies of OLGO
one identifies corresponding points of M. using as chart of a boundary point the union of the chart of OU30 in the half space {x(=-In: xn>0} together with its reflection into {xn<0} ).
Assuming the metric tensor hjk E C-(0U3O) , we also may double h3k i.e., reflect it at the boundary. However, the resulting extension in general is no longer continuous at 30 On the other hand, it is easy to arrange for a smooth transition, ,
accross 30
,
by modifying the reflected h3k in a strip along O
"below O " (cf. also the proof of III,thm.2.2) Furthermore, if a potenSimilarly for a measure dp on 0 tial q is given on 0 (normally singular on 30), it also may be reflected and suitably modified in a (two-sided) neighbourhood of 30 in the doubled manifold, called 0^ , to obtain a q^ E C-(92^) In this way, if a triple {0,dp,H} is given, on the above 0 .
.
it is possible to satisfy all assumptions of thm.4.1, regarding the imbedding into 0^ , with a sub-extending triple {O^,dp",H^}. We must require that q satisfies (4.3), and that Vq/q = o(l) in in order to Q\K , with a compact neighbourhood of O in cu50 ,
insure that (4.1) can be satisfied.
This shows that the case of a manifold with boundary is naturally included in our discussion.
Example (F): Finally we point to the case of 0 _ In, with du=dx, and a (globally defined) comparison operator H, under the assumptions of IV, thm.1.9 .
V.S. One dimensional problems In this case we will assume S2"
=
S2
= In
,
and q"=l
.
Then we
again have all assumptions of thm.4.1 satisfied, for the classes A#
,
D#
.
Again, the structure of the corresponding symbol space
seems to be unexplored. 5. A discussion of one-dimensional problems.
In this section we get restricted to the case of a 1-dimensional manifold 0. That is, our comparison triple assumes the form {O,dp,H}
(5.1)
{I,Kdx,K-1(-d/dx p d/dx + q)}
_
with a finite or infinite open interval I = {a < x < g} # 0 and globally defined Cam-functions K(x) p(x) q(x) , x E I ,
K(x) ,
p(x) >
0
,
as x E I
,
,
.
In this case the two normal forms of III, prop.'s 6.1 and 6.2 may be further simplified by also introducing a special independent variable . A simple calculation,left to the reader, gives the following result. Proposition 5.1. a)If new dependent and independent variables (v,y) instead of (u,x) are introduced by setting (5.2)u=yv, Y=(pK)-1/4 y(x)=JX
dt, a-=limx;ay(x), R-=limx}6y(x) 0
with any given x0 E I
{S2-,du-,H'} _
(5.3)
with (5.4)
then the triple (5.1) assumes the form {1-,dy,-d 2/dy2 + q-}
I` _ (a`,B-), and q"' of the form, with Y-=Y-1=(pK)1/4(x(y)),
q-(y) = (q/K)(x(y)) - (d2Y /dy2/Y )(y) = Hy/y(x(y))
.
b) Let y E C'(I) be a positive solution of the differential equation Hy y (which exists according to III, prop.6.2, since I is noncompact, and the minimal operator H0 has the lower bound 1) Then, if new dependent and independent variables are introduced by setting (5.5)
u = Yv
,
y(x) = JX K(t)y2(t)dt 0
then the triple (5.1) assumes the form (5.6)
{I-,dy,H-1 _ {(a-,s-),dy,-d/dyp-d/dy+1 }
,
150
V.S. One dimensional problems with p'(y) = (Kpy4)(x(y))
(5.7)
,
a- = limx+ay(x)
limx+sy(x).
Generally a transformation of dependent and independent variable of the form u = yv
(5.8)
y =
,
fi(x)
,
4'(x) > 0
,
with a positive Cm-function y and an invertible C'-map :I + Imay be applied to the triple (5.1) as well as to any first order differential operator F = bD + c
(5.9)
,
b,c E C_(I) , D = d/dx
and the Hilbert spaces H and H1
,
.
In view of proposition 5.1 it is no loss of generality to depart from a triple in Sturm-Liouville normal form (5.10)
I = (a,8) , du = dx , H
D2 + q
,
:
Dx= d/dx
The proposed change of variables will carry (5.10) into I- _ (a",$-) dp-
(5.11)
H"'
= K-dy
,
a- = 4(a+0)
,
b_(y) =
_
,
K_(Y) =
-K--1 DyK-p-DY + q-
p"'(y)
S- _ (6-0)
,
,
F`
= b-Dy + c- , Dy = d/dy ,
q-(y) =
c-(y) _
(c + by'/y)(4(y))
.
All this assumes that the transformation amounts to an isometry H « H`=L2(I`,dp"'), carrying the equations Hu=f F-v"'=g"'
Fv=g into H-u'=f"',
, with the transformed functions u-,V",f-,g- of u,v,f,g,
respectively. While we will work with the triple in the form (5.10) we
also note that our conditions on the coefficients of H and F will not remain invariant under a transformation (5.8). Hence we shall k
tend to repeat the construction of a C -algebra with compact commutator of the preceeding sections,while trying to optimize these
151
V.5. One dimensional problems
152
conditions, or else trying to bring them onto a simpler form. First the requirement of condition (w) is invariant under (5.8) as well as that of (s) or (sk because (5.8) amounts to a unitary transformation U: H + Htaking the operator H into H- = U-1HU, and C_(I-) . An application of IV, thm.1.8 ,
to (5.10) yields the following Proposition 5.2. a) If a = -- , $ = -, then the assumed semi-boundedness of H alone implies that (s) = (so) holds. b) If a or S (or both) are finite then there exists a nondecreasing sequence nk , k = 1,2,... , such that (sk) holds whenever (5.12)
q(x) > nk(x - a)-2 near a, or q(x) > nk(x - S)-2, near B
,
respectively. (Both for 2 finite ends). Specifically (w)=(s1)
holds for q with (5.12), for nl = 12 , and (s) _ (s,) holds for (5.13) (x-a) 2/q(x)=o(1)
,
as x+a
,
or, (x-a) 2/q(x)=o(1)
.
as x-5,
respectively. (Both conditions, if a and $ both are finite, but if only one end is finite then the infinite end never requires a special condition,except the semi-boundedness of the operator). From (5.11) and (1.7) we know that the operator FA , with A = H-1/2 is bounded whenever, for some y>0, y E C-(I) ,
b = 0(1)
(5.14)
c + bY'/y
,
= O(/q--ywTy)
.
This condition may be simplified by introducing the function (5.15)
6
= Y'/Y ,
6'
+ 62 = Y"/Y
,
Y = Y(x0) exp(JX 6(t)dt) 0
Proposition 5.3. The operator A = (FA)** is in L(H) whenever there
exists a real-valued function 6 E C'() such that (5.16)
b = 0(1)
,
c + b6
= O(V`q-77 67)
The proof is immediate. Note that change of coordinates does not give an improvement of conditions for b. To evaluate the second condition (5.16) we assume that the operator bound, hence the 0(.) -constant in (5.16) is 1 , as always can be achieved by multiplication with a positive constant. Then (5.16)2 is equivalent to the Riccati type differential inequality (5.17)
6' + 26 Re(cb) + (l+Ibl2)62 < q -
Icl2
.
V.S. One dimensional problems For example, consider the case of b-0 , where (5.17) yields
0 < icl2 < q - 6' - 62
(5.18)
.
For 6 = 0 (5.18) gives the trivial condition IcI A more general 6 will improve this only if (5.19)
6'
over the interval I
+
62
<
< 0
or at least near one of the endpoints. (or in a sequence of intervals clustering at an endpoint), in view ,
of the fact that a C_(I) -coefficient b or c always gives a bounded operator FA In fact, q0= -(6+62) also is the improvement of .
the potential q under the change of variable (with 6=Y'/Y) Notice that a real-valued 6 satisfying (5.19) in some open interval J must be a decreasing function of x which can vanish at On the other most once.At its only possible zero we have 6' < 0 hand for all x E J with 6 9 0 , (5.19) amounts to .
(5.20)
y'
= (1/6)' = - 6'/62 > 1
,
for y(x) = 1/6(x)
.
Consider a right endpoint B=co , for example. If 6(x0)<0
for some x0
,
,
then 1/6 , having slope > 1 , by (5.20), must get
zero after a finite x-increase, leading to a contradiction. Hence 6>0 for all x>x0, x0 sufficiently large. Then (5.20) yields 1/6 - 1/60 > x-x0 Hence, for all x>x0 , with 60=6(x0) ,
(5.21)
0
so that 6-0
,
.
< 6(x) < 60/(1+60(x-x0))
,
for all x > x0
,
as x- . One then concludes that also
x+h (6'+62)dti
(5.22)
< 60/(1+60(x-x0))+h60/(1+60(x-x0))2 , h>0.
X
Accordingly, while the possible improvement of q may be large at some specific point, it will become small in the average, over a finite interval (x,x+h) only if q+0
,
as x-*0
.
,
as x-*
, and can be significant
Since we get q_l in suitable coordinates,
it follows that q, in the average cannot be too different from 1 Accordingly the improvement of the boundedness estimates by coordinate changes may be of little use, if the interval is (--,+°) On the other hand, if the right end B of the interval I=S2 is finite, then one may think of possibly significant changes of
q by change of coordinates.
153
V.5. One dimensional problems Example 5.4. (5.23)
The choice 6(x) = (B-x)A
y(x) = exp(-(B-x) X+1/(X+1))
,
0
, near B
x < B
,
gives the additive improvement (5.24)
Similarly 6(x) = -(B-x)X the even faster increasing (5.25)
(B-X) 2X
q0(x) = X(B-x) X-1 -
-1<X<0 , near B , resulting in
,
(B-x)2a
q0(x) _ IXI(R-x)X-1 +
x.B
= A(B-x) A-1
IAI(B-x)A , x
=
, near B.
However, cdn.(w), if installed by IV,thm.1.5, requires q>y(B-x)-2, which again makes the possible gain by either (5.24) or (5.25) appear insignificant.
Coming to some more explicit examples in 1 dimension, let us examine examples (A) and (C) of sec. 4 in this connection. Both give some typical 1-dimensional examples. While (A) already is ,
in Sturm-Liouville normal form, (C) is not. Its Sturm-Liouville transform is very similar to that used in the proof of thm.4.4, in the case of B>1 We will take up the discussion of the corresponding algebras C in VII,'+. Also there we will discuss certain more .
general examples of Sohrab, around the quantum mechanical harmonic and anharmonic oscillator equation in one dimension. Finally, in this section, consider the following: Example (G): Define du=dx
(5.26)
,
H=1-Dx2 , Dx=d/dx
Let A# be the class of all C'(1)-functions (5.27)
a=b+c, b,cEC'(1n), where bECS(1), and c is 2ir-periodic
Here CS(I) denotes the class of continuous functions over I with limits at +- , and at -- , where the limits need not to be equal. Also, D# is the class of all folpde's with coefficients in A# Here we want to show that the corresponding algebra C(A#,D#) does not have compact commutators. Indeed, consider the commutator
(5.28)
[ elx,DA]
_
[ elx,s(D)]
,
s(x) = x/(x)
If we conjugate with the inverse Fourier transform (5.29)
Fs(D)F-1 = s(x)
F-
FeixF 1 = e-iD = T-1 ,
we get
154
V.S. One dimensional problems with the translation operator T-1u(x) = u(x-1) [e1X,DA] = (V-1s)(D)e1X
(5.30)
,
.
Therefore,
V-1s(x)=s(x-l)-s(x)
eix This is not a compact operator, since multiplication by is a unitary map of H=L2(1) , while the function V-Is(x) does not
vanish identically, so that V-1s(D) cannot be compact. Similarly we get [e'j',DA]
(5.31)
(for j=0,±l,±2,... V-js(D)e1jx
=
where Vjb(x)=b(x+j)-b(x)
.
(cf.[C1]
ch.III
,
,
[e13X,A] =V-jX(D)e13X
.
On the other hand, for aECS(E we get
[a(x),DA] E K(H)
(5.32)
,
, with X(x)=(x)-
,
[a(x),A] E K(H)
,
or also by application of sec.3, above.)
Proposition 5.5. The commutator ideal E of the present algebra C is the norm closure of the algebra of all operators of the form X-()lj=-Na-.(D)e13X + K, where
X
(5.33)
ajE CO(s), K E K(H), X1 for ±x?y »1, =0 for ±x
one concludes that every operator a(D)e1JX is in E
,
for arbitrary
aECO. Similarly E contains the entire compact ideal, by lemma 1.1. This completes the proof.
In VIII,l we will look at this algebra again, trying to determine its symbol space and ideal chain. 6. An expansion for expressions within reach of an algebra C.
In the present section we shall prepare relating a differential expression L to a comparison algebra in a manner which will allow us to connect an invertibility or Fredholm statement of a certain realization of L to a corresponding statement involving a bounded operator in the algebra. The general technique appears to be common in theory of singular integral operators (cf. Calde-
155
V.6. Expressions within reach
ron and Zygmund [CZ2], for example). Given a comparison triple {D,dp,H) on a manifold D
,
and
any subalgebra A of L(H), H=L2(D,dp). Let us assume cdn.(s) for all of the present section. Definition 6.1. A differential expression L of order < N (for some integer N=0,1,...,) is said to be within comparison reach (or shortly 'within reach') of the algebra A if the unbounded operator L0AN , with the minimal operator of L and has a continuous extension A to H, where A E A More precisely we should speak of an 'N-reach'
A=H-1/2
.
,
since the
condition obviously may depend on N. We will omit this distinction, however, since the choice of N always will be clear. In particular we note that cdn.(s) implies that the space dom LOAN = A-NC-(G) is dense in H Indeed, for even N=2k we get A-NCO = HkC' = im H dense in H , since H is > 1 and essentially .
self-adjoint, using I,thm.3.1. For odd N=2k-1, if fEH, (f,A-N0=0 for all uECO, then we set f=A-lg , using that A-1 is self-adjoint and >1, hence im A-1=H . It follows that (A-1g,A-Nu)=(g,A-N-1 u)= =(g,H0u)= 0 for all uE dom H0 , hence again g=0 and f=0, so that A-NCO again is dense. A-NCO are dense it follows that L of order Since all spaces N is within reach of L(H) whenever we get, with a constant c, (6.1)
HLup < cIIA-Nul for all u E C_(G).
It is trivial that all 0-order expressions (i.e. multiplications) aE A# and all folpdes DED# are within reach of the comparison algebra C= C(A#,D#), by definition of C. In VI,3 we shall see that every differential expression of compact support is within reach of the minimal comparison algebra. The general question for expressions within reach of a comparison algebra C is vital for our applications and will be studied in IX,3 and X,2. For a differential expression L of order N within reach of a comparison algebra C there is a natural realization Z defined: Definition 6.2. Let AEC be the continuous extension of L0AN to H requested in def.6.1. Then we set (6.2)
dom Z = im AN = dom A_N , Zu = AA-Nu , u E dom Z
i.e., Z=AA-N, in the sense of I,(1.3).
156
V.6. Expressions within reach Remark 6.3. The space im AN = dom A-N= HN is commonly referred to as the (L2-) Sobolev space of order N of the triple {D,du,H} (assuming cdn.(s), as we presently do). Here HN is a Hilbert space under the graph norm I,(2.9) of the (closed) self-adjoint operator A_N Since A-N>l it follows that the graph norm is equivalent to .
(6.3)
IIuIIN
11A-Nun
=
,
u E HN ,
called the Sobolev norm of order N . With this norm the operator A-N:HN. H becomes an isometry between the Hilbert spaces H and H Then the unbounded operator Z E P(H) may just as well be regarded as a bounded operator of L(HN,H) There exists an inverse Z-1 E .
L(H,HN) if and only if A-1 exists in L(H) Similarly the unbounded operator Z admits a bounded closed inverse if and only if A E L(H) is invertible. We also set HZnHN and note that all L and .
ANtake H.->H.. For more details cf.VI,3 and X,2.
In order to facilitate a later discussion of expressions within reach and their realization Z we next will discuss a certain Taylor type expansion with integral remainder, valid for certain expressions within reach.
Definition 6.4. A (C--)differential expression L of order
(6.4)
(ad H)0L=L, (ad H)OL = [H,[H,...[H,L]..] , with j commutators,
are within reach of L(H) (as expressions of order N+j).
Proposition 6.5. Let again R(a) _ (A2-a)-lE L(H), and let L be H-compatible. Then the operator A=LA NE L(H) satisfies the identity
R MA - AR(A)
a3-'((ad H)OL)A23+N+2Rj+1(A)
(6.5)
+ XMA2R(a)((ad H)M+1L)A2M+N+2RN+1(a) for all M=0,1,2,....
,
x E r
,
.
Proof. We get (6.6)
R(a)Lu - LR(A)u = A2R(X)[H,L]A2R(a)u
as used before (cf. sec.3)
,
,
u E im(l-AH0)
and may extend this to u E H.
.
Note that (6.6) amounts to (6.5) for N=O. Suppose (6.5) holds
157
V.6. Expressions within reach for N>O then write A2R(A)=1+AR(A)
,
158
and LN+l = (ad H)N+1L
for a moment. For u E im(1-AH0) we get, A2R(X)u = (1+XR(X))LN+lu (6.7)
= LN+l(1+R(A))u
+ X(R(a)LN+lu-LN+lR(X)u) = LN+lA2R(X)u+AA2R(A)LN+2A2R(A)u
,
and again we may extend this to H. . Substituting (6.7) into (6.5) yields (6.5) for N+1 q.e.d. ,
Proposition 6.6. In the notation of lemma 3.3 we have (6.8)
Jr A2JR]+1(A),s/2da
=
2rti(-l)j+l(s,2)As
with binomial coefficients norm convergence of L(H)
,
s>0, j=0,1,...,
Here the integral converges in
as an improper Riemann integral. Proof. Integrate by parts in formula (3.6), using that Rk+l(a) d/dXRk(A) = k ,
.
Proposition 6.7. With the notations of prop.6.5 we have (6.9)
AsA
(-1)3(s/2+j-1)
=
((ad H)OL)As+2j+N + RM
0
where RM = i/27r JrA2R(a)((ad H)M+lL)A2M+2+NRM+1(X)Xs/2+MdA
(6.10)
Proof. Multiply (6.5) by i/(2n)as/2 , and integrate over r Note that the integral may be interchanged with the operators (ad H)3L = Lj , which are preclosed. Then use (6.8) to evaluate some of the integrals.
Theorem 6.8. For all (real or complex) s with -2M-2
(6.11)
where Lj
=
+ SM,s
(ad HA E LN+j, and the 'remainder' SM's is given by
-2'rtiS M
(6.12)
M=1(-1)3(s/2]3-1)(LjAN+20)
s = fl, XC-1dXSl(a)(ALM+aAN+M+l)S2 s(a) >
>
SM s(a) = AM+1/2+s/2-eAM+l-sRN+l()
,
0<e
V.6. Expressions within reach
Here the two L(H)-valued functions Sj(A) of A
,
A
,
(and s) are
bounded and regular analytic for A E r\{0} -2M-2+2e < s < M+l as e is kept fixed. The integral in (6.12) converges in L(H), ,
as an improper Riemann integral of a continuous integrand, even if the constant operator AAM+lAN+M+l is replaced by an arbitrary operator VEL(H) (6.13)
.
Moreover, we even have the functions
SM s(a)A s-M-1 2
,
as s>0
,
2 and SM s()A 2e-M-1
>
as s<2
>
bounded and analytic on the (A,s)-set specified.
We should emphasize that, by writing (6.11) we intend to ASAA-s
imply that the (product of unbounded) operator(s) is bounded, and that its continuous extension to H should be taken in its place.
Remark 6.9. It is an immediate consequence of thm.6.7 that the remainder SM
s is a holomorphic function of s (with values in L(H) ), defined in the strip -2M-2 < Re s < M+1 and that even ,
AE-M-1 , the operators SM's AS-M-1, for 0<s<M+l for and SMM's S A -2M-2+e<s<e are in L(H) . Moreover, it should be observed ,
,
that the two factors S1(A) and S2
(A) both are functions of A hence commute with every function of A The proof of thm.6.8 is a matter of analytic continuation
of the identities (6.9), (6.10) from the positive real s-axis onto the s-strip specified. First we get (6.11) for 0<s<M+1, by Next we note that all multiplying (6.9) from the right by A-S terms at right are entire functions of s except the remainder. .
On the other hand, the remainder may formally be written, as specified in (6.12), where the function Si are analytic. Also, boundedness of the functions S!
, near their singularity a=0 is
a consequence of estimate (3.9). Thus one concludes that indeed the remainder integral admits an analytic extension into the strip -2M-2< Re s < M+l Finally, we at least have the complexvalued function ((ASAA-S - A)u,u) _ 4(s) analytic in the strip, for every fixed uEH. . One thus concludes that (6.11),(6.12) .
holds in the same inner product sense. This, in turn, implies for s in the H-boundedness of the pre-closed operator ASAA-s
strip. The remaining estimates follow as part of the above proof, or are trivial consequences.
Remark 6.3'. Note that the above definition of Sobolev space is meaningful not only for integers NO but also for general s>0 .
159
V.6. Expressions within reach
Similarly, for s<0 one defines the Sobolev space H
as the comples
IIA-sup (note that now A- ELM). For a more detailed investigation of the spaces Hs cf.IX,l.
tion of H under the norm Ilups
=
Here we only note that the spaces {Hs:seU} define an HS-chain in the sense of 1,6. Then thm.6.8 above has the following consequence Corollary 6.10. The operator LOAN, for an H-compatible differential expression of order
Also, for every s,N there exists a constant c and an integer M>0 such that the operator norm in L(Hs) of As is bounded by (6.14)
c
I
IILkAN+2k II
,
Lk = (ad H)kL
0
for all H-compatible expressions L of order
show that IIL0ANups = IIA-sL0ANup < cllups = cIIA-sull
,
A-s-NCO v=A-su , IIA-s(LOAN)Asvll < cllvp for v E That, exactly, follows from our expansion (6.11) if only we choose M large ,
,
enough to include the real number s in the strip of analyticity, defined in the thm. In particular we find that L.AN+j E L(H) and the remainder S
Ms s
as well, in view of the properties stated.
160
CHAPTER 6. MINIMAL COMPARISON ALGEBRA AND WAVE FRONT SPACE; SOBOLEV SPACES ON COMPACT MANIFOLDS.
Starting a general investigation of the symbol space of a comparison algebra, (i.e., the maximal ideal space of the commutative algebra C/E) , we first will focus on the minimal comparison algebra JO
,
introduced in V,1 (It is generated by the classes
of a and D with compact support). We know that it always has the commutator ideal K(H)
Its symbol space will be called the wave
.
front space , and be denoted by W . We first will show that the
wave front space is independent of the choice of H and dµ (sec.l). In fact, it proves to be homeomorphic to the bundle of spheres of radius infinity compactifying the cotangent space T Ox at a point x GO (sec.2). For simplicity one commonly identifies these spheres
with the corresponding unit spheres (0.1)
{
(x,F)
1
:
} C T*Ox
Thus W then is called the 'co-sphere bundle' of 0 (cf.[AS],[Se2]). The wave front space is found naturally imbedded in the symbol space of every comparison algebra C meeting our general assumptions (cf. VII,l). The result described above is due to Seeley [Se21 in the case of a compact manifold, and to Gohberg [Gbl], in the case of It
n
(cf. also [CI], Herman [H1]).
For a compact manifold the minimal comparison algebra is the only comparison algebra, evidently. For noncompact manifolds one will ask the question about points of the symbol space M , not contained in W This will be the focus of our attention in .
ch.VII
.
In fact, it will be found that such points are linked to
the essential spectrum of the comparison operator H
.
In sec.3 we show that all differential expressions of compact support are within reach of the minimal comparison algebra This leads to a result about the Fredholm property of the realization of V,6 (thm.3.4) on compact manifolds. In fact, we JO
.
even get a Green inverse, i.e., an inverse modulo 0(-co) (thm.3.7).
VI.l. Local invariance
This fact is exploited in sec.4 to obtain the familiar Sobolev estimates on compact manifolds. 1. The local invariance of the minimal comparison algebra. In this section we focus on the minimal comparison algebra JO introduced in V,l Let Am = C0 (U) and Dm = {D=b38 j+p: b3 and .
x
p are C' , and of compact support } be the corresponding sets of
functions and folpde's. Essential self-adjointness of H0 is not required in this section.
Theorem 1.1. The algebra J0 remains unchanged if the measure dp
,
and the comparison operator H are modified over a compact
subset U C 0
.
That is, if {O,du,H} and {Q,dv,K} are two triples
and if there exists a compact set U c 0 such that H=K and dp=dv outside U , then both triples have the same minimal comparison algebra J0
Remark. Notice that the two Hilbert spaces L2(O,dp), and L2(Q,dv) coincide as sets, and have equivalent norms, but in general have different inner products. Accordingly there also are two different adjoints. Our proposition states that the two corresponding algebras J0 of bounded operators coincide. They necessarily are C algebras with respect to either adjoint. The proof of thm.l.l will require some preparations. In order to use methods similar to those developed in V,3 it will be necessary to have condition (w), or even (sk) for ,
larger k, available. Since we focus on a and D with compact support in this section, it will be convenient to relate the results to a modified comparison operator H° = H+q° , where the nonnegative function q° is chosen such that {O,dv,H°} satisfies (sk)
for suitable k
.
First of all, by III, prop.6.2 we may change the dependent variable, and then obtain a constant nonnegative potential q= const >1 (1.1)
.
q°
In this representation define = X(nkT)-2h1kTlxjTlxk ,
T
= jj=12-3ejwj
with our partition {w.} of app.A, where e.}0, as j}-, and 1 3 3 < (Max{l, sup{IVwj(x)I 0 < £j x E nj}}) Also, nk denotes the positive real number specified in IV, thm.l.l, and X denotes :
.
162
VI.l. Local invariance any C-(Q)-function with 0<X<1
X=l outside some compact set.
,
In particular x may be made to vanish in some given compact set, so that H=H° there. Then a calculation confirms that the expression H° satisfies the conditions of IV, thm.l.l, for m=k we have (sk) satisfied for H°
so that
,
Finally we may switch back to the
.
old dependent variable, which does not disturb (sk), and leaves the function q° unchanged as well. We have proven: Proposition 1.2. For every comparison operator H and k=1,2,... there exists a nonnegative function q° E C'(c) such that He=H+qs satisfies condition (sk) of V,1 . Moreover, q° may be chosen to vanish in any given compact set of D .
Remark. It is easy to also obtain a q° such that H° satisfies (s), For example, one just might replace the constant nk in (1.1) by a positive C°(c2)-function n(x) satisfying limx+.n(x)=0
Proposition 1.3. Let A=H-1/2 and
A°=H°-1/2
For every function
.
a E Am and folpde D E Dm we have (1.2)
aA - aA° = AC = C'A
,
DA - DA'
= C"
,
C,C'5C" E K(H)
.
Proof. For u E C-(SI) write (1.3)
D(H°+A)u = (H+A)Du + Dq°u + [D,H]u = (H+A)Du + Mu
with a second order expression M of compact support. For any A >0 (H+A)-1
write v = (H°+A)u , and multiply (1.3) by
,
to get
(1.4) R A)Dv=DS(A)v+R(X)MS(A)v=DS(A)v+(A-1R (X))AMA(A-1A°)A°-1S(A)v with R(a) _
(H+a)-1
,
S(a) = (H°+a)-1. By (w) the space (H°+A)C'
is dense, so that (1.4) may be extended continuously to v E H
.
and A'IS(A) are bounded and 0((l+a)-1/2) as easily seen, while IIA-1A°II < 1 follows from the inequality In particular A-1R(a)
(1.5)
nA-1u112 = (u,Hu) < (u,H°u) =
IIA°-1u112
,
u E C'(cl)
Now a resolvent integral may be employed again. Let us use the resolvent of the operator H , not the resolvent of A2
,
for
a change. We have the formula (1.6)
H-s
S-1 = m-lsin rs /(M-1
J'(R(A))mam-s-lda,
0
s>0
, m=1,2,...
163
VI.l. Local invariance
with binomial coefficients (m_i) , where the denominator may have a simple zero, for certain integers s , but then the numerator sin its vanishes also, and (1.6) holds with the limit of the quotient, as s approaches such points. The proper and improper Riemann integrals exist in norm convergence, as usual. Formula (1.6) is a direct consequence of V,(6.8). One simply must introduce the new resolvent and make a substitution p=-1/A of the integration variable. Also the integration path must be deformed into the positive real axis, chosen as slit of Am-s-1
the branch of , and then the integrand must be evaluated along the slit. Finally an analytic continuation in s must be made, using the spectral theorem for the self-adjoint H We use (1.6) on H and H° for s=1/2 to obtain A = 1/7 J-R(A)A-1/2dA
,
A°
=
1/7r
0
J-S(A)A-1/2dX
,
0
Integrating (1.4) it follows that (1.7)
(AD)
= DA' + 1/Tr
N(A)dW 0
with (1.8)
C(A)=A-1/2(A-1R(A))(AMA)(A-lA°)(A°-1S(A))=O(A-1/2(l+A)-1),
as is derived, using the fact that A-1R(A) = 0((l+A)-1/2) AA-1S(A)
,
and
Also C(A) is compact for all A > 0 and
O((l+A)-1/2).
=
norm continuous in A again. (One may write (A-1R(A))(AMA) = (A-1-ER(A))(A1+EMA) , where the first term is bounded,the second is compact, due to the compact support of M .) It follows that the integral in (1.7) is a compact operator. Taking adjoints in (1.7) we get DA = (A°D)
+ C
,
C E K(H)
This may be combined with the
second formula V,(3.17), applied to H° and A°
for the second relation (1.2). For the first relation we assume the expression D ,
of order 0 (i.e., bi = 0) Then M will be of first order only, and we may repeat the argument with AMA replaced by AM or MA .
while the remaining factor A (or A°) may be taken out of the integral, either left or right. This completes the proof. Proposition 1.4. The minimal comparison algebras J0 of {D,du,H}, and {O,du,H°} are identical. This proposition is an immediate consequence of V,lemma 1.1, and the second formula (1.2) We will need (1.2) in a slightly more general setting, .
164
VI.l. Local invariance expressed by the corollary,below.
Corollary 1.5. Formula (1.2) also holds if H and H° are two general comparison operators of the form V,(1.1) satisfying V,(1.2), perhaps with respect to different measures dp and dUA , as long as the two Hilbert spaces over 0 coincide (i.e., the function * =
du/dpA is bounded and bounded below), and as long as H and H° have the same principal part (i.e., H-H° is a first order expression). not necessarily for H , and (1.5) must Also (w) must hold for H° ,
hold, at least with an additional multiplicative constant at right Proof. The important point is that (1.3) is still valid, i.e., D(H°+a) = (H+X)D + M , with a second order expression M of compact support. Also the resolvent integrals are still well-defined, and satisfy the estimates required, so that the proof just may be repeated, q.e.d.
It is clear now that, for the proof of thm.1.1 it may be assumed that both triples {52,dp,H} and {S2,dv,K} satisfy (sk), for an arbitrary k (or even (s)=(sue) ). Also we may assume that dp=dv.
Indeed, let c = dii/dv
is a positive C-(S2)-function
Note that
.
equal to 1 outside the compact set U K=-(CK)-12
We get
K°=-K-18 jKh^jkB k+q+y Kh^OkB k+q=K°+D° x x x a k - y having compactly supported prin,
xj
_ -C
with DA
.
Jxj
x
cipal part, while the constant y may be chosen such that K°
Clearly H and K° now are self-adjoint with satisfies V, (1.2) respect to the same inner product, while K and K° generate the .
same minimal algebra JO
in view of cor.1.5. If necessary, prop.
,
1.4 must be used again, to get (s) or (sk) for K°
.
Now theorem 1.1, will be a consequence of the following
result, first published in [CS] , ch.V,lemma 4.10. Theorem 1.6. Let condition (s) hold for H and K , and assume that dU = dv
,
so that we only have one adjoint. Then the operators
Vt=(HtK-t)**
are in L(H)
,
for every t E I
denotes the coset mod K(H) of Vt, then {Ut
.
If Ut= Vt+K(H)
tEg} is a 1-parameter
abelian subgroup of positive self-adjoint elements of the linear group GL(L(H)/K(H)) in the C*-algebra L(H)/K(H). Let us first show that thm.l.6 implies thm.l.l. For a moment write J0(H) and J0(K) for the minimal algebras of H and K. We conclude that V1-1 =
HK-1-1 =(H-K)K-lE J0(K), because G=H-K has com-
165
VI.l. Local invariance
pact support, and may be written as a finite sum of products DF , DK-1/2FK-1/2
with folpde's D,F E Dm. (Then DFK 1 = E J follows from V, lemma 3.6.) Thus U
+
DK-1/2CK 1/2
E J /K , and for the 10 coset U-1/2 of V1/2=(H-1/2K1/2)** we also must have U_1/2-lEJ0/K Indeed, U-1/2 as positive hermitian element of LIK must be the ,
unique inverse positive square root of U1, since the group property of Ut gives U1 = (U-1/2)-2 This means that U_1/2 may be obtained as a resolvent integral .
Jr(U1-a)-1X_1/2dA
U-1/2 = i/2,
(1.9)
with a curve r surrounding Sp(U1) c (0,co)
, and staying in the right half-plane Re X > 0 In particular a1/2 denotes the branch of the root assuming the value 1 at X=1 It is important that 0 is not in Sp(U1) since U1 is invertible (its inverse is U-1 .
.
,
by thm.1.6). Write U1=1+J J E J0/K. One may select r such that 1 is inside (it will belong to Sp U1 anyway). Then ,
(1.10)
U_1/2-1 = i/2n Jr ((J+l-x)-1
(1-a)-1)X-1/2dX
-
by the residue theorem, where the integral converges in norm of L/H Clearly, .
(J+1-a)-1-(1-a)-1
(1.11)
= -J(1-a)-1(J+l-a)-1 E J0(K)/K, X E r
Since J0/K is closed it follows that U-1/2-1 E JO(K)/K. Thus also V-1/2 = (H-1/2K1/2):;* = 1+J^
(1.12)
Now from the generators a E Am ,
,
J^ E J0(K)
DH-1/2
, D E Dm , of JO(H) the first type trivially is in J0(K) as well. On the other hand,
(1.13)
DH_ 1/2
=
DK-1/2(K-1/2H1/2)
=
DK-1/2(1+J^x)
as well, by (1.12) so that J0(H) C J0(K)
.
E JOCK)
For reason of symmetry
we then get the opposite inclusion, hence J0(H) = J0(K)
,
q.e.d.
Notice that the full strength of thm.1.6 was not used in the proof of thm.l.l. In particular, only the classes U+1 and U+1/2 were ever used, together with the group property (U-1/2)-2 = U1 Thus, while stating thm.1.6 in its elegant form, we only discuss the following weaker result (for full proof of thm.1.6 cf.IX,S) Lemma 1.7. Assume only (s2), for H and K Then we have Vt e L(H) for all ltl<1. The cosets Ut mod K are hermitian and positive in .
166
VI.1. Local invariance L/K , and we have (Us/2)2 = Us
,
UsU-s=U0=l for all lsl<1
167
.
Proof. We already proved that HK-lE K(H), so that U1= HK-1+K(H) is well defined. Similarly KH-1E L(H)
so that U_1
E L(H)
is well-defined. Moreover, using I,lemma 5.1, we now
HtK t
and
KtH-t E L(H)
defined for all ltl
thus also
V-1 = (H 1K)** _ (KH 1)
(1.14)
get
,
0
,
,
and hence get Ut well
Also we get
.
HK-1 - (HK-1)* = GK-1 - (GK 1)
, G = H - K
The compactly supported second order expression G is a sum of products of < 2 folpde's. We get DFK-1 = DK-1/2FK 1/2 + C
,
CEK
as already used above. Accordingly DFK-1 - (DFK-1)* E K, by V,thm. 3.6, and U1 = U1 follows Similarly U-1 = U-1 .
Next we need a commutator relation. Proposition 1.8. For 0<s<1/2 we have (1.16)
Hs[H-s,K-s]Hs E K(H)
, and H2s[H-s, Ks] E K(H)
Proof. With the resolvents R(X)=(H+a)-1, S(A)=(K+A)1, and integrals as in (1.6) (and either (a,T)=(s,t), or (a,T)=(2s,0)) write (1.17) Ha[H-s,K-s]HT =
(sin 2Trs)/Tr2
10
1-x-s dAU-sdpHa[R(A),S(p)]HT, 0
(cf. (1.6) for more detail ) where (1.18) [R(A),S(p)]v = R(A)S(p)[K,H]S(u)R(A)v
,
v E im(H0+X)(K0+p),
by a simple calculation.
We now need condition (s2)
,
at least, to insure that (cf. prop.1.9, below).
im(H0+A)(K0+u) is dense for all A,u > 0 The expression J = [K,H)
.
is of order 3 , and has compact support.
Accordingly we may write J as a finite sum of products of at most 3 folpde's with compact support. Accordingly, for evaluation of the term at right of (1.17) we will substitute [K,H] in (1.18) by a product DEF of 3 such folpde's
,
to indicate the procedure.
We then will have to commute S(U) with D
(1.19)
VD = [S(p),D]
_
:
(K1/2S(ir))(K-1/2[ DFK] K-1/2)(K1/2S(u))
(using (w)) , where the middle bracket is bounded (and compact, K- E if multiplied by , with arbitrarily small e>0 ) , while the
,
VI.l. Local invariance
two outer brackets decay with p (1.20)
,
168
so that [S(p),D] is compact, and
VD = [S(p)5D] = 0((l+u)-1)
.
Moreover, the commutator VD not only is bounded, but even GVD with an arbitrary folpde G of compact support, is bounded. Therefore we now can write R(X)S(p)DEFS(p)R(X)v = R(X)DS(p)ES(p)FR(X)v (1.21)
+ R(X)VDES(p)FR(X)v - R(X)DS(p)ESF,R(X)v - R(X)VDEVFR(X)v
,
where the right-hand side contains only bounded operators, so that a continuous extension to v E H is possible. The remainder of the proof consists of just balancing all the various effects occurring in the integrand X-sp-SHaR(X)S(p)DEFS(p)R(X)HT
(1.22)
where we focus on the first term at right of (1.21)
.
The integral
(1+X)s-1-d(1+11)s-1-d
needs a power
with 6>0 to converge. That , KVS(p)=0((1+p)v-1) is, in view of Hs-g+l we can absorb a total power (left and right) , using the For 0<s<1/2 this means that both a=T=s and two factors R(X) v=2s, T=0 give a converging integral of compact operators, as long HVR(X)=0((1+X)v-1)
.
as two of the folpde's D,E,F can be absorbed by the factors S(p). That can be done, because it still leaves a factor (l+p)-1 , hence a total power p-s(l+p)-l to generate a convergent integral. For s=1/2, on the other hand, one may split K 1/4 from the second S(p) and H-1/4 from the last R(X) for a term K- 1/4 FH- 1/4= (K-1/4H1/4)(FH-1/2+[H-1/4,F]H-1/4) by V, lemma 3.6, while E L(H) 0(X-1/2(1+X)-3/411-1/2(l+p)-3/4). The we still have the integrand ,
other three terms in (1.21) are similar, although easier to treat. This completes the proof of prop.l.8. Then relation (1.16)1 (K-sHs)*-
HSK-s
E K is hermitian, 0<s<1/2 so that U From the second (1.16) we get (HSK s)2H2sK 2s+H2s[H-s,K-s]HsK-s = H2sK-2s+C , CEK, so that Us2=U2s This also gives Ut=(Ut/2)2 positive hermitian, for all 0
,
.
yields the same relations for -1
.
Proposition 1.9. If condition (s2) holds for K then it also holds
VI.l. Local invariance for H , and we have im(H0+A)(K0+u) dense in H , for all 0
Proof. Let fEH , and (f,(H+A)(K+p)u)=0
,
for all u E CE (O) . Write
this as (f,(K+p)2u) = (f,((K-H)+(u-X))(K+p)u)
,
u E C_(O). Observe
that (H+X)(K+p) is an elliptic, hence hypo-elliptic expression. Hence Thus the distribution solution f of (K+X)(H+A)f=0 is C_(O) .
we can write (f,(K+p)2u)=(((K-H)+(p-a))f,(K+p)u). Due to selfadjointness we may write g=(K-H)f+(u-a)f=(K+p)h, with hEdom(K+u). i.e., f=h, (K+p)f-(H+X)f=(K+p)f, or Conclusion: (f-h,im(K+p)2)=0 ,
(K+p)f=O
,
f=0
,
so that indeed im(H0+a)(K0+u) is dense, q.e.d.
Remark: A proof of prop.l.9 avoiding hypo-ellipticity is possible, using the same arguments as in the proof of IX, prop.5.1. 2. The wave front space.
In this section we will discuss the structure of the symbol The space W is space W of the minimal comparison algebra J0 defined as the maximal ideal space of the commutative CC-algebra .
J0/K(H) (note prop.2.1, below). This space will be seen to be the bundle of unit spheres in the cotangent space of 0 , while the continuous function aA:W*U assigned to a generator A=a (or A=DA) turns out to be the formal principal symbol of the (ze'o-or firstorder) expression D or a Especially, for compact manifolds 0, this result is well .
known (cf. Seeley [Se2], also Gohberg [Gb], Cordes [CI], in case of the (non-compact) Rn). (Note that for a compact 0 the algebra J0, hence the space W, is entirely independent of the operator H or measure dp used, within the general restrictions made. This is an immediate consequence of thm.l.l.)
Proposition 2.1. The algebra J0 contains the compact ideal K(H) and has compact commutators.
Proof. It is sufficient to discuss compactness of commutators, since J0 3 K(H) was seen in V, lemma 1.1. Moreover, we may apply prop.l.2 again, to conclude that, for the commutator of two geneone may assume H replaced rators with compactly supported a or D by H° satisfying M. Then,of course, V,thm.3 .1 and V,thm.3.4 give the desired compactness of commutators, q.e.d. The theorem,below, gives complete control of the wave front ,
space tive
,
as we shall call the maximal ideal space W of the commutaJ0/K(H), henceforth. We use the notation 'symbol'
169
VI.2. The wave front space
170
(or symbol function) for the continuous function over W associated to the coset A+K(H) of an operator A E JO by the Gel'fand map. The space 1 is homeomorphic to the co-sphere bundle
Theorem 2.2. S
S2 of Q. Here S S2 is the collection of all (x,>;) in the cotangent
bundle T*S2 of S2
(i.e. x E =Q , hjk(x)EjEk
E n ) with =1
Moreover, in this interpretation of W the symbol functions of the + p E D# algebra generators a E A# and A = DA , for D = b3 X3j
are explicitly given by the formulas (2.2)
aa(x,l;) = a(x)
ib0(x)i;j
,
E S*S2 = w
,
Proof. First we observe that JO contains the commutative C*-subal-
gebra CO(D) of all continuous functions a(x) with limx,a(x) =0 The maximal ,
which is obtained as closure of the algebra Am C JO
ideal space of CO(D) is equal to S2 itself
.
.
The imbedding CO(D) -
JO generates a corresponding 'associate dual map' i:W - D , which must be surjective (cf. [C1], App.AII,5) (In particular we use .
that a multiplication operator u-au is never compact unless asO so that also an injection CO(D)-JO/K is induced.) Later on we will * recognize i as the bundle projection in S D as defined. ,
Let Slj
,
j=1,2,..., be a locally finite atlas of D. Assume has compact closure
that every S2
QJlos
C :2-i
,
where {c''
} is an
atlas again. Define J. as the closed 2-sided ideal of JO containing all operators with symbol = 0 outside of i-1Q. Proposition 2.3. The ideal Jj is generated by K(H) and all a, DA with a E Am , D E Dm having compact support, contained in Q. Proof. Evidently all the above generators are contained in Jj since for a E C-(D.) we have o a () = a(i(m))=0 , as m E W\1-152 using the inherent properties of the dual map i Similarly, for D with compact support, contained in D. a function X E C;(52.) .
may be found with X = 1 on the support of D , so that DA = XDA aDA = aX.aDA We have aX(m) = X(i(m)) again, so that aDA=O .
for all m with i(m) E D\D.
,
follows as well.
Vice versa, let A E Jj. We choose functions X1E C0 (Q
1=1,
2,..., with 0<X1
sequence of compact sets with U K1 =
52j
.
Such a sequence may be
constructed, since Stj may be regarded as an open subset of In with
VI.2. The wave front space compact closure. Since aA(m) is continuous and =0 outside 1-1QJ we conclude that Ia(1-Xl)A(m)I=(1-Xl(t(m)))IaA(m)I<sup{IoA(m)I:1(m)EOj\Kl}.0, 1--,
in view of the compactness of aQj , and uniform continuity of aA
It follows that the coset mod K of A1=(l-X1)A converges to 0 as 1}- . Or, there exists a sequence {Ci } of compact operators such that IIA-X1A-C1H
-
0
,
1}°°
.
However, it is clear that X1A+C1
E J , since Ai can be approximated by a finite sum C + JP with CEK , and P = D1AD2A.... DMA , D.E Dm , so that 7
X1P = XlD1AX1+1D2A...Xl+N-1DNA + C', C'E K
(2.3)
where Xl+k-1Dk have support in
.
,
we used prop.l.2,
For (2.3)
0.
and Xl = X1X1..1.. .X1+N-1
,
of course. The proof is complete.
Note that a partition of unity 1 =
wi
,
supp wi C Qj , may be
J
used to verify that J = +.J3 .
.
That is, every A E J may be obtai-
ned as a limit (for m-'") of finite sums IjAm = Am, where Am E J.
Again it is clear that each Jj
is a C -algebra without unit
(except in case c._0 compact) containing K, and that J./K has maximal ideal space ffi
1-l0j
C W
With this identification the
functions in CO(ffi7) correspond to the cosets A'
for A E Ji
= {A+ K} of Ji /K
Clearly we get W =U.ffiY due to 0 = U,Oj
.
For the description of W it therefore suffices to discuss ffi0
and the homomorphisms
J.
-* J-1K i
Since O^.D njlos is a
defined by a homeochart, we have global coordinates xl,...,xn C In , mapping fj onto OTC Oi O' An explicit morphism Q^ .
set of generators mod K of Ji then is given by (2.4)
{
xkA ,k=l,...,n ,: f E C'(O4)
}
.
(Here we break with our usual convention to silently identify Q with the open subset 03 . of
In
Proposition 2.4. The space ffi3 .
(2.5)
{(x,E) E 0i X In
, for reason of clarity.
is homeomorphic to ,
1
the set }
,
equipped with its natural topology as a subset of I 2n . Moreover,
171
VI.2. The wave front space
172
under this homeomorphism, we have (2.6) Q _ fi(x)
'
QA,(x'E) = (x)Ej, for Ai _ -i4a
A, J=l,...,n.
xi
I
Proof. For an m E ffi let x = t(m)
k = QA (m) , where
,
E C'(c2.)
k
(real-valued) satisfies fi(x)=l near x=t(m) , 1=1,...,n. We claim that this defines the desired homeomorphism. Indeed, first of all we note that (x,C) is well-defined both are = 1 near if $ and x = t(m) and if B. = -i'Pa then we get A7-B. = (1-X)(Aj-Bj ), :
,
,
x
with a C0--function X
,
equal to 1 near t(m)= x , hence aA (m) 3
aB (m), showing that
Cj
is well defined. Also
= oA *(m), where I
is real, due to
C = -i$3
x
using (1.2) and V, thm.3.4
A + C', C,C' E K,
xi
so that 7l = l . Furthermore, with For u,v E AC' we then get
,
as above we note that h3k¢ E Am
.
(u,A(-a
k)Av) =
x - (u,A(D 3 + q¢3)Av)
ACD
Since
x
.
is dense we conclude that A(D 3+0 3)A =
(2.7) Ai
$3+C ,
C E K
using (1.2), V. lemma 3.3, V, lemma 3.6, and that AHAu=u5 uEACO Taking symbols one obtains
the relation
hJk(x)EiEk = 1
(2.8)
,
as x = t(m)
.
With the above we have defined a map u:ffii (2.9)
M.
_ {(x,E)
:
x E 0i
,
E E
n
-> Mi
, with 1
}
.
We shall prove that u defines a homeomorphism onto Mj. Assuming this to be true let us observe that the proper transformation
law is valid,since they will inherit the co-variant tensor property of the 8
.
x]
To verify that u is 1-1 we recall that the maximal ideal mEffi.
is characterized by its corresponding homomorphism J -' M
,
given
VI.2. The wave front space
by A - aA(m) , A E J J. Let m, m' be such that i(m) = t (ml) = x, and as above. Note that this implies a4)(m)=a4)(m'), and aA (m)=aA (m') J
for all
= 1
E C0-(S2j), regardless whether 4)(x)
J
or not. Since a
,
is a homomorphism, and since the m and Aj listed form a set of generators of J., modulo K , it follows that aA(m) = aA(m') for ,
all A E,J Thus the homomorphisms corresponding to m and m' are identical, which means that m = m' Thus the map u is injective. .
Finally we focus on surjectivity of u
above every x GO
.
It is clear that,
there will be at least one point (x,l;) E u(ffij). j
For the map i maps onto I
so that some mE M with x=t(m) can be found. For this m we then get the l;j by our above construction. To show that every (x,n)
,
, for general n E
construct an automorphism of Jj
n is an image, we
, leaving K invariant, which
carries (x,F) into (x,n) , in form of a coordinate transform. Let M=((m.k)) be a constant real nxn-matrix with M>;=n, M*J2M=J2,
where J=((hJk(x)))1/2 (i.e., JMJ-1 is orthogonal). Such a matrix exists for every i;,n E gn and one even may choose det M = 1 ,
,
so that JMJ-1 is a rotation, and can be smoothly deformed into the identity matrix. Denoting x=x0 , for a moment we design a diffeo-
morphism e:Q.- 52. as follows: Let R(T), 0
for O2e. Define 6(x)=x0+JRT(IJ(x-x0)I)J-1(x-x0), x E Q.. By construction 0 maps J
each (Euclidean) ellipsoid T2=IJ-1(x-x0)I2=h.k(x0)(x3-xj)(xk-xk) J Thereonto itself, and is equal to the identity as lJ(x-x0)1>2e .
fore, for O<e sufficiently small, 0 defines a diffeomorphism of 9j onto itself, and 0 is linear near x0 ty near a12j
.
and equal to the identiIn particular we may extend 0 to a diffeomorphism
a-0 equal to the identity outside S2j
,
.
Using 6 we define the unitary operator 0 (2.10)
:
H -y H by
Ou(x) _ (det(a0/3x))1/2u(0(x))
.
Then A-A°=0-1A0 is a *-automorphism J.+ Jj leaving K(H) invariant. Indeed this automorphism trivially leaves K invariant, and
a =0-1a0 = ao 6-1, where ao 0- E CD(c.) if a E C0_(52j ). Similarly, (2.11)
0-1u = (u/v)oe-1
and, for b E CD(S2j)
Dr = ba
Xr
,
, we get
(2.12) D A u =0-1b3 r®u =(boo-1)((.(v r x
v = (30/ax)1"2
r/v)oe-1)u+(01
Ix
xr
6-1)a lu), x
173
VI.2. The wave front space
which again is an expression with compact support, contained in s0j. Accordingly, the automorphism 0 leaves the generating sets of J. invariant. J
In fact, near x=xO
,
the expression Dr=-i43 r goes into x
Dr° = -i4mr]ax. = mr3Dj , modulo a zero order term. We now must find out how the operator A changes under conjuClearly the differential expression H goes into
ation by 0
.
HA = 01H0 =-K°-1a (2.13)
xi K°
= Koe-1
K°hAjka k + q° x
h'jk=(hlmOjIx10klxm)oe-1
,
0=0-1
Indeed, since 0 is unitary (i.e., (2.14)
(u,HAu) =J
,
(hJk(v(uoO)) S2
du°
= K°dX
,
q°=(Hv/v)oe-l ,
), we get for u E CD(S2),
IXj (v(uoe)) Ixk +IgoOI2v2)du
For the first term we use the formula (2.15)
(v(uoO))
(uoe) + v(u
= v Ixj
Ixj
1oe)e1 IXj
IX
and a partial integration similar to that of 111.(6.7), to get (u,H°u)=J(hlmej
(2.16)
Ix
1
ek
Ixm
((u
k )oe)+(Hv/v)Iuoel2)v2d u
u Ixj
Ix
Using that v2=1ae/ax) we may transform the integral backward. Then the hermitian form may be polarized to to get an expression for (u,H°v)
.
is uniquely determined for all u,v E C , and (2.13) results.
The second order expression ft
by the values of (u,H4v)
,
Clearly H° = H and du = dp° outside a small compact neighbourhood of the point x0 . Accordingly we may apply thm.l.l and {Q,dp°,H°}
the triples {1,dv,H} and generate the same JD . In fact, by definition of A and A° as inverse positive square roots of H and H° = 0-1HO (where now H and H° mean the self-adjoint closures of the minimal operators) it is clear that A° = 0-1A0 ,
We thus get
0-1a0 = ao O-1 = a°
, O-1DAO = D° A°
,
(2.17) D°
= 0-1DO = ( (b1ej
Ix
1)o 0-1)a
xi
+ (Dv/v)o e-1
.
Now consider the associate dual map m + m° = 0-1m0 of the By construction we get i(m) = i(m ) , when-
automorphism A -)- A°
.
174
VI.2. The wave front space ever t(m)=x0
,
since a°(x0) = a(x0) for all a
Therefore for our
.
(x0,E) we have m° E 1-1(x0) as well, and we now will show that u(m ) _ (x0,n)
Indeed, aA (m') = aA e(m) , by the pro-
.
r
r
perties of the associate dual map m-m° Ar°
(2.18)
We have seen that
.
= mr1DIA' + C = mr1A1(A-1A°)+ C
= Dr A°
Accordingly, a Ar e(m) = m
r
1
1a
(A-s lA ) (m)
= nra
we are only left with showing that a
(m)
A
(A-lA
lA ) (m) (A-, =
1
.
.
Thus
. However,from
)
lemma 1.8 we know that the coset U-112 of A-1AA is the inverse square root of U1 =
we get aU
=
H°H-1
=
1
.
Zhr1ArA1 + C , near x0. Thus, over x0
This completes the proof.
3. Differential expressions within reach of the algebra J. In chapter X we shall exploit the information on the symbol space of the general comparison algebra to its full extent. Presently let us discuss differential expressions within reach of the minimal comparison algebra. For the special case of a compact manifold D we then will discuss the implications of our theory, such as existence of a Fredholm and Green inverse, the Sobolev estimates etc.
First let 0 be general, not necessarily compact. Consider a given triple {G,du,H} on G , with its Hilbert space H=L2(Q,dp). Let J0 be the corresponding minimal comparison algebra. Using prop.l.2 (and the remark following it) and prop.l.4 we may modify
the potential q of H such that cdn.(s) holds, without changing J. Moreover, using thm.l.l we also may modify H and du on compact subsets of G (In fact, applying later results of algebra surgery (cf.VIII,thm.3.1) it will follow that J0 remains unchanged as long as the measure du, hence the Hilbert space H is kept the same outside some compact set).In particular, in the following, it is no loss of generality to assume that cdn.(s) holds.
Specifically for a compact manifold there is only one Hilbert space H, and the choice of H and dV will not affect J0 at all. Then we have cdn.(s) for just any H, regardless of the potential q
,
by IV,thm.l.l.
175
VI.3. Reach of the minimal algebra Proposition 3.1. Let L be an N-th order differential expression with C"-coefficients and of compact support on 0. Then the unbounded linear operator LAN with domain A-NC-(Q) admits a continuous extension A to all of the Hilbert space H , and we have AEC=JO In other words, every differential expression with compact support is within reach of the minimal comparison algebra JO Proof. For NO and N=l the statement is trivial, since the opera.
tors aA0 = a and DA , for a multiplier a or a folpde D belong to For the general case it is essential that the generators of C=JO every L of order N and of compact support can be obtained as a finite sum of (not more than N) first order expressions. Indeed, .
refer to some finite open chart cover of the compact set supp L a subordinate partition of unity, and explicit form of L in the local coordinates (3.1)
supp L NS2,, 1=1w, on supp L, L=L3=
a33(x)Dxa on 52j,
JaT
and write (3.2)
L = 1w.L.
_ .
3
3
7
(w.a3)IIn
3 a
a'f
IIa
v=1 p=1
(w.
D
3vp xv
using the auxiliary functions w,vpE C0-(cr.), =1 near supp wjCcr..
Here the expressions wjvpD x
v
may be interpreted as folpdes in Vm
and, for jaI=N, we may multiply the factor (mia3a) into the last
such folpde. Accordingly each summand in (3.2) is a product of at most N folpdes. Trivially we may assume exactly N folpdes in each product. Accordingly is is sufficient to prove the statement for products L =
( 3 . 3 )
......
,
Dj D E Dm .
Notice also that A-NC0(S2) is dense in H for every N, as implied by cdn.(s), as seen in V,6, for example.
Therefore it is sufficient to show that LANu for uEA-NC0, can be written as a sum of terms A1...ARu , with generators ,
Ai of J 0
.
For an induction proof consider K = D1D2 , M = D3...DN. Assuming that an operator CJKA2 , CEL(H) exists, we get
(3.4)
KMANu = CMAN-2u + CI H,M] A Nu ,
uE
A-NCO
(11)
(Confirm in details that CHMANu=KMANu and CMAN-2u=CMHANu.)
176
VI.3. Reach of the minimal algebra Note that the commutator is an expression of order
,
so that
[H,M]AN=[H,M]AN-lA E L(H), by induction hypothesis. Thus we only must prove the statement for N=2 to make the induction complete.
On the other hand, for a twofold product DF , with folpdes D,F, we write (with exactly the same reasoning)
DFA2u = DA2Fu + DA2[H,F]A2u
(3.5)
,
u E A-2C'(O)
Again the first term DA2F = (DA)(F*A)* is a product of generators, while [H,F] is of order <2 , hence a sum of products of at most two folpdes. It follows that the second terms operator is a finite sum of terms (3.6)
(DA)(AD')(F'A)A
E C
,
with folpdes D', F', where F'=1 may occur as well. This completes the proof. Next let us ask for the symbol of AJLAN
,
as in prop.3.1.
We find that A D Dl..... DNAN has symbol (3.7)
aA = RJ=IaA
,
A
= DMA
J
Indeed the second terms in both (3.4) and (3.5) carry an extra power of A which may be written as XA , mod K(H), with XE C- since there always is a expression with compact support in the
same term. Recall that XAEK(H), by III,thm.3.7. Also we get DA-AD E K(H) by V,thm.3.4, so that aDA
= aAD Now we may combine (3.7) and the product representation (3.2) of an expression L and apply thm.2.2 to obtain the symbol
of A D LAN. The proof of prop.3.2 below is a formal verification left to the reader: Proposition 3.2. Let L be as in prop. 3.1. Assume that, in local
coordinates of O'C 0 L has the form (3.8)
a(x)DX a , x E 0'
L =
Then the symbol of the with coordinates x of the chart O'C 2 operator LAN C A E C on the set i-1(0') C W _ M = SCO is .
,
,
explicitly given by the formula (3.9)
a 'AN 'O =
I
ac=N a
E 1-1(D')
Notice that the expression at right of (3.9) is commonly
177
VI.3. Reach of the minimal algebra referred to as the principal part polynomial (or the principal symbol) of the expression L Since we know that oA for AC-C, is a continuous function over the subset S*c of the cotangent space, .
,
it follows that also the right-hand side of (3.9) is independent of the coordinates used, as also easily confirmed directly. In particular it sometimes is more convenient to use tensor notation instead of multi-index notation: We may write a-il...3NEj aa(x)i;a =
(3.10)
...Ej
Ja=N
N
1
with summation convention and a unique symmetric contravariant tensor a- of N variable indices. Remark 3.3. Observe that the expression L is elliptic over c'ai if and only if the symbol of A J LAN (with respect to the algebra J0 does never vanish on the part of the wave front space t-1(0') .
From now on consider the case of a compact manifold Q only. Then the algebra J0 contains the identity operator of H and its wave front space W is compact. As a consequence of prop.3.1 every differential operator on 0 with C"-coefficients is within reach of the unique (minimal) comparison algebra C=JO. Then let ADLOAN and Z=AA-N be the operator of V,def.6.1 and realization of V,def. 6.2, respectively.
Theorem 3.4. The operator A D LAN has an inverse modulo the class F(H) C L(H) of all operators of finite rank if and only if L is In details, there exists an operator B E L(H) with elliptic on 0 .
(3.11)
AB =
1+F
, BA =
1+G ,
F, G E F(H)
,
if and only if L is uniformly elliptic. Moreover, we then get (3.12)
dim ker A < -
,
dim H/im A < -
,
and B may be chosen as operator in JO such that -F and -G coincide with the orthogonal projections onto the finite dimensional spaces * ker A and ker A , respectively. On the other hand, we instead may choose BEJ0 such that for an arbitrarly given e>0 we get (3.13) IBH
II
A
a (x)Ea in local coord's. +e, a (x,E)= A a =N a
Proof. For a bounded operator A an inverse modulo F(H) exists if and only if A has an inverse modulo K(H). Indeed, since F(H)CK(H), we only must show that (3.11) for a BEL(H) with F,GEK(H) instead
178
VI.3. Reach of the minimal algebra
of F(H) implies (3.11) for some other B=C. By Rellich's criterion we get orthogonal projections P
(3.14)
Q with finite rank such that
,
IIF(1-P) II < 1/2 , IIG(l-Q) II
< 1/2
Then we may write AB = (1+F(1-P)) + FP
(3.15)
,
BA = (1+0(1-Q)) + GQ
where the first terms at right are invertible (an inverse is given by the norm convergent infinite series
j=0
J_(£(P-1))3
,
in
the first case, for example). Thus we have (3.16)
A(B(1+F(1-P))-1) = 1 + F'
,
F'
= FP(1+F(1-P))-1 E F(H),
= B(1+F(1-P))-1 is a right inverse mod F of A Similarly we use the second (3.15) to obtain a left inverse. It
showing that B'
.
is known that existence of both a left and a right inverse implies that both, left and right inverse are inverses, so that indeed A is invertible mod F This shows that we only must discuss invertibility of A D .
LAN mod K(H) . But K(H) is a closed 2-sided *-ideal of the C*algebra L(H) . Moreover, C = JD is a C*_subalgebra of L(H) containing K(H)
. It therefore follows that L' = L(H)/K(H) and C" C/K(H) are C*-algebras, and that C"C L', using a well known result
on C-algebras (cf. [RlI or any general book on C-algebras). * Moreover, using another standard result on C -algebras, C" is inverse closed in L". That is, A"= A+K(H) E L"is invertible in C" if and only if it is invertible in L" (cf.[C11,AII,L.7.15).
It follows that A D LAN is invertible mod F(H) if and only if its coset A A+K(H) has an inverse in CV . Here we finally use the fact that the commutator ideal E of C=J0 equals K(H), and that C" is isometrically isomorphic to C(M). Since the cosphere *
bundle ID=S 0 is compact, a function aEC(W) is invertible if and only if it never vanishes. Hence A D LAN is invertible mod F(H)
if and only if L is elliptic, by remark 3.3 above. This completes the proof of the first statement.
Observe that ker A C ker(l+G) and ker A C ker(l+F ), where both right hand sides are finite dimensional. Hence dim ker A <-, dim ker A
(l+G)uk+Bv .
.
Notice that in A is closed: For Auk+v , k+. get
If wk is the component of uk orthogonal to ker(l+G)
179
VI.3. Reach of the minimal algebra then we must have wk-+w and BAw = By . Hence v=Aw + z Then let pk=uk-wk (Ecer(1+G). We get Apk;z
.
,
zEker B
If qk denotes the
component of pk othogonal to ker A C ker(l+G) then we also have Aqk+z , hence qk->q
, since A is invertible in the finite dimensional space ker(l+G)n(ker A)1 Then v=A(w+q), so that im A is closed
hence im A = (ker As*)-L. Thus dim(H/im A) = dim ker A*
have (3.12) confirmed.
It then is clear that A0 = Aim A constitutes an invertible map im A ; im A. Let its inverse be called B and extend 0
B0:im A ; in A to an operator B':H;H by setting B'=0 on ker A (im A)'L. It follows that B'EL(H) and that AB' abd B'A are the orthogonal projections onto in A and im A , respectively. Thus 1-AB' and 1-B'A project onto ker A* and ker A, resp. Since B' is an inverse mod K(H) of AEJ0 we conclude that also B'EJ0, using again that inverses in Lv of elements in C' must be in C' .
Finally it is found that '1/2' in (3.14) may be replaced by Then the conclusion leading to any arbitrary positive number 6 .
(3.16) gives an inverse mod F(H), called B", for a moment such that IIB"II
(3.17)
= inf{IIB+KII: KEK(H)} = y
0
A Since B+K is a K-inverse whenever B is, we can find a K-inverse B with IIBf
This completes the proof of thm.3.4. An operator AE L(H) satisfying (3.12) is called a Fredholm operator. An inverse B of A mod F is called a Fredholm inverse of A . For more details cf.X,l The unique Fredholm inverse satis.
fying (3.11) with the orthogonal projections onto ker A and ker A will be called the distinguished Fredholm inverse. Next it will be our intention to show that the distinguished Fredholm inverse has the properties of a "Green inverse", intimately related to the generalized Green's kernel of the realization Z .
Proposition 3.5. Every Cs-differential expression of order
, whether compact or not, is H-compatible in the sense of V,def.
6.4.
The proof is evident, since indeed the successive commutators L0=L , L1=[H,L] . ..... Lj+1=[H,Lj] all are compactly supported and of order
.
180
VI.3. Reach of the minimal algebra
181
For an H-compatible expression of order N we now may apply V,thm.6.8 and its corollaries. Proposition 3.6.For ADL0AN of prop.3.1 we have ASAA-s
(3.18)
- A E K(H)
ASAA_S
E L(H), and
.
In other words, for every < N-th order differential expression L
with Cm-coefficients on a compact manifold 0 the operator ADLOAN is of order 0 and is K-invariant under H-conjugation. The proof is immediate, in view of the above mentionned thm. since all terms at right of V,(6.11) are compact: Their differen-
tial expressions are compactly supported and there is an excess of A-powers. Theorem 3.7. For an elliptic differential expression L on a compact manifold 0 the realization Z of V,(6.2) allows a 'distinguished Green inverse' G
, defined as an operator GEO(-N) (in the
sense of 1,6) such that (3.19) ZG = l-Q Z G = 1-P with the H0-orthogonal projections P and Q onto ker Z and ker Z* ,
respectively. We have ker Z
,
C H , and P, Q are conti-
ker Z
nuous operators H_.-)-H. of finite rank.
Proof. Using thm.3.4, if L is elliptic, then the distinguished Fredholm inverse B of ADL0AN is well defined. This implies that the operator A=A0 is Fredholm. Moreover, from prop.3.6 above we know that AEO(0) and that A is K-invariant under H-conjugation. Thus I,thm.6.6 may be applied and we get ker A C Hm, ker A C H_.
Accordingly the distinguished Fredholm inverse B gives a distinthen we get LG=AB=1-Q, guished Green inverse of A . Let G=ANB , with P of thm.3.4. One as stated. Also GL=ANBAA-N = .
1-ANPA-N
then confirms that P' onto ker L
ANPA-N
=
is the H 0 -orthogonal projection
= ker(A AN). In particular (A AN)0
is Fredholm and
K-invariant under H-conjugation, so that ker L*CH. again. Q.E.D. 4. The Sobolev estimates for elliptic expressions on compact 0
.
In this section we again consider a compact manifold under any H and dp whatsoever. We have condition (s) and therefore the HS-chain of Sobolev spaces as in sec.3 above. First we note: Lemma 4.1. For a compact manifold we have H. = C_(c2)
and we get Lu = Zu for all uE H_ and every expression L and the realization ,
VI.4. Sobolev estimates Z of V, def.6.1.
This follows at once from the well known Sobolev imbedd* lemma: For s>n/2+k the space Hs is (compactly) imbedded in C k (S2) (cf. IX, prop.l.5). Here we shall discuss a somewhat weaker result also implying lemma 4.1. (For a Sobolev imbedding of LP into Lq cf. IV,prop.4.1.)
Lemma 4.2. For s>Ln/2]+k+l the space Hs has a natural imbedding defined as follows: The class of squared integrable functions (with any two coinciding outside a into the space Ck(12)
,
set of measure 0) contains precisely one function in Ck(S2) Moreover, there exists a constant cs k independent of u such that .
(4.1)
p up C
k
< cs'k
(c)
,
II uU s
for all u E HS , sj n/21 +k+l
.
Proof. We use Taylor's formula with integral remainder: A function q(t) of one variable t, infinite differentiable on I , and with derivatives of all orders (including 0) vanishing at 0 satisfies t It
(4.2)
_
(t-T)N-1/(N-1)! (N)(T)dT
0
with the N-th derivative 4
E
,
since all terms of the Taylor series at 0 vanish and only the remainder in integral form remains. We apply (4.2) to the function 4(t) = u(x0+(l-t)z0)X((l-t)z0) ,
where u EC_ (In) is arbitrary and X E C,(]kn), X>0, denotes a cut-off
function at 0; we have X=l near 0, X=0 for Ixl>1/2. Also, x0,z0 E 2
1n are arbitrary, but IzOI=IzO
=1
.
N>0 denotes an integer to be
determined. With a change of integration variable 1-T-Tt we get 1
(4.3)
dt tN-1/(N-1)!dN/dtN(X(tz0)u(x0+tz0))
u(x0) = 0
Now we estimate, writing 1
(4.4)
Iu(x0)I
0
where
as X and
Iu(6)(x0+tz0)I
dttN-1l,(tz0)
< c
I0I
denotes another cut-off functions with the same properties IX(0)I
for all x and Iai
to the variable z0 , and the surface measure dS . Using that dx = rn-1drdS , with r=lx-x01 , we get (4.5)
Iu(x0)I
<
Iu(9)(x)I
cJdxlx-x0IN-n,p(x-x0)
10l
182
VI.4. Sobolev estimates Now we set N=[n/2]+1 and use Schwarz' inequality, noting that Ix-x012(N-n)
dx = c0<- , with this choice of N. Thus,
11x-x 0I<1
(4.6)
Iu(x0) I < c
I0I
where c is independent of x0
I2dx}1/2, x0E,,n
{J .
We now may implant (4.6) onto a compact manifold P. Choosing a finite atlas such that each chart S2' is contained in a chart S2" in such a way that dist(S2',a52")>2 we find that
(4.7)
Iu(x0) I < c INLj,x uN
, X0 E 52 ,
u E C°°(R)
.
0
where the sum is finite and the differential operators L.
of
JSX0
order
ANN < c
IIL.
c independent of x0 and j
,
0
(Also their number is independent of x0 ). The verification of the latter fact requires some calculation involving decomposition of Ljxo into sums of products of folpdes, similar as in (3.2), left to the reader. Combining (4.7) and (4.8) we get
(4.9)
(lull
for all u E C°°(c)
,
IIuNIN
.
L°°(S2)
For an arbitrary u E HN , N=[n/2]+1 u E C00(c) such that Ilu-uk0 k
->.
0
,
k+co
.
there exists a sequence We then conclude uniform ,
convergence of uk (i.e., convergence in C(3) ), applying (4.9) to the difference u.-uk E C°°
.
Hence there exists a limit of uk in
C(S2). It is clear that the limits in C(S2) and in H must coincide
whenever they both exists. Thus indeed uEC(1)
,
i.e., HNCC(S2)
Similarly, for N=[n/2]+l+k we may apply (4.7) to wu(a) IaI
a neighbourhood of the complement of the local chart used. Clearly the expressions Lj,x w3X are of order
may be replaced by (4.10)
w3XAN+kN
IIL.
< c
,
c independent of x0 and j
0
This implies the result for general k. Q.E.D.
Corollary 4.3. If 0 is not necessarily compact, but the triple
183
VI.4. Sobolev estimates
{c,dp,H} satisfies cdn.(s), then we still have the imbedding
, as s>[n/2]+k+l , although the estimate (4.1) holds only locally, with gu II and c s,k replaced by gu B , and Hs + Ck(S2)
Ck(S2' )
Ck(S2)
es,k,Sl,
, respectively, for each S2'c 0 with compact closure.
All above proofs may be carried over to prove cor.4.3. A simple consequence of thm.3.7 is the result below. Theorem 4.3 (The Sobolev estimate). Let the expression L on S be elliptic. For every s E R there exists a constant c s such that (4.11)
flullN+s< cs(fLull s+ Bugs)
,
for all uEH.
Proof. Using thm.3.7 we get (4.12)
1 (1-P)vll
s
=
BBAv1I
Here we may introduce v=A-Nu
,
s
<
013 11
u E C0
s
BAvll
.
s,
vEHm.
It follows that
BuNN+s = RA-Null s< BBBsHZuHs+ IIPA-Nulls.
(4.13)
Finally we use the fact that P E 0(-°) , so that Therefore (4.11) follows from lemma 4.1, q.e.d.
BPA-Nlis<...
Remark 4.5. Note that the term csllulls at right of (4.11) may be
replaced by cs tfu't
,
for just any tEl. Indeed, the expression
PA_N of (4.13)'still belongs to 0(-oo)
,
so that fPA-NB5,t<m for
any pair s,tE 1 Remark 4.6. It is possible to prove the following stronger result For every a>O there exists a constant c(e) such that for an elliptic expression L of order N we have (4.14)
Bugs+N < BQA lg
)HZugs + c(e)Bu1s
uEH
L-(Q)+6
To prove this we apply thm.3.7 to the self-adjoint elliptic expression L L It is found that the bounded self-adjoint opera.
A0A0 (in H0) is Fredholm and admits compact conjugation. Similarly all the operators ADA6-A = AN (L L-AHN),AN , as long as X<m2 = L
(inf{laAI:(x,i;)EW})2. We conclude from there that A A 0 has only 0 discrete spectrum below mL . Moreover, all eigen functions to
eigenvalues less than mL are in H. . From this fact one concludes the existence of a bounded self-adjoint B0 with norm BBlo<mL'+e such that
184
VI.4. Sobolev estimates A0AB = BA0AO = 1-R ,
(4.15)
with a finite dimensional orthogonal projection R onto a subspace of H. , i.e., the span of all eigen spaces to eigenvalues < mL/(l+emL)
.
In particular B is positive self-adjoint. Now rela-
tion (4.15) may be used instead of the earlier BA=1-P to derive the improved estimate for s=O
.
Similarly for general s, using the
operator As( s) As , with the Hilbert space adjoint A(s) of A in the space Hs
.
185
CHAPTER 7. THE SECONDARY SYMBOL SPACE.
In the present chapter we turn our attention to the set it\W of the symbol space it of a general comparison algebra, consi-
sting of all points not contained in the wave front space. As pointed out before, the wave front space W (i.e. the symbol space of the minimal comparison algebra) is found naturally imbedded for every comparison
as an open subset of the symbol space M
,
algebra C meeting our general assumptions. This now will be discussed in detail in sec.l, below. If the manifold Q is compact, so that the minimal comparison algebra is the only comparison algebra, then we have it=W. In that case every differential expression on 0 is within comparison reach of the algebra J6 , as was shown in VI,3. For an elliptic expression L then the realization Z-X=(A-AAN)A-N
Z of V,def.6.2 has the property that is Fredholm for every XET. In other words, there is no essential spectrum, in the sense of [CHe1]). This fact is in direct relation to the fact that the set it\W is void. Essentially it follows that the latter is the origin of the essential spectrum of elliptic operators within reach of a compaset, or, rather its interior its = it\(Wclos)
rison algebra C, (assuming that E =K(H))
,
.
Accordingly we now engage in a detailed discussion of the set its , which is called the secondary symbol space of the algebra C In this respect it may first be noticed that so far, .
as long as only the minimal comparison algebra was under discussion,
we did not require any geometrical assumptions on the space 0 under its Riemannian metric induced by H whatsoever, except for the structure. On the other hand, for the investigation of the secondary symbol we will have to impose assumptions at infinity. We choose to work with a set of analytically formulated conditions (cf. sec's 2,3,4 below) but will make little effort to translate these conditions into geometrical properties. Notice that an extensive study of the space its for a variety
VII.O. Introduction of comparison algebras over
n was done in [CHe1] and [C1],III,IV.
It appears natural from these results to seek the space ffis as a
subset of the boundary of a suitable compactification of the
cotangent space T 0 of S2 To be more specific, we first note that the multiplications a(x), for aEA# , alone generate a Csubalgebra C # of C
,
isome-
trically isomorphic to the function subalgebra of CB(S2) generated by A#
.
The maximal ideal space M# of C# will be a compactifica-
tion of the manifold Q
.
Actually, this is the smallest compacti-
fication of S2 onto which every function of A# allows a continuous extension, as well known (cf.[Ke1]). If a E EOCA# then Xa E EnJO=K(H), for X E C0(2), hence a=0 as a C*-subalgebra of the commutative C A algebra C/E Then the associate dual map of the injection CA# + CIE defines a continuous surjective map is ffi } CA#
we also find C
Thus
.
already used in ch.6 for the minimal comparison algebra JO Now one may assign a formal algebra symbol TA to each of the generators of C T
DA
by setting Ta (x,E) = a(x)
,
(x,C) _ (Wb0()E.+p(x))/(hOk(x)
E D#
.
jk
,
for a E A#
+q(x)) 1/2, for D=-ib08
xj +p
Clearly the formal symbols are bounded complex-valued
Cm-functions, defined over the cotangent bundle T*S2 of 0
.
Thus
again, a smallest compactification & S2 of TtO exists onto which
all formal algebra symbols can be continuously extended. Again *
*
*
1 0 is the maximal ideal space of the C -subalgebra of CB(T Q) generated by the formal algebra symbols. Again one obtains a conas associate dual map of the A since CA# also is a subalgebra of the
tinuous surjective map 7:1 S2 *
injection CA# - C(& O)
,
-
M #
,
formal symbol algebra.
We are interested in the boundary 31 S1 compactification 152
.
=
P*S2 \ T0 of the
Notice that i-1 (0) is the bundle of cosphe-
res of infinite radius, recognized as homeomorphic to the wave front space W in VI,2 and VII,1
.
In case of the algebras over In mentioned it proved that the above homeomorphism extends to a homeomorphism between all On the other hand, in case of certain of ffi and all of 8PTh .
algebras over the cylinder Slxg one obtains no secondary symbol
187
VII.O. Introduction
space at all, although the space a&*SZ\W is nonvoid. In each of these two cases, on the other hand, I may be found (naturally For a third example, identified with) a compact subset of aF S2 however, - i.e., the half space algebra of [C1],V - this no longer .
is true: certain one-dimensional "filaments" no longer are subsets
of aP 0 In [CS],VI a result, identifying the entire M with a compact subset of 9&*0 , wa's discussed under a number of special assump-
tions, and for a compact commutator algebra. The proof was simplified in [CM2] . McOwen [MO] , [Mi] extended this result to a larger class of Laplace comparison triples and algebras. Here we adopt some principles of McOwen's discussion by imposing a set of conditions (m,) , naturally satisfied in McOwen's case. In essence, we request that the pointwise 'inner product'
{D,F} = hjkVjck
(0.1)
defined for folpde's D=b3ax.+p defined over aMA# = MA#\S2
.
,
+ pr/q
, xE52,
F=cJaxi +r E D# , also is well-
In other words, the function {D,F)(x)
of (0.1) must be in C # In addition, certain separation condiA tions are required. One finds that such a condition has the effect .
that, over each point x0EM # , the set R-1(x0) retains the vector A space structure of the cotangent space, or that of a lower dimensional space. This, in essence, makes the indicated result possible: IN is found as a compact subset of the space 31*0 In sec.2 we specify cdn's (mj) , and first show a local
result, identifying t-l(x0)
,
for x0EM
#
, with a compact subset
. In sec.3 we add more conditions and then find 1-1(x0) as a compact subset of a hemisphere. This also allows a proof of
of a Tk
* IN + a& S2
(sec.3)
.
On the other hand, in sec.4 it is found that
the compactification aE*S2 is not unique over aM #
,
insofar, as
A the expression (0.1) may be replaced by the corresponding one for any subextending triple {O,du",H"} and then the formal symbol by the corresponding expression involving the coefficients of ,
H" . Then one obtains an imbedding Bi +
are right.
(a&*Q)" if the conditions
188
VII.l. General symbol space
All of this is discussed for a variety of examples, espe. In particular
cially those of V,4 and V,5(cf. sec.2 and sec.4) some examples of Sohrab [S1]
,
[S2]
,
[S3]
, are discussed and
partly generalized.
In general we do not assume that the algebra C has compact
commutators. 1. The symbol space of a general comparison algebra.
In this section we will discuss a general comparison algebra C = C(A#,D#) of a triple {G,du,H} with generating classes A#, D# satisfying cdn.'s (a0),(a1), and (d0) of V,1. Again no essential self-adjointness of H0 will be required. In general we cannot expect the commutators of C to be compact. For example let 0 = fin, dy=dx , H=1-A , with the Euclidean Laplace operator A
Let us choose A# as the class of a=b+c E C°°(ln) with c E CO°°(jn) _ .
{a E C"(In): a(a)(x)=o(1) for all a} , and with b an n-fold periodic function with periods ej=(d class of folpde's bJ3
j+p
)
Let D# be the
.
jl 1=1,...,n , with coefficients
b], p E A#. Then
x
it is easily seen (along the proof of V,prop.5.5, discussing the 1-dimensional case) that not all commutators of operators in C are compact.
Generally we denote by E the smallest norm closed 2-sided ideal of C containing all commutators of elements in C .
Lemma 1.1. We have [A,B] E K(H) whenever A E C , B E JO Proof. It is sufficient to give the proof for the case that A and .
B are generators of their corresponding algebras. For a E Am D E D# let X E m CD be such that XD = D . Then aDA = aXDA =_ DAaX =_ XDAa (mod K) (we used V, thm.3.1 twice here). For so that again [a,DA] E K
D E D# we get [a,DA] E K by V,thm 3.1.For a E A#
DED#
m
,
F E D# we get
,
DAFAv = ADFAv + Cv = AFDAv +C'v = AFXDAv
+ C'v = FFXDAv + C"v = FAXDAv + C"'v = MDAv + C"'v
, for all
vEAC', where we used V,lemma 3.6, V,lemma 3.7, V,lemma 3.6, and V,thm.3.1, in that order. Thus we also get [DAFFA] compact. Note that not even condition (w) is required, for each commutation, in view of VI,(1.2), since we always were commuting expressions or functions with compact support. This completes the proof.
Lemma 1.2. Let XjE C_(G), 0<Xj<1, and Xj=1 in DJ U...UGj, for j =1,
189
VII.l. General symbol space 2,..., referring to a locally finite atlas {sL } with properties as
in VI,2 again. Then an operator A E C is in the minimal algebra J0 if and only if there exists a sequence of operators C1E K with A = lim1;.(X1AX1+C1) in operator norm convergence.
Proof. If A E JO then its symbol vA (with respect to JO/K) is a function in CO(W) , so that we must have liml..(1-Xl)6A = 0 in uniform convergence over W . This directly implies (1.1). Vice versa, for any A E C we have X1AX1 E J0. Indeed, it is sufficient to show this for the generators of C , in view of the and the commutator property of X1 =
relation
lemma 1.1. But for a E A we trivially get aX12E JO . Also for A DA , D E D# we get X1DAX1 E JO as well, q.e.d.
Theorem 1.3. We have
E 0 JO = K(H)
.
Proof. Note that T = E n J0 is a closed 2-sided ideal of the algebra JO , and we observed before that T contains nontrivial compact operators, while JO was seen to be irreducible in V,1. It follows that T J K, by V,lemma 1.1, so that we only must show that TCK. Let A E T. There exists a sequence A.-*A such that A. is a finite sum of terms X=B[C,D]E , where B,C,D,E E C Also, for a sequence of functions X1 with the properties of lemma 1.2. we must have .
lim1.(X1AX1+C1)= A in norm convergence, by lemma 1.2. Combining these two facts, we get (1.2)
1 uA-C1-J7_iXrXljXrf
Ni < 11A-C1-XrAXrIi+gXr(A-l=1X1.)Xrp < e
as r and j are sufficiently large. This means that we only must show that terms of the form XXX = XB(C,D]EX are compact, where B,C,D,E E C , X E CO , due to closedness of K in norm convergence. Applying lemma 1.1 we get XXX =_ BX[C,D]XE (mod K). Hence we only must show that X[P,Q]X E K for every P,Q E C In turn then .
this may be reduced to the same statement where now P and Q are generators of C , again using lemma 1.2. Let again w E CO , wX=X
For a E A, D,F E D
we have X[ a,DA] X = X [ wa,DA] X E K
, again by
lemma 1.2. Finally, X[ DA,FA] X = X[ wDA,FAw] X , where FAw = FwA + C, by V , (3.12) again. Accordingly, X[ DA , FA] X =_ X[ wDA , FWA] X E K, which completes the proof of thm.1.3. The significance of thm.1.3 is realized when we now start discussing the symbol space BI of the algebra C . We noted before
190
VII.l. General symbol space that CIE =
C"
,
as a commutative CC-algebra with unit, is isome-
trically isomorphic to an algebra C(BI) of continuous complexvalued functions over a compact Hausdorff space Bt, by the Gel'fand
Naimark theorem. The space BI is explicitly given as the collection Its topology is the weak topology of all maximal ideals of C" .
of the function class induced by the elements of C" The space and M was called the symbol space of the comparison algebra C .
,
the continuous function corresponding to A E C the symbol aA of A. We will recognize the wave front space 01 of the corresponding minimal algebra as an open subset of BI Consider the injection J0+ C . The commutator ideals of J0 .
and C are K(H) = K and E , respectively, by V, lemma 1.1. We have K C E , by thm.1.3 , hence a map JO~+ C" is induced
by A" = A+K + A+E ,(i.e. we have the inclusion A+K C A+E, so that the map is independent of A(=-A' , hence well defined). Now by thm.1.3 this map is an injection. For if A,B E J0 and A+K , B+K both are subsets of A+E , then B-A E J0fE = K so that A~ = By ,
This means that inclusion of cosets defines a natural injection J0 -. C' , which is a *-isomorphism, and will be considered as an identification of J0" with a subalgebra of C"
Proposition 1.4. J0 is a closed two-sided ideal of C
.
Correspon-
.
dingly J0with above identification, is a closed two-sided ideal of C°
.
Proof. We have A E C in J0 if and only if A = liml
(X1AX1+C1)
in norm convergence, with some C1E K , and with the sequence X1 of For B E C we get B(X1AX1+C1) = X1BAX1+Cl' Cl'E K by lemma 1.1. Therefore BA = liml(X1BAX1+C1*) E J0 , q.e.d. For a closed ideal 1 of a commutative C -algebra A one defines the hull h(I) as the collection of all maximal ideals containing I. The hull is a subset of the maximal ideal space M of A and a closed subset, since we clearly have lemma 1.2
.
,
h(I)
(1{
xEI
{m E M:4 (m)=0) x
with the symbol homomorphism 4:A + C(.M)
,
so that h(1) appears as
an intersection of closed subsets of M
The complement R = M\h(1) is an open (locally compact) subset of M
and it follows that the restriction 1I of the symbol homomorphism m to I constitutes an isometric isomorphism I + C(M), ,
separating the points of R, but vanishing outside of R
.
One thus
191
VII.l. General symbol space
may identify the functions x with their restrictions to R, arriving at an isometric isomorphism from I onto CO(R) which must be the symbol homomorphism of the commutative C*-algebra I . In other words, the space R may be regarded as the maximal and the symbol homomorphism of I appears as ideal space of I the restriction of the symbol homomorphism of A to R and I In our case of I = JO° C C" the maximal ideal space of JC was the wave front space W , so that we have a natural identification of l with an open subset of the symbol space I of C On the other hand we again may consider the C -function algebra C # obtained as norm closure of the algebra generated by A# A (either in Lw-norm of functions over 0 or in operator norm of ,
.
L(H)
, which give the same, (cf.[C 1 ], III,(8.14)). In the case
of J0 (i.e., A#=Am) we had C #=CO(O). For general A# the class A the algebra of all bounded C # will be a subalgebra of CB(O) A ,
continuous functions over 0
.
The maximal ideal space M
will be a certain compactification of 0
A# of C A#
(For more details about As in VI,l we get such compactifications cf.[C1],IV,lemma 1.5) an injection CA#- C. Using that CA# n K = {0), we also get a cor.
.
CIE = C(I) That injection again defiA #+ nes a corresponding associate dual map t:N -' M # , and i again A is surjective, similarly as in VI,1 in the special case of J0
responding injection C
.
Now we have the following result. Theorem 1.5. The symbol space El of a comparison algebra C contains the wave front space W of the corresponding minimal algebra J0 as
an open subset. Moreover with the map t:1+M # defined above we get A W = i-1(a)
(1.3)
.
Proof. The only thing left to prove is (1.3)
.
Note that we have
m E w if and only if there exists an operator A E J0 with aA(m)#0.
Using lemma 1.2 once more we get A = liml(X1AX1+C1) , hence (1.4)
aA(m) = liml
(Xl(1(m))2aA(m)) = a
But then we must have i(m) E 0
,
since the compactly supported
192
VII.2. Secondary symbol space functions xl vanish at each point of3M # . Hence we get A
Gict-1(SZ).
Vice versa, for m E 1-1(S2) let A E C be an operator with aA(m)#0
and let the function x E
be 1 at x0= i(m) E 0 . Then lemma
1.2 implies that XAX E J0 while we get axAx(m)=aA(m)x(t(m)) aA(m)# 0
, hence m E W . This completes the proof.
Note that the wave front space was completely characterized as the cosphere bundle S*0 of the manifold 0 (cf. VI, thm.2.2) We now denote the closure of 91 in M by ffip , and call ffip the
principal symbol space of the comparison algebra C
.
The comple-
ment Ms = ffi\ffip will be called the secondary symbol space of C
.
Our next effort will be aimed at the structure of the secondary symbol space. Sec. 2, below, will give some answers, in that respect.
2. The space M\W , and some examples.
In this section we will look at the secondary symbol space ffis. We will obtain some information about ffis , but under certain
restrictions on 0 and the triple at infinity, and not as complete as for the wave front space. Again no assumption on essential self-adjointness of HO will be made.
In addition to cdns.(a0),(a1),(d0) of V,1 the classes A# and D# now will be assumed to satisfy the following.
Condition (ml): For each pair D=bJ3xi+p , F=c
j3xj+r of folpdets
in D# the bounded C-(0)-function {D,F} defined by (2.1)
{D,F} = hjkFjck + Fr/q , x E S2
,
is a function in C #, the norm closed subalgebra generated by A#. A Condition (m2): A# and D# are complex vector spaces, and D# is a left module of A#.(i.e., the pointwise product aD belongs to D#,
for all a E A# and D E D#)
.
A condition like (m1) was first used by McOwen [M1] in his thesis written with the author. Also the approach, below, to use (2.1) as a local inner product is due to McOwen. For any given D,FED# the function {D,F}:O
admits a conti-
193
VII.2. Secondary symbol space
nuous extension to the space M # which is a certain compactificaA tion of 0
.
Then let us focus on a given fixed point x E M #, fin0 A
ite or infinite. The value Q(D,F)={D,F}(x0) at x0 of the function as D,F range over D# , defines a positive semi-definite sesquilinear form on D# , given explicitly by (2.1) at finite {D,F)
,
points, but only by continuity at points x0E M #\D A D# x
(2.2)
=
{DED#
The set
.
Q(D,D)=0}
0
defines a linear subspace of D# Indeed, one obtains Schwarz' 12< Q(D,D)Q(F,F) for the semi-definite form Q, inequality IQ(D,F) .
which implies that Q(D,F)=O whenever either Q(D,D)=0 or Q(F,F)=0. Then we get Q(cD+c'F,cD+c'F) = 0 whenever Q(D,D)=Q(F,F)=0
.
It then is easily seen that Q induces an inner product in Clearly, for finite x, the space the quotient space D#/D# = S x .
0
Sx is just the local Mn+l spanned by (b1(x),...,bn(x),p(x)) , with inner product given by (2.1). Proposition 2.1. The space Sx is finite dimensional for x E M #, A and we have 6x=dim S < n+l (n= dimension of Q). Moreover, 6x is a lower semi-continuous function of x, defined on 2MA#=MA#\0 (That is, lim inf x}x
6
0
x > 6x
0
.)
Proof. If D1,...,D6 E D# is (a set of representatives for) an orthonormal base of S then {D.,Dl}(x0) = 6j1 j,1=1,...,6=6x0 x
,
0
But the 6x6-matrix V=(({Dj,D1}(x))) has continuous entries over the space M # , hence will stay invertible in some neighbourhood A of x0 In fact, a nonsingular 6x6-matrix W(x)=((w )) may be ji .
found such that W*VW = ((6jl))
.
Then Vj=Jwj1D1
normal set of Sx , so that dim Sx=6x? dim Sx
= 6x 0
lim infx3x 6x > 6x 0
defines an ortho.
Thus we get
0
, or 6x is lower semi-continuous. In particu0
lar, due to the basic properties of A# and D# of V,1 it is clear that 6x = n+l for all x E 0 , since A# and D# contain all C--functions and folpde's.
194
VII.2. Secondary symbol space
We now arrive at the following characterization of the secondary symbol space (rather, of the space ffi\W of all points of
M over infinite x). Assume given an orthonormal base D1,...,D6 of Sx 0
at x0E M #, and a set of Sx folpde's DjE D# with {Dj,D1}=6j1. A 0
For a maximal ideal m E 1-1(x0) let us define n(m) _ (aD A(m),...,aD A(m)) = (nl(m),...,na(m))
(2.3)
1
.
S
Theorem 2.2. The map n: 1-1(x0) i Md
,
is continuous and
6=6X 0
injective from 1-1(x0) C i\W onto a compact subset of fs . Moreover, for M E 1-1(x0) we have the symbol of the generators defined by aa(m)=a(x0)
,
aDJA(m)=nj(m) , a(DJA)*(m) = Fj(m)
,
(2.4)
*(m) = 0 for all D E D# 0 (DA) Proof. The continuity of n.(m)=aDA (m) is evident, since every
aDA(m) =
.
Cr
J
J
con-
aA:ffi->M, for fixed AEC, is continuous over ffi, hence
tinuous. The injectivity of n is proven just like Herman's lemma in (C11,IV,2 Suppose for two maximal ideals m,m- E 1-1(x0) we have n(m)=n(m-) This means that aa(m)=aa(m-) for all a E A# .
.
(due to aa(m) = a(i(m)) = a(x0) = a(i(m")) = aa(m") by (2.3) also aDJA(m) = nj(m) = nj(m-) = ODJA(m"') ,
, and
)
Now let us
.
first assume the following. Proposition 2.3. We have (2.5)
aDA(m) = 0 for all D E D# x
and m E 1-1(x0) 0
Assuming prop.2.3 it follows that aDA(m)=aDA(n
)
=
0 for all
DEDxu. Hence we conclude that the symbols of all the generators of C coincide at m and m"' (including the symbols of the adjoints (DA)
, which are just the complex conjugates of aDA
In particular we find formula (2.4) confirmed. Also, the two homomorphisms CIE - T
defined by A-aA(m)
and A-aA(m"')
,
respectively, for A E C , must coincide, since they coincide for the generators. But these homomorphisms determine their correspon-
ding maximal ideals uniquely, so that we must have m = m- , proving the theorem.
195
VII.2. Secondary symbol space
Proof of prop.2.3. For D E D#
we have {D,D}(x0) =
0
.
Thus there
0
exists a neighbourhood Ne of x0 (in M #) such that I{D,D}(x)I A e2/4, x ENe, for every e>0, by continuity of {D,D} on M # . There A exists a function a E C(MA#) with 0
:
x ES2})1/2, Let b be in the algebra,
finitely generated by A# with Ib-al
(2.6)
a(x0)=1 , a=0 outside of
,
< e/2M . Then we get
{bD,bD}1/2<{(b-a)D,(b-a)D}1/2 + {aD,aD}1/2
,
at each x E Q The first term at right is bounded by e/2, at each x, as follows from (2.1). But the second term also is bounded by .
e/2
,
since a(x)=0 , hence {aD,aD}(x)=0 outside Ne , while < Ilall 2{D,D}(x) < e2/4 there.
{aD,aD}(x)I
Accordingly we conclude that (2.7)
sup {I{bD,bD}(x)l
:
x E S2}
< e
Then, using (2.6) we conclude that also (2.8)
IbDAI < e
Finally we observe that
(2.9) sup {IabDA(m)l
:
E E E) < IbDAI < e
mGM) = inf{IIbDA+EII
using that the Gel'fand isomorphism of a C -algebra is an isometry. Also, ab(m)= b(x0) , by construction, and Ib(x0)-lI = (b(x0)-a(x0)I< Ib-all
< e/2M , which implies ab(m) > 1/2 ,for
sufficiently small e . Thus (2.9) implies e> Iab(m)IIODA(m)I < 2e Since e>0 was arbitrary, we > IaDA(m)1/2 , or, IaDA(m)I conclude that aDA(m) = 0 , q.e.d. With thm.2.2 we now have the .
ID
following model of the symbol space M
,
from the structure of the inver-
M
MsI
s
I
se images 1-1(x0), x0EM 0(fig.2.1).
w
A
is a certain compactification of
M
A# SZ
.
aM #
am
All functions in A# extend conti-
A
A Fig. 2.1.
196
VII.2. Secondary symbol space nuously to M Then
t:
A H - M # A
Q is dense in M # A
,
we write
am # A
is a continuous surjective map
=
M #\
S2
A
.
The inverse image of the manifold 0 , i.e., the set W=t-1S2
, called the wave front space, is (homeomorphic to) the
bundle of unit spheres in the cotangent space (or better the spheres of radius - ). In particular the sets t-1(x0), for x0EQ are n-l-spheres. On the other hand, over any x0E am # the sets A I
-1(x0) are compact subsets of M6 , with 6=6(x0) the dimension of
the local space Sx 0
Since the topology of H is the weak topology, determined
by the set CM, an explicit knowledge of the sets n(1-1(x0)), for all x0EaM # , and of the functions n , will characterize the A topology of I as well. In particular we have the following. Corollary 2.4. For a point x0E am A
#
assume the dimension Ix 0
for all
locally constant (i.e., 6x=6x
A#
0
neighbourhood Nx
nNx 0
, with some
of x0 ). There exists a compact neighbourhood 0
Ox E Nx naM 0
A
0
#
of x0 in am # such that the map
t-1(Ox0 )-0x
0XM6
A
, defined by
6=6x
m + (t(m),n(m))
(2.10)
,
, defines a continuous injection of t-1(0x
with the map (2.3)
0
xM6
onto a compact subset of 0x
Here the map n(m) is defined
0
with an orthonormal base of Sx
which still is a base at xEOx 0
, although not necessarily an orthonormal one.
for x E 0x 0
Proof. The continuity again is evident. For the injectivity we observe that t(m)=t(m-) implies that m,m-Ei-1(x) , where x=t(m) Then if also n(m)=n(m-) we may repeat the conclusion of thm.2.2, to show that m=m" {Dv}
.
In particular, D# is independent of the base
, and the orthogonality of the base is not essential. Q.E.D.
Remark 2.5. We observe that 6 =n+l whenever xESt. Therefore the x
197
VII.2. Secondary symbol space
assumption that, for some x0EaM # the dimension dx is constant in A a neighbourhood of x0 relative to M A
that dx=n+l
,
#
, not only to 8M #
implies
,
A
in such a neighbourhood. In this case an analogue
of cor.2.4 for a neighbourhood 0x
relative to M # is valid. 0
A
However, in this case we can make a stronger statement under slightly stronger conditions, as will be discussed in sec.3. Remark 2.5. If we have
Sx=d=const. in some subdomain BCBM
A#
'
then every point in B has a compact neighbourhood N such that t-1(N) is homeomorphic to a compact subset of NxM6 In many con.
crete examples this property may be translated into a manifoldlike structure of the subset t-1(B) C !\W
.
Let us now examine some explicit examples. Example (A) of V,4. We obtain the algebra s0 of [CHe1]. with the classes As, DS, as follows. A# is the function algebra finitely generated by
a(x)=(x)-1
, sj(x)=xja(x)
,
j=l.... n
D# is the class of all folpde's with coefficients in A# s s A# and Ds satisfy (m1) and (m2) . We get (2.11)
S3b3
{D,D}
1
J
+ pp
bJa
, D = nj In
In [C1],IV we have seen that MA #
.
Clearly
+p bJ,pE A#s xj
is the 'directional compac-
s
tification' of In , corresponding to the closure in In of the unit
ball 5(1n) , with the map s : x->x/(x) taking In onto { x
Do = 1 , Dv=-ia x
v
, v=l,...,n , for all x E Bn
Thus the dimension 6x=n+l is,constant, we get (2.13)
A0=D0A = A
Aj=Sj=DjA ,
, nj(m)=oA
(m)
,
j=0,...,n
.
J
The Aj are self-adjoint operators, hence nj are real-valued. Also A0 is a positive operator, hence n0>0. A calculation shows that (2.14)
IV=0 Aj *A
1
198
VII.2. Secondary symbol space so that 0< no
(1'Yv=1In.12)112 .
Thus we have the following.
Proposition 2.7. The space M\W is homeomorphic to a compact subset of the set fn={,EBn+l.
DEnxEn , where
(2.15)
It is clear that
Iv_olnvl2=1
,
no>o)
Bn is topologically equivalent to the ball
En (of infinite radius). In [CHe1] is was seen that
MW = aBnxBn = aBnxBn
(2.16)
.
From V,4 we know that the algebra s0has compact commutators, as a subalgebra of the algebra obtained in example (A) there. Similarly, for the algebra N of V,4, example (A) we get ffi\ID = aMA # X En c
(2.17)
.
Example (B) of V,4. Let us use the function algebra A# and xEO=M3k=gjxTk in
folpde-class D# introduced in V,4,(B). Write
the form x=(x',x") , with x'EB3 , x"ETk . A function a(x',x")EA# must also have its x"-derivarives decaying, as x(i.e., as Ix'I
- ). This implies that functions in A# must be nearly constant on the k-tori x'=const., Ix'I large. In VIII,2 we will consider a larger class A# which, however, will involve non-compact commutators for the corresponding algebra C. The compactification MA c
pjk
#
will look similar to the Stone-Cech compactification of M3k
One may use exactly the same basis Dv , v=0,...,n
as defined in
(2.12) (with D. , v>j, now acting on a periodic function). Again the Aj=D.A are self-adjoint, and A0>0 - . Again we get (2.14). 13
Accordingly it follows again that MW C aMA #xIn
.
However, in the
c
present case, we do not have equality. Rather, the secondary symi.e., to the characbol space MW)int is homeomorphic to B3xZk ter group of the group Mj,k (cf. thm.3.1 of [CS],ch.5 ) We postpone the discussion of other examples of V,4. Here let us first discuss an example of different type. Bn Example (H): Let 0 be an open subdomain of , with smooth smooth boundary, compact or not. Assume the boundary ao to be a smooth n-l-dimensional submanifold of En. We will focus on two ,
.
199
VII.2. Secondary symbol space special case only: either QUaO is compact, or 0 _ l+={xeRn:xl>0}, referred to as "cases (a) and (b)" respectively. Consider the Euclidean Laplace-triple,in either case, with the classes A# and V# of restrictions to Q of the functions (folpde's) of As and Vs of example (A), above. Let C be the coresponding comparison algebra. Again we may use the orthonormal base Dv of (2.12),for every xEaMA#
.
and MA#=B+=(g+)clos
(We have MA#=cU2c2 in case (a)
with the closure of In in En .) The dimension is 6=n+l everywhere. However, an investigation shows that the operators Aj no longer are self-adjoint, although we still have (2.14). In particular, focusing on case (b), the operator Al , involving the derivative normal to the boundary, is not even self-adjoint modulo E
. Accordingly the set n(t-1(x0))
, for a point x0 with x0=0
only is a subset of the complex unit sphere 1v=O1nv12=1, nEWn+1. For a precise description of the space M cf.[C11,V,thm.10.3. The description there is in a slightly different representation, to be translated into the present form. Over points of the finite boundary of 0 one only gets the (still well-defined) cosphere bundle, and, in addition certain 1-dimensional 'filaments'. Over points lxl=00, xl=0, one obtains the ball JEJ0 , and again some 1-dimensional filaments). For other points
Ixl=° we of course get the same as for D=Yn
,
A# , VS again.
In [C1],V we also investigate the commutator ideal E of case (b) in detail, and thus obtain a complete Fredholm theory for the algebra C. We have not analyzed the space ffi\T in case (a)
, but
expect a very similar result.
Note that the last example pertains to the case of a genuine boundary problem, where cdn.(w) is never true. In [C1] and [CE] we also show that the elliptic boundary problem with 'LopatinskyShapiro'-condition is solvable with our algebra C 3. Stronger conditions and more detail on ffi\W
.
It should be realized that our assumptions on A# and V# as well as on the triple {O,du,H} are minimal, so far. In sec.2 we have studied some examples which allowed a more complete description of the space ffi\W
.
We now first will look at conditions to
200
VII.3. Stronger conditions allow a description of the sets i-1(x0) C I\W as compact subsets of a real upper hemisphere ffl
1, just as in the examples of sec.2,
although only locally.
the following observa-
For the Rn-related examples of sec. 2 tion may be useful.
Remark 3.1. Suppose near a point x0E3M # the manifold Q 'looks A neighbourhood Nx like Rn,, in the following sense: x0 has a 0
relative to M 4 , such that = U is a chart. In particular, QnNxO A assume that, if U is considered a subset of In , then either x0 is a point of a smooth boundary piece of U or JxoJ=- , and ,xl for all xENx \R . Suppose then that, in the coordinates of U , 0
the coefficients hjk(x)
, q(x) extend continuously onto Nx
, and 0
still define a positive definite (n+l)x(n+l)-matrix, at all points of N x0 Suppose also that D# contains folpdes D v such that .
(3.1)
D0 =
1
, Dv = -i3 v
,
v=l,...,n , for xEU .
x
Then we have dx=n+l in Nx
, and it is possible to use the folp-
0
de's (3.1) as a base of Sx, although not necessarily orthonormal. Accordingly cor.2.4 applies in the entire Nx
,with n defined 0
by (3.1). For a generator A =a(x), or =DA have the value of aA(m) at some mEt-1(x0)
(3.2)
oa(m) = a(x0) , aDA(m) = b3(x0)nj
or =(DA)* of C we then , x0EaM #
,
given by
' a(DA)*(m) =
b3(xO)nj.
for D=InbvDb3Dj+p, summation convention used from 1 to n,only for non-greek indices.
In thm.3.2, below, we will show that, in this case, we get (3.3)
hJk(x0)njnk + q(x0)1no12 = 1 , n0>0, for nj = nj(m)
.
Therefore we may write (3.2) in the form (3.4) aa(m)=a(x0),
aDA(m)=(b3(x0)Cj+p(x0))/(hjk(x0)Z* jCk+q(x0))1/2
201
VII.3. Stronger conditions
using the coordinates j=nj/n0 , with a E E ]En
.
Here the vector _ is not always finite. It is assumed to be a point in the directional compactification In of Cn , defined similarly as
n above. If, in addition, it can be shown that all nj are real, then we have E,n (but we should ,
recall example (H) of sec.2,
of a comparison algebra on a manifold with boundary, where we do not get all nj real). In this representation, the space N x$n appears as a part x
0
of the compactification of the cotangent space T*c2 of 0 , generated by the bounded continuous functions (3.4) over 0 . Moreover,
the representations of t-1(x0) C M of thm.2.2 are obtained by a change to 'projective coordinates'. In the following we will seek to recover these features. While we return to the general case, assuming only (a0),(a1),(d0), and (m1),(m2), (and not (w)) we impose (some or all of) the following additional conditions. Condition (m3): For each x0EM # and open neighbourhood N of x0 A there exists a function a E A# with 0
,
xj
Condition (m4): There exists an c (3.5)
A-1-e[a,A]
E C
,
>
0 such that
A-'[D,A] E C
,
for all a E A# , D E D#
where the last expression should be interpreted as A-'(DA-(AD)**), as a product of unbounded operators, to be continuously extended. Condition (m5): We have A E C Condition (m6 )x The dimension 6x of S#=D#IV# is maximal (i.e., we have 6 = n+l). .
x
We define (m6) = U{(m6)x :x E 3M #} = U{(m6)x :x E M #}, A
A
i.e., cdn.(m6) (without subscript x) means that (m6) holds for all x E M # (it is always true for xEO) . A Note that (m3) is a separation condition comparable to the condition describing completely regular algebras (cf.[C1],AII,5). On the other hand, (3.5) may be compared to the statements of V, lemmas 3.3 and 3.6. Using the techniques of V,3 one may reduce (m4) to explicit conditions on a E A# or the coefficients of DED#, but we will not make a systematic discussion here. Note that
202
VII.3. Stronger conditions (m5) holds if 1 E D#. For a weaker condition cf. IX,5
.
Theorem 3.2. Assume condition (mj)
, j=1,2,3,4,5,6, and let Dj, j=l,...,6, of thm.2.2 be formally self-adjoint. Then the image
set of the map n:i-1(x0) - M6 is contained in the sphere {n E ]Rd
(3.6)
:
1v_lInll2 = 11 C In
Proof. Since the dimension 6(x) is lower semi-continuous, it has Thus we conclude
to be constant near x0 if it is maximal at x0 that dx= n+l near x0 We have Parseval's relation in S
.
.
D1,...,D6
.
x0
for the orthonormal set
For every D E D# we get {D,D} _ 1d_ll{Dv,D}I2 , at x=x0
(3.7)
Due to continuity of {.,.} as function of x this implies that I{D,D}(x) - Ea=1I{Dv,D}(x)I2I < E{D,D}(x)
(3.8)
as x E Nx
,
0
for a suitable neighbourhood Nx
of x0
.
Indeed, we already used
0
in prop.2.l that D-v(x)=Xwvu(x)Dm , for suitable W=((wvu)) close
to C(&
)), is orthonormal at x, for every x E N
we will have (3.7) at x E Nx plies (3.8)
,
X
Since 6=n+1,
.
0
for D-v instead of Dv
by a simple estimate, as Nx
.
This im-
is chosen sufficiently 0
small. (For more details cf. proof of IX, thm.2.1)
Proposition 3.3. Let a E A# equal 1 and 0 outside Nx =Nx 0
v=l,...,6 (3.9)
,
.
as x near x0 ,and x
E, respectively, and let 0
,
0'
Then for all u E AC0'(c7) we have the estimate
IJ,,dv(a2(hJk(Au)
x
(Au) k+gIAul2)- fIaAvul2)I < EQu02 x
v
This proposition follows from (3.8) by a calculation.
Actually, (3.8) implies a corresponding local estimate, and then we may use that Bul12 = (u,Hu) _ flA-lufl (cf. V, (1.4) ) .
Using (m3) we now may bring the factor a2 of the first term in (3.9) under one of the derivatives(Au)lj In detail we get .
203
VII.3. Stronger conditions
204
J = J,,dpa2(hJk(-)Ij(Au)1k+gluI2) into the form
2J = (a2Au,A-lu)+(A-lu,a2Au)-2((Aa(DaA)+(DaA)*aA)u,u)
(3.10)
with the inner product (.,.). Apply the first relation (3. 5) for a2A
(3.11)
=
A(a2+aC+Ca+C2)
,
C = A-1[a,A] E C
.
It follows that the first two terms can be written as (a2Au,A-lu)+(A-lu,a2Au)
(3.12)
2(u,(a2+C')u)
=
C'=aC+Ca+C2
,
Finally (3.9) with (3.10), (3.12) may be polarized for
a av
2
1((a +C'-(Aa(D A)+(D A) aA)- JAa2Av)u,v)j
(3.13)
V
valid for all u,v E ACS . All operators in (3.13) are continuous,
hence we may extend to all of H Denote the self-adjoint operator at right by Z and introduce Zu=v , for .
IIZuf2 <E(fluu2+11Zu112) , u E H ,
(3.14)
which results in a - J(aAv)*(aAv)-C'-(Aa(DaA)*+(DaA)*aA)=Z
(3.15)
fZR<E/(1-E)
,
v
Accordingly we also get loZ(m)I
< E/(1-c)
maximal ideal m E 1-(x0) we get aa(m) =
1
.
However, for a
, and aaA (m) = aA (m) v
V
. Also the symbol of all other terms of Z vanishes at m Indeed, we get Da E D# , by (m3), while Da=O near x0 , by (m3) = n v(m)
Hence one may find a function X E A# equal to 1 near x0 and 0 on supp Va , and set w=l-X E A# , wDa = Da
.
It follows that aD A(m) a
w(x0)aD A(m) = 0, which makes the symbols of the last two terms a
vanish. The same may be used for C' = A-1[a,A]
. With a resolvent
integral we get (3.16)
C'
= 1r-l
J-A-1(H+A)-1[H,a](H+A)-1X-1/2da 0 *
Using V,(2.24) we get [H,a]
=
= Da
A-1(H+a)-1
.
(Note that (H+A)-1
,
Da , where we again use wDa ,
A-2(H+A)-2 E C
that the integrand is in C and has symbol vanishing at m
so
,
.
The
.
VII.3. Stronger conditions integral exists, by the standard estimates. Thus ac,(m) =
0.
As a consequence of (3.15) we therefore get (3.17)
I1 - Iv=11nv(m)I21 < e
This estimate holds for all e>0 1v=l1nv(m)I2 =
(3.18)
so that we get
,
1
.
Note that self-adjointness of the Dj so far has not been used. This condition will be essential if we try to show that
n(m) is real. Clearly the complex conjugate of nj(m)=aA (m) is J
aA *(m) . Also A
(ADj)** , using that DD . By the
(DjA)*
second relation (3.5) we get (3.19)
(DjA)*
= DjA + ACC
,
C E C
If the maximal ideal m considered has the property that aA(m)=0 then (3.19) implies at once that nj(m) is real, since the symbol of the last term vanishes. On the other hand, we get (3.20)
[D.A,A]
_ ((DjA - (ADj)**)A E E
,
**
so that Y = DjA - (AD gives
satisfies aA(m)oy(m)=0 , which again 0 Thus
aY(m) = 0 (,i.e., n(m) real ) whenever OA(m) 9
.
we indeed have proven our theorem. Remark 3.4. Note that we have shown above that relation (3.5) 2
and (m5) imply relation (3.21) imply (3.21)
(3.21)2
below. Similarly, (m4) and (m5) 1
:
[D,A1 E E
,
A-1[a,A] E E , for all D E D#
,
a E A#
Indeed we showed that the symbol of [D,A] vanishes for all m
Similarly, A (A-1[ a,A]) = [a,A] E E
so that ac , with
.
.
C=A-1[ a,A],
vanishes whenever aA90. But C=ACC', C'E C, which implies aC=O also if aA=(a C) 1/e=0, or, again aC=O for all m it, It, i.e., C E E.
Corollary 3.5. Under the general assumptions of thm.3.2, if we only require (m.), j=1,2,3,4,5, but not (m6) then we still have (3.22)
n(i-1(x0)) C B6 = {xER6
:
IxI
.
205
VII.3. Stronger conditions
206
Moreover, if also the assumption of selfadjointness of the expressions Dj is dropped, then we still have (3.23)
n(t-1(x0)) C {xEad
:
Ix1<1}
.
Proof. With the present assumptions we can substitute (3.8), (3.9) by the weaker estimates
(3.24)
and
(1+E)10du(a2(hJk(Au)Ixj(Au)Ixk+gIAu12) > Xa=1IIaAvul2
Here the left hand side may be treated exactly as above. One finds that it is of the form (1+E)(u,Vu) where VEC has symbol aV(m)=1
.
Write Z=16=1Aa2Av , and conclude from (3.24) that (1+E)V-Z>0 From this hence (1+E)V-Z=T2 with a positive self-adjoint T E C we conclude that (1+E)aV-aZ=((y T)2>0 , which yields In(m)I2=az(m) .
< (l+e)aV(m)=l+e tement (3.22)
.
Since this holds for every a we get the staNow (3.23) is trivial, q.e.d. .
Let us now come back to a representation of the symbol space IN as a compact subset of a certain compactification of the cotan-
gent space T O of 0 as discussed for examples earlier in this section. For our generators a, DA we define formal symbols similar ,
to (3.2) by setting (3.25) ta(x,E)=a(x), j+p E D#
for all (x,E)ET*D , where D = -ib32
We require (m4)
x
and (m5), hence can be assured that (AD)**=DA+E , EEE , by remark 3.4, so that (AD) need not be taken as generators, mod E Note that the formal symbols are bounded complex-valued
CW(n)-functions over Q. Hence the class of functions (3.25), for all aEA#, DEV# determines a smallest compactification of D, called *
R
, with the property that each function (3.25) admits a
continuous extension to
*
Q (cf. Kelley [Ke1])
.
This is
just the maximal ideal space of the algebra of bounded continuous T*O functions over generated by (3.25). The injection of the algebra generated by A# into that algebra induces a surjective
associate dual map n: &0 Let aIE*sl = E*R\T*D .
M # again, defined just as the map i A
VII.3. Stronger conditions
207
For the theorem, below, we will require a somewhat stronger form of cdn.(m1), as well as cdn.(m7), below. Condition (m1')
:
, and, in addition, p2/q E A#,
We have cdn.(ml)
for all D=b33 .+p E D# X3
At each x E aM # there exists a folpde D E D#, A such that {D,D}0(x)#0 but {D,D}i(x)=0, with {D,D}j of (3.26). Under cdn.(m1') we do not only have {D,D}(x) well defined Condition (m7)
:
,
for points xE3M # A {D,D} _
(3.26)
extends to 3MA#
,
but even the decomposition
{D,D}0+ {D,D}1
,
{D,D)O=IPI2/q
,
{D,D}l=hjkFibk
Then (m7) requests a nontrivial DESx which
'emphasizes the 0-order component'.
Clearly it makes sense to request (m1') or (m7) only at Then we will speak of (ml') x or a specific point x E 3M # .
0
A
again.
(m7)x 0
Theorem 3.6. Assume that (m1'), and (mj)
,
j=2,3,4,5,7, hold.
Then the maximal ideal space fi of the comparison algebra C is
homeomorphic to a compact subset of the space aF*Q. Under this homeomorphism the wave front space W is mapped onto the bundle 7r-1(S2) of co-spheres of infinite radius. Moreover, if we regard It as a compact subset of aF S2, by virtue of this homeomorphism, then
for each mEM, and every generator a E A#, DA, DED#, the symbol is the restriction of the formal symbol
.
That is, aa(m) and aDA(m)
equal the values of the continuous extension of a and TDA (3.25)
)
to ff*0 at m
('of
, respectively.
Thm.3.6 was first proven in [CS],ch.6. (cf. also [CM2])
understronger assumptions (cdn.(14),of IX,3). The idea of the present proof is due to McOwen [M1], where the result was established for the Laplace comparison triple of a complete Riemannian manifold, and the sets A# and D# of functions (folpdes) with covariant derivatives (of the coefficients) vanishing at .
McOwen assumes the additional condition that the dimension 6x0 is constant at each end of 0, and that E = K(H)
.
VII.3. Stronger conditions Proof. We only must provide an injection ffi-.21 *SZ with the proper-
ties of the theorem. This will define a continuous map since the topology of M as well as that of 2B*S2 is determined as the weak topology of the same function class (3.25).
One finds that, for x0EO,the set7r-1(x0)rig B*O is a sphere there is one boundary point of 2F 0 in each direction
T-*-,
IEI=1. By VI,thm.2.2 we may define the injection over x0 as the * map E-E from the co-unit-sphere to Tr-1(x0)r12B 0 Therefore we .
only must focus on points x0EaM #. First assume cdn.(m6)x . Then, A
0
for a formally self-adjoint orthonormal base D as in thm.3.2, we v SS-1 may define the injection map n of that thm., with t-1(x0)+ ,
:
S6-1
the unit sphere
of (3.6)
Under the present assumptions we
.
may choose the base Dv such that DS=Dn+1 is a folpde with 1 , just by using a multiple of the assumed D#0 with {D,D}1=0 Then a conclusion like that in cor.3.5 may be used to show that
nn+l(m) = oD A(m)>0 (where, of course (m3) is essential). Accord H6-1
dingly, the map n now takes t-1(x0) into the hemisphere {HES6-1:n6>0}
=
.
Now we claim, that also n-1(x0) is homeomorphic to (the entire)
HS-1 .
For pGTr-1(x0) let
with
(3.27)
(P)=TD A(p) V
Observe that TD A is real-valued
and that 1v=1Cv2(p) =
,
by self-adjointness of Dv
, again follows from Parseval's relation.
1
Thus we get C:7T -1(x0)-S6-1 again, and the injectivity of C may be
proven as in thm.2.2 again. In fact, we also have r,>0 C again maps into
,
so that
HS-1
To show the surjectivity, let us introduce 'projective coordinates', for (x,E)ET S2 , with x near the point x0 , by setting
j=nj/no , n0>0 . We get (3.28)
(bjnj+pno)/(hjknjnk+gn02)1/2
Here we introduce n0=r(x)/q(x)
,
nj=hjk(x)ck(x) , with the (real)
coefficients of some folpde F=-icla
j+r, into the right hand side. x
1/ Then it will assume the form {F,D}/{F,F}2
.
For j=nj(x)/Jr(x)J
208
VII.3. Stronger conditions
209
we thus have ±{F,D}/{F,F}1/2
(3.29)
In (3.29) set D=Dv . Also assume Dv orthonormal in some neighbourhood of x0 , and let {F,F}(x)=l , near x0 , without loss of gene-
rality. It follows that (3.30)
TD A(x,C) _ ±{F,Dv}
where the same sign holds for all v
,
,
since the sign is determined
as the sign of the coefficient r(x) If F runs through all unit vectors of the 6-dimensional space, at some x , then the 6-tuple .
({F,D
v=l,...,d
v
runs through the entire unit sphere. Thus, for
any point XES6-1 we have either X or -X assumed by (TD A(x,&)) The same property then must hold for the values gy(p)
,
.
as p runs
H6-1 But we already know that (p) E Thus it 0 (H6-1)int follows that all points of must be s assumed. Since in Ha-1 is closed, we then conclude that im = i.e., is sur-
through it-1(x
)
.
.
,
jective.
It then follows that the map p = v(m) (3.31)
,
for mE1-1(x0)
is well-defined. Let
We get
.
Ta(p)=a(x0)=aa(m)
,
TD
A(p)=Cv(p)=nv(m)=oD A(m)
U
so that indeed formal symbol and symbol coincide.
On the other hand, if (m6)x
is false, i.e., 6x
we have seen that n maps into the ball
0 BS-1={EEI6:I&I<1}
Again we may assume that D6 satisfies
(cor.3.5)
at x0, and then con-
clude that nd>0 so that the map n really goes into the half-ball B+d-1={nEBS-1:nd>0} ,
.
But if 6x dim Sx 0
= ax 0
,
0
for all x-'Q, and every neighbourhood of x0 contains such points.
Repeating the above conclusion we find that the vector (TD V
now assumes either C or -C, for every
CEBa-1
.
The same again is
true for the vectors c(p), as p runs through w-1(x0) Thus, in B+6-1, this case, the map also goes onto and we again may define .
the map v(m)
.
Thus we have the same statement, and thm.3.6 is
VII.4. Structure of ffi\W; Examples
210
established.
4. More structure of M.
,
and more on examples.
Remark 4.1. Before we attempt application of the results of sec.3 let us point out that thm.3.2, cor.3.5, and thm.3.6 all have 'local' generalizations, in the following sense: Under cdn's (m3)x
only, for some given point x0
and (m6)x 0
0
E 3M # , we still have the statement of thm.3.2 true for that A point, regardless of all other points, assuming the other (mj), of course. Under (m3)x alone we still get cor.3.5, at x0 .
0
Again, under (ml')x '
0
(m3)x
,
with the other (mj) required for
0
thm.3.6 we get the homeomorphism of thm.3.6 at least between
C1(x0) and r-1(x0) , although perhaps not between ffi and 3F SZ Here, by (m3)x we mean that the function a of (m3) exists 0
for that x0, and its neighbourhoods. By cdn.(mi')x
we mean that 0
(ml') holds 'near x0', i.e. {D,D} and function in A# near x0
.
each coincide with some
Similarly (mi)x
, which may be substi0
tuted for (ml)
,
above.
We shall refer to these local generalizations as thm.3.2x
,
0
etc. Note that there also is a thm.2.2
x0
Let us point out again, that, speaking in terms of thm.3.6, examples have been given which show that the set i-1(t ) , for some x0E3M # , may range from the minimum (the set Wclo n -1(x0) A Tr
to the maximum (all of ff-1(x0))
.
In particular, example (B) of
V,4 shows that cases between these two extremes occur: For the Laplace comparison algebras of the polycylinders G3k we may get more than Wclo:h n-1(x0) but not all of n-1(x0) , rather, only some sub-surfaces of the hemisphere Ha-1 are contained in I
.
As a partial answer to the question for the set i-l(x0) we prove the result, below. Theorem 4.2. Under the assumptions of thm.3.6x , if there exists 0
VII.4. Structure of 81\W; Examples a function aEA# with a(x0)90 , and aD.0AE E , where D0 denotes the
folpde assumed in thm.3.6, with {D0,D0}={DO,DO}0#0 we have
t-1(x0)
(4.1)
at x0 , then
,
t-1(x0)f'$Tclos
=
Proof. Let mEHts = ffi\Wclos = ffi\,s , and let 2(m)=x0. Since aDDAEE
we conclude that 0 = aaD A(m) = a(x0)OD A(m), or aD A(m)=0, since 0
A
d
by assumption. Therefore the statement is an immediate consequence of cor.4.3, below, q.e.d.
Corollary 4.3. Under the assumptions of thm.3.6x , with the 0
operator D=D0 of thm.3.6, we have (4.2)
t-1(x0)nIDclos={mE1-1(x0):aD A(m)=0}={mE1-1(x0):TD A(m)=0} 0
.0
Proof. First notice that TD A(m)=0 for points of Wclos, Indeed, 0
we know that, for an (x,F)EW we have
so that the formal
symbol assumes the form (4.3)
T
DA
(x,E) = b0E?/(hOkE0E0)1/2
,
3 k
1
for
1
01=1
With the technique of the proof of thm.3.6 we express this as TDA(x,C) _ ({D,F}1/({F,F}1)1/2)(x)
(4.4)
,
where F is a suitable folpde. (Note that the sign of the 0-order term no longer is essential.
)
Since we have {DO,D0}1(x0)=0 , we
conclude from (4.4), that TD A(p) =
0
,
for all
p E Tr(x0)
whenever p E Wclos, which proves the above. Now, again in the terminology of thm.3.6, we must show that the restriction of the map p to the set {mEt-1(x0):r) 6(m)=0}
maps onto the sphere (or ball) ns=0 However, for xE12 near x0 the values of the vector (aD A(m)) , mE1-(W), are exactly the .
v
values of (TD
(({Dv,F}/({F,F}1)1/2)(x))
,
taken over all
folpdes F, defined near x. From this one concludes that also in the limit x-x0 every vector of the sphere or ball is assumed. This completes the proof. Remark 4.4. Let us emphasize again, that thm.3.6, and cor.4.3
211
212
VII.4. Structure of ffi\W ; Examples
require conditions (m4) and (m5), which imply that, for a basis (mod A#) of self-adjoint folpdes DIV the corresponding generators DvA are self-adjoint mod E . If these conditions are violated,
as in the case of a comparison algebra associated to a boundary of section 2), then N no longer must problem, (cf. example (H)
be a subset of the space 3P 0 In [C1] and [CC1] we discussed such an example, where it turns out that certain additional .
'filaments' of N occur, which cannot be found from the values of the formal symbol. Remark 4.5. As another feature, illuminating the role of the let us observe that, formal symbol and the compactification 3&*sl ,
in all of the results of sec's 2, 3, 4, involving the sesqui-linD,FED# , we may use the form ear form {D,F} {D,F}^(x) of any ,
triple {O^,dp^,H^} (c> {O,dp,H} , defined as (4.5)
{D,D}^ = h^jkFibk+IPI2/q^
,
and by polarization for {D,F}^ . Actually, one may use just any similar sesqui-linear form, positive definite over each space Mn+1 of n+l-tuples (Vu(x),u(x)), at every x E 0 , as long as the estimate V,(1.6) remains satisfied with {D,D}^ instead of {D,D}. It is clear then that we get corresponding conditions
(mj=1,6,7, as well as (ml')^, and the corresponding local etc. (Note that cdn's (mj), j=2,3,4,5, do not conditions (mj)^x depend on the form {D,F} ). Here it is essential that (ml implies that {D,D}^
substituted for the ordinary ones we then again get thm.2.2, thm.3.2, cor.3.5, thm.3.6, thm.4.2, cor.4.3 true again, with exactly the same proofs, and including all the local versions. However, the formal symbol changes with the form {D,F} now must take
.
We
T^DA(x,E)=(b3Ej+p)/(h^]kEjEk+q^)1/2
(4.6) *
instead of (3.25). Accordingly, the boundary 31 0 of the compactification F 0 of T 0 also changes. In fact, one easily verifies, *
*
that there exists a surjective continuous map 8&^ 0 - 2F 0 , which * also is injective in WQ F^ 0 and acts as a homeomorphism between the two images of W . Over the complement of W, on the other hand,
points may split into several points. In other words, the representation of IN as a subset of 3r 0 changes with the form {D,F}
213
VII.4. Structure of M\W; Examples Now let us return to the discussion of explicit examples again. First we look at the comparison algebra of V, thm.4.1. In details, we have 0 given as a subdomain with smooth compact boun-
dary M of a complete Riemannian manifold 2^ without boundary with aQ is an n-l-dimensional suband the tensor hIk is the remanifold of Q^ . We have dp=du^IQ striction of h^jk to N , but q is not necessarily a restriction of metric tensor h^3k and measure dp^
,
and {D^,du^,H^} (c> {S1,dp,H} in the sense of q^. We have q > q^ V,3. For formal reasons it is convenient, however, to impose ,
the following. Condition (ql)
:
We have q(x) = q^(x) in N^\N , with some neigh-
bourhood N of 30 To satisfy V,(3.13) and (w) we again require (as in V,4) that .
q(x) > Y1(dist(x,3c))-2 , and V(log q^) = o(1) (in R^)
(4.7)
Now it turns out that the classes A'
,
Do of V,4 in general
will not satisfy conditions (m.), j=1,2. To discuss an example only let us work with the form {D,F}^ of the above h^jk and q^ and the following more restricted classes called A# and Do AC is the class of all bounded functions a E C'(c) such that Va = o,^(q^E/2)
(4.8)
,
for some e, 0<E<1
.
(The difference to the class A# is that the functions a E A# must for a E Ac may have their derivative bounded near aQ , while Va ,
not remain bounded, but only must be o(qC/2), O<e<1, near a2 DC is the class of all D=bJa j+p with bounded {D,D}^ and x
{D ,D }^ (bounded only over N) satisfying the following stronger version of V, (3.16): There exists an e , 0<e<1 , such that (b'lk)=o0^(q^e/2)
, b3(log
q)lj=oN^(1)
,
(4.9) V((bJ01j/(c)/q^))=0S1^(q^E/2)
.
Again the requirements are distinctly different near - of V, Near - the covariant derivative balk stays bounded, and near 3Q Near 3N the last two conditions just near - it must be o(qe/2) :
.
amount to (4.10)
Vp=0(1)
,
V((b0C)lj/)=0(1)
VII.4. Structure of ii\w; Examples
Clearly we have A# c Ac
,
214
D# c D# . A calculation, not dis-
cussed in detail will confirm the following. Proposition 4.6. Let cdn.(gl) hold for a triple as described above
satisfying (4.7). Then the classes A# and D# satisfy (m1)^, (mj), j=2,4,5,( as well as
(miMoreover, A even is an algebra,
and DC satisfies the condition of thm.3.6^, for every xE3MA#, with the expression D=l, at every point. 3.3
,
In particular (m4)^ is an immediate consequence of V, lemma and V, lemma 3.6, since K c C .
We now consider the comparison algebra C = C(A#,D#) algebra clearly has compact commutators, and contains K(H)
.
This . More-
over,it also satisfies the conditions of thm.2.2, so that we get the sets 1-1(x0) imbedded in M6 , for each x0 E 3M # A Note that the functions of A# must have continuous extensions to 52U812, since they have their first derivative bounded near 8S2 , by (4.6). Also A# contains all restrictions to 0 of functions .
in C_(52Uac)
This shows that 352c8M #
.
,
In fact, we find that 3M #
A
A#
is a disjoint union of 3Q and a set 8MA# 'at , where both sets are separated, i.e., have disjoint closures. Again, it is easily verified that (m3)x holds for all points x0E352 , since 3S2 '
0
is smooth. The corresponding property (m3)x
for points in 3M #,
,
A
0
will be verifiable only under additional conditions on the Riemannian space (2
,
as well as on the potential q
.
For a more detailed
investigation of M\W in case of the Laplace comparison algebra of a complete locally symmetric Riemannian manifold cf. McOwen [M1]. For a complete characterization of 1 in case of a Schroedinger expression on In, for certain (spherically symmetric) potentials
q ,
cf. Sohrab [ S1] ,
[ S2]
,[S31.
Theorem 4.7. Under the above assumptions, with the decomposition
3MA# = 352 U 3MA#
(4.11)
, we have 1-1(352) n m s = 0
That is, any possible points of the secondary symbol space His _ fl\(Wclos) are over infinity, not over 30
.
Proof. We just apply III,thm.3.7 to conclude that aA E K(H), for
VII.4. Structure of M\W; Examples every aEC_(OUdO)
with a(x0)#0
.
.
For a point x0E3O one may find such a function holds, we have the con-
Since E=K(H),and (m3)x 0
ditions of thm.4.2 satisfied for {.,.}^ indeed is no point of ffis over x0
,
.
It follows that there
q.e.d.
The above discussion translates literally to the language used in V,4(E) . Details are left to the reader. On the other hand, just to illustrate the variety of problems encountered, let us shortly look at example V,4(D). For the classes A# and D# used there one finds that the form {D,D} is .well defined, but{D,D}^ is not generally defined for all DED# . Moreover one finds that D# is not an Accordingly, none of the results of sec's 2, 3, 4
applies for these sets Ac
,
D# , although the algebra
C(A#,D#) has compact commutators and a well-defined symbol.
For example, one now might well expect nontrivial parts of the We have not tried to apply the secondary symbol space over 3O .
techniques described, although this should be easy and only a formal task.
Similarly for the examples V,4(C) and V,4(F), which will not be discussed in detail. Finally let us discuss some 1-dimensional examples, extending an investigation by Sohrab [S2], where a complete knowledge
of the symbol space will be obtained. These are examples of a Sturm-Liouville-triple V,(5.1). Applying the Sturm-Liouville transform (V,prop.5.1(a)) we may assume V,(5.1) in the form V,(5.3). In that form we assume the which amounts to the condition that
interval I- to be s
(K/p)1/2dx does not exist on both ends a,s
(4.12) a
We assume the triple in the normal form {1,dx,q-3 X} , where
q(x)>1 on I
.
In order to reach the generality of [S2] all con-
ditions imposed below must be translated back to the general case.
In order to have only one point of M # at each end of the interval we select A#=algebra finitely generated by C_(I) and s(x)=x/(x) , and D#=span {DO=/q, D1=-i3x}(mod A#). We want E=K(H), hence we require V,(3.13) and V,(3.15), V,(3.16), with q=q^. A
calculation shows that this requires V,(3.13) (4.13)
q'
= o(q)
,
i.e.,
215
VII.4. Structure of M\Q1; Examples
as only additional condition. (Actually, we could allow the larger function class A#:A# of V,4, and still would get E=K under (4.13) only, but this would give us many points in aM # over ±o'.) A Note that (4.13) will not allow'potentials of exponential growth, while potentials of arbitrary polynomial growth often are acceptable. In particular, the potentials q(x)=l+x2 and q(x)=1+x2 +x4
of the harmonic and the anharmonic oscillator in quantum mechanics satisfy (4.13).
All conditions (m.), (ml') hold, as well as the assumption of thm.3.6 (with
We will show that
(4.14)
IN
=
3P*(R)
,
where the form {D,D} of hll=1 and q=q^ is used. We trivially have W= Ix{-oo} u Ix{-} c1. For the secondary symbol space Sohrab looks at the operator T = observing that the range of p2(m), in the set 1-1x0 , for either one of the infinite points x0=±. with a cut-off
coincides with the essential spectrum of function X=l near x0 , = 0 near -x0 .
Formally we get T = L-1, with a self-adjoint realization L associated to the expression (in R = (--,o) ) (4.15)
32q-1/2+1
L
= -
3xq-1 ax +1-q-1/2(q-1/2)
Transforming L onto Sturm-Liouville normal form again we get
x _ (4.16)
L` _ -a2 + l+q"/(4q2) -5q'2/(16g3)
y
,
q
-cc
Let us assume that the potential in (4.16) has the limits l+B+ at Then we have limit point case and the self-adjoint realization L'"-1/2 is equal to the interis unique. The essential spectrum of
val [ 0, (l+B)-1/2] , where B=Min{B+} (cf.[ Wel] , [ Ne2] , [ DS3] ). Moreover, even the essential spectrum for a suitable realization over one of the half-lines R± of an expression modified near 0 is for [0,(1+B+)-1/2]. Here we refer to L±-1/2 where L+=L"'+c0w/x2 ,
example, with a cut-off function w=l near 0, wECp(I ), and a sufficiently large constant c0 .
216
VII.4. Structure of ffi\j; Examples
217
It follows easily that the essential spectra of X,/T and of L±
(with the back-transform of L±) coincide. On the other hand $+=0 follows if S+ exists. Indeed, (4.13) implies q'2/q3 + 0
If S+#0 this implies and lim(q-1)" _ -46+ , with y > 0 for every large (small) x Or, q=O(x-2 1q-1Lyx2 at that end. This contradicts q>1. ,
.
,
.
This argument shows that n2(m) has range [0,1], over each end ±- of 1. Accordingly 1nl(m)1=(1-n2(m))1/2 also has range [0,1]
This may be used to show that, at each end ±- , the half-circle H1 = {g12+g22=1 , n2>0}
(4.17)
is in in n
,
so that we get (4.14). Indeed, if n2(m)+inl(m) = z
where z is any value of the circle JzJ=1 , Re z >0
,
,
then the ope-
rator Tz= X(D0+iD1)A -z = X(Vq+ax)A -z is not Fredholm, since its essential spectrum contains zero. Here X denotes any real-valued cut-off function: XEC'(E, 0<X<1, X=0 in (-o,x0), X=1 in (x1,o). This means that there exists a sequence ujEH
,
IIujII=l
, with
L ]ITzu.II+O
, u.E (ker Tz)
.
But then the conjugate-complex sequence
uj will satisfy the same, with respect to z , since D0+iD1 and A both are real operators, left invariant by complex conjuga=ax+/-q
tion. It follows that TZ is not Fredholm as well, so that (-nl(m),g2(m)) also must be assumed by the map q , as another maximal ideal of course. We have proven the following. Theorem 4.8. Assume that q>l, that (4.13) holds, and that exist. Then the comparison algebra C (with compact Topologically
commutator) has symbol space It given by (4.14)
this symbol space is a circle.
.
CHAPTER 8. COMPARISON ALGEBRAS WITH NONCOMPACT COMMUTATORS.
In this chapter we consider some examples of comparison algebras with noncompact commutators. The most important example, perhaps, has been discussed in detail in [C1],ch.5., in connection with the half space R++1 = {(y,x):xE1k n, y,-R, y>0} . If the manifold 0 has a regular boundary, onto which the expression H, and as well as A# and V# may be extended, meeting similar ,
measure du
conditions as for interior points, then commutators are no longer compact. However, both quotients, CIE as well as E/K(H) , turn out to be function algebras. The maximal ideal space 1 of C/E looks similar to the spaces encountered for complete manifolds like except that it has certain one dimensional 'filaments' over the boundary of 9
n
,
.
On the other hand we get (0.1)
E/K(H)
__
C(N,K(h))
,
with the Hilbert space h=L2(R+), and the symbol space ID of a well If the E-symbol investigated Laplace comparison algebra over In .
of an operator AEL(H) does not vanish, then A may be inverted mod E , which corresponds to the first step of the common approach of solving a boundary problem: That of using a fundamental solution (or parametrix) of the differential equation to solve the differen tial equation, though not yet the boundary condition. This then must be followed in common theory by solving another system of equations- for example a singular integral equation over the boundary, in case one uses the well known boundary layer approach. The latter corresponds to the inversion mod K(H) of a (system) 1+E , with E E E .
Presently we do not study such manifolds with boundary, but rather turn to a pair of different examples. First (sec.l) we consider an algebra over I , where some of the generators are periodic functions. (It was seen in V,5 that a commutator [ e1X,DA]
is
VIII.O. Introduction not compact)
.
It is found that again a 2-link chain results. Now
we get
E/K(H) E C(N,K(t2))
(0.2)
with the Hilbert space t2= L2(Z)
,
i
S1 U S1
=
and two disjoint copies S1+ of
,
the circle S1
In direct relation to the above special importance of L2(Z) this algebra is invariant under translations by 2kir . Again there
is a tensor decomposition of H involved, together with a certain unitary map of H. Results are related to those discussed in Gohberg-Krupnik [GK), ch.l1, (cf. also Sarason [Sail
,
and [CMel1
where we expand on the present discussion).
Next, in sec.2, we look at a product manifold O=1n'>B, where B is a given compact manifold. The simplest example would be the circular cylinder XxSl , with the circle S1 We studied a compaXxSI rison algebra on in example (B) of sec.4, using an algebra .
A# of bounded Cm-functions having all derivatives tending to 0, at ±That including the derivatives in the 'S1-variables' algebra had a compact commutator. In sec.2 we admit more general .
,
generators, and obtain noncompact commutators. Again a 2-link ideal chain is found. Again the second quotient is of a similar form: E/K(H) __
(0.3)
C(ffi,K(h))
,
where I is the symbol space of a certain algebra of singular integral operators over in In case of n=l we get I = R U I , .
In each case we have h= L2(B). a disjoint union of 2 copies of I Again a tensor decomposition of H is involved, after a unitary .
transformation. The ideal E is identified as a certain algebra of singular integral operators with compact operator valued symbol. In section 4 the corresponding is done for a manifold with finitely many cylindrical ends. To make this discussion possible we introduce the technique of 'algebra surgery' in sec.3.
This simply means that we cut out the portion of a comparison algebra over some manifold 01
,
corresponding to an open subset
U C 0 , and compare it with the corresponding portion of another
algebra C2 belonging to another manifolds 02 , but with the same subset U
.
Of course algebra surgery also may be performed if com-
mutators are compact.
It should be noticed that sec.3 only discusses the basic
219
VIII.O. Introduction
method of surgery, at the example of a 'compact cut', - i.e. the boundary of the cut-out portion is assumed to be compact (not the cut-out portion itself). It is not hard to remove this assumption, at the expense of additional conditions near the infinite parts of the cut. We find that these additional conditions are too complicated to state, except for very specific geometrical models. For a discussion of surgery on a manifold with polycylindrical
ends we refer to a[ CDgl]
,
sec.5.
In all three examples considered here the symbol space of C (i.e. the maximal ideal space of CIE) is a compactification of the wave front space. In other words, the secondary symbol space is void, fon each of these examples. The theory of sec.4. has an application to a problem on In, studied first by Nirenberg-Walker[ NW1J, then by Cantor [Ctl],
Lockhart [Lk] and McOwen [ M1] (cf. also [ LM] ). This involves differential operators with coefficients constant at infinity and homogeneous symbol in certain weighted Sobolev spaces, where derivatives of different orders have different weights. Details of this application will be discussed elsewhere. We note that there are numerous other approaches to singular boundary problems on manifolds with cylindrical or conical ends (cf. Agmon-Nirenberg [AN1]
,
Bruening-Seeley [BS1,2] '
Melrose-Mendoza [ MM]
,
Schulze [ Schu1]
). Also let us point to
a direct extension of our present theory discussed in [CFb] and It turns out that the E-symbol can be extended to the
[ CMe1]
.
entire algebra C One thus obtains Fredholm criteria without involving a chain of two inversions. .
1. An algebra invariant under a discrete translation group. Let us consider the Laplace comparison algebra C on I of example (G) of V,5 with generators V,(5.27). We have discussed the commutator ideal E in V,prop.5.5. Now we want to obtain more details about the ideal chain C D E D K(H) and the quotients CIE and E/K(H). First we study the symbol of the algebra C .
Proposition 1.1.
The compactification M A
#
of the function class
A# described by V,(5.27), as a point set, is given by collapsing in the compact each circle {(x0,y):yESl}, for a fixed x0EI infinite cylinder [-o,o]xSl , into a single point. The points of ,
220
VIII.l. Periodic coefficients the two circular 'caps'
1(-co,y):y ES1} and {(co,y):y 61} correspond
to the maximal ideals {a(x): limk}-.a(2sy+2kir)=0} (and with the limit k}+oo, respectively, for integers k) of the algebra C A#
The topology is the weak topology induced by CA#. In particular, the relative topology induced on the two caps is equivalent to the Euclidean topology of the circle S1 . Every neighbourhood of one of the points 6 of the circle caps contains an open neighbourhood of the point 0+2km, for all sufficiently large (sufficiently small) integers k corresponding to the cap at respectively. ,
The proof is left as an exercise. Theorem 1.2. The symbol space ffi of the algebra C generated by A#
andD# of
(5.27) is homeomorphic to a compact subset of M #x[-co,+wl. Using the homeomorphism as an identification we have A ,
(1.1)
ffi
That is,
ffi
= M
A is the closure of the wave front space in the above
product. The symbols of the generators are given by QA(x,E)=a(E)
(1.2)
,
cDA(x,E)=s(E)
, ca(x,E)=a(x),
where the functions A , s and a must be continuously extended. Proof. We use the argument of Herman's Lemma again (cf.[C1],IV,2): ,
The algebra C has the two commutative subalgebras C0 = C # and A C# = algebra span of the operators S=s(D) and A=a(D). These alge-
bras have the maximal ideal spaces M #, and (the interval) A respectively. Clearly both of them together generate the algebra C.
be the duals of the injections C0
Let r0:M-M # and iT #:M A
iC and C# + C
.
Then u:ffi + MA#x[-co,+co]
,
defined as u=TrOxTr#
defines a continuous map onto a compact subset of the space at right. This map must be 1-1, hence a homeomorphism, because 7T 0(m) _ 70(m') and n#(m)
=
7#(m')
,
for m,m'E ffi implies that the
homomorphisms hm:C+M and hm,:C-M corresponding to m and m' coincide on C0 and
C#, hence on C
.
Then hm=hm, implies m=m'.
Now N must contain the set $x{-o,+oo}
,
identified as the
221
VIII.l. Periodic coefficients
wave front space, by VII,thm.l.5. Indeed, the map n0 is identical
with i of the proof of VI,thm.2.2 over interior points of O. The cosphere at each x0E I contains exactly two points which must 70-1(x0) Since N is closed, it must agree with the two points of i.e., the space (1.1). On the other contain the closure of W , hand, N cannot contain any (x0,E0) in the product set having 0 Then f(D)EE finite. Indeed, let 0ECO(I) be such that Using the associate dual map one finds that the and also E C On at m=(x0,E0) symbol of A=O(D) assumes the value the other hand aA=O since AEE (by V,prop.5.5), a contradiction. This shows that N coincides with the set (1.1). Then formula (1.2) is a consequence of the fact that aa(m) = a(s0(m)), aECO and af(D)(m) = (x#(m)) , O(D)EC , by virtue of the properties of the associate dual map (cf.[C1],AII,5). Q.E.D. Next we examine the quotient E/K(H). In fact it will be possible to obtain a much more precise control of the ideal E. First, in that respect, it is useful to look at the C -subalgebra F of E with generators .
.
(1.3)
j=-Naj(D)eiTx E E , ajECO(l)
E
, N=0,1,25...
.
Note that the equation (1+E)u=f , for such an E, is equivalent to the linear 2N-th order finite differences equation (for the inverse Fourier transforms u' and f') (1.4)
u' (x) +
j=-N
a.(-x)u'(x+j) = f' (W )
,
x E I
J
Here (1.2) relates only the values of the vectors (1.5)
u'=(u )=(u'(x-j)) i=O,±l i
for each fixed x
,
O<x
.
...fA=(fj)=(f'(x-j))j=0,±1,... I
In fact, for each such x we get an
infinite system of equations (1.6)
uj+Lk..
= fj
Najkuk
,
j=O,±l,... , with ajk=aj-k(j-x)
Note that the infinite matrix A=((ak))J. set a J.
=0, as lj-kl>N
space Z2= L2 (Z)
.
,
J,k=0,tl.... ' where we defines a compact operator on the Hilbert
(Such a matrix, with the property that all
entries outside a finite number of diagonals vanish, is called a Toeplitz matrix.) The present Toeplitz matrix has the additional property that the elements in each diagonal have limit zero, as
222
VIII.l. Periodic coefficients JjJ-°
,
since aieCOW . This implies compactness of A . (To prove
this, observe that the matrix obtained from A by replacing all entries ajk , with JjJ,JkJ<M , by 0 has norm <e , as M is large, ,
by Schurs Lemma [C1],I,l. This means that A differs from an operator with finite rank by an arbitrarily small amount,in norm. Thus it must be compact.)
Notice that the vectors u° x E I
,
f° of (1.3) vary with x
,
,
as
and that we have the 'periodicity condition' u°(x+l) = Y-1u°(x)
(1.7)
f°(x+l) = Y-1f°(x)
,
,
with the shift operator Y-1:C2i.22 defined by Y_1(uj) _ (uj_1)
Correspondingly the matrix A is a function of x and we have
A(x+l) = Y-1A(x)Y1
(1.8)
x E I
,
.
In order to convert (1.7) and (1.8) into true periodicity one may 0
of
2 .
For example one may define
(1.9)
Y t
t
((u(j-k+t))).
,k=0,±1,...
,
tEl
,
with the entire analytic function u(T)=e17rTsin7rT/1rT, T90, V(O)=l.
It follows easily that Ymu°
_
(uj+m)
,
so that indeed Ym shifts
the components of u°=(uj) by m units. Moreover we get Yt+T=YtYT, as t,TEI (i.e.,Yt is a 1-parameter group). Also Yt is norm continuous and even C0(l,L(t2)) and unitary. All this follows from the fact that Yt, in essence, may be obtained as the conjugation by
the Fourier transform of Z of the family of multiplications (of d function in L2([0,1])) by e2 rtit0 The Fourier transform of Z .
is defined by
(1.10)
u°
_ (uj) -* u-(8) _
Zj=__uje27ii8
,
8 E [0,1]
Introducing u° (x)=Y-xv° (x), f° (x)=Y-xg° (x), A ° (x)=YxA(x)Y-x, we find that (1.7) and (1.8) go into
(1.11)
v° (x+l)=v° (x)
,
g° (x+l)=g° (x) , A(x+l) = A° (x)
In other words, v(x) and g°(x) now may be regarded as functions on S1
,
the one-dimensional circle obtained by identifying the
endpoints of the unit interval [0,1) . In fact we get v°, g° E L2(S1,.22)
, and, moreover, using that Yx:t2+22 is unitary,
223
VIII.l. Periodic coefficients (1.12)
uuu2=JRlu(x)I2dx
=
J 1dtlly°112
.
S
Thus we have proven:
Proposition 1.3. The map u
u° + v° defines a unitary map Z:H+H°,
with H=L2(it), H°=L2(S1,L2) . Moreover, for any EEE of the form
(1.3) the operator E°=ZEZ-1 E L(H°) is a 'multiplication operator' (1.13)
(E°v°)(t) = A°(t)v°(t)
,
v°(t) E H°
with a continuous function A°(t) E C(S1,K(t2))
,
.
Proposition 1.4. Conjugation with the operator Z-1:H°+H takes the algebra FCE onto the class C(S1,K(t2)) of all multiplications of the form (1.13)
.
Proof. This is an immediate consequence of the Stone-Weierstrass theorem for algebras of matrix functions: (i) At each point tES1 the algebra of all 'values' of functions A°(t) obtained from E of 2
the form (1.3) is dense in K(Z ), since by a special choice of the functions a.(x) it always can be arranged that only the coefficient a°jk , for a given fixed index pair j,k, is different from zero, while all other coefficients of the matrix A°(t) , for the given t, vanish. (Note that conjugation by Yt does not change this since Yt is unitary.) Also, (ii) F is invariant under multiplication by b(D) , with bEC"(R) , b periodic with period 1. Such operation translates into a multiplication of A°(t) with the function bEC(S1)
.
This proves the statement.
Proposition 1.5. A multiplication u°(t) + A(t)u(t) by an A° (t) E C(S1,K(t2)) is in K(H°) if and only if A° (t)EO Proof. Let A(t0)#0
.
.
There exists wEl2 with uwp=l and fA°(t)wp
Then the class of all > p>O in It-t0I<E , for some p>O, E>O E C0'((t0-E,t0+E)) defines an infinite dimensional subu°= w .
space of H° on which we get
II A° u° ll p u° II .
Hence A° (t) cannot be a
compact operator, q.e.d.
In order to obtain full control on the ideal E it now appears natural to investigate operators of the form B°= Za(x)Z-1 , where
aECS(1g). These operators will only appear in the form B° E° or E° B° , with an E°EZFZ-1
.
Getting an explicit form for B°
implies that
we can carry out all investigations on E by looking at E°=ZEZ-1 only.
Note that a(x)E E K(H) for all a E CO(s). Also we may get
224
VIII.1. Periodic coefficients
225
restricted to the special case a(x)=s(x)=x/(x) 8S(l). Then we have a(D)=(s'*) equal to the convolution operator with kernel s'=F-1s The properties of s' are well known: Using [C1],II, for example, one finds that s'ESad
.
The singularity at 0 has a homogeneous We have s'EC-(1\{0}),
expansion mod D with leading term c1/x
.
and it equals a function in S outside lxl
. More directly, s' may be expressed by modified Hankel functions, showing the
same feature, and even that it decays exponentially with all deriUsing a cutoff function XEC'(E), X=l near 0,
vatives, as lxl+oo
.
X=0 for Ixl>1/4 we write s'
+ w', where 1=Xs' , wv=(1-X)Q' It follows that w'ES , hence wESCCO . Accordingly s(D)=p^(D)+w(D). where one may neglect the second term for reason mentionned above.
In fact one may split again 1(x) = using the asymptotic expansion of s'
i-(x), where p-^ECO, .
This discussion shows that it suffices to assume (1.14)
Tu'(x) = a(D)uv(x) =
ip(z)
= X(z)/z
with a cut-off function X as described above.
Proposition 1.6. For the operator T of (1.14) and T'=(ZF)p(ZF)-1 we have (1.15)
T'v°(x) = J*(x-y)Yx-yv'(y)dy
,
v =(vj(x)) E
C'(S1,L2),
where Yt:t2+L2 denotes the operator of (1.9), and where we interpret v° and T'v° as periodic functions (1) defined on I
.
Proof. Given v '(x) _ (vj(x)) , where the v. are assumed defined
over I and periodic (1) (u'(x-j)) = Y-xv (x) (1.16)
.
.
Let u = Z-lv
.
We then have u '(x)
Or, using the periodicity of vk
u'(x-j) _ Ik u(j-k-x)vk(x) _ Ik u(-(x-j)-k)vk(x-j)
We replace x-j by x and then apply the operator ' of (1.14), for (1.17)
Finally let g°
f 1(x) _ 'u'(x) = Jdylk (x-yWu(-y-k)vk(y).
= (gj) = Zf
.
We get g° (x) = Yx(f'(x-j)). Or,
(1.18)
Note that near each x,y only one term of the infinite series 11 can be 90, since supp V C {IxI:1/4}
, by construction of 1 , hence
VIII.l. Periodic coefficients
interchange of sum and integral in (1.18) is justified. Using once more the periodicity of vk we shift the integration variable, for gj(x) = Jdy*(x-y) lklu(j-l+x)u(1-k-y)1Pk(y)
(1.19)
Note that I1u(j-l+x)u(1-k-y) is the (j,l)-matrix element of the product YXY-y = YX-y , using the properties of Yt of (1.9). Thus, (1.20)
gj(x) = Idy lp(x-y)yku(j-k+x-y)vk(y)
,
where the integral extends over I but may be extended from y=x-1/2 to y=x+1/2 only, due to the properties of supp Q.E.D. It is convenient to introduce the periodic matrix kernel ,
.
(1.21)
'Y °(z) = rV(z)Yz
as JzI<1/2 , periodic (1) elsewhere.
,
Then (1.15) assumes the form
(1.22)
Y'°v° (X) = J
1
`Y° (x-y)v° (y)dy
Moreover, using that the group Yt is not only continuous but even CW(&) , in norm convergence, and that Y0=1, we conclude that
`Y° (z) - Trcotirz = (z) E C(S1,L(k2)).
(1.23)
Proposition 1.7. For every E° E F° we have
(1.24)
'Y° E°
- TrH° EA E K (H°)
where (1.25)
H°u(x) =
J
1cotTr(x-y)u(y)dy
denotes the Hilbert transform of S1 Indeed, an operator E° is a multiplication by a compact operator valued function E(x)E C(S1,K(t2)) , by prop.1.4. Thus
the right hand side of (1.24) is an integral operator with norm continuous kernel taking values in K(.62) Such an operator must .
be in K(H°)
,
q.e.d.
At this point it will be useful to recall the topological = L2 (S1,22) . Defining
tensor structure of the Hilbert space H°
h = L2(S1) we find that
(1.26)
H°
= hg.C2
226
VIII.l. Periodic coefficients
with the topological tensor product of the two Hilbert spaces, as defined in [C1],V,8 (or [BC2]). On Sl we introduce the C*-algebra SL of singular integral operators, i.e. just the unique Laplace comparison algebra of the compact space S1
.
Clearly SL
has compact commutators and its symbol space ffiSL coincides with
wave front space, since S1 is compact. Note also that the space since the unit sphere ffiSL consists of two disjoint copies of S1 ,
in I consists of two points only. Theorem 1.8. The C*_subalgebra E° = ZEZ-1 of L(H°) coincides with
SL. = AK(t2) . Moreover, the quotient E°/K(H°), and hence also the quotient E/K(H) is isometrically isomorphic to the function algebra C(ISL,K(t2)) This isometry is such that an operator E° = (multipliation by) x°(x) = ((ejk(W))) has the symbol For an operator E of the form ((ejk(x)) on both copies of Sl the algebra
.
.
(1.3) the symbol on ffiSL = S+ U S1
IE(x) = E°(x)
(1.27)
,
is given as
x E S1+
,
E°
Moreover, for an operator A = a(x)E, EE=-F
,
=
ZEZ-1
we get
YAW = a(±')E°(x) , xE S1+
(1.28)
We shall offer only a skeleton proof. For details cf.[CMe1]. It may be shown that the generators V,(5.33) transform into terms of the form + K , X+ = 1/2(1±iH°)
X+E°T + X-E°
(1.29)
,
with the Z-1-conjugates E°+ of the two terms E+ E F occurring in V,(5.33). Here it must be verified by a calculation that the Z_ conjugates of E+(x) equal the terms X+ , modulo a term W+ such W+E°+ E K(H) This calculation was prepared by prop's 1.6 and 1.7, but its details are left to the reader.
that
.
t2
with j-th component =1. From Let e3 be the unit vector of Using prop.l.4 we then get all V,lemma 1.1 we get K(HA) C E° .
multiplication operators a(x)M(ej)(e3) contained in E°, where jEZ and aEC-(S1) are arbitrary. Then also H°a(x)M(ei)(ej) (=prop.l.7
.
E°
, by
Now it follows that all operators AN(e3)(e3), for A E
SL are contained in E°
. Repeating the same conclusion with
ej
replaced by e3+e1 we find that also AM(e3)(el) , for AESL, j,lEZ, Thus, letting KN denote the set of all infinite matriis in E° .
ces CEK(k2) with components zero outside IjI,Ikl:N , we find that
227
VIII.l. Periodic coefficients
AOKEE° for all AESL , KEKN . In other words, SLIKKN C E° . In [C1], V,8 (or [BC2]) we have shown that SLR = SLIRK(Q2) is the closure of SMUKN) . Since E° is closed we get EADSLO, and equality follows since the opposite inclusion is trivial. Finally, the statements about the quotients are immediate consequences of an investigation for abstract algebras with symbol in [BC2], sec.6. Q.E.D. 2. A C -algebra on a poly-cylinder. Next we look at a Laplace comparison algebra with noncompact commutator on a poly-cylinder Q=Rn xB. Here B denotes a compact Riemannian space of dimension n" with metric dp2=gjkdxjdxk. r
Accordingly, for the metric and the Laplace operator of 1 we get (2.1)
ds2= dt2+ dp2
,
jgjka
Ax=(Vg)-la
A = At + Ax
,
where Ax is the Laplace operator on B
k
x
xi
In (2.1) we are using the
'Euclidean
metric dt2=dt1 +...+dtn
.
of Bn ={t=(tl,...,tn ):t.EX},
and the Euclidean Laplace operator At=j8 j2
.
The summation con-
t
vention often will be used from 1 to n", as will be clear from the context. We set n=n'+n", so that Q is n-dimensional. Let H be the Hilbert space L2(0) = L2(c2,dS) , with the sur-
face measure of the metric (2.1). Let C C L(H) be the smallest C*-subalgebra containing the (5 types of)operators
(2.2)aEAB, sj(t)=t/(t),
A=(l-A)-1/2,
jA, DxA, DXED#, j=l,..,n'.
a
t
Here (t)=(l+t2)1/2; we write AB=COO(S) while DB denotes the collection of all Cam-vector fields on B . Also A is the unique posi-
tive inverse square root of the (unique) self-adjoint realization (1-A) of the Laplace differential expression A There is a unique such self-adjoint realization because 0 is a complete Rie.
mannian space (cf.IV, cor.1.7). The functions aEA# and sj(t) in in (2.2) represent the corresponding multiplication operators on functions u(t,x) E H. Correspondingly, we denote by A# and D# the algebra finitely generated by C_ and the first two kinds of generators, and the left module of A # spanned by {1, 8 tj
,
D x},
228
VIII.2. Polycylinder algebra
229
Au and DXAu are well defined for
respectively. The operations a t
uE A-1C0(S)), a dense subspace of H (cf.V,l) and have continuous extensions to H (cf. also (2.6), below). We notice that H = h Rh is the topological tensor product x
of the Hilbert spaces ht= L2(in ), and h = L2(B) (cf.[C1],V,8). x Let us write Lx = L(hx), kx = K(hx), Lt = L(ht), kt = K(ht). It^may be observed that the topological tensor products Lt'Lx=Ltx' K(H) , , k 9L = Kt are all well-defined C -subLt9 algebras of L(H) , where K(H) C Kx , Kt C Lxt C L(H) all are pro-
xx
tN x
per inclusions. In fact, Kx and Kt are proper closed two-sided ideals of Ltx , and, of course, K(H) is a proper closed ideal of
all the others. Evidently, Ltx (but not Kx or Kt) contains the identity operator I
.
Note that we may write H = L2(In ,hx) as the space of all functions over In ' with values in h = L2(B) such that x
(2.3)
11u02
n,dtUu(t,.)N2 <
)
R
by Fubini's theorem. Correspondingly, if CB(In',kx) and CB(Rnl,Lx) Ins
denote the classes of bounded norm continuous functions over taking values in kx and Lx, respectively, then a function P E CB( ln ,Lx) has a natural interpretation as an operator in L(H), defined by u(t,.)-; Vt)u(t)
. Moreover, this operator is in Kx
whenever 4D E CO(I,kx) (cf.[C1],V,B). (We indicate a proof of this
fact in cor.2.4.) This establishes an isometric *-isomorphism of CB(l;Lx) (as a Banach algebra with norm (2.4)
sup{NA(T)N
:
T E 11
,
is the norm in Lx) into L(H). In the following we will use this interpretation of functions as operators in L(H) where IIA(T)II
In order to find the commutator ideal E of the unital Calgebra C with generators (2.2) we conjugate the generators with the Fourier transform in the t-direction. In detail we have (2.5)
Ftu(T) _ (2n)-n'121
n,e-itTu(t)dt
which defines a unitary operator of ht
,
.
T E Ins
,
By conjugation with Ft
we, of course, mean conjugation with Ft0Ix, where Ix denotes the identity operator. (However,we will write this as Ft-AFt ,for l ) First consider the Ft-conjugations of A, D .A, D A E L(H). A t
X
:
VIII.2. Polycylinder algebra
A-(T) _ ((T)2-
(2.6)
Ax)-1/2 ,
TjA-(T) , DxA-(T)
,
with Dt.=-iat. and above interpretation of a function as operator. Since B is compact we have A"(t) E K(hx), for all T (cf.III, cor.3.9). Moreover, we even get A"'(T) E CO(lnt,kx), where CO CCB
denotes the class of functions with limit 0 at infinity. In fact, A -(T) even is analytic, in norm topology of L(hx) , and we have
IIA- (t)II
< (T)as well as IIA-(0)-lA-(T)II
so that
,
T.A-(T) J
DxA-(T)=(DxAx)(A-1 A-(T)) are functions in CBU nt ,Lx)CL(H) , confirming the boundedness of the last two types (2.2). Also, writing Ax=A-(0), the generators (2.2) correspond to T(T)=(l+T2Ax)-1/2
(2.7)
a(x), p.*
,
,
AxT(T)
,
(iTjAx)T(T)
,
(DxAx)T(T)
with Vj=Ft1s. and ut*= convolution in ht Proposition 2.1. The operator functions
AXT(T) and
ITI1-CA
xT(T)
for each fixed s , O<E
Proposition 2.2. All commutators of the Ft-conjugated generators and their adjoints are contained in the algebra (2.8)
G^ = CO(]Rnt,K(hx)) + K(H)
C K x
Here a function C(T) E CO(&nr,K(hx)) must be interpreted as an operator in Kx in the manner described above. Proof. We will use the resolvent integral technique used in V,3. Let the operators ( 2 . 7 ) be denoted by G1,...,G5, in the order lis-
ted (we write t=t3, at=atj, T=Tj, etc). Clearly [ Gi,G2] _[ G3,G4] =0. We get (DxA-)*-DxA-=[A ,D*]ECO(k n',kx), by (2.14) below, and
adjoint invariance of DB). All other generator(classe)s are selfadjoint. Hence the adjoint generators need no special attention. For the commutators [G.,Gl],j#2, we use the well known resolvent integral representation of A-=G3: Let R(s)=(s+(T)2-Ax)-1 Then we have
230
VIII.2. Polycylinder algebra G3=A-(T)=AT(T)=((T)2-A
(2.9)
0
(cf. VI,(1.6)), with a norm convergent improper Riemann integral, in the algebra L(hx) . We get (2.10)
[G3,G5]=[A-,Dx]A-, [G4,G5]=[A-,Dx](TA°), where A-,TA-ECB,
so that it suffices to show that [A-,D x] E CO(R,kx) Similarly, [Gi,G4l=(TA»)(A"'-1[ax,A"']),
[Gi,G3]=[ax,A`], (2.11)
px=[ax,Dx]EC'(B).
[Gi,G5]=pxA-+(DxA -)(A--1[ax,A-]), with
Accordingly we also must show that A"-i[ax,A-] E CO(R,ftx)
Both these facts are consequences of prop.l.3, below. Before we discuss it we turn to the commutators [G2,GI]. There we find it practical to work without the Fourier transform (2.5), writing (2.12)
A
=
S(r) _ (r+1-At-Ax1
1/Tr 0
Instead of 'diagonalizing the t-variable by using the Fourier transform' we will consider the x-variable diagonalized later on. Let Aj =FtGi Ft-1 be the generators (2.2). Then we have (2.13)
[A2,A4]=ptA+(8tA)V, [A2,A51=(DxA)V, V=A-1[A 2,A31, PtEC
Note that ptECO(RnT) hence p+AEK(H) by a well known result (cf. III,thm.3.7). We claim that all 3 commutators [A2,A11, ,
1=3,4,5, are in K(H). Again this is a trivial consequence of prop. 1.3 below, so that all of prop.l.2 has been reduced to prop.1.3. Proposition 2.3. For c with 0
A"-1-E[ax,A-] E COOL n',hx)
(2.15)
A-1-E[sj(t),A] E K(H)
,
,
A--E[Dx,A-] E CO(Rn',hx)
si(t)=tj/(t)
.
Remark. As a consequence of prop.2.3 the algebra C satisfies condition (m4) of VII,3 as required later on (cf.thm.2.8). For the proof of prop.2.3 we use (2.9) for
(2.16)
Li=[ a(x),Ax] .
A --1 E[ Gi,G3] =1/Tr 0
231
VIII.2. Polycylinder algebra
232
The integrand in (2.16) is a norm-continuous (even analytic) function F(s,T), with values in kx Indeed, one may write .
(A--1-ER(s))(LlAx)(A-1R(s)//-s)
F(s,T) _
(2.17)
and use the estimates (2.18)
IIA_"R(o)I<(1+s+T2)p/2-1,
x
IIA`-nR(s)1<(1+s)n/2-1,
0
easily derived from the spectral decomposition of the self-adjoint operator -Ax>0 of hx. Note that L1 is a folpde on B independent of s,T, so that the second factor in (2.17) is a constant in L(hx). Analyticity of the first and third factor is a consequence of ana0((l+s)(E-1)/2)
lyticity of the resolvent R(s). These factors are and
0(s-1/2(1+s)-1/2),
uniformly for all TEIR
respectively. Thus F(s,T)=0((1+s)-1+E/2/yrs-)
. Also both factors are in kx since B is
compact, insuring the compactness of the resolvent R(s) of the Laplace operator Ax (III,thm.3.1). This implies existence of the improper Riemann integral (2.16) in norm convergence of L(hx) and uniformly so , for TEX Thus the integral is in For more detail in such a proof cf. V,3).Moreover, since 0<E<1 .
is arbitrary, one may use this for E+6<1 , with some 6>0 This gives a factor A-6ECO, whence the first (2.14) is CO, not only CB. .
Similarly we may use (2.9) for (2.19)
A--'[ Dx A-]
=1/7J(A` 0
where the integral exists for analogous reason. (Now the second factor contains the second order operator [Dx,Ax] over B so that again the factor is a constant in L(hx).)
For (2.15) we use the other resolvent integral (2.12), for (2.20) A-1-E[ s(t),A] =1/rr
P
M1=[ s(t),At]
0
Now, with respect to an orthonormal base of the Hilbert space hx consisting of eigen-functions of the self-adjoint operator Ox, the operator S(r) is diagonalized, with respect to hx
in
the tensor product decomposition of H. Its diagonal components are (2.21)
S.(r) _ (r+(A.)2-At)-1
with the eigenvalues X of the positive self-adjoint operator on B. In analogy to (2.18) we conclude that, with
Ax
At=(1-pt)-1/2
VIII.2. Polycylinder algebra IIAtnS.(r)II<(l+r+A])fl/2-1,
(2.22)
BAt-nS.(r)II<(l+r)'1/2-1
,
Aj=(1+X2+Ot)-1/2
for j=1,2,..., and At=(l-At)'1/2 Notice that the entire relation (2.20) is 'x-diagonalized', i.e., decomposes into a set of countably many relations, involving At and S. instead of A
, j=1,2..... . Again one may write the integrand Fi(r) as a product of three factors:
(At-l-esj(r))(M1At)(A-1Sj(r)/V-r)
F.(r) _
(2.23)
where the second term is a constant in L(H) 0((l+r)(e-1)/2)
third term we get estimates 0((aj)-d(1+r)(b-1)/2r-1/2),
(2.24)
IIF.(s)II
.
For the first and
and
for any 0<6
=
3
3
The right hand side is integrable, as long as e+6<1
The integral
.
fFj(r)dr = Cj is an operator in K(ht), and we get
(2.25)
HCj II
= 0(( aj) - d)
.
Accordingly the operator (2.20) corresponds to a diagonal matrix ((Cj6kl))k,1=1,2,,,,
. with diagonal components converging to 0
in norm. This indeed implies that (2.20) belongs to K(H). (It is limit in L(H) of the sequence of diagonal matrices Tk obtained by setting all Cj
,
j>k, equal to zero, while the matrices Tk are
in K(ht)HF(hx) C K(ht)ZK(hx) C K(H) , with the class F(hx) of
bounded operators of finite rank over hx .) This completes the proof of prop.l.3.
Corollary 2.4. The C*-algebra G^ and its Ftl-conjugate G are
subalgebras of Kx = Ltt kx Proof. It is sufficient to show that
C Kx
.
For any
orthonormal basis TOi:j=1,2.... }, and the orthogonal projection Pn onto the span of TO,.... ,O } we get uniform convergence
PnC(T)Pn+C(T), TEl ,for every C(T) E CO(ln ,K(hx)). But the operator PnC(T)Pn is a finite sum of operators E CO(s)
.
Thus PnC(T)Pn E kx
Proposition.2.5.
cjl(T)
and the limit C(T) as well, q.e.d.
The commutator ideal
GO of the C -subalgebra of
L(H) generated by the operators of the form G1,G3,G5 contains the
233
VIII.2. Polycylinder algebra
234
algebra CO(1nt,hx).
Proof. For a moment consider only the C -algebra B generated by By virtue of the isometric isomorphism mentionned G1,G3, and G5 .
initially in this section, the generators of B belong to the function algebra CO(Int,L ). Hence the algebra B may be interprete x
,Lx). For a fixed T=T0E gnu the values
as a subalgebra of CO(I'
BT ={A(T0):A(T)EB} form a *-subalgebra of Lx
It is clear that
.
0
is a *-subalgebra of the Cat-subalgebra CT
BT 0
of Lx generated by 0
the values of the functions Gj, j=1,3,5 at TO
,
i.e., by the
operators ax
(2.26)
A-(T0)=(-Ax+(T0)2)-1"2
,
,
DxA-(T0)
,
where ax and Dx run through all the functions (folpdes) over B . Also C is just Moreover, C evidently is the closure of B TO
TO
TO
the minimal comparison algebra in the sense of V,1 generated by the triple {B,-Ax+(T0)2,dS} , on the compact manifold B By .
V,lemma 1.1, it follows that CT
and even its commutator 0
ideal contain all of hx
.
But commutators in CT
are compact,
since B is compact. Therefore the commutator ideal of CT
equals 0
h
x
.
Since that commutator ideal is the closure of the commutator
ideal ET
TO
0
of the finitely generated algebra, we must have ET
dense in hx
TO
.
On the other hand E0
clearly is contained in the 0
, and even in the localization
commutator ideal of the algebra BT 0
at TO of the commutator ideal GO
GOT
.
Thus we conclude that the
0
algebra GOT of 'values' of GO at T is dense in k.x, for all
TEXn'.
We also find that the algebra B contains A_(T) , hence also , by the spectral theorem and the Stone-Weierstrass theorem. Hence B con-
contains every f(T,Ax), for a general fECO(]Rntx(0,11)
tains all functions *(T)Ex, with lECO(Xn') and the projection operators EX of the spectral family of Ax-1 . Note that EX are of finite rank, and that EN3l , strongly, as N+-. Since GO is an ideal of B , it follows that GO contains (2.27)
GON = {1P(T)ENA(T)EN
:
A(T)E GO, ip(T)ECO(Xn)}
.
VIII.2. Polycylinder algebra
But GON is a self-adjoint algebra of (finite) jNxjNmatrixvalued functions, separating points in the following sense. For e>0,and jNxjN matrix P there exists K(T)EGON such
every
that K(T2)=0 and IK(T1)-Pl<e (This follows from the above). By the matrix version of the Stone Weierstrass theorem this implies that GON=CO(In' L(T
JN
)). Since this holds for all N we find that
GO contains all these matrix algebras. But for a general iN
A(T) E CO(In' kx) we get AN(T)=ENA(T)EN E CO(En' L(C )). Also AN(T)-A(T) ; 0, in C0(lntkx) (with the norm (2.4)), since A(T) is compact and EN converges strongly to 1 Since GO is closed we conclude that A(T)EGO , so that indeed CO(l,kx) C GO , q.e.d. .
Returning to our task of describing the commutator ideal E of the cylinder algebra C we conclude from prop.2.5 and V,lemma 1.1 that E contains the C*-algebras GO' = FtGOFt1 and K(H).
It is convenient again to work with the Ft-conjugated ideal E^ Ft1EFt , containing the sum GO+K(H)=SO (which in turn contains all the commutators of the generators G., j=1,...,5)
Notice that SO is invariant under left and right multiplication with the (functions in CB(I,kx)) G1,G3,G4,G5, but not Accordingly E- must be properly under multiplication with G2 .
larger than SO.
Specifically G2=sj(Dt) is a singular convolution operator with Cauchy-type singular integral and kernel p.=sj', so that a product K(T)G2
for K(T)ECO(ln ,kx), appears as an infinite matrix of singular integral operators on Ins, if we introduce ,
some orthonormal base of hx.
Theorem 2.6. Let SQ be the C -algebra of singular integral operators over ht generated by the multiplications in CO(In ), and the operabrs a(M)s .(D), aECO(e'), j=l,...,n'. Then the ideal E^ coincides with the J topological tensor product SQ. = SQNkx .
Proof. Notice that SQ coincides with the minimal comparison algebra of the triple {in',dt,l-ot} (cf. ch.Vl). Thus it contains the compact ideal kt of L(ht), and SQ. contains ktNkx=K(H). Therefore it is trivial from the above that SQ. contains all the commutators j,1=1,...,5 and that it is a closed *-ideal of C Hence we have E'CSQ.. To show equality we introduce a fixed orthoEGj,G11
,
.
normal base l'2' . of the space hx and first consider K(T)E GO
235
VIII.2. Polycylinder algebra such that K(T) takes span
236
into itself and its ortho-
gonal complement to 0, for all T. Thus the infinite matrix vanishes outside its first N rows and columns. Let CON denote the subalgebra of CO(I
,kx) of all such finite matrices, for a given fixed N. Clearly CON is isometrically isomorphic to CO(ln ,L(CN)). Now we observe that K(T) and L(T)s(Dt) , with K,LECON , belong to E^ , and generate the algebra SQN = SQNL(&N) for each N=1,2.... . Also we again find that S9, is the norm closure of USQN Therefore indeed q.e.d. ,
We now come to our main task: The description of the symbol chain of the cylinder algebra C First let us look at the ideal quotient E/K(H) In that respect we observe that the algebra .
.
SQ is a subalgebra of the algebra SI of singular integral operators on In' generated by Sj=sj(Dt)=(pj*), j=1,.... n', and the multiplications with functions in C(Bn ) , with the 'ball compactifiIns
Ins
cation'
of
, having one infinite point in each direction
ItOl=l (cf.[C1],IV,l,problems). The special comparison algebra SI C L(ht) has symbol space MI = {(t,T)EC(1n'xEn')
(2.28)
:
ltI+ITI=0}
(cf. ex.(A) of V,4, or [CHe1]). The subalgebra SQ of SI consists precisely of all operators in SI with symbol vanishing at , as follows from the Stone-Weierstrass theorem, looking at the symhols of the generators. (Note that the generators are
ITI
written as multiplications by functions of the variable T and convolutions by functions of T as well, since we consider the Ft-conjugated ideal E^ . Accordingly we have t and T reversed, compared to the normal notation for space and momentum coordinates.) It follows that SQ/kt is isometrically isomorphic to the function algebra CO(f) with the locally compact space compact space 1 = NQ = {(t,T)E NI
(2.29)
:
Itl=o'
,
ITI-} = 2Bn'xgn'
.
In case of n'=l this space is a disjoint union of the two sets {oo}x1K=1- and {_co}x$=1+ Both E± are copies of I , with the variable T running over 1. In the general case n'>l the space I is connected, and is a product of the infinite sphere 3Bn'=%n'\ln' with in Ins
Clearly this also is just the wave front space
,
but
with t and T interchanged. We have proven the following result:
VIII.2. Polycylinder algebra Theorem 2.7. The quotient algebra E/K(H) is isometrically isomorphic to the function algebra CO(L,kx),sothat E is a C*-algebra with (compact operator valued) symbol, with symbol space F In .
the special case n'=1 we have CO(E,kx) = CO(]E-,k x) (D COW,kx) .
(2.30)
The symbols of the generators of E are given as compact operator valued functions of (t,T), for tE3En' (i.e., t0Mnl, , as follows: Let A1, j=1,...,5, be the generators (2.2) of C, in the order
ltol=l) and TEIn
listed (so that Gj are their Fourier transforms). Then [A1,A2]=0= IA3,A4]. The symbols Y(A3'As]' YIA4,A5] are independent of t, as 1E311 ,the value given by the terms of (2.10), respectively, where A-(T)=AXT(T) , while the commutator IA"',Dx] is obtained from the resolvent integral from (2.19). Similarly 1[A1,A7], j=3,4,5, are independent of t, as ltl=°°, and, for TEIn1, their values are
given by (2.11) and the resolvent integral (2.16). the symbol of a product Aj(Ak,Al] or [Ak,A1]AI Also, for is obtained by multiplying tion G. of (2.6)
Y[Ak Al]
with the corresponding
func-
, from the left or right, respectively. Also,
for j=2, the symbols of these products equal the product of ](T) with the value of the function s(t) (extended contiY[Ak' A 11
nuously to 81n1) at t. More generally, for every function bEC(Zn1 the operator b(t)IAk,Al] E E has the symbol (2.31)
Y = b(t)y(A ,A
1](T)
, k,192
.
k
Next we turn to the discussion of the quotient C/E
,
i.e.,
of the symbol and symbol space of C *
Theorem 2.8. The C -algebra CIE is isometrically isomorphic to the algebra C(ffi) of continuous complex-valued functions over a compact space ffi, called the symbol space of C. Here M is (homeomorphic to)
the bundle of cospheres with infinite radius of the compactified poly-cylinder InfxB considered as a compact C--manifold with boundary (i.e., the product Bn1xB of the unit ball Bn1={tEgn1:ltl=l} with the compact manifold B ). in In1
Let A1,...,A5 be the generators (2.2) again. Then the C-sym-
237
VIII.2. Polycylinder algebra
bols aA =0A J
i.e., the functions in C(ffi)
, associated to
J
Aj by the above isomorphism, are given as explicit functions of (t,x,T,E) as follows: a
(2.32)
Al = a(x)
iT/(T2+I
aA4
, a A2
X2)1/2
,
= t/( t) '
a
A3
=
0
aA5 = (b3Ei)/(T2+IEI2)1/2
f
with (2.33)
ICI
Dx = b3(x)3 j+p(x)
=
x
in local coordinates of B , where tE,n , xEB , while (T,E)ESt'x, the co-sphere at (t,x) with infinite radius. (Actually the last two symbols are the limits of the full symbol quotients (2.34)
T/(1+T2+IM I2)1/2
,
and p-). as (T,E) is replaced by Proof. We may just apply the general results of ch.VII, first verifying their assumptions. Let us shortly summarize these facts. First of all every symbol space of a comparison algebra contains
the wave font space W , normally identified with the bundle of unit spheres in the cotangent space T*c of the underlying manifold St (cf.VI,thm.1.5). The space W is an open subset of
1
It
.
precisely coincides with the set of points (x,t,T,g), with ItI- . Essentially this follows whenever C can be shown to contain all Cs-functions and all operators DA for a general first order ,
differential expression with C--cofficients and compact support. In order to study the points at ItI=0 we require the compactification P*Q of T*0 induced by the formal symbol quotients aEAB and t/(t) , and (2.34) together with the functions a(x) (in other words, by the formal symbols of the ,
,
(1+T2+II2)-1/2
5 types of generators (2.2)). (That is, P 5
is defined as the
smallest compactification of T*(1 onto which all above functions
can be continuously extended.) It is readily verified that
P*St
*
is given as the compactification of T S2 obtained by adding the infinite sphere {oo(T,E)
:
ITI2+IEI2=1} to each fiber T
*
x of the Ste=PnxB
cotangent bundle T*cc
,
of the compactification
*
of S2
In other words, P a is the disjoint union of all balls {(t,x)}x1n of infinite radius, as (t,x)ESt c
238
VIII.2. Polycylinder algebra Moreover, S2c coincides with the compactification M of 52 A# defined by the functions a(x), t/(t)EC"(c), introduced in ch.VI,
and the algebra C satisfies conditions (ml'), and (mj), j=2,3,4, 5,7. (We noted before that (m4) is implied by prop.2.3. Cdn.(m5) is trivial: We have AEC. Cdn.(ml'), first used by McOwen, requires that the functions lp(x)I2 and gjk(x)Fj(x)bk(x)
(2.35)
,
are in A#, for every Dx=bJ(x)8 j+p(x)ED#. This again is trivially x
true, since A# contains all of C'(B)
.
Furthermore the conditions
(mj) involve some separation conditions which can be satisfied by enlarging A# and D# in such a way that the generated algebra C remains the same. Details are left to the reader. As a consequence we may apply VII, thm.3.6. The conclusion is that the symbol space N of C is a compact subset of the
boundary 31 0 = ff SAT *S2 of our compactification & X52
,
containing
the wave front space W , i.e., the bundle of cospheres of infinite
radius over Q. Since N is compact, it must contain all points of the infinite cosphere bundle over3nc as well. Thus it remains to be In partishown that no other point of 31 S2 is contained in ffi .
cular none of the points Itl=co, T2+IEI2<_ can be contained in N
This, on the other hand, is a consequence of VII,thm.4.2. To indicate at least the idea, we find that AEE , for our present algebra, while the formal symbol of A is
(1+T2+I02)-1/2,
i.e., is tO for the latter type of points. Hence such points can not be in IN
,
since the symbol of AEE must vanish, while VII,
thm.3.6 implies that symbol and formal symbol coincide for the t
points of 3P 12 which are in N
This completes the proof of thm.2.8. 3. Algebra surgery.
Let us assume cdn.(w) (at least)
,
in this section.
We already started discussing the relation between the minimal comparison algebras of different triples, although so far we only allowed a change of H and dp over a given compact set K C 2 In this section we will look at the comparison algebras of
239
VIII.3. Algebra surgery
triples over two different manifolds 01 and 2
.
Suppose two
2
triples {c21,dp1,H1} , and {12,dp2,H2} are given, and assume that
an open subdomain U1 of Q1 is diffeomorphic to a subdomain U
C 02, 2
in such a way that the two triples correspond to each other over In order to have simple notations we will regard the two .
U1-.U2
sets Uj identified by the diffeomorphism. This will bring a configuration as sketched in fig.3.1. We are given a manifold U and two different exten,
sions 01 and 02 of U. The two triples coincide over U . That is, we have du1IU = dp2IU H1JU = H2IU, Also we will assume that the boundary ,
or 1
3U of U is compact, although U itself may be noncompact. The last assumption may be removed, at the expense of additional
U (noncompact)
02
conditions at the infinite ends of 3U. Fig.3.1 For a discussion of manifolds with polycylindrical ends involving such noncompact cuts cf.[CDg],sec.5. We shall discover a 'common part' of the two comparison
algebras C=C(0,D!), induced by classes A
,
D! over fj
,
j=1525
whenever the classes of restrictions of functions (folpde's) to U coincide, i.e., AU = A#JU = A2IU , DU = V
(3.1)
U = D2JU
We introduce the Hilbert spaces H1=L2(c1,du1), H2=L2(22,d112), and the space H=L2(U,dpj) . Any CD(U) function will also be considered a C0-(Q.)-function (with support in U ), without change in
notation. Similarly the functions u E H1 with support in U will also be considered functions in H or in H2, and vice versa. Also, for A E L(H1) and Q E C0-(U) we may regard A as an operator in L(H2,H1) and similar.
Clearly we will get corresponding maximal ideal spaces M Al
and MA2# , and corresponding associate dual maps il:Il- MA # and 12:M2; M
#
. Note that the restriction to U of an a E A# induces
A2
an algebra homomorphism C(MAl#) i C(MAU#) which identifies the closure of U U 3U in MA # i
with the space MAU#
,
by virtue of its
240
VIII.3. Algebra surgery (injective) associated dual map. Similarly for M
#
,
so that
A2
MA # is found as a closed subspace of both spaces M U
# and M
#
A2
Al
i.e., as the closure of U U 3U Note that the common restriction {U,dp,H} , dp = dujJU
(3.2)
H = HjjU , j=1,2
,
,
represents a comparison triple of its own. As we know, this triple cannot satisfy (w) , unless 3U is void. However, as long as we use sets A# , D# of functions and folpde's vanishing near 3U , we may use the techniques of VI,l and V,2 to achieve (w) or more by modifying the potential near 3S2 .
Now, for the remainder of sec.3, we assume the following separation condition.
Condition (m3)U
:
For every aEA# and DED# with a=O near 3U and D=O
near 3U we have XUa E A#
,
XUD E D# , with the characteristic
function XU of UCc.
Notice that
(m3) U is always true if U U 3U is compact, and
that (m3)U follows trivially from (m3)
Accordingly we now define classes A# and D# on U by setting (3.3)
A#,O= {aEAU
:
a=0 near 3U}
,
DU,O= {DE=-DU
:
D=O near 3U}
and then consider the comparison algebra C = C(AU 0'VU 0) C L(H) We notice that AUTO in general does not contain the constant functions. Therefore this is one of the exceptions to the general In general C will not have a unit (except only in rules of V,l case where 3U=0 ). However, as a consequence of (m3)U it follows (al') of V,l. Also, we conclude that, for that AU 0 satisfies a,E AU D E DU every the trivial (i.e.=0) extension to S. .
0'
is contained in Aj
0
, respectively. We claim that C is the 'common part' of C1 and C2 ,
or Dj
,
in the
following sense.
Theorem 3.1. Let V. C C. denote the closed two-sided ideal of all A E C. with symbol]aA having support in the set tj-1 (M A #). Then U
we have (3.4)
V.
= C + E.
,
C = XUVjX3
241
VIII.3. Algebra surgery
with the commutator ideal E. of C. , and the characteristic function XU of the subdomain U C Q.. Here in the first relation (3.4)
an operator AEC must be interpreted as AXUEL(Hwhile in the second relation (3.4) we mean the restriction to its invariant subspace H of XUBXU for B E V. ,
Proof. First we notice that a statement like VI,prop.2.3 is valid, with exactly the same proof:
Proposition 3.2. The ideal V. C C. is generated by E. and all a, DA. with a E A,,0 , D E DU,O (extended zero outside of U ). After this prop. we must use prop.3.3, below. From (3.6) it follows that COXU C V. , for the finitely generated algebra CO corresponding to C
.
Taking closure we even get CXUCV., using
that (3.5)
,
1AXUfL(H.) = BAIIL(H)
for all AEL(H)
Accordingly, C + Ej C Vj Vice versa, for AEVj let AkEC? be an approximating sequence, RA - AklIL(H )}0 , where Ak is a finite .
sum of finite products of the generators of prop.3.2. Using prop.3.3 again we substitute each DA. in the products of Ak by J DA', and obtain an operator AkEC=CXU with Ak=A0+ Kk , Kk E Ej .
Therefore IIA - AkXU KkII ; 0 , k-. Applying XU from both sides
we conclude that Ak + XUKkXU 3 XUAXU in L(H)
where AO = XUAXU and Kk = XUKXU now stand for the restrictions of these operators ,
to their invariant subspace H . We claim that Kk E E
.
Indeed,
let yECp(sL ), Y=l near 3U. From VII,thm.l.5 we know that YKk, Kky E K(H.) Thus, with 6=1-y, Kk=SKk6+Ck, CkEK(Hj) Now KkEE fol.
.
lows from thm.3.4, below, and from the fact that XUK(H)XUCK(H), In since XU acts as a projection, hence a contractive map Hj+ H we get AO = XUAXU E C C Cj , and the second (3.4) the limit k.
follows.
Finally consider A-AO=KECj. For w E A with supp w E 2j\BU we get wKw = wAw - (wXU)A(wXU)
.
Using that AEVj has symbol with
support in UUDU we conclude that awKw (W01 )2aK = 0. Since this holds for all such w we find that aK(m)=0 for all mE t(M # 3U). A However, the set t-1(3U) C Wj is nowhere dense in ffi
,
since this
242
VIII.3. Algebra surgery
is just the co-sphere bundle over Q . Hence the continuous function aK:ffi-C, zero in i l(M #), must vanish identically. This A proves that A-A0 E Ej , and completes the proof of thm.3.1.
The existence of the potential q' a consequence of IV,thm.5.1.
,
in prop.3.3, below, is
Proposition 3.3. Let H'=H+q' satisfy (w), where q'>0,q'=0 outside Then we have
a given neighbourhood N' of 3U , and let A'=H'-1/2
DAj - DA'XU E K(H
(3.6)
whenever supp D C USN'
.
C Ej, j = 1,2,
.
The proof will again use resolvent integrals. Let R(A)
=(Hj+a)-1, R'() = (H'+A)-1. For u E CD(Qj) write (3.7)
Iu = DAju - DA'XUu = D(Aj-A')Xu + Cu
,
C E K(Hj)
where V,lemma 3.3 was used, as well as the fact that K(H)XUCK(H.) Also we wrote D=DX with X E C-(U), 0<X ql X=1 in supp D, =0 in N D N,clos Then .
(I-C)u = W-1
(3.8)
f-X-112dADX(R(A)-R'(A))Xu 0
where we write (3.9)
X(R-R')X = [X,R]X - (XR'-RX)X
.
For v E im(H0+a) we get (3.10)
XR'v - RXv = R(H+A)XR'v-RX(H'+X)R'v = R[H,X]Rv .
In view of (w) for H' we may extend continuously to H for [X,H]
, using (w) for Hi
X(R-R')X = R[H,X](R-R')X
(3.11)
.
Similarly
. Accordingly we get ,
which may be used in (3.8) for (3.12)
I = C +
i-1
f'A-112
dXDR(a)[H,X](R(X)-Rt(A))X
0
The integral in (3.12) has a compact integrand, since the folpde [H,X] has compact support. Also we use an estimate like V,(3.9)
again, to show that we have a norm convergent improper Riemann q.e.d. integral. This shows that I E K(H) Next let us compare the commutator ideals. ,
243
VIII.3. Algebra surgery
Theorem 3.4.The commutator ideal E of C is given as algebraic sum E = XEjX + K(H)
(3.13)
with any cut-off function XEC0(U)
,
0<X<1
,
X=l outside some com-
,
pact subset of U U 3U . Here again we consider X extended zero to Q.\U , and, for EEE, interpret XEX as an operator in L(H) Proof. Note that the following operators form a set of generators (with the finitely generated algebras C?) (mod K(H)) for Ej (3.14)
A[B,C]
:
A , B
Similarly for E , where now A , B
C E CY
,
,
C E CO
,
and where the A' of
prop.3.3 may be used for CO, instead of A Now such a generator of E is contained in Ej as a consequence of prop.3.3, so that .
,
ECEj
, all by virtue of the interpretation E = EXU E L(Hfor
EEE, of course. Also we get K(H)CK(Hj )CE
,
so that Ei DXEX + K(H).
For the converse we first note that trivially XKX E K(H) , for every KEK(K.) . Also, for a generator E. of the form (3.14), conHere we use the fact that there exist sider XEiX = XA[B,C]X 'wider cut-off functions' X'E C-(c), 0<X'<1 , X'=O near an, with .
XX'N=X , for every N=1,2.... with every generator a , DAB
.
,
Note that X' commutes mod K(H of Ci Indeed, the gra(ADD) .
dient VX' has compact support, so that V,thm.3.1 applies to a commutator It follows that while trivially [X',a] = 0 .
we can write (3.15)
XEjX = X(X'(AO[BO,CO] + K)XI)X
,
where A0 has been constructed from A by replacing in the finite sum of finite products of generators a, DAB, every a by aO=X'a and every DAB by D0A'
, with DO=X'D and A' of prop.3.3. Similarly for B and C , going into B0 and CO Then we have X'KX' E K(H) , and X'A0[BO,CO]X'E EO , proving that XEjX C C. Q.E.D. ,
.
Corollary 3.5. We have (3.16)
EXU = Eon (CXU)
That is, an operator AEC has AXU E E3 . Proof. We know that EXUEEI .
if and only if AEE
, by (3.13). Suppose AXU E Ej for some
AEC . With the decomposition 1=y+6 used in the proof of thm.3.1
we get A = 6A6 + K , where KEK(H) C E . Also 6A6=(6XU)A(6XU) E E,
244
VIII.3. Algebra surgery by (3.13), since AEEj
,
q.e.d.
Theorem 3.6. The quotient algebras Vj/Ej and C/E are isometrically *-isomorphic under the canonical map A+E -, A+Ej , AEC The symbol space of C/E is homeomorphic to the set i.. (MA #) C ffij, U
m-, for m E C/E
with the map m
or m= m U E.
.
m"'E Vj/Ej given by m = m- n C
,
For AEC the C-symbol equals the restriction QAlM
of the Cj-symbol to ffl
(MA
i.
U#
The proof of thm.3.6 is evident, after the above discussion. In particular, every *-isomorphism is an isometry ((C1],AII,thm. 7.17, or [R1], IDxl]
).
Remark 3.7. Let us give a special consideration to the case where
U U M is compact. In that case, clearly, we have AU 0 = C0(U) and DU
is the class of all folpde's over U with compact support.
Hence, in that case, the algebra C = CU coincides with the minimal comparison algebra JD of the restricted triple {2,dVJU,HJU} Its commutator ideal is K(H), and its symbol space is the restriction of the cc-sphere bundle to U The homeomorphism of thm.3.6 .
is just the injection of that restriction into the co-sphere bundle of 0.
Remark 3.8. Now consider the case where U U 3U is non-compact. Then either 3MA1# = aMAIU#
,
or 3M A,# = aMAIU# U 3MA,V#
,
where
V = c\(UU3U) is the interior of the noncompact complement of U A similar alternative holds for j=2
.
Now the homeomorphism of
thm.3.6 will put the cosphere bundle parts ti l(U) of ffij, j=1,2,
into correspondence. But it also will establish a homeomorphism between the spaces tll(aMAI #) and 1-1(amA2 #) , under which the U
U
symbols of generators remain invariant.
Remark 3.9. We also get relation (3.17), below, as an equiK(H) and K(H.) valent of cor.3.5 for E ,
(3.17)
K(H)XU = EXUn K(H
in fact (3.17) remains true if we substitute L(H) for E ). This will be useful in the nextfollo(The proof is left to the reader
;
wing section, where the algebra C/E turns out to be a function algebra again.
245
VIII.4. Cylindrical ends
246
4. Complete Riemannian spaces with cylindrical ends. In the present section we consider a noncompact (infinitely differentiable) Riemannian manifold 0 of dimension n , with a disjoint decomposition into a finite number of subdomains (fig.4.1
)
0 = S20 U S21 U ... U S2N U 3 3 1 U . .. U a S2N , where the 12j are mutually disjoint with compact boundaries aS2j
(4.2)
aS20 = 3S21U ...Ud0N
each aS2j
,
aS2j n aS21 = 0 ,
j,l > 0
,
,j=1,...N being a (connected) n-l-dimensional submanifold
of 0 . Moreover, SZ0U3O0 is compact and, (4.3)
0 U 30. = B.xg+
,
R+
_
[0,oo)
, with aS2j
= Bjx{0}
.
j
In other words, using the terminology of [KG 1J
,
the manifold 0
has N ends, and is a half cylinder Bjxg+ at each end
, where
Bj
is a compact n-l-dimensional Riemannian space, and where B.x1+ carries the product metric (4.4) ds2=dp2+dt2, dp 2=metric of B,1
with Euclidean metric on R On such a manifold 0 we now just consider the Laplace comparison algebra C for generating sets A# and D# satisfying (a0), (a1), (d0).
Fig. 4.1
In general we cannot expect compact commutators. For the investigation of C=C(A#,D#), and its commutator ideal E=E(A#,P#) it appears natural to use algebra surgery of sec.3 to link the algebra C with corresponding C., j=1,...,N, where C. live on the full cylinders RxB.= 00 Notice that for this type of surgery we indeed get 'compact cuts', i.e., the cylindrical end Q. is a common subdomain U. of 0 3
and
with compact boundary aS2j
.
3
For the corresponding task
involving a poly-cylinder of sec.2 , having n'>l, (i.e., some noncompact part of 0 is identified with a subdomain of a polycylinder) we get a noncompact boundary, however. Such type of
algebra surgery is attempted in [CDgll. First of all it is clear that the wave front space T of C is just the cosphere bundle of 0 , and the restrictions of the
VIII.4. Cylindrical ends symbols of generators to W are the functions a(x)
(4.5)
,
bJ(W )j
aEA#
,
D=bJ3 j+p E D# x
,
for a multiplication operator a , and for DA , respectively.
The important question to be answered is about the i.e., we ask for points of BI
secondary symbol space M\01
,
over infinite points of the compactification M # . At each end A the open subdomain 0, of the manifold S2 is identified with an open subdomain Uj
Bjxl I =(0,oo) = of the cylinder 0. Thus we have the configuration of thm.3.1 , assuming that
the generating sets A# A#
,
D# on S2
3
J
.
,
,
D are matched with corresponding sets In that respect let us choose A# and D# in accor,
#
3
3
# J
dance with (2.2) and the proposed sets A D following (2.2), chosing n'=l and B=BJ . , j=1,.... N This and (3.1) describe the ,
.
D# uniquely, since in any part of 0 away from its ends we just get all Cm-functions (folpde's). sets A
,
Next we look at the commutator ideals E. Laplace comparison algebras C.
of the These
=
j=l,...,N
=
were studied in sec.2. In fact, thm.2.6 gives a precise desciption of the Ft-conjugated ideal E.^ of E. , where Ft denotes the Fourier transform (2.3) in the t-direction of 0
.
.
J
Remark 4.1. The ideal E. contains the operator A. 3
(l-A.)-1/2
=
1
0
where 4j = D2 + Ax3j denotes, the Laplace operator on Stj , with
the Laplace operator A
on B x,j
.
Indeed, we get
J
Aj^ = F.t1AjFt = (1+T2-Ax,j)_1/2
(T)-1
Also the operator valued function S(T) _
(1+T2_A
(4.6)
-+
0
x,j )
lTI takes values
in K(hx'j) because the manifold B. is compact. hx'j = L2(B.) Thus we indeed get Aj^E CO(l,hx'j) C Ej^ or, A.E E. ,
,
The above remark shows that VII,thm.4.7 may be applied to each of the cylinder algebras Cj , showing that the secondary symbol space is empty. (In particular we have MA #
= Bas
I
easily seen. One thus confirms easily that all assumptions of VII, thm.4.7 hold.) In view of remark 4.8 we have the following result. Theorem 4.2. The secondary symbol space of the Laplace comparison algebra C(A#,D#) is void. The symbol space Iffi of this algebra coincides with the compactification of the wave front space 1
247
VIII.4. Cylindrical ends
under the symbols (4.5) of the generators Next we look at the commutator ideal E(A#,D#) Here we may apply thm.3.4, which will link E with E. . Again 0 and have .
the common subdomain Uj=2j=BjXE+ Using thm.2.4 we get a complete description of the commutator ideal E. Then we may gene.
.
rate the comparison algebra - here called C. - of the subdomain U.C. SZwith the classes (3.3) and call its commutator ideal E.-. J
J
J
Note that Cj" and Ej- also live on the manifold 0 , where they again are generated by the common subdomain Uj=cj of 0 from restrictions of the classes A# and D# defined on 0 For each j=1,...,N choose a cut-off function 1(T), depending .
on t only, and meeting the requirements of thm.3.4. Then, we get E C'(N)
For any EEE we have 0E ,
E K(H)
, by
VII,thm.1.3, and [Pj,E]EKr(), by V,thm.3.1 and ECC. Accordingly (4.7)
E =
jj=1pjEipj (mod K(H))
y* EP +
Using thm.3.4 this implies the proposition, below. Proposition 4.3. The quotient algebra E/K(H) allows the direct decomposition (4.8)
E/K(H) = E1-/(H1
® ...® EN"/(HN')
On the other hand, we now look at E
,
HL2(O.)
as a subalgebra of Ej,
again using thm.3.4 and remark 3.9: Theorem 4.4. We have, with !z (4.9)
.
K(hj)
E3.°/K(H.-) = C(Ekx'j)
,
hj=L2(Bj)
Ii = E
,
j=1,...,N.
Proof. We apply thm.3.4 and conclude that, for EEE ., we have J
(4.10)
E + K(Hj) =
K + K(Hj)
*jEj*j + K(Hi)
,
with some EEEj, and H=L2(0). Since K(Hj-)CK(Hj) with the imbedding of thm.3.4, we get a map u:Ej-/K(Hj")} Ej/K(Hj) , which is an isomorphism, by virtue of (3.17). On the other hand, thm.2.7 implies that E./K(Hj) = C(E+,kx'j)®C(E-,kx'j). Also, from (2.31) we conclude that the symbols of operators of the form 4) jEj vanish identically on E . Again, checking on the symbols of all
248
VIII.4. Cylindrical ends
the generators, as they are listed in thm.2.7, one finds that there are enough symbols of this form to generate the entire C(tk ), by the Stone Weierstrass theorem. We still introduce x,j
the notation Ij = E + _ I for each j=1,...,N, and find (4.9) established, and our proof is complete. ,
We now summarize the results about the commutator ideal E below. Remaining proofs are standard and will be left to the reader.
Theorem 4.5. For the quotient algebra E/K(H) we have (4.11)
EIK(H) = C(&l,fzxej) a) C(E2,hx,j) ® ... ® C(EN,kx,N)
I
where each I. is homeomorphic to I , and where fzx'j are the compact ideals of the spaces L2(B.)=h3.. Moreover, E (mod K(H)) is generated by (4.12)
:
$jax,jlj*j, *jDx,jAj,pj
with the classes
and
Dx'jED#j, j=1,...,N,
ax'jE
of of (2.2), for B=B.
.
In (4.12)
may be chosen arbitrafor arbitrarily large R.
the support of the cut-off functions rily far out - i.e. within t>R The generators (4.12) have symbol ,
(4.13)
a
D x,jA2x,j T?(T), x,jAx,] T.(T), J J
TEE =1 ,=0 on E J
,
k'
k#j
with (4.14)
A
x,j
= (1-A x,j )-1/2
T.(T) = (1+T2A2 i
)-1/2
x,j
Thm.4.5 has an application to a singular elliptic boundary problem on In considered by Nirenberg and Walker [NW], which we shall not discuss in detail.
249
CHAPTER 9. Hs-ALGEBRAS; HIGHER ORDER OPERATORS WITHIN REACH.
In the present chapter we attend to two different, but related problems. First, we propose to also study comparison algebras in L(Hs) , where Hs is an L2-Sobolev space on 0 of order s. Here the spaces Hs always are understood as spaces of the HS-chain induced by the (self-adjoint) Friedrichs extension H of HO5 for a a given triple {O,dp,H} (cf.I,6). In that respect we must assume cdn.(s) in order to be in agreement with the customary definition of Sobolev spaces.
Second, we will focus on more general N-th order expressions within reach of a given comparison algebra C (cf. V. def.6.2). Every compactly supported expression already is within reach of the minimal comparison algebra, as we know from V,6. However, fm a
general algebra C on a noncompact 0 the general expression L no longer is within reach. Thus we will ask for criteria to decide whether a given L is within reach. The relation between these two tasks is discussed in sec.l below. There we also discuss the organization of the present chapter. In particular we point out that some of the theorems have to be 'recycled', in the following sense. The first application only applies to the minimal comparison algebra J in L(H) 0 (or in L(Hs) ). That brings certain higher order differential expressions 'within reach' of JO , hence of larger algebras, so that, in turn, the same theorem now may be applied again, to get larger classes within reach, etc. Unfortunately this makes it necessary to allow as assumptions, in many theorems, a variety of special cases, so that the theorems look complicated.
IX.l. Sobolev comparison algebras 1. Higher order Sobolev spaces, and Hs-comparison algebras.
In this section, and in the following we assume cdn. (s): All operators Hm are essentially self-adjoint, for m=1,2,...
.
Then it becomes possible to work with general Sobolev spaces Hs as already introduced in V,rem.6.3 (and rem.6.3'). We recall the definition:
For s > 0 we define Hs = dom A-s (u,v)s = (A-Su,A-S V)
(1.1)
,
(lulls
=
,
and
(u,u)1/2 , u,v E Hs
For s < 0 we define (lulls and (u,v) s by (1.1) for all u,v
E H = dom
A_s
and then Hs as the completion of H under ilulls The spaces H. and H_. are defined as the intersection and .
the union of all Sobolev spaces: H. = n{H
(1.2)
:
sell
,
H__ = V[Hs
:
s E 11
.
In particular H. is considered a Frechet space under the s E 1) The space H_. is provided
topology of the class {Il.lls
:
.
with the inductive limit topology. For details cf. [C1],III,(l.23) (but this will not be required in the sequel). Notice that the collection (H:Isl
By definition Hs consists of special L2-functions, as s>0 For s=l and uEC0 the Sobolev norm Ilull1 coincides with a (weighted)
L2-norm of the (n+l)-vector-valued function (u;Vu), by V,(1.3). Correspondingly, every norm Ilullk , k=2,3,..., may be expressed as a weighted L2-norm of the 'jet' (u(a):Ial
of order < k . We shall discuss this in sec.6, where covariant
derivatives will be used, for a global representation, under some assumptions (cf. also [C1],III,(2.7), (2.8)). In VI,cor.4.3 we already discussed a weakened version of the Sobolev imbedding lemma, stating that H C Ck(S2) for s>[n/2]+k+l s
Accordingly Hs consists of smoother functions as s grows larger,
and H.C C(). For s<0 the general uEHs no longer will be an L2-function, but still may be interpreted as a distribution. Indeed, for s<0
and u E H
,
v E H-s = don A s , we get
251
IX.1. Sobolev comparison algebras (1.3)
I(u,v)I =
I(u,v)ol =
I(A-su,Asv)I <
'lullsllvll_s
Accordingly, the sesqui-linear functional (u,v)
,
.
extends conti-
nuously to a sesquilinear functional over HsxH_s , which again will be denoted by (u,v) , for the moment. For every u E Hs the (u,v)=lu(v) defines a continuous linear functional over
map v -
Hs , and we get
Ilulls , by (1.3) and "_" for v=A-2suEHs This shows, that Hs coincides with the set of linear functionals over H-s , under the induced pairing HsxH_s 3 M In other words, u E Hs , s<0, may be characterized by the values of the linear functional (u,v) v E H_ Note that the above pairing is identical to the pairing lllull
=
.
,
S.
I,(6.13) introduced for abstract HS-chains. By lemma 1.1, below, the functional v+(u,v) is characterized by its values over CD(Q). Moreover, the restriction to C_ of the functional is a distribution over N
as an immediate consequence of VI, cor.4.3.
,
In this sense it is true that all the Sobolev spaces Hs Isl
are subsets of V'(1)
<
over 0
,
the space of all distributions
Note also that we have Hs C Ht , as s > t (including In fact the real-valued functions s } (lull for a fixed .
,
s
u E V'(N) are defined and nondecreasing on the half-line (-=,s0) if u E H s This is a trivial consequence of the definition. .
0
Unweighted L2-Sobolev spaces over In have been introduced in [C1],ch.III, using distributions and the Fourier transform. The same approach is used in [C3] as well. The present definition is easily related to the earlier one if we observe that 1-A = F-1(l+lx12)F in case of the triple {,n,dx,l-A} , with the Fourier transform F, and its inverse F-1 Note that this triple and that (s) is quite essential in proving satisfies cdn.(s) equivalence of the above definition with that of [C1],V,(2.1) ,
.
,
The lemma below is an easy consequence of I,prop.6.1 and the remarks on density of A_NCD(c2) in H following V,def.6.1.
Lemma 1.1. The space C_(c) is dense in Hs for --<s-After the preliminary introduction into Sobolev spaces of 1,6 (from an abstract view point) and VI,4 (mainly directed to compact manifolds) we will have to make a more systematical investigation for noncompact spaces now. In particular we will show independence of Hs of the choice of the comparison triple in a sense similar to that derived for comparison algebras: One may
252
IX.l. Sobolev comparison algebras change du and H on a compact set without changing Hs
.
253
Also, if
the manifold is changed outside a region U C St with compact boun-
dary then Hs does not change over U , in a manner to be discussed (thm.1.2 and thm.1.3 below, to be proven in sec.5). However, our most important task will be the discussion of comparison algebras Cs C L(Hs)
.
Such algebras will have a similar application as
the algebras CC L(H) discussed in the previous chapters. We will use them to characterize the Fredholm properties of realizations of differential expressions within their reach. For the concept of
algebra to be studied next
D we again require classes A satisfying (a0), (a1) (or (al')), and (d0) of V,1. In addition to (s), and (m2),(m3) we now must
impose stronger conditions on A# and D#
.
We choose to work with
a subselection of the conditions (lj), j=1,2,3,4,5, below. (Note that (11) implies (m2) and (13) implies (m3). ) ,
Condition (11): A# is an algebra under the pointwise product of functions, and D# is a Lie-algebra under the commutator product [.,.] of folpde's. Also, D# is a two-sided A#-module, as under (m2), and A#, interpreted as a set of zero order differential expressions, is a subset of D#. Also, for aEA#, DED# we
have [a,D]EA#
.
Condition (12): D# contains the formal adjoints of its folpde's.
Condition (13): Condition (m3) holds, and, in addition, for any a E A#, satisfying ago at a closed set Q C M# A
there exists a function b E A# with ab=1 near Q Condition (14): The expression H can be written as a finite sum of terms of the form aDF , with a E A#
,
D,F E D#
Condition (15): For every aEA# and DED# we have [a,H] E D#
.
, and
[D,H] is a sum of finitely many terms bEF, bEA#, E,F E D
In addition (with one exception) we will either impose (m5), or even the following formally stronger condition. We have 1 E D# Clearly (m5') implies (m5), but (m5') means that A even be-
Condition (m5')
:
.
IX.l. Sobolev comparison algebras
longs to the generators of C , hence is in the finitely generated algebra CO This will be formally useful for the Hs-comparison algebras, below. .
The exception, mentioned above, is the minimal algebra J. It is clear that its generating classes A# Dm satisfy (1.), ,
j=1,2,3,5, while, on the other hand, (m5) will not generally hold. The case A# = A# , D# = Dm , nevertheless, must be admitted in all results of sec's 3f. Actually, we know that J0 is a compact commutator algebra, and it will be seen that (m5) will not be required if the algebra C has compact commutators. Note that the other exceptional case, not satisfying (a1), and giving a non-unital C -i.e. the classes AU 0' DU 0 of VIII,3 needs no special consideration here. For, in order to achieve cdn. (s) in spite of the truncation boundary 3U , one will have to work with a potential q tending to infinity at 3U. This implies (m5), (cf. prop.2.3). For a pair A#
,
D# satisfying (m5), and (1j), j=1,2,5,
(in addition to (a0), (d0) and either (a1) or (al')), we introduce the new notation A!=A#, and revised family DI=D#+{al}, where now A! D! generate exactly the same algebra C, but (m5') is valid for A! and D!. Clearly A!, D! also satisfy (1 .), j=1,2, ,
J
since the constants commute with all of D! and A!. However, (al') might be violated for A! , D! , although valid for A# D# (In ,
.
case where the minimal algebra J0 is under consideration, we will use thm's 3.8, 4.2, and 4.3 with A! = Am , DI = Dm, since then the adjunction of the constants to D# might give a larger algebra.) show
It is clear that (m5') follows from (a1) and (11). We will in sec.2 that (a1), and Cl) j: j=1,2, imply (m4), and we
will see that (m5) even follows from (al') and (11) alone, if 'q grows large where D# gets scarce', as to be specified (prop.2.3). It follows by a calculation that {(11):j=1,2,4} implies (15) while {(lj):j=1,2,5} does not necessarily imply (14) .
In thm.2.1 we will show that (14) is a consequence of (m6) assuming some of the other conditions (m.) and (1.) . Actually, we 3
3
will show that (14) is equivalent to a 'global Parseval relation' of the form (1.4)
{D,F} =
LN,p=lave{D,Dv}{DU,F} for all D,F E D#
with a symmetric positive matrix ((avu)) of functions avuE A#
254
IX.l. Sobolev comparison algebras and a system D1,...,DN E D# follows locally from (m6)x
The same relation (1.4)
.
also
for an orthonormal base Dj,j=1,... 0
..,n+l, at x0, and then can be pieced together globally, using the compactness of M # A
.
It should be emphasized that condition (14) may amount to a at infinity. It implies restriction of the Riemannian space that the metric tensor hIk can be globally written as a finite sum javbvcv , with bounded functions av and tensors bv, cj still
satisfying a set of conditions, as principal parts of D,F E D! (cf.[CS1,VI,L.2.5, [CM11, [CM21, and sec.3, below ). To say this again in different words: We presently depend on (15) which is difficult to achieve for classes A!
,
D!
, nontri-
vial near infinity, except through (14). But (14) often cannot be satisfied. However (15) holds trivially for the generators Am Dm of the minimal algebra JO , regardless of any other properties of the triple or the space The above difficulty will in part be removed later on, as follows: After proving our results for .
the algebra JO (and its Hs-equivalents) we will be able to substitute (15) with a more general condition, but will have to go several times through the same chain of arguments, under slightly varying conditions. This forces us to make split assumptions. Also, all the results, below, concerning our above Sobolev spaces, will only require the higher order theory for the special case of the minimal algebra JO
.
Theorem 1.2. Let two triples {c,dp,H}
,
{c,dv,K} on the same
manifold be given, where H=K dp = dv holds outside some compact Then we have cdn.(s) valid for {Q,dp,H} if and only if set M C Q ,
.
it holds for {c,dv,K}. For each sE [-co,o1 the Sobolev spaces Hs of The corresponding the two triples coincide as subsets of DI(Q) .
Sobolev norms are equivalent (although the inner products and Hsadjoints in general are different).
Theorem 1.3. Let two triples {Qj,dvj,Hj} j=1,2, be given, where and 02 both are extensions of a manifold U such that U appears 1
as an open subdomain with compact boundary 3U of 2 1 and 22 ,each. Let condition (s) be satisfied for both triples, and assume that the restrictions to U of the two triples coincide:
255
IX.l. Sobolev comparison algebras (1.5)
dp = dp1IU = dp2IU
,
H = H1IU = H2IU .
Then, if the Sobolev spaces of the triples are denoted by Hs then for every open set V C U with Vclos C U we have HSIV = H2 IV
Hs IV = {uIV
for all s E I
,
:
u E Hs}
Moreover, if 2U is a smooth n-l-dimensional (compact) submanifold of Qj
, then the statement also holds for V=U . (Recall that the restriction uIV of a distribution u E V'(2) to
the open subset V is defined as the restriction of the linear functional to the testing functions with support in V.) The proofs of thm.1.2 and thm.1.3 depend on techniques involving higher order commutators to be prepared in sec.3. We will discuss these proofs in sec.5. For a given triple on a manifold 1 consider a chart U as a common subset with the manifold In . Assume that U is a subset with compact closure of a larger chart V Then all assumptions .
of thm.1.3 hold, and it follows that the subset of u E Hs with support in U coincides with the subset of Hs(Rn) with support in U Here we first must use a measure du and expression H different .
from the surface measure and Laplace operator of
In ,
on some
but we then may use thm.1.2 to compact subset containing U remove this and obtain the Laplace comparison triple of In ,
Thus we have the following.
Corollary 1.4. At each point x0Ec there exists a neighbourhood N of x0 such that the set HsIN of restrictions uIN of u E Hs to N if related to In by the local coordinate map, precisely consists
of all restrictions of u E H(In) to N Indeed it is a direct consequence of thm.1.3, that HsIN H`sIN , where H-s is formed with the manifold In and a triple equal to the Laplace triple outside a compact set, containing N .
but coinciding on N with the given triple on Q. Then one uses thm. 1.2 to show that H-s is just the ordinary Sobolev space Hs(Rn) This corollary connects our present Sobolev spaces with those of In , discussed in detail in [C1],III. In particular we obtain the Sobolev imbedding, generalizing VI, cor.4.3:
256
IX.l. Sobolev comparison algebras Proposition 1.5. We have Hs C C(O)
,
as s>n/2 , and H C Ck(Q), s
as s>n/2 + k
.
In particular, for any triple over D2, we have
C' (D) C H
C C0(Q)
Note that we are not attempting an investigation of the spaces Hs at infinity. For example, for S!=Xn it is known that the
functions of H. have all derivatives vanishing at infinity. More generally one could ask how Hs will be influenced by a change of hjk the metric tensor or the potential q. Discussions of this type will not be made here.
Assume now that we are given a function algebra A! and a Lie-algebra D! satisfying (m5'), and (1.), j=1,2,5, as well as (a0),(al)(or (al') for D# only, D!=D#+{al}), (d0), (or also, we admit the case of the minimal classes).
Under these assumptions we will show that the operators
(1.9)
a ,
DA
, AD ,
for a
A!
, DED!
belong to L(Hs) (cf. thm.4.2). Accordingly, for s E B , the Banach algebra Cs = C5(A!,D!) obtained as norm closure in L(H5) of
the finitely generated algebra C0 of the operators (1.9) will be called the Hs-comparison algebra of the triple {S2,du,H} , and the generating sets A! and D! Similarly we introduce the algebra C. as closure of C0 in 0(0) For a precise definition of the opera.
.
j+p in terms of distribution
tcrs (1.9) one may think of D=bJB x
derivatives. Or else, one also may think of the closures (in Hs) of the unbounded operators ajHs, D0,(AIHs), (AIHs)D0, respectively.
For technical reasons we introduce a second type of finitely generated algebra, called CO , obtained from the generators (1.10)
a
,
Clearly we get C0 C
DAj
,
AJD
,
for a E A!
,
D E D!
,
j=1,2,....
while C = Co whenever (m5') holds,
since then we have A E CO, hence DA3= (DA)Aj-'E CO, and similarly A2D E CO. If (m5') does not hold, then (1.9) and (1.10) still generate the same Banach-subalgebra of L(H), (and of L(Hs) or 0(1) as well, as will be seen later on). Indeed, from (al') we get existence of a function X E A! such that XD=DX=D, for any given D E D!
,
and with OX of compact support. Then we get DA=(DA)(AX) 2
257
IX.l. Sobolev comparison algebras
+D[x,A2], where the last term is in KO), by V,lemma 3.2. By (11) so that the first term is contained in CO Also, C D K(H), by V,lemma 1,1, thus it follows that DA2 E C. A similar conclusion implies DAB E C, j=3,4,... , and AIDE C, j=2,3,... In thm.4.2 we also will show that, under above conditions, the algebra Cs is a C -subalgebra of L(Hs), while the ideal chains C and Cs are in agreement, and the symbol spaces remain the same. In fact, under suitable additional conditions, such as (14), even we have x E D!
,
.
.
the symbols of the generators remain independent of s. 2. Closer analysis of some of the conditions (1.) and (m.). In this section we assume (a0),(d0),(al)(or (al')), and more cdn's as stated. In thm.2.1 we show that (m6) implies (14), assuming some other (mj) and (lj)
. Actually, we show that (14) and
D=b03x.+p E D# implies p E D#, and D0=b33
(**)
D#
.
imply a 'global Parseval relation' of the form
{D,F} = JN,=lava{D,Dv}{D,,F} for all D,F E D#
(2.1)
with a symmetric positive ((a,X)), avXElk, and a system D1,..,DNED# while (2.1), even with avxEA , and without (**), but with inver-
tible ((avx)) implies (14). Thus (m6)x
gives a local (2.1),i.e., 0
H = 16,X-1DvaVADX
(2.2)
DvED# ,
,
near x0,
from which a global (14) may be pieced together, if (m6) holds. Theorem 2.1. Assume cdn's (m1),(lj),j=1,2,3. Then (m6)x
implies 0
(2.2), while (m6) implies (14), and even (with certain DEED#) 6=6(x)=n+l
H =
(2.3)
with a partition {xj:j=l,...,N} of MA#, xjE A#, au,=a,u=ai.E A#. Vice versa, (14) and (**) imply (2.1) with certain DvED#, avxE I Proof. Assume (m6)x
for x0E 3M #. With some orthonormal base {D.} A ((a.l(W))) invertible for xE N=Nx , where 0
we get
of Sx 0
(2.4)
0
ajl(x) = {Dj,D1}(x)E A# , aj1(x0) = Sjl
258
IX.2. The conditions (1)
259
Using (13) on det((a J.l)) we get X((ava(x)))-1=X((avX(x))), with ava E A# , with a cut-off function X E A#, in accordance functions
with (m3), for x0 and N
We introduce ((bvx)) _ ((av,A))1/2, and
.
find that Dv=lXbvXDX , v=l,...,5, is an orthonormal base at every xE N (Perhaps D )"'E D#, but still D v"E S ). Use Parseval's relation x for the DJ J. Define the local matrix H"= ((hv,X-))v
n by
a=0
hvA =hvX for v, a>1, =q-1 for v= A=0, =0 for v=0, a>l, and v>1, u=0, for a moment. Also let D,-= Jbva j+b0
, and then define the matrix
x
B=((b)) v=0,...,n,a=1,. .6 X
,
with'v=row-index, u=column-index.
The above is valid in local coordinates, near points of Q. dense in M
Observe that then Parseval's relation can be written as
A#
(2.5)
b*H-b
=
b(H-BB*H"')b
,
for all b E !):d
This relation may be polarized, and then yields
,
xE NO
H"'-1=BB4 .
Or,
(2.6)
which is valid for all x E N
,
u E Md
.
If we substitute u0=u(x),
.(x), for some u E C' (O) , and multiply by X(x) , we get
uj= u IX
1b_1IIXDv""uII2
J0dpX2(hJku
(2.7)
u IxJ
k+gIul2)
=
,
u E C .
Ix
Now we get Dv = XcvXDX- , with the matrix ((avX))1/2
.
((cva))=((bvX))-
As a consequence the right hand side of (2.7) may be
written in the form (2.8)
Lsa=iJ0dpX2avXDvuDXu
Finally, assuming (m6) U(m6)x, use (13) to construct a partition of unity {X1,...,XN} on the compact space M # , where we A require that XjE A#
,
that (2.7) with (2.8) Dv = Dvj (2.9)
.
j=1,...,N , while JX? = 1 on M # A
,
and such
hold for every Xj with suitable
Taking a sum over j we then get
(u,u)1 = LN=l1 v,a=1(XjavADvju,DXju) for all u E CQ(O)
IX.2. The conditions (1)
Here we finally may polarize and change notation i.e., the first half of the theorem.
for (2.3),
Vice versa, let (14) and (**) hold. Accordingly we can write (2.10)
H = Iv=lavDvFu , av E A#
,
D')
,
Fv E V#
where all coefficients may be assumed real-valued, since H has real coefficients. Using (11),(12),(m1) one may rewrite this as (2.11)
IN
v=1D
H =
Fv
with changed, but still real-valued DV,FV Let Gay , X=1,...,R, be a basis of span {D,,, D,,, F,,, F :v=1,...,N}, with the principal parts DV , FV of Dv Fv We may express Dv and Fv as linear combinations of the G. and (2.11) takes the form .
,
.
,
H = iR,X=lavXGv*GX
(2.12)
' a,,=
a,X= a.X E R
Now use that H is self-adjoint, so that H = 1/2(H+Hh) (2.13)
H =
.
We get
IR
X=lavXGv*GX
where again the avX have been changed. Now the matrix ((avX)) is real and symmetric. Returning to our old notation we have proven Proposition 2.2. If condition (14) is valid, then we have (2.14)
H = IR,X=lavADv*DX , with avX=aX=aXv E A# , Dv E V#
where all functions and folpde's are real. Now we write (2.15)
Dv = bj3xj+b
,
B =
n,v=1,...,N
where B will be considered an (n+l)xN-matrix , i.e., j indicates the rows and V the columns. It may be assumed that DV, v=1,...,M, have bR=0 , while D
v=M+1,...,N, are of order zero. Now let
denote the sum (2.14) taken only over v,X with at least one >M While g' formally is of first order, it really is of order zero,
0 0* _ (DVV 0+D 0* )b v 0
0 0 due to bXDv + (bXDv)
0 0 + [bX,Dv], where the right hand
side is sum of products of two zero-order terms in V#. We thus may
260
IX.2. The conditions (1)
261
repeat the above proceedure once more on 1', to arrive at a new matrix ((aV), )) with a.VX=O as v<M , X>M or V>M
,
X<M
.
Comparing
coefficients in (2.14) we then find that, with H- of (2.5) we have H--1 = BAB*
(2.16)
Then (2.16) implies H_=(H-B)A(H-B)* which translates into {D,F} = IN
(2.17)
X=lava{D,DV}{DX,F}
for all D,F E D#
,
s
This completes the proof of thm.2.1. Next we will look at condition (m5)
.
This condition is a
consequence of (11), if (a1) holds. On the other hand, under (al') only, prop.2.3, below, shows that (m5) follows for the algebra C of the 'truncated classes' AU,O' DU,0 of VIII,3. (We, of course, then must work with a potential q going to - at 2U as was used ,
in VIII,prop.3.3.)
Proposition 2.3. Suppose D# contains a sequence of functions (i.e. zero order expressions) X. j=1,2,..., such that ,
0 <Xj <1 , Xj
= 1 in Qj , where UQj = 0 , while
limj,(inf{q(x): x E S2\Q}) = - .
(2.18)
Then condition (m5) holds. Proof.
(2.19)
Let Aj = XjA E C 0A1
-
A'1
=
.
From V,(1.7) it follows that
0(1-Xj)AII
<
0
, as j--
Hence A E C , q.e.d.
Proposition 2.4. Conditions (15),(m5) and (w) imply cdn. (m4)
.
Proof. First, in the special case of the minimal classes, we have a compact commutator algebra, for which (m4) was proven in V,3 (since a and D have compact support, V,lemma 3.3 and V,lemma 3.6 apply ). Similarly, in the general case, we again use a resolvent integral of the form (and with the contour of) V,(3.6). Since we (A2-X)-1 assume (m5) we have A2 E C , hence the resolvent R(X) = is in C for all X E r\{0), with the path r used in (4.6). (If C is non-unital, we at least get AER(X)EC, which is enough). We get
(2.20)
Ia,R(X)]=AR(X)[H,a]R(X)A2 2
,
ID,R(X)I=A 2 R(X)IH,DIR(X)A
2
IX.2. The conditions (1) [a,R]A-1-e
Using (15) we get [H,a] E D!, so that
,
A-1-e[a,R] E C.
[D,R]A-e
A-£[D,R] E C , due to (15) again. Moreover, the integrands are norm continuous (even analytic). Using V,(3.9) Similarly,
,
we get estimates like those used in V,lemma 3.3 or 3.6, so that the integrals converge in norm. This completes the proof. 3. Higher order expressions within reach of C or Cs. In this section we first assume cdn's (s),(1.) j=1,2,5, and either (m5'), or else the minimal classes Am, Vm. Later on we C(A!,D!) C L(H) shall replace (15) by a weaker condition. Let C
and C0 be the comparison algebra and the finitely generated algebra introduced in sec.1, repectively. Our first use of (15) will be to identify larger classes of expressions L within reach of C or C. , as specified in V,def.6.1. By (11) the class D! is a Lie-algebra, under the commutator product, and a two-sided A!-module. Let L' denote the corresponding universal enveloping algebra, defined as the algebra of all differential expressions of arbitrary finite order, of the form j
L =
(3.1)
X.
1a.
II1=1
Djl
,
a,'= A!
, Dj1E D!
Clearly this defines an algebra under A! since D! is a two-sided A!-module. We may set a =1 except for Ni = 0 ,
,
.
j
By Lm we denote the class of all operators in L of the form (3.1), with NJ < m , for all j=1,...,M
nition (or else D! = Dm )
.
Since 1 E D! , by defi-
we may assume exactly Nj=m terms in Also we except for m=0 where L0 = A! ,
each product of (3.1) get L1 = D! , while, for m=0,1,2....., Lm is a linear space of ,
.
m-th order differential expressions over 0, containing its formal adjoints, and all m-th order Co-expressions with compact support. Note that (14) amounts to the condition that H E L2. Also (15) means that [a,H] E L1, [D,H] E L2 , for all a E A!, D E D! Let A!0 and D!0 denote the ideals of A! and D! generated by the commutators [a,D] E A! , and [D,F] E D! , respectively. Let L,,o C L,, LN,o C LN denote the subsets of all L of the form (3.1), where at least one factor of each product is in D!0. Here it is no loss of generality to assume that always the first (always the last) factor is in D!0.
To be more specific, we set L0
0,0
of L1
1,0
= D!0 is given by {a[D,F]
:
= A!0 . A set of generators
aEA!, D,F E D!), or also by
262
IX.3. Higher order expressions {[D,F]a
:
aEA!, D,FED!} , or by {a[D,F]b
:
a,bE A! , D,FE D!}
For m=2,3....., a set of generators is given by D.,D,FED!}, or else {[D,F]Dl...DN-1: D.,D,FED!}. .DN-1[D,F]:
{D1"
In particular, L. 0 again is an algebra invariant under multiand adjoint invariant. For the minimal classes plication by A! ,
it follows easily that A!0=A!=Am , D!0=D!=Dm Lm O=Lm . Therefore, since a general pair always contains the minimal classes, ,
we find that also a general Lm o contains all Cam-expressions of s
order m with compact support.
The following modification of condition (15) will be useful: Condition (15'): We have CH,a]E L1,0 , CH,D]E L2
a E A!
for all
2,0
,
D E D!
.
Note that (14) not only implies (15), but even (15'). Also, not only (15) Some of the need the stronger condition (l5') instead of (15).
the minimal classes satisfy (la') results
below
,
.
Proposition 3.1. The class L.O is a two-sided ideal of L_ for L E Lm LIE= Lm, , and 0 .
e
Moreover, we have LL', L'L E Lm+m,
,
0
s
(3.2)
[L,L'] E Lm+m' 1 0 for L E Lm , WE Lm?
.
The proof is a calculation, and is left to the reader. For every differential expression L E LN consider the two linear operators (3.3)
(L.0 AN)4 ,
)e.* (ANL 0
formed as closures of products of unbounded operators over H
,
The closures are well defined with the minimal operator LO of L since AN is bounded, and L has a well defined formal adjoint. .
Proposition 3.2. The operators (3.3) are in L(H), and, moreover, In fact, they they even are in the comparison algebra C(A!,D!) .
even are in the finitely generated algebra C9 of sec.l. In particular, using the terminology of V,def.6.1, every expressions LELN
is within reach of C and C , as an expression of order N Remark: Please observe that C and C, have the generators (1.10), each algebra with its own way of generation. In particular, the ideal of compact operators appears on its own, and is not needed among the generators.
Proof. Note that the set of generators (1.10) contains its
263
IX.3. Higher order expressions
adjoints, and that also LN is adjoint invariant. Thus we have (A NLO) ** =(L"A N) * E C if we can show that (LOA N ) and it E C. is sufficient to consider the first kind of operator (LOAN) ** ,
Accordingly we essentially may repeat the proof of VI,prop.3.1, noting that cdn.(l5) just supplies the essential property. It is clear that the assertion holds for N
O** a=(aA)
,
since
and DA=(D,0A) 1** just are the generators of C
,
For
.
a product DF E L2 we get, as in the proof of V,lemma 3.2, DFA2u = DA2Fu + DA2[H,F]A2u
(3.4)
,
for all u E im H0
.
Using (l5) we may substitute [H,F] by a finite sum of products D'F'
,
for which the second term (3.3) assumes the form * All these terms involve opera-
DA2D'F'A2u = (DAM' A) (F'A2)u
.
tors of C0 only. Since in HO is dense, we thus get ((DF)OA2)** E CO , proving the assertion for N=2. (The first term, of course, gives (DA)(D*A)* E C..) Assume now the statement proven for LN ,N>2 product LM
(3.5)
,
LEL2
, MELN-1 C LN
For uE
.
A-N-1CO
.
Consider a
we get
LMAN+lu = LA 2MA N-lu + LA 2[ H,M] A N+lu
since AN±lu E C_ . Using (15) and prop.3.1 we get [H,M]ELN,0 C LN+l 0
,
so that the second term represents an operator in C..
(It may be written as (L0A2)**([H,M]OAN+l)**u.
) The space
A-N-1C0
is dense in H in view of cdn.(s). Accordingly (3.5) Since the general expression of LN+l implies that LMAN+1 E C_ .
is a sum of expressions LM of the above type, the prop. follows. In order to simplify notation we will, for the present section, use the notation L for the realization Z of V,def.6.2, corresponding to an expression LELN In other words, the expression L and the unbounded (differential-) operator L:HN+ HO=H will be .
denoted by the same symbol L
.
(We recall that HN = dom A-N =
im A N. For u e HN we write = ANv , and the define (3.6)
Lu = LANv = (LOAN)**v
which is well defined, by prop.3.2.) Note that this notation already was in use for DES! , where we used to write DA , interpreting dom D = H1 = im A . Also we observe that L C L0** The .
264
IX.3. Higher order expressions
265
notation is ambiguous, due to the fact, that an expression L of order N also has every order R>N . However, all these various operators, denoted by L , have the same closure L0
Proposition 3.3. For L E LN 0 we have ALA
N
N
L)**A
E E Proof. Again it is sufficient to focus on the first kind of operator, using that (ANL)**A = (AL*AN) Looking first at Am=A!, Dm D! , we find that then the operators listed are compact. Indeed, ,
(A
.
.
from prop.3.2 we conclude that LAN E CO is a finite sum of products of terms (1.10), with compactly supported a and D. But then Aa, ADA3, Aj+1D all are compact, by III,thm.3.7, so that ALAN is compact as well. Since K(H) c E , the statement follows. In the general case we always have (m5). For N=0 we must look at A = A[a,D]. Note that [DA,a] _ (DA)(A-[A,a])+[D,a]A E E By prop.2.4 we have (m4) valid as well. Thus VII,rem.3.4 implies
that the first term at right is in E , hence [D,a]A E E * E E as well. Note that we have (m4 and A[D,a] _ (A[a,D]) again by prop.2.4. To discuss the case of N=1, let L=a[D,F] aEA!, D,FED! Now we get, with proper interpretation of [D,A], .
(3.7)
[DA,FA]
_ [D,A]FA + A[D,F]A + [A,F]DA E E
.
We already noted that [D,A] E E, [A,F]EE, by VII,rem.3.4, since we have (m4) and (m5). Therefore (3.7) implies A[D,F]AEE. But we get
(3.8)
Aa[D,F]A = ([A,a]A-1)(A[D,F]A) + aA[D,F]A
where all terms at right are in E for N=l
.
,
Therefore the statement holds
.
For N=2 the typical term of L is of the form F[D,E]. We get
(3.9)
AID,E]A2u = (AF)(A[D,E]A)u + (AF)[A,[D,E]]Au , uE H2
where the right hand side operator is in E
,
,
by the above.
the typical term in LN+1,0 may be Then we may write (3.5) written as LM , with LFEL2 2,0 , M E LN-1 again, multiplied by A from the left. Again we get ALA2MA N-]EE Finally,for general N>2
,
.
ALA2[ H,M]AN+l E E
, completing the proof.
In the following we introduce more general algebras of diffe-
rential expressions of arbitrary order, within reach of an algebra C or Co. The purpose is to replace cdn's (15), (15') by weaker conditions.
IX.3. Higher order expressions
266
N=OPN be a graded algebra of differential expresLet P = sions, invariant under formal adjoints, and a 2-sided L.(A!,D!)-
module. In details, we assume the expressions in PN of order
(3.10)
P P
N M C PN+M' PNLM C PN+M' LNPM C PM+N' N,M=0,1,2....
.
In addition let the following conditions (pj), j=1,2, hold: Condition (pl): [H,L] E PN+l for all L E PN , N=0,1,2,..., Condition (p2): Every L E PN is within reach of L(H) .
Notice that (pl),(p2) imply that all LEPN are H-compatible, in the sense of V,def.6.4. We shall say that an algebra P. with above properties is within reach of an algebra A C L(H) if every P E PN as an ,
Also, if P. is expression of order N is within reach of A within reach of C_(A!,D!) for some classes A!, D!, with C.CO(0) ,
.
of sec.1, then we say that P
is within general reach of A!
For P0, within reach of C5(A!,D!)
for all LEPN, N=0,1,...
,
,
D!
if we even have ALA NE Es(A!,D!),
we shall say that P. is of principal symbol type (in C ), - the latter only for s E I s For PW with above properties define the classes QN = PN+LN' N=0,1,... Then Q. = N=0QN clearly again is a graded algebra ,
.
.
of differential expressions and P. is an ideal of Q. We will ask whether Q,, again is within reach of C(A!,D!), given that P,, is within reach of C(A!,D!)
.
Cor.3.4 below gives
an affirmative answer, using the following amended condition: Condition (15)(P,,): We have [H,a] E Q1, [H,D]E Q2, for aEA!, DED!. In case of an algebra Pte, satisfying (pj), j=1,2, and the
above, and of principal symbol type, we also formulate a relative cdn.(l5') , as follows (setting QN,0 = PN+LN,0' Q- = U QN ): Condition (15')(P):Wehme [H,a]EQ1,0, [H,D]EQ2,0, for aEA!, D(=-D!. Notice that (15)(P.) implies that Q,, = UQN satisfies (p.), j=1,2, by the corollary, below. In fact, this even is true without the condition that P. is within reach of some comparison algebra, by the same arguments. In the corollary, below, the special case of the minimal classes Am, Dm is excluded. (Since Am , Dm satisfy (15) the corollary would be useless.) ,
Corollary 3.4. Under the above assumptions on P,,, if cdn.(l5)(P,,)
holds (but not necessarily cdn.(l5)), then the graded algebra QW
IX.3. Higher order expressions
267
is within reach of C(A!,D!). Moreover, if in addition P0, is of principal symbol type (in C0= C)
then the algebra Q,
,
0
= P+L_
0
also is of principal symbol type (in C).
Proof. Since P E PN already is within reach of C we only must show this for 1. E LN again. For the latter one may repeat lite-
rally all arguments of the proofs of prop.3.2 and 3.3, using the ideal property of Pte.
In (3.4), for example, the commutator now is
a sum P2+L2 , where L2 is treated as before, while P2EP2 gives the
term (DA2)P2A2EC. Similarly for (3.5), also in view of prop.3.3. 4. Symbol calculus in Hs
In this section we keep all assumptions of sec's 1 and 3 valid. Particularly we assume cdn's (s), (1j), j=1,2, and (15) (or (15')), and either (m5) for a triple {O,dp,H}, and classes A! A#, D! = D#+{al), or we use the minimal classes A!=Am, D!=Dm Or else, we assume (15)(P0,) (or (15')(P0,)), instead of (15), (or (15'))
, relative to some algebra P_ within comparison reach of
C(A!,D!)
,
satisfying (pj),j=l,2, and (3.10) (and of principal
symbol type (in C)). In the latter case we exclude the minimal classes, and always assume (m5')
.
In the interest of a unified notation, let us always at least formally work with an algebra of expressions P. , where we set PN = {0} in the case where no preconceived such class is
given. Clearly this defines the trivial algebra P_
,
satisfying
all assumptions. It is within general reach of every A!. D!, is of principal symbol type, gives an algebra P0,, and an L0,-module,
satisfies cdn's (pj), j=1,2, and (15)(P.)
(15')(P0,)
,
,
for the
trivial algebra P0,, coincide with the ordinary conditions (15) and
(1b'). We appoint that for the minimal classes Am , Dm only the trivial algebra P, is to be used. We then always define QN=PN+LN QN,O=PN+LN,O ) so that QN LN vial algebra P.
'
QN,O = LN,O
5
in case of the tri-
.
Let us recall the order concept introduced in 1,6 for gene-
ral HS-chains. Here we use the chain {H:sE[--,+-]} of Sobolev spaces, of course. We shall adopt an attitude similar as in theory of pseudodifferential operators to not distinguish in notation between the various extensions in L(HsHs-m) of a given operator A E 0(m). If AjC is given by a differential expression L then we
IX.4. Symbols over HS-spaces
even use the same notation L for all above operators as well as the realization Z of V,6
This will bring us away from the unbounded operator notions of ch.l, and will be practical in view .
of the fact, that there is a unique such extension, for every s, and a unique closed realization, all in view of cdn.(s).
Focusing on the algebra Q_ = P + that every L E QN is within reach of C in the sense of V,6, by cor.3.4 and the Thus we find that L defines an operator Proposition 4.1. For A=LAN, B=(ANL)*
AsAA-s-A = CA=AC'=E
(4.1)
where E, E',C, C', C", C"'E C
,
L. of expressions, we find and even is H-compatible, remarks preceeding it. in 0(N), by V,cor.6.10.
, with LEQN , we get
AsBA-s-B =C"A=AC I"=E'
may depend on s
require (15')(P-) instead of (15)(P-)
.
,
sEl
Moreover, if we
, we even get E,E'E E
.
Proof. For a given s we apply V,thm.6.8, with M > Isi . Also we use that Lj=(ad H)2L EQN+j (or EQN+j,O, in case of (15')), so that
(4.2)
LjAN+3E C , and LjAN+j+1=ALjAN+j+ [LAN+j,A] E E,
by cor.3.4, correspondingly, depending on the assumption made. (For the minimal classes we must use XA for A
,
in the commutator
with suitable X.) Similarly the term ALN+lA11+1 in V,(6.12) is
or of the form CA, CEC, respectively, hence also the reminder gives a term of the proper form for V,(6.13). (For the in E
,
minimal classes Am, Dm the resolvent R(?) only is in
C + {al} but C is an ideal of C+{al} so that the integrand of V,(6.12)
still is in C or E resp. This gives the first (4.1) while the second follows by taking adjoints, q.e.d. For thm.4.2 below we recall the natural isometry Ar:Hs+Hs+r (Actually, Ar stands for the extension (Ar)s of Ar to Hs, but the subscript 's" can safely be dropped). Note that
(4.3)
L(Hs+r) = Az'L(Hs)A-'
,
K(Hs+r) = ArK(Hr)A-r
,
For the remainder of this section we assume, in addition to the conditions mentioned earlier, that P- is within general reach of the classes A!, D!, and of principal symbol type in Cs, for sEl.
Theorem 4.2. The generators a, DAD. ADD, aEA!, DED! of (1.10) are in L(Hs), for every sEl, (and in 0(0)), so that all comparison
268
IX.4. Symbols over HS-spaces algebras Cs (including C,, ) are well defined norm closed subalge-
bras of L(Hs) (or 0(0)). We have LAN, ANL E Cs LEAN . Every Cs
,
-°°<s<-
for all
,
sEl, contains its HS-adjoints, i.e., Cs is a C -subalgebra of L(Hs). For and every A in the algebra ,
finitely generated by C,,O and {LAN, ANL: LEAN, N=1,..} we have, (with the commutator ideal Et of Ct)
A-SAAB - A E Ct, and 'E Et' in case of cdn. (15')(P-),
(4.4)
Also,
Cs+r =
(4.5)
AsCrA-s
,
Es+r = ASErA-s
,
for all s,r E 1
In particular, the class Q. is within general reach of A!, D!
and Q
is of principal symbol type in every C (A!,D!)
ANL E 0(0), hence DA , AD LAN EL(Hs), justifying the definition of Cs, also for s=-. To show Proof. Prop.4.1 implies that a LAN, A N LEC5
,
,
one repeats the argument of prop.3.2, as. in cor.3.4,
using that P is within reach of CS, --<s<°°. Inspecting (3.4) and (3.5) one indeed confirms AECs Similarly, in case of (151)(P,,), .
one confirms that Q_ 'O is of principal symbol type (in Cs ), sEX. A2sA*A-2s,
S For an operator AEO(0) the H-adjoint is given by
with the H-adjoint A*, since (u,Av)s= (A2sA*A-2su,v), for u,vEH,. We get A2sA*A-2s E Cs , by (4.4) with s,t replaced by -2s and s. Thus the adjoint invariance of Cs follows from (4.4). Next we note that (4.4) for t=O implies A-sCQ,OAs C C
(4.6)
,
for all s E I
However, if A. converges to A in L(H) , then A-sA.As E L(Hs) A-5AAsE L(Hs), and vice versa. Therefore we may
converges to
take closures in (4.6) and get
(4.7)
Cs C AsC0A-s This conclusion may be repeated with CO=C replaced by Cs, and A-SC SAs
As by A-s, using (4.4) for t=s.Then we get C=C0C equality in (4.7). Thus we have the first relation (4.5)
,
or,
,
for
r=0, and the second follows since, of course, the commutator ideal is invariant under a conjugation with an isometry. Similarly for general r.
Accordingly, (4.5) and the adjoint invariance both depend on proving (only the first part of) (4.4).
269
IX.4. Symbols over HS-spaces However, (4.4), for t=0
and s E Id
is a consequence of
,
as far as the generators are concerned, and it then follows trivially for all A E C-0. Similarly for A finitely gene(4.1)
,
rated by ANL, L E 2N. For general t we cannot use prop.4.l, but must go back to V,thm.6.8 instead, showing that all terms at LAN,
right of V,(6.11) are in Ct. Indeed, we get LjEQN+j, hence LjAN+2j E Ct , as already shown above, using that P. is within reach of C. . Also the integrand of S m,s is in C t again. In fact, we have
ALM+1AM+jE Ct. In case of (m5') the functions S1, SM Ct take values in since AECt Therefore the integrand is in Ct
s
also
and S3 are functions of A
,
.
On the other hand, in case of
the minimal classes, the resolvent is in Ct+fal}
,
hence the S1
are in that algebra. But Ct is an ideal of Ct+{al}, so that again the integrand is in Ct. Thus we only must prove convergence of the integral in L(Ht) .
For that we may look at the L(H)-convergence of the integral
V,(6.12) with LM+IAM+N+l replaced by At(LM+IAM+M+l)A-t= Vt, since At and A-t commute with SJ(1) But we get V E L(H) , by another .
t
application of the expansion V,(6.11). Accordingly the integral V,(6.12) with Vt indeed converges in L(H), by V,thm.6.8 and we E Ct get SM'-s as stated in (4.4). Finally, for the second part of (4.4), assume that (l5')(P.) holds. We may assume (4.5) true now, which only depends ,
on the first (4.4). Thus, in order to show that the right hand side of V,(6.11)
t
(for -s), is in E , we must prove that Wt_
,
A-t LjAN+2jAt E E, as j>l, and A-tS
M,-s
At E E , where we know that
From cor.3.4 we get LjAN+2j E E. Again we use LjE 2N+j,0 V,(6.11) on Lj , and for -t instead of s, to express W. by .
Lj+kAN+2j+2k
and another remainder. The first terms are in E as posted. The remainder is an L(H)-convergent integral with inte,
grand in E
.
Thus WtE E
,
and L.AM+2!E Et
.
The A-powers may be
taken into the remainder S and will give an integral V,(6.12) M+N+l M,-s, with ALM+1A replaced by V. above. We just showed that VtE E (we have Vt=Wt, for j=M+1). Thus also Sm (4.4) for A = LAM
.
s
E Et
,
and we get
Similarly for the other generators B = ANL.
The same follows trivially for AEC.0, or for A finitely generated N by LAN, A L , with LEAN and the proof is complete. ,
In thm.4.3, below, we now summarize the consequences of thm.
270
IX.4. Symbols over HS-spaces
271
4.2. These facts will be pursued in closer detail in X,4, where it will be seen that the Fredholm theory of an Hs-comparison algebra is similar, or even the same, as that in the corresponding H-comparison algebra. The assumptions are the same as for thm.4.2. Theorem 4.3. The quotient algebras Cs/E5
,
s e I
,
all are isome-
trically isomorphic, and also the quotient algebras Es/K(H5), SEEM,
all are isometrically isomorphic. In each case a *-isomorphism Cs/Es } Ct/Et and a *-isomorphism Es/K(H5) - Et/K(Ht) both are induced by the natural isometry A - ArAA-r , r=t-s Accordingly, the maximal ideal spaces Ms and ffit of Cs/Es and .
Ct/Et are homeomorphic under the associate dual map of the above isometric isomorphism, and, if E/K(H) is a function algebra
CU,K(h)), as in the examples of VII,1,2, and 4, then also Es/K(Hs) = C(Es3K(h))
(4.8)
,
where all spaces Is are homeomorphic to E = E0
.
Moreover, under condition (l5')(P.), if the homeomorphisms with M _ MO then the operators in the algebra finitely generated by CO and LAN, ANL , for used to identify all the spaces ffi
,
LEQN, N=0,1,..., have their Cs-symbol independent of s The proof of theorem 4.3 is evident. We will continue this
discussion in X,4. Note that a dependence of the Cs-symbol of a (finitely generated) operator A on s is possible, if only (15)(P00)
, not (15') holds. We shall not discuss such problems
here.
Finally let us shortly look at the (Frechet-)algebra C_ already mentioned. Note that the algebra C.0 C 0(0) also has a closure C, in the natural topology of 0(0) i.e., the locally ,
convex topology of all operator norms
(4.9)
IlAlls
= sup{llAulls: Hull s<1)
,
s E IR
.
By the Calderon interpolation theorem this is a Frechet topology; the class {IIAllj:jEZ} defines the same topology (cf.I, thm.6.3)
.
If we assume (15')(P,) then a C.-symbol a:C.+C(ffi), with the common space M=ffis of thm.4.3 still may be defined as the restriction to
C. of the Cs-symbol (which proves independent of s) (cf.X,5).
In the special case of a certain comparison algebra for it was shown by E.Schrohe and the author [CSch], that Laplace-fin
IX.4. Symbols over HS-spaces
the symbol of this Frechet-algebra still maps onto C(M). Moreover, the same is true for the (unique) Frechet comparison algebra of a compact manifold Q (cf. Schrohe [SchI]). (In both cases we have Es=K(Hs), so that the symbol of the second kind is trivial.)
For a more detailed discussion of the abstract elements of the Frechet comparison algebra Cam, and the Hs-comparison algebras Cs cf. X,4.
5. Local properties of the Sobolev spaces Hs.
In sec's 5 and 6 we will look somewhat more closely at the spaces Hs of sec.l, which, so far, were little more than the domains of the powers of the self-adjoint operator A=H-1/2. Here we will discuss the proofs of VI,thm.1.6, and thm. 1.2, thm.1.3, concerning local properties which were postponed so far. In sec.6 for an integer k>O, with a weighted L2-norm of the 'k-jet' of u First let the assumptions of thm.1.2 hold: We are given we will focus on equivalence of Ilufk
(5.1)
(O,dp,H)
,
,
{O,dv,K)
on the same manifold 0, coinciding outside some compact set. Only the minimal sets A# and D# are needed in this discussion. We assum m me condition (s) for the first triple. The sets Am , Dm , with the first triple, then qualify for all results of III,3,V,3, V,6 and will give us a minimal algebra of expressions- explicitly known as the set of all C'(O)-differential expressions of compact support, with LN consisting of all such expressions of order
.
Although the spaces 0(m), for the first triple (5.1), are established, we will not use them, or the notation of sec.4, until thm.1.2 is established, for reason of clarity. However,
once we have thm.1.2, we will have a unique H. and Hs , and it follows that all classes 0(m) for H and K coincide. Thus the notation of sec.4 then will be clear and convenient again. Note that y = {dp/dv}1/2 defines a non-vanishing C'(O)-func-
tion, with y=l outside a compact set. We have dp=y2dv
,
and a
change of dependent variable u=yv will take the second triple onto the form (5.2)
{O,dp,K'}
,
K'
= y-1Ky
,
which has the same measure and inner product as the first (5.1).
272
IX.5. Local properties
273
Proposition 5.1. The second triple also satisfies condition (s). Proof. Clearly (s) holds for {S2,dv,K) if and only if it holds for {ft,du,K'}
Thus it is sufficient to prove (s) for K`
.
,
i.e. one
may assume that dp=dv
Note that
with
KPH-3
Pj =
(5.3)
A=H-1/2
_
(H] + (K3-H3))H 3
=
1 + LJA2J E J
, where Lj=K!-Hi has compact support, hence L3EL2j
This operator is Fredholm, since we have aP
(5.4)
=
((k01(x)EjEl)/h31(x)EiEl))3
J
by VI,thm.2.2, since C=J has E=K(H) , while its symbol space ffi is the 1-point compactification of the cosphere bundle. Moreover,
is always positive, the Fredholm index must be zero,
since aP
since the symbol can be connected by a continuous non-vanishing family of symbols to the identity function (cf. X,thm.1.9(A))
.
(Notice the symbol is =1 for all x ouside a compact set.) Also,
(Hiu,P.u)
(5.5)
=
, then also for all u E H , by conti-
follows first for u E HjC
nuous extension. Thus Pju=O implies u=0. Since PI is Fredholm and has index zero, it must be invertible.
Now, to show that the minimal operator Ki is essentially self-adjoint, assume that fEH, (f,KJu)=O for all uEC (Q). We may write this as 0 = (f,P.H3u) _ (Pi f,Hju)
,
u E CO
.
But since A
0
is essentially self-adjoint, we have in H3 dense in H P. f =
0
.
,
so that
Since P. is invertible in L(H), we conclude that f=0. Since K3 > 1 we conclude that K3
Therefore in K3 is dense in H
.
is essentially self-adjoint, by I, thm.3.l, q.e.d. Denoting the Sobolev norms of the triples (5.1) by 11.0 and I1.11 s K
(5.6)
H =
Iluus
where 11.1
ss H
for a moment, we note that
,
IIH-s/2uII
,
Iluns
is the norm of L2(S2,dp)
=
K .
IIy-1K s/2ufl ,
as uEC'(c)
For a proof of thm.1.2 it is
sufficient to show that both norms (5.6) are equivalent, for every s E I .
Proposition 5.2. For a given sEI the two norms (5.6) are equiva-
IX.5. Local properties
lent if and only if the operator (H-s/2 bounded (in L(H)).
Ks/2)'3d
and its inverse are
Indeed, C. is dense in both Sobolev spaces, by lemma 1.1, (since we have (s) for H and K), we get equivalence if and only if
cl'ull ls H
(5.7)
<
IIu11
s ,K
<
c2
11u11for all u E C s ,H
Setting H s/2u=v , and using that H-s/2C0 is dense in H we conclude that (5.7) holds if and only if (5.8)
c111v11
< Ily-1(H s/2Hs/2)* vll
< c211v11
,
for all v E H
,
which means that (H s/2 K s/2 ) *' is bounded and has a bounded inverse since y is bounded and bounded below, and by symmetry. Next we apply I,lemma 5.1 :The operators H and K+ are selfadjoint with respect to the same inner product, and we have K`1H-! E J C L(H), while the inverse HjK-j exists in J C L(H) as Hs/2K.-s/2 well. Hence we get K`S/2Hs/2 bounded for s>0, and bounded as well. Taking the adjoint of these operators as well, -s/2 1 s/2 E L(H), we conclude that (W YH ) = Y (K and that these operators have bounded inverses. Finally we apply V, thm.3.8 for A=y AS=K-s/2, noting that CK-S/2 Since Y-1E , with C E K(H) the remainder is of the form ,
.
L(H), we get y-1Ks/2y = Ks/2 + CKs/2
,
C E K(H) =
(1+C)Ks/2H-s/2=Y1Ks/2YH-s/2EL(H)5
(5.9)
with compact operators C,C' PSKs/2H-s/2
hence
and
.
Clearly l+C
Hs/2K-s/2Qs
E
.
Or,
Hs/2K-s/2(1+C')
,
E L(H),
1+C' are Fredholm,
, with orthogonal projections
PS, QS onto spaces of finite codimension are bounded. Set JS = K
Hs/2-s/2
.
This operator and its inverse are preclosed. If
is not bounded, there exists a sequence u.EH with u.-O, IIJsujll=1.
We must have PSJsuj-0 , hence vj=(l-PS)Jsu. is bounded in a finite dimensional space. It may be assumed convergent,by compactness of closed balls in im(l-PS)
.
Thus uj+0 Jsu.->v
,
Ilvll=l
.
Since
Js is preclosed, this gives a contradiction. Similarly for JSl This completes the proof of thm.1.2. We also have made progress in the general proof of VI,thm. Vt-1 was 1.6, insofar as the boundedness of the operators Vt ,
shown for all tel. To confirm the statements about the cosets Ut Vt+K(H), we must extend VI,prop.l.8. As mentioned above, the nota-
274
IX.5. Local properties
tion of sec.4 now is in order: all operators in 0(m) are considered as operators H_-H_ , but also are identified with their continuous extensions to any Hs, as maps Hs;Hs-m. Specifically we have H S, Ks E 0(2s) .
Proposition 5.3. For 0 < s,t,a,T E I we have (5.10)
Ho[Hs,K-t]HT E K(H)
, whenever o+T < s+t+1/2
Proof. With R(X)=(H+a)-1, S(p)=(K+11)-1 as in the proof of VI,prop.
1.8 we set up a formula like VI,(1.17) again, but now we will use identity VI, (6.8) to express H-s and K-t by integrals over higher powers of the resolvents. Instead of the commutator formula VI,(1.18) we need formula (5.11), below. Proposition 5.4. Under cdn.(s) we have (5.11)
[Rm(A),Sl(P)] = Rm(a)S1(U)L(X,P)S1(P)Rm(a)
for all X,p >
0
, and all l,m=1,2,..., where L(A,p)=L1 m(A,p)
denotes the polynomial (with rPq=2(m+l-p-q)+3 (5.12)
)
=lp=llq=lLpgaP-l11 q-1,
L(X,p) =[(H+X)m,(K+11)1]
LPgELr
,0
Pq
where we again use the classes A!=Am , D! = Dm
The proof is a calculation, noting that R(a) S(p) E 0(-2). One has Lpq = cpq[Hm-p+lKl-q+l] , using the binomial theorem, and ,
the fact that [HP,Kq]=0, as p=0 or q=0. Also, a straight-forward extension of VI,prop.1.9 is required. We indeed get LpgE Lr
,0.
P9
Now we prove prop.5.3. Using VI,(1.6) for H-s and K t write (5.13)
Ho[H-s,K-t]HT = lP=1Fq=l JdA JdPIpq(a,u) 0
0
with Ipq=cpgm+p-s-211 1+q-t-2 HoRm(a)S1(11)LpgSl(P)Rm(A)HT,
(5.14)
For given s,t,a,T >0
,
cpoER.
if 1 and m are chosen sufficiently large,
then the integrand will be norm continuous and
0((Xp)-1-e)
,
e>0,
so that the improper Riemann integrals exist in L(H) . Also we get Ipq(X,P) E K(H) so that (5.13) is a compact operator. ,
Indeed, the existence of the integral at a=0 or P=0 just requires s<m , t
,
since p,q>l. Write
275
IX.5. Local properties AaRmHoK6-aS1PY)(Ko_6LpgKT-6)(UYS1K6-THTRmaa)
Ipq=cpq(Au)-l-6( (5.15)
=cpgJpq , Jpq=A.B.C
,
with the three parentheses called A, B, C , where a=(m+p-s-l+e)/2
(5.16)
,
y=(l+q-t-l+e)/2
while $ must be chosen such that B is bounded. From m>s
,
1>t
we also get 0
so that fAll=O(1) whenever O aa+$+y-m-1=6-(m+l-p-q_+2-2e+s+t)/2, or,
2S < m+l-p-q+2-2e+s+t = rpq/2 + 1/2 - 2e +s+t ,
(5.17)
with the order rpq of the expression Lpq
.
For the operator C we
obtain exactly the same estimate, (i.e., CE L(H) if (5.17) holds. On the other hand we get BEK(H) if 2s-a-T>rpq/2. Accordingly, 6 satisfying both estimates can be found if rpq/2+a+T<rpq/2 -2e +s+t +1/2 In other words, we must require (5.10). Then it also follows that the integrand is norm continuous, q.e.d. .
The proof of VI,thm.l.6 now only requires the use of prop. 5.3, to establish the group property and adjoint invariance of the cosets Ut , since the boundedness of Vt was shown. In VI,prop.l.7 we have demonstrated how to do this if t is small. We now may
use prop.5.3 instead of VI,prop.l.8 to do this for general t. Details are left to the reader. Next, in this section we focus on the proof of thm.1.3. We will use all the notations introduced in VIII,3. Specifically let
Aj=Hj-1/2 .
Since this is a matter of Sobolev spaces which
require cdn.(s), with our definition, we will generally assume that both triples {Oj,duj,H} satisfy cdn.(s). On the other hand, the prop's below clearly all are local. Thus cdn.(s) may be avoided, using a method similar to the one in early VI,l. We need a few preparations. Proposition 5.6. Under the assumptions of thm.1.3 let ¢EC'(U)
4=0 near 3U , =l outside some neighbourhood N of 3U (5.18)
es
for all u E CE(O)
.
.
,
Then
276
IX.5. Local properties
Proof. It is sufficient to prove (5.18). Indeed we may write
Aj-s(W = Aj-s+2m(lHm.u) = AjtPv, with
H = H1IU = H21U, , and also t=-s+2m , where j satisfies the same assumptions as
, and where the integer m=1,2,..., may be choosen such that
V¢ _
Then (5.18) for all v, i and t instead of u, imply our statement for the general case.
t>0
.
Let XEC_(U)
,
0<X
,
and -s will
X=0 near 8U , Xc= , and let w=l-X
We get
HwAlt$ull = J1+J2 ,
(5.19)
where we apply V,prop.6.7 to the term J2
.
(Here we work with the
operator A = SAO involving the H-compatible expression
of order
zero.) The terms J. in (5.19) will be treated separately.
Proposition.5.7. With , w, satisfying the above assumptions, there exists a constant y = y such that r,t,4,w
(5.20)
u E CD(S21)
where rEl is arbitrary.
Proof. It is enough to prove (5.20) valid for every sufficiently large integer r=2N. Thus let N>t>0 . Again, we may replace ¢u at
For if that is true for all such ,w then with - = 1 and replace u by ¢u in for the the inequality achieved. Also we may replace u by H2Nu right of (5.20) by u
.
we use it for 4- instead of
,
equivalent relation (we used cdn.(s))
yuull2 , u E C'(S22)
(5.21)
Again in (5.21) we may replace u by u with some function i with V = 1 , and then get (5.22), below, equivalent to (5.20).
yllull2
(5.22)
,
u E C1(S22)
But this is an immediate consequence of V,(6.9) and (6.10). = 0, and also, There((ad H1)1w) = 0
For we may assume
i
.
fore, if the expansion V,(6.9) is applied to A=w, with N=O, s=t, and the adjoint is right-multiplied by , then the remainder is the only non-vanishing term of the expansion. By choosing M large the reminder term can be made L2-bounded, even compensating the H1N yQ,Pu112 q.e.d. power
.
Hence we get IIwA1H1N1ull1 <
,
(Note that V,(6.9)-(6.10) are valid since w is H-compatible.)
277
IX.S. Local properties
Proposition 5.8. For every p,r,t>O we have XA2X)Ak-r
(5.23)
E K(H)
,
],k=1,2
Proof. This is a matter of the following identity, where we write Ry(a)=(Ai2-a)-1. We have, with Mi=(ad Hk)jX XA12R1(a)X
-
XA22R2(A)X
,
for N,N'=0,1,...,
=
(5.24)
(-l)NXN+N'AI. 2NRiN(a)(MNAl2Rl()MNt- MNA22R2()MN,)A12N'R1Nt(M) For N=1, N'=0 relation (5.24) coincides with VII,(3.11). For more general N,N' it follows by induction, using that (1-AHD)-1 = Tj
MNT1-1 - TJ-1MN
(5.25)
so that AMN+l =
,
,
MN(1-XH1)
-
i.e.,
2 AAi 2Ri(a)MN+lA12R1(A) = Ai R (a)MN - MNA12R1(a). i
Relation (5.25) serves for an induction proof of (5.24). On the other hand, (5.23) follows from (5.24) with the same resolvent integral procedure used over and over, q.e.d. Finally, to complete the proof of prop.5.5, we write (5.26)
XA24u + (XAt1 x -
XA2t
where the first term is
and the second term as well,
due to (5.23), q.e.d.
We now may approach the proof of thm.1.3. The second statement is easily reduced to the first. If DU is smooth, then the diffeomorphism identifying the two subdomains U. of Qj to constitute the open subdomain U may be extended beyond 3U such that ,
01 and Q2 contain the commom subdomain W , with compact boundary, while UL U C W . Perhaps then the triples are different over W\U. However, we then may use thm.l.2 to remedy this, at least in a subdomain of W , still satisfying the conditions. Then (1.6) holds for U as a subset of W. ,
To prove (1.6), let V C U be an open subset, with VclosC U. Since 3U is compact, there is a compact neighbourhood N of 3U relative to UUBU such that U\NnV=O Then we may select a func,
.
tion
meeting the conditions of prop.5.7, with q=l outside N
,
278
IX.5. Local properties so
particularly
=l on a neighbourhood of Vclos
applied to the 0-order operator find that
E C-(St
.
By V, prop.6.7,
with [H,q,] E Dm , we
is a bounded operator of L(H1.)
, hence qE L(Hs).
Now let u E HS. There exists a sequence ukE C0(S21) such that
We have u E Hs as well, and ctuk4 u in HS. However, since u may be interpreted as the product of the distribution u and the C"-function , and since =l near C U V . Also, the distribution u has support in U since supp Accordingly, we may interpret $u as a distribution on S22 as well, and also we get cu k- u in D' (02) , i.e., in weak convergence of Also, by (5.18), the sequence uk is distributions over S22 a Cauchy sequence in H2 , hence converges in H2 to some vEH22 Since this also implies distribution convergence (on St 2), we find that v=4u E H2 This shows that every restriction ujV of some s u E H1 may also be obtained as a restriction to V of v=mu E H2 uk-r u in Hs
.
we get uIV =
.
Since the assumptions are symmetric, this implies the statement of thm.1.3.
6. Sobolev norms of integral order. In this section, we look at Sobolev norms Ilullk for the case Clearly, with the inner product (.,.) of H , of an integer k>0 .
Iluuk2
= (A-ku,A-ku) = (u,Hku)
u E C0'(S2)
,
expressing Ilullk in terms of derivatives of u. In tin, where H has
constant coefficients, one may integrate by parts for (6.2)
Ilull
k
2
=
a
llu(a)02
,
I
with positive (multinomial coefficients) ya (cf.[C1],III,(2.11)). In order to derive a similar formula under more general conditions, let us assume a triple {St,dc,q-0} in Sturm-Liouville
normal form, i.e., du = dS = Fdx , h= (det((hjk)))-1
.
(From
III,prop.6.1 we know that this is no loss of generality, but it should be kept in mind that cdn (q3), below, refers to this form.) We assume {S2^,dp^,H^}={P,du,H}
, and q>l on 0 . Moreover, we will use the Riemannian co-variant derivatives induced by the metric tensor hjk (cf.app.B.).
Under these general assumptions we impose the following. Condition (q2): The Riemannian curvature tensor R (cf.(B.10))
279
IX.6. Integral order is bounded on n
,
and its co-variant derivatives of
all orders also are bounded over 0 Condition (q3)
.
The potential q satisfies the estimates 4kq = k O(ql+k/2) , k=1,2..... x E S2, where V q denotes the k-th order co-variant derivative (tensor) of q. :
,
Our problem formally suggests introduction of the norms {IqN/2u12+Iq(N-1)/2Vu
(u)N = (6.3)
N =
I92(
u) 2du = IIgN/2u112
+
.. .
,
+ IIVNuII2
with the formal covariant tensor 4k of the k-th covariant deriIIt" vative, and with Itl for t=(t. ), defined by 11...ik ,
,
ItI2= h11]1...h1kOk t.l
(6.4)
1
IItu2=
..1
3
l"
ItI2du
Theorem 6.1. Under cdn.s (q2), (q3) the Sobolev norm 11-11N
,
on
the dense space C' C HN, is equivalent to the norm <.>N. That is, clN < IIuIIN < c2N
(6.5)
where c1
for all u E C0_(S2)
,
c2 are positive constants independent of u
,
.
Proof. The second estimate (6.5) essentially is formal: For N=2m, an even number, one writes (6.1) in the form Hull N2= IIHmu112.
In order to simplify the formalism introduce, for a moment, the (n+l)x(n+l)-matrix ((gvu)) 9jk=hJk , g00=q n goj=gj0=0 and its inverse .((gvu)) 0 Use summation conven,
,
.
tion from 0 to n (over greek indices only, one up, one down). Also define a0k=-hjk, j,k90, a00=q, a03=a00=0, j90 , and, with formal tensor notation, (greek indices from 0 to n), [u]k the 'tensor' with components uv V u=ulv = VV ...V u, with 70u=u ' ., v V k 1 k 1 ,
as V #
0
.
Recall that the metric tensor hjk acts like a constant; it has all its covariant derivatives zero. Suppose, for a moment, that also the co-variant derivatives of q vanish (i.e.,q=const.). Then we may write umvm
u1vl ( 6 . 6 )
H
...a
u
u1v1...umvm
This is just the contraction of the tensors (a)x(a)x...x(a)
280
IX.6. Integral order with m factors (a)= (avu)
and [u) N . Thus it is a matter of
,
Schwarz' inequality that IHmul2<(guvgoTauoavT
(6.7)
)m(givi...guNvNuu u 1..
uN
uv 1
..
V
N
The first term at right is a constant. Integrating over 0 we get the second estimate (6.5). The second term at right of (6.7) may be written explicitly as (1)gION-lul2+
...
IVNul2+
+ (I)gJIVN-7uI2 +...+ gNIuI2
(6.8)
(u) N
<
However, in general the derivatives of q will not vanish. Thus (6.7) must be replaced by a more general representation,
containing additional terms reflecting the interchange of q with co-variant differentiation. The point then is, that the estimates of cdn.(g3) give
v ., vN
Hmu = a i
(6.9)
u vl...vN
(avi...vN)
satisfies the same estimates as
where the 'tensor'
(a)x(a)x...x(a), above, so that (6.7) still follows. (If a then we will get one more derivative 'lands' on q instead of u ,
power
but one less derivative on u, precisely as required). The for an even N This proves the second relation (6.5) ,
,
.
proof for an odd N is similar. One will set N=2m+l and work with Hmu as before. Then, however, IIuIIN2 = (Hmu,Hmu)1 so that one ,
more derivative must be worked under the coefficients of Hm There is no new point, however. Details are left to the reader. For the first inequlity (6.5) we require the prop.,below. Proposition 6.2. Under cdn's (q2) H(uIi
(6.10)
.i
1
)
_
,
(q3) we have
(Hu)Ii1
i
N
+ O((u)N N
Proof. This is a matter of pulling out the covariant derivatives from the expression H with V il.. 0. iN
ture tensor R (6.11)
=
,
.
Clearly the interchange of A = hjkV.Vk
0"' will give additional terms involving the curva-
and its derivatives up to order N-1
D4' - 0"A _
j
IN p-1
pti 1..
iN 1
j
p
u
.
We get
Ijl ...j p
281
IX.6. Integral order
with bounded tensors pt
in view of (q2) On the other hand, Leibniz' formula gives ,
.
4-(qu) = q0-u + jy(V'q)(4"u)
(6.12)
,
where V', V" are covariant differentiations for complementary subwith jl...jr and kl.. sets (i 31. ..i ), (ik ..ik ) of (i1...iN)
1
3r
$
..ks in increasing order each, and where the sum is taken over all such splittings of (i1...iN) , with certain individual constants y for each term. Again (q3) must be used; it will give From (6.12) we get the corresponding qV-u = V-(qu) + O((u)N-1) for Au , where it must be used that q>l Q.E.D. For the first inequality (6.5) we introduce N, the .
.
form obtained by polarizing the hermitian form (6.8), integrated over 0 (i.e., N is equivalent to (N)1/2 ; only the positive binomial coefficients distinguish (6.3) and (6.8)). An integration by parts gives
=
gulvl...guNVN u
N+1
(6.13)
ul...11N
((v ul...uN
= N + 0(NN) = + O(IIuONIIvIIN)
by induction. Iterating (6.13) , we get N+l = N-1+ O(IIuUN2 +
(6.14)
= N-l+ O(IIulIN+l2)
pull N-lIIHuIIN-1)
.
if Continuing on we either get N+l = IIHmUII2 + O(OutIN+l2) N+1=2m is even. Or else, the first term at right takes the form ,
if N+l = 2m+l is odd. In either case (6.5) (Hmu,Hmu)1 follows, q.e.d. ,
Note that thm.6.1 is applicable to both of our typical cases, that of a regular boundary, with a singular q , and the case of a complete manifold with q=1
or q slowly increasing. For example, the expression H = x-2-d2/dx2 in the interval ,
,
(0,°) has a q satisfying (q3)
,
as easily checked.
282
IX.7. Examples 7. Examples for higher order theory. The secondary symbol.
In this section we will consider examples of classes A! D! satisfying the general conditions imposed. Clearly we must insist on cdn.(s), for each example considered. As we know, the conditions Cl.) in particular give us the possibility of looking at operators A=LAN as members of a comparison algebra. It is easily checked that many of the classes A# , D# singled out so far do not give an algebra or Lie-algebra. For example, regarding A#, Dc AC, DC of V,4 and VII,4 one finds that Ac, AC are alge,
, DC are not in general Lie-algebras. On the other hand, some of the classes used for In and Jk
bras, but DC
have this property. For example, let us look at IC1 and define A! = A , with the class A defined there. (In detail, A
= A! is the class of all bounded C(I)-functions having their partial derivatives of all orders vanishing at infinity. ) Let D! denote the class of all folpde's with coefficients in A , with the standard coordinates of In, of course. Then it is easily seen that this pair A!, D! has all the properties (lj) In particular D! is a Lie-algebra. Also, the universal algebra L_ is completely known, in this case. LN is just the algebra of all differential expressions of order < N, with coefficients in A , and L_=ULN On a general manifold Q we always have complete control of the minimal classes Am , Dm , and their enveloping algebra, which .
consists of all expressions with compact support. The generated the coralgebra will be the minimal algebra JO or, in L(Hs) Now, thm.6.1 will imply responding minimal algebra Js C L(Ns) ,
,
.
a simple generalization of V,(1.8) to higher order expressions, allowing us a better control of operators A=LAN in our algebras. Lemma 7.1
Assume H in SL-normal form, with (s),(q 2)1 (q,). Let an
L = NaVN + N-laVN-1 + ... + 1aV + 0a
(7.1)
,
with (locally Lw-) symmetric contravariant tensor fields ja =
(jai1
J) defined over 0 , where jaVJ denotes the j-fold
contraction with the formal tensor V
V.
:
J
J
For such expressions let a norm 'ILIIH be defined by
11
...V. 1j
283
IX.7. Examples IILIIH = JN j=
(7.2)
Ohq-(N-0)/2j.ajiI 3
L '(Q)
with the local norm I.aI(x) of the tensor
a. There exists a con3
stant c, independent of u and the choice of tensors ja, such that IILANuUI
(7.3)
< cUILIIHPull
for all u E A-NC (O) C H
,
Proof. Clearly (7.3) is equivalent to (7.4)
IILvh
< cIILIIHIIvON
,
for all v E C0'(c2)
On the other hand (7.4) simply is a matter of Schwarz' inequality in view of (6.5), q.e.d. Theorem 7.2. Assume H in SL-normal form with (s),(g2),(g3). Then the minimal comparison algebra J0 contains every operator A=LAN, with L of the form (7.1), with continuous tensors ja, satisfying (7.5)
1 j al
= o (q
(N-J')/2
0
s
j=0 ,1,...,N
Moreover, let R. = URN, where RN denotes the class of all expressions of the form (7.1) with C-(O)-coefficients satisfying Vk(ja)
(7.6)
o2((/)N-J)
=
, j=0,1,...,N , k=0,1,...
.
Then the collection R.= N=ORN is an adjoint invariant graded algebra satisfying
(pl),(p2), and (3.10) for the minimal algebra J0
Also, R. is within reach and of principal symbol type, for J0. Furthermore we have A=LANE O(0),and AAE K(H5), for LERN, sEi. Proof. We know that every A corresponding to Co-tensors a with compact supports is in JO (cf. VI,prop.3.1), since such L has compact support. In fact, such A even is contained in the finitely generated algebra C0 corresponding to the minimal classes, hence AEJ5 sEI For a general continuous a satisfying (7.5) there exists an approximation under the norm IILIIH ] by compactly supported tensors, and the corresponding sequence of operators converges to ,
.
A in L(H), so that A E J0. Next a calculation confirms that R. is invariant under formal adjoints. In view of V,cor.6.10 it is clear that the remaining statements of thm.7.2 depend on the verification of cdn.(p1) for the class R. defined. (Note that (p2) is a consequence of the H-boundedness of A, already shown.) In other words, we must verify that [H,L] E RN+l for LERN. This is shown by a calculation which we shortly indicate: Let
284
IX.7. Examples L.
= a.V
, with a j-tensor a. Using all rules of covariant dif-
ferentiation we get IH,LjI
(7.7)
= IH,al.V3 + a.IH,oi1
,
where the second commutator already was considered in prop.6.2. , j=2,3 , one finds that the same arguments give a.IH,VII E RN+l, using (7.6) for a=ja On the other hand,
Using (qj)
.
(7.8)
[H,a]
i ...i 0)Ik Vk + (Aa) = IA,a1 = 2(a l
,
with the contravariant derivative 'Ik,
(i.e., covariant, raised with the metric tensor). Again it is evident, looking at the estimates (7.6), that IH,a1.V
E It is clear that R_ = URN of thm.7.2 is within reach of J0
For J0 the classes LN and LN
0
coincide. Every A=LAN, for LERN was
seen to be a limit in norm of a sequence L AN, with compactly supk
ported Lk, i.e., LkE LN
O.
Thus ALAN= limk+-6ALkANE K(H)=E, q.e.d.
It is clear that R_ also is within reach (and of principal symbol type) with respect to any arbitrary comparison algebra C(A!,D!) with the property (n4) A!RN , RNA! C RN
D!RN e RND! C RN e (which implies (3.10)), since J0 CC and K(H) C E are always true. D! for which (15)(R_) or even This fact proves useful for an A! 5
,
(15') can be confirmed, as in ex.7.3, below.
On the other hand, while thm.7.2 did not state that R. is within general reach (or princial symbol type) for the minimal classes, it did assert LANEO(O) , ALAN E K(Hs), for LERN .
Therefore, while thm.4.2 and thm.4.3 do not apply in their present form, they will apply if the operators LAN , ANL , for all LERN are added to the generators (1.10) of Cs, the other conditions granted, as the reader is invited to check. Example 7.3. For a triple in SL-normal form satisfying (qj), j=2,3, consider the following classes A!
D!
,
.
A! is the class of all bounded C°°(O)-functions with all (Riemannian) co-variant derivatives o0(1)
.
D! is the class of all C-(c2)-folpde's D=b4 + p with b=0(1) and p= (7.9)
and
Okb = o0(l)
,
4kp =
A calculation confirms that, for these classes, we get (°0*)
285
IX.7. Examples
above, as well as cdn's (11), (12), and also (7.10)
[H,a] E 1 , [H,D] E R2
,
for all aEA!
,
D(=-D!
In other words, A! and D! also satisfy (151)(R_) . Accordingly, cor.3.4 applies C(A!,D!) C L(H)
:
All operators in Q.=R.+L. are within reach of
,
We also obtain (slightly enlarged) comparison
algebras in L(Hs)
and in 0(0), containing LAN and AN, for LEAN' RN + LN give operators Also all expressions L E QN 0 = 0 A=LAN in K(Hs), sER. We observe that the class D! used here is less restricted ,
in the first derivatives of the coefficients, than the class D# of V,4, since E
Following the example of let us look at the above pair A!, D!, in the special case of the Laplace triple of a complete Riemannian manifold 0. The corresponding comparison algebra was discussed by the author in [CS],IV,11, under cdn.(14) (which poses a rather severe restriction), and, almost in the present ,
generality, by McOwen in his thesis [M0]. In fact, our discussion here uses some ideas of McOwen in [M0], in a generalized form. Example 7.4. We are given the Laplace triple {l,ds,l-A} of a complete Riemannian manifold I , with A!, D! as in example 7.3. This triple is in SL-normal form, with q=l satisfying (q3). cdn.(s) follows from IV,thm.l.8. We have the additional requirement (q2), i.e., the Riemann curvature tensor and all its derivatives are bounded. A! and D! consist of all bounded functions (folpdes) with covariant derivatives of coefficients 0 at Notice that A! C AO , D! C D# , with the classes of VII,4. .
Recall that the algebra C =C(A!,D!) has compact commutators, hence gives the simplest ideal chain. Also, cdn's (11), (12) hold trivially, for A!, D! (we assume Q^ = R , of course). We still observe, that although the algebras C and Cs of the present example are well defined on a general space Q with bounded curvature tensor (having bounded covariant derivatives), we get (7.11)
limx,.,,,b1Riljk
=
limx+.(b1Ijk-biIkj)
0,
for every tensor (bi) occuring as principal part of some DC-D!
as a consequence of app.B. This shows that the curvature is
286
IX.7. Examples
restricted at -, if folpdes DED! exist which are not in R1
For example, on a manifold 0 having a chart {xE In lxl>l} near °° if there exist n vectorfields with linearly independent limits at .
:
w , occurring as principal parts of folpde's in D!, then we must have limx.,R = 0 R being the curvature tensor. ,
Note that even on In (or Mjk) the comparison algebra C(A!,D!) is inconveniently large, for some discussions. Its symbol space HI exhibits some features of the Stone-Cech compacti-
fication. This is why we often prefer to work with the smaller
algebra S of [CHe1], corresponding to the functions and folpde's having directional limits at infinity. Similarly, working on a noncompact Riemannian space one might tend to restrict the geometrical structure of the space at infinity, and in addition the generating classes.
From example 7.4 we conclude that compact commutator algebras which are 'free at infinity', in the sense that their symbol space contains unrestricted cotangent spaces also over only can be achieved on spaces Q which are (asymptotically) flat at infinity. On the other hand the few examples of non-compact commutator algebras with a 2-link ideal chain we have discussed in VIII,l, 2, 4, and in [C1],V, may be offered for examination, in view of the following points.
(a) The longer ideal chain may reflect a more complicated structure of the compactification M # at °°, as in the first examA ple of VIII,l. Or, (b) it may result from a nonvanishing curvature tensor at °°, as in VIII,2. Or, (c) it may reflect the necessity of boundary conditions (at °°) as in [C1],V
.
In every case examined, existence of a finite set of generators of D! (mod A!) gives a simpler theory. In case of example 7.4 - Laplace comparison algebra of the
bounded functions and folpdes with covariant derivatives vanishing at °°, on a complete Riemannian manifold, we now want to
supply a criterion, allowing the construction of the secondary
A = LAN , for LELN Above we have discussed validity of (s),(lI),(l2),(15)(R.) It is found that (m1'), (m4), (m5), (m7) hold trivially, but we now require the separation condition (m3). Then VII,thm.3.6 is applicable and gives the secondary symbol of DA, for DEED!. On the other hand, for L E LN
,
the operator A = LAN first must be writ-
287
IX.7. Examples
as a sum of products of such DA
288
(mod E), before its secondary
,
symbol can be obtained. For the principal symbol we know, of course, that only the principal part of L matters, and that o A1W
equals the principal part, restricted to S S2 Theorem 7.5. Let all above assumptions be satisfied. (a) The operator A = LAN
for a product
,
L = D1D2..... DN
(7.12)
DjE D!
e
, j=1,...,N
is contained in C , and its symbol is given as the restriction to I of the product (7.13)
TD
ATD2 A " " TD A
N
1
(b) Let x0E 2MA! have a neighbourhood N in M
A!
with the
following property: There exists a cut-off function X with the properties of cdn.(m3)x
such that, in local coordinates any-
where at NnD we have Oa L = IN j=0
(7.14)
1
where (with
rb
,
11...1.
11...1. 0V1 ...41
j E C 'M
ja 0
k depending on j, of course) for all j=O,...,N,
(7.15)
with finite sums, and with bounded tensors (kb) of one variable index, defined over all S2 such that ckE A! ,r(rb13 1) E D! x
Then we have A=XLANE C. Moreover, the secondary symbol of A over x0 is given explicitly as the restriction to the (possibly void) set t-1(x0) c &*S2 of the continuous extension of the
function (7.16) (which is contained in CB(T 0))
...C1 )(hkm(x)ekEm+1)N/2
(7.16) 1
0
(c) For any LELN,O, and, more generally, for LE RN , we get A=LANE C., and the secondary symbol of A is zero. Moreover, we have A E K(Hs), for all sE 1, whenever L E RN is of order
so
,
and follows by induction:
Write (7.17)
D1....
DNDAN+1=(Di.... DNAN)((DA) + (A-N(DA)AN-(DA)))
,
IX.7. Examples
where the first term's symbol can be written in the form (7.12), by induction, while the last term is compact, by cor.3.4. The key to (b) is found in (c): Any commutator [D,F] has coefficients vanishing at infinity, by V,(2.13), where the derivatives may be written as co-variant derivatives. Hence the formal symbol vanishes there too. Accordingly, using (a), it is clear that the secondary symbol is zero for any LAN , where LEAN, since for such L there is at least one commutator in each of the N-fold products. It thus follows that, for the construction of a local decomposition of LAN into N-fold products the operators DA may be treated as if they were commuting. Hence the polynomial decomposition corresponding to (7.15) gives the proper symbol. Also, since LAN , for LERN , has vanishing secondary symbol, it is suf-
ficient to calculate modulo RN
.
This completes the proof.
Remark 7.6. Notice that thm.7.5 will be valid for the Cs-symbol
ANL for LER_ , are added to the generators (1.10). Also we observe that McOwen's result has (m3) as well, if the operators LAN
,
,
replaced by another condition - that the dimension 6x of VII,2 be constant on connected components of BMA, A local version of thm.7.5 is valid too, requiring only at some x0E BMA, Also, one of the other representa(m3)x .
,
.
0
tions in VII,2,3,4 may be applicable, if (m3)x
fails. 0
Example 7.7. Consider the comparison algebra C of VIII,4 again.
Here we have a Laplace triple on a complete Riemannian manifold again, but the classes A! and D! of example 7.4 appear as too restrictive. The algebra C of VIII,4 corresponds to a different choice of A! and D!
, which we discuss in X,5. These new classes
contain functions (folpde's) with covariant derivatives not tending to zero at infinity (only the t-derivatives must vanish). On the other hand, there exists a limit function (or folde), at each end, which is not in general true for example 7.4. Example 7.8. For the one-dimensional example of Sohrab, discussed in VII,thm.4.8, we trivially have cdn.(q 2) satisfied, since R=0 and then must impose (q3) (7.18)
, which simplifies to
q(k) = dkq/dxk = 0(q1+k/2)
assuming SL-normal form, of course.
, k=1,2,...,
289
IX.7. Examples
Covariant derivatives are ordinary derivatives, in this case. Accordingly, LN consists of all N-th order expressions aj=O(q(N-j)/2), a,(k)=o(q(N-j)/2
with
(7.19)
k>0
Control of the secondary symbol is easy, in this case. We get (7.20)
for all LELN Example 7.9. Consider again the triple of V,thm.4.1 (and VII,4), now considered with cdn's (q2), (q3), and without a sub-extending triple. We require cdn.(s), hence VII,(4.7) valid for every positive y
,
not for yl only, and then use the classes A!
,
D! of
example 7.3.
Here we only notice that the conditions imposed are consistent. For example, consider the 1-dimensional example (7.21)
sl
_ (0,o)
,
du = dx
,
H = -d2/dx2 + l+x-3
.
A calculation shows that (q3) holds as well as (s) (IV,thm.1.5.). Accordingly we will get a special case of example 7.3. The discussion of classes of differential operators within reach of the generated comparison algebra should follow that of example 7.3. Also, of course, we will have to characterize the secondary symbol space of this algebra. This will not be investigated here.
290
CHAPTER 10. FREDHOLM THEORY IN COMPARISON ALGEBRAS.
In this final chapter we are going to discuss the main application of theory of comparison algebras to the derivation of Fredholm properties, and essential spectrum properties for differential operators within reach. Note that the general concepts were developed in [C1], and
they were demonstrated there in relation to elliptic differential operators on In or on the half space R++l There is no change at all when we apply these abstract concepts to general manifolds. A more detailed introductory discussion is given in sec.l, below. In particular we choose to discuss this abstract theory ,
after our full scale investigation of the symbol spaces of a comparison algebra, above, to direct the focus of attention accordingly. Also, many proofs will be omitted. Note that we shortly touched the concepts of Fredholm operator, Fredholm inverse (and Green inverse) in VI,3, while results on compact manifolds were discussed, requiring this terminology. In sec.l we discuss the general facts around Fredholm opera-
tor, Fredholm index, and C -algebras with symbol. In sec.2 we discuss some facts concerning the Fredholm properties of differential operators within reach of a comparison algebra, and about essential spectra. In sec.3 we discuss comparison algebras with compact operator valued symbol, and operators acting on crosssection of vector bundles over a noncompact manifold. In section 4 we discuss abstract Fredholm theory for a C*algebra with a two-link ideal chain. In sec.5 we turn to the
Frechet algebra of operators of order 0, bounded over all Sobolev spaces, reviewing some abstract general facts, discussed in [C1]
X.1. Fredholm operators in C -algebras
292
1. Fredholm theory in C C L(H).
In this section we will continue the abstract discussions of ch.I, looking at the concept of Fredholm operator in general, bounded or unbounded, and, especially, at the theory of Fredholm operator and index in a comparison algebra CCL(H)
,
H=L 2 (Q), and
for differential operators within reach. This material has been
discussed by the author and his associates in [C 1 1, [BC 1 1, [BC [ CM1] , [CM 1, [ CHe1] , [ CS] . Some slightly different approaches 2
21,
are given by Coburn [Cbi1, Douglas [Dgj], Barnes et al.[BMSW] Accordingly we will omit some detailed proofs, referring to [C1], where our views were presented in close detail. In these notes we have focused on the Hilbert spaces Hs s E 1, and the Frechet space H, only. Accordingly we restrict our review to Hilbert spaces only, except in sec.4, where the Frechet algebra C, and differential operators within its reach are studied In Hilbert space many features become simpler, and discussions are less costly. However, most features extend to bounded or unbounded operators in Banach spaces as well (cf. [C1],App.AI) On the other hand, a general Fredholm theory in Frechet spaces seems to have pathological features. For example, the linear group of all invertible operators of L(X), for a Frechet Therefore it perspace X , no longer needs to be open in L(X) .
.
haps is even more interesting that the Fredholm theory of C. still
remains intact. It looks like that of a C-algebra (cf.sec.4). In fact, C. is a 4-subalgebra of L(H), in the sense of Gramsch [G11. As a subalgebra of L(H) it contains all its inverses in the larger algebra, just as this is known for C -algebras Moreover, in [CSch], [Schl] it was seen that, at least for some important special cases, the symbol homomorphism of C. still is a surjection onto C(ffi) (This property is known to be generally false for comparison algebras in LP-spaces or other Banach spaces.) Perhaps, in the interest of a more systematic approach, the
present discussion should have been in order at the begin of ch.5. However, we wanted to establish the Banach algebra structure of a comparison algebra as the focus of our attention. Thus we were pushing aside the important motivation of this theory from theory of regular and singular elliptic boundary problems in partial differential equations. The latter will proceed now.
X.1. Fredholm operators in C -algebras Returning to abstract theory of (unbounded closed) linear operators in Hilbert space of I,1 we recall that a closed operator A: dom A -> H
,
in an abstract (separable) Hilbert space H, is cal-
led a Fredholm operator if (i) in A is a closed subspace of H and (ii) we have (1.1)
dim ker A < -
,
and codim im A = dim H/(im A) < -
.
It is not hard to see that (ii) implies (i), but it is useful to keep (i) in mind, as an important property of Fredholm operators. In particular, for operators with closed range (with im A closed) the second condition (1.1) may be stated by requiring A).1_-
For a Fredholm operator one defines the that dim(im Fredholm index by setting .
(1.2)
ind A = dim ker A - dim(H/im A) = dim ker A - dim ker A".
The concept of Fredholm operator is nontrivial only for infinite dimensional spaces. This may be the reason for the fact that they were systematically studied only in the 1950-s, although the concept is quite simple. Special classes of Fredholm operators(integral operators and differential operators) were studied in the late 19-th and early 20-th century (cf. Fredholm [Fh1] and F.Noether[Nt1], who showed that Fredholm operators with
nonvanishing index occur in the theory of the 2-dimensional oblique derivative boundary problem).
We summarize the principal properties of a Fredholm operator in thm.1.1, below. We will express the statement that A is Fred', or that ' A has property
holm by stating that 'A is
'.
Theorem 1.1.(a) Property (D and ind A are invariant under small is bounded and arbitrary compact perturbations; (b) Property t adjoint invariant, and ind A = - ind A ; (c) The product AB of closed Fredholm operators A and B is always a well defined closed and operator; we have (AB) = B A ,
(1.3)
ind(AB) = ind A + ind B
(d) A closed operator is
an isolated point of Sp(A)
;
if and only if either OE Rs(A) or 0 is ,
and an eigenvalue of finite multipli-
city for A and A Remark 1.2. Point (a) of thm.l.l may be expressed as follows.
293
X.1. Fredholm operators in C -algebras (a): For a closed A with preperty we have A+C E as well as A+cB E (
,
and B E L(H)
,
for small c
ind A = ind (A+C) = ind (A+cB)
(1.4)
,
,
C E K(H) and
.
The same statement holds for unbounded operators B and C which are 'A-bounded' and 'A-compact', respectively, i.e., BI(dom A) E L(dom A,H)
(1.5)
, CI(dom A) E K(dom A,H)
,
where dom A is regarded as a Hilbert space of its own under the graph norm I,(2.8) .
For a proof of thm.l.1 cf.[C1] (for Banach spaces)
,
[CLa]
for Hilbert spaces, with generalized perturbations, also [Nbi]
for generalized perturbations in Banach spaces. For a discussion of bounded and unbounded Fredholm operators between different Banach spaces cf.[C1l,AI We also will consider Fredholm operators between different .
(separable) Hilbert spaces. However, since any two separable Hilbert spaces H1
H2 are isometrically isomorphic (with an iso-
,
morphism U defined by mapping j -"Vi for any fixed pair {4i },
of orthonormal bases of H1 and H2) we simply define A E Q(H1,H2) to be Fredholm if =
ind(AU-1)
.
AU-1
is a closed D-operator in H2 , and define
This reduces our discussion to the case of 0-opera-
tors in Q(H) , H=H2. The key to our present theory of Fredholm theory of compari-
son algebras is the following fact stated for bounded Fredholm operators. Let O(H) denote the class of bounded c-operators.
Theorem 1.3. (a) An operator AEL(H) is Fredholm if and only if there exists an operator B E L(H) such that
(1.6)
AB =
1+E
, BA = 1+F ,
where E and F are operators of finite rank (with finite dimensional image) .
(b)
An operator A E L(H) is Fredholm if and only if its
coset A = A + K(H) mod K(H) is an invertible element of the C*-algebra L(H)/K(H)
(c) Let [Al be the homotopy class of q in the group These homoGL(L(H)/K(H)) of invertible elements of L(H)/K(H) topy classes form a group Hd under operator multiplication. There exists a group homomorphism u:HcD - Z such that ,
.
294
X.1. Fredholm operators in C-algebras ind A = u[A-]
(1.7)
Remark 1.4.
,
for all A E VH) .
An operator B E L(H) satisfying (1.6) is called a
Fredholm inverse (or (D-inverse) of A . The homomorphism u of
(1.7) is called the index homomorphism (of the C -algebra L(H)). Clearly a Fredholm operator will have many Fredholm inverses. Among these we have focused on the special Fredholm inverse in
[C1],AI and, with respect to an HS-chain, the distinguished Fredholm inverse (VI,3). for Banach spaces. For a proof of thm.1.3 cf.[C1],AI ,
9e
Now let us recall the following property of C -algebras, not normally shared by other Banach algebras: Proposition 1.6. Let A be a
invariant sub-algebra of the C -
algebra 8 , where both A and B have (the same) unit. Then, if
A E A C 8 is invertible in 8
,
For a proof cf. [C1] , All
its inverse A-1 is contained in A ,
or [ Rl]
.
Combining thm.1.4 and prop.l.6 we now get a result useful for Fredholm theory in comparison algebras. containing the identity operator and the compact ideal K(H). Then an operator
Theorem 1.7. Let A be a COY-subalgebra of L(H)
,
A E A is Fredholm if and only if its coset Av = A+K(H) is an invertible element of the quotient algebra A- = A/K(H)
Indeed, we then have A/K(H) a C -subalgebra of L(H)IK(H) both with the unit 1+K(H)
,
so that prop.l.6 applies.
Remark 1.8. Under the assumptions of cor.1.7 we also may replace the homomorphism u by a homomorphism from the group of homotopy classes of GL(A/K(H)) to 1 (cf. [BC1]
).
Now, if A = C is a comparison algebra, as studied in chapters V - IX, then the algebra C/K(H) has a very simple structure at least in the following two cases. Case (A): The commutator ideal E is
K(H)
(Such comparison alge-
bras were investigated in V,4 and VII,2).
Case (B): The commutator ideal E is not K(H), but has the property that E/K(H) is an algebra of compact-operator-valued functions over a locally compact space (such as the two examples discussed in VIII,l and VIII,2.)
295
X.1. Fredholm operators in C -algebras Note that other cases are possible: Not every comparison is either in case (A) or (B)
.
In particular there may be ideal
chains longer than the two-link C D E D K(H) , with other proper ideals fitting between E an K(H). First consider case (A) symbol space
.
Then C/K(H) = C(81)
, with the
. A continuous complex-valued function aA(m) over the compact Hausdorff space M is invertible if and only if it I
never vanishes. Therefore we have the following result. Theorem 1.9(A). Under cdn.s (a0),(a1),(d0) only let the comparison algebra C have compact commutators. Let HI denote the symbol
space of C (cf.VII,l), and aA the symbol of AEC. Then an operator AEC is Fredholm if and only if its symbol aA(m) , mEI , does never vanish. There exists a group homomorphism u from the group f1(ffi) of homotopy classes [a] of non-vanishing continuous complex-valued functions a over M such that
ind A = u([aAI) for all A E MOM .
(1.8)
The proof of thm.1.9(A) is an almost trivial consequence of our above discussion. For a more detailed derivation cf. [BC1]
For a partial discussion of case (B) we refer to sec.4 below. Two inversions, modulo the two ideals E and K are needed,
in this case. We also refer to [ CPo] , [ CDgl] and [ CMel] , where we treat a useful generalization for the special 2-link-algebras of VIII,1,2,4
,
and for more general such algebras. Instead of the
two inversions we there have only one symbol, assuming singularintegral-operator-values at certain points of the symbol space. 2. Fredholm properties of operators within reach of C. Note that thm.1.9(A) gives a very simple necessary and sufficient criterion for the Fredholm property and index of operators in the algebra C provided that we know details about the ,
symbol space 1I of the algebra, and also know how to explicitly
obtain the symbol of an operator AEC
That precisely was the intent of our effort, in the preceeding sections. .
The present thm.1.9(A) only represents the simplest non-trivial case of a chain of similar results. Later we will discuss a similar result for case (B) as well. Also some cases involving longer ideal chains will not be discussed. Also we will have to
296
X.2. Differential operators within reach deal with the case of a more general matrix A = ((A.k)) of J
operators AjkEC, where even infinite dimensional matrices will be admitted. Also, this will lead us into comparison algebras acting on L2(O)-crossections of a vector bundle on c
.
Let us use the present example to explain the principle of comparison: Often one will not be interested in property (or the index) for operators in C, but rather, for an unbounded operator of the form L = AQ-1 , with A, Q E C
(2.1)
Example 2.1. Let Q = A = H-1/2 triple {O,dp,H}
.
, with the expression H of some
and let A=DA , with some DED. Assuming cdn.(m5),
,
it t2en follows that A and A both are in C However, the operator A-1 = H1/2 is an unbounded closed Fredholm operator of H , with .
since it is self-adjoint, positive definite, and >1 (thm.l.1(d)). Suppose now that A is Fredholm. For example, assume
index 0
,
,
that C has compact commutator, and that aA(m)#O for all mE ffi. Then
apply thm.1.9(A)). Then AA-1 is an unbounded Fredholm operator, and ind(AA-1) = ind A
,
by thm.l.l(c),(d)
,
since ind A-1 =
0
However, we find that (2.2)
Du = bJu
AA-1u,
.+pu =
for all uE dom(AA-1)=dom A-1 = H15
1x
with the first Sobolev space H1. In other words, Z=AA-1 is a realization of the first order differential operator D, with domain H1, in the sense of V,1
.
Thus,if aA does never vanish on M
,
then we have found a realization Z of the differential operator D which is Fredholm. In fact, since A-1:H1-H acts as an isometry, (cf.V,l) one concludes at once that the realization D constructed if and only if A is ' i.e., if and only (with domain H1) is ,
if aA(m)#O
,
mEffi
.
The above conclusion is not very far reaching, since there are much simpler methods available, to solve the single first order partial differential equation Du=f, with or without boundary conditions. (In fact, looking at the symbol aA = bJ(x)Cj over it is clear at once that the condithe wave front space W C ¶ ,
tion aA90 on W cannot be satisfied, unless n=l.)
This changes, however, if we either consider a matrix of operators Aj1E C or an operator linked A=((Ajk))j,k=l,...,m to a higher order expression L E LN as in VIII,3. We look at ,
,
297
X.2. Differential operators within reach matrices of operators in C in sec.3, below, although this is the simpler problem. Here we next look at an LELN Example 2.2. Consider the case Q=AN, for some integer N=1,2,..., with a differential expression LELN. Here we must ask for cdns. (s),(m5),(lj), j=1,2, (apart from (a0),(a1),(d0)), and (15)(P.) for a suitable class P., within reach of A!, D!, so that IX,prop. 3.2 or cor.3.4 hold. (Note that (s) may be replaced by (sk), for sufficiently large k, as easily seen.) Also, cdn.(m1) or a cdn. (m1)^, for suitable q^, may be useful. Since compactness of commutators is required, we assume a triple as in early V,4, with a large constant y
of V,(5.3) to get cdn.(s ), not only (w), and
cdn.(g1), with classes A!CA#, V!CD#. Or else, we assume the case of IX,example 7.3, with cdn's (q2), (q3). In any such case we get
(2.3)
A = LQ = LAN E C
.
Thus the conclusion of example 2.2 may be repeated. We have im AN = dom A_N = HN, with the Sobolev space of order N of IX,l. Formula (2.4) defines a closed realization of the expression L (It was already discussed in V, def.6.2 , and called Z there. Here we find it more convenient to use the same symbol for the expression L and its realization.) (2.4)
L = AA-N , dom L = HN = im AN This realization L is a closed '-operator, if and only if aA(m)#0 for all mEffi
(2.5)
.
We again are interested in checking on (2.5) explicitly. In that respect note that formula VI,(3.9) indeed gives the symbol on the wave front space. We get (2.6)
as IfI=h3kEjY1,
where L=a(x,D), in local coordinates. In other words, aAIW is just the principal symbol of the expression L, at the unit co-sphere, in the conventional sense, and the condition (2.5), restricted to g!clos, amounts to the statement that
(2.7)
IaN(x,E)I
>
Ti
>0 for all
E ICc? = W ,
i.e., that the expression L is uniformly elliptic on 0, with res-
298
X.2. Differential operators within reach pect to the metric tensor hjk
.
It is important to observe that we are not yet in control of the full symbol, except in a case where the secondary symbol space is empty. However, we have the results of VII,2, 3, 4, and IX,7 to investigate the secondary symbol. For example, under (ql) of VII,4, and (m1'), (m3), (m7), or the corresponding "'-conditions, we conclude from VII,thm.4.7 that there is no secondary symbol space over aQ , although there may well be points of the secondary symbol space over DMA#\aQ=DMA#
,
as our examples of V,4,
VII,2, VII,4, and IX,7 show. Even if the space ffis is completely known, as in case of VII, thm.4.8,
for example, the secondary symbol of A=LAN, for a higher
order expression L is known only if a representation of A as a sum of products (D1A)(D2A)...(DkA) is explicitly given. For the factors DA of such a product we may invoke VII,thm.2.2, thm.2.3, thm.2.6, etc. Again, under (q2), (q3), we may refer to IX,thm.7.5, given its assumptions.
In some cases, however, a complete knowledge of the symbol is not required, while significant statements still can be made. Note that the differential operator L of example 2.2, regard-
less whether Fredholm or not, always is a restriction of the cloA-N sure of the minimal operator L0 , since (A-NICO)clos = due to cdn.(s). Assuming the expression L to be formally selfadjoint, the operator L will be hermitian, and a restriction of t'he closure L0
of L0
.
In fact, we now have the following.
Theorem 2.3. Assume cdn's (a0), (al), (d0), (s), (11), (12), (m5), (15)(P,) with respect to a suitable class P, within reach
, and of principal symbol type, and assume that C has compact Then commutator. Also assume that a0 = a N # 0 on all of M
of C
.
n
for every self-adjoint uniformly elliptic expression LELN (i.e., is a self-ad-
L satisfies (2.7)) the realization L of (2.4) joint operator.
Remark 2.4. The condition'
c N#0
(or aA#0) on Ms
VII,cor.4.3, if q=l for large x of QUMM
,
'
follows from
since then D=D0=1 may be
taken in VII,thm.3.6. N/Proof of thm.2.3. Let
us define B = A2LA N/2
,
where, for a
299
X.2. Differential operators within reach moment we use the convention of IX,4 (i.e., regard all operators and expressions as operators on H.). It follows that (the H-closure of) B defines a bounded self-adjoint operator of H. Moreover,
fromIX,prop.4.1
we conclude that
A - B = A - AN/2AA_N/2 = AC, with C E
(2.8)
so that also BEC
aA=O on W
,
since AEC
.
C
,
In fact, we get aA=aB on ID
,
since
.
Since B is a self-adjoint operator, its symbol must be realvalued. Since we assumed L uniformly elliptic, it follows that aB = aA is bounded away from zero on W. On the other hand we assumed aA00 on Ms = M\(,clos) Since A also is self-adjoint, aA is .
real-valued as well. Hence we conclude that (m) # 0 for all m E H1
(2.9)
(B+ieA N ) since aB90 in a neighbourhood N of pjclos , while ffi\N C ffis is compact, so a N bounded away from 0 on ffi\N , and the two terms cannot A
cancel each other, one being real, the other purely imaginary. Conclusion: B+iCAN is Fredholm, hence L+ie= A-N/2(B+iEAN)A-N/2
also is Fredholm, whenever e90 as follows from thm.l.l(c). From the same thm. it also follows that ,
(2.10)
(L+ie
=A
-N/2
(B+iCAN )A-N/2
=A-N/2 (B-iCAN)A_M/2
= L-ie ,
in the precise sense of adjoint of unbounded operator, using that both operators A_N/2 and (B+ieAN) are Fredholm. Hence it follows that L
L
,
or that L is self-adjoint, q.e.d.
For a general, not necessarily self-adjoint, but uniformly elliptic expression we can derive knowledge on the essential spectrum of the realization (2.4) from the knowledge of the secondary symbol. Here the essential spectrum of an unbounded closed operator AEQ(H) is defined as the collection of all AEtC such that A-A is not a Fredholm operator. From thm.2.l it follows
that the essential spectrum consists precisely of all points A E Sp(A) which are not isolated point-eigenvalues (of A or A*) of finite multiplicity. (Note that other definitions of the concept 'essential spectrum' are in common use (cf. Kato [K2], for example
)
Theorem 2.5. Assume cdn's (a0), (a1), (d0), (s), (11), (12),
300
X.2. Differential operators within reach (m5), (15)(P0), as above, and that the algebra C has compact commutators. Let oQ=oAN#0 on all of Ms
.
Let LELN be a uniformly
elliptic expression (satisfying (2.7)), and assume that the set {oA(m)/o N(m)
(2.11)
:
mEM 5}
A
leaves out at least one complex number. Then the realization of L defined by (2.4) is a closed operator (and the closure of the minimal operator L0). Moreover, the essential spectrum of this realization is given by the set (2.11). Proof. In essence we repeat the arguments of the proof of thm.
2.3 in the following simpler form. First, the realization L Next, we write again is a restriction of L0* .
L-A = (A-AAN)A-N
(2.12)
where AA=A-aAN E C , and aA X =CA # 0 on Mp=wclos on 1%
,
due to oA=O
, and (2.7). There exists at least one XoEM, not contained
in the set (2.11). We conclude that oA
=oA-Xoa N#0 on all of ffi Ao
A
so that AXo is Fredholm. Accordingly L-ao , hence L is closed.
Also L-Xo is Fredholm, by thm.1.1(c). The argument may be repeated for any XEM not assumed by the set (2.11), showing that L-A is Fredholm for every such A. Hence the essential spectrum is a subset of the set (2.11). On the other hand, if A is contained in the set (2.11), then we know that A-aAN is not Fredholm. But the operator A-N:HN=dom AN + H is an isometry between the Sobolev spaces HN and H as we know. Relation (2.12) may be ,
interpreted in two different ways, either the product is a product and of two unbounded operators, or a composition of
A-N:HNH
.
Both interpretations describe exactly the same map,
(even though not topologically). With the second interpretation we AAA-N find that is not a bounded Fredholm operator HN-3.H , hence
or both. But the bounded either codim im = co or dim ker = operator HN+H and the unbounded operator of H with domain don. L ,
= HN have exactly the same image and kernel. Therefore L cannot be Fredholm as well, q.e.d.
The above examples were designed as a demonstration, showing how a (singular) elliptic problem may be handled with the 'comparison techique'. We selected the simplest nontrivial case for this demonstration. Results like thm.2.3 and thm.2.6 have
301
X.2. Differential operators within reach straight generalizations to the case of a (finite or infinite) matrix of differential expressions, as discussed in sec.3, and to expressions formally acting on the type of vector bundles of sec.3 as well. We shall not discuss details. Perhaps we should also point to a possible use of resolvent integrals for realizations of the form (2.4). A criterion like thm.2.5 gives very precise control on the Fredholm domains of the analytic operator family LT = L-X They are just the connected components of the complement of the (closed) set of (2.11). By a result of Gramsch [G2] it is sufficient to know existence :
of the resolvent in only one point of a Fredholm domain, to conclude existence of the resolvent as a meromorphic function defined in the entire domain, with poles clustering at the boundary only. Such knowledge then allows definition of functions of L as resolvent integrals.
Let us finally note that one needs not necessarily choose the operator Q of (2.3) as a power of A. Clearly, Q-1 should be Fredholm, and perhaps should have other properties. It could be self-adjoint (and positive definite), for example. Note that the condition aQ#O on ffis, required in thm's 2.3 and 2.5, does not hold
for the problems of Sohrab (VII,thm.4.8).There we have aA the entire symbol space, including ffi
=
0 on
.
s
Our technique for those thm's depended on the fact, that the principal symbol space coincides with the set where aQ=O
.
In general one could redesign the results, replacing the set ffip
by ker aO Note also that we have been using the comparison technique in [C1],V,14, with an operator Q (called T there) of a different kind. Each T used there represents an elliptic boundary condition (of Lopatinskij-Shapiro type), and we were solving a regular (not
boundary-value problem.
302
X.3. Systems and Vector bundles 3. Systems of operators, and operators acting on vector bundles.
We continue our review of Fredholm theory in comparison algebras, and now discuss operators with compact operator-valued symbol, as in IC1],V,8 Recall that such algebras and [BC2] already were used in VIII,1,2,4. In particular it was found that ,
.
the commutator ideal E
if not compact, often has a natural compact-operator-valued symbol. Here, however, we work from the start in a Hilbert space H- which is a topological tensor product H_ = hiH
(3.1)
,
H = L2(0)
,
where h is a separable Hilbert space. (We normally assume h=k2 = L2(Z+)
,
Z+ = {1,2,3.....}). In [BC2] the space H also is an
abstract Hilbert space. Here we consider L(H) as the home of our comparison algebra C, acting on the complex-valued functions of H = L2(M. We now want to consider the C' -algebra C' =
(3.2)
kffiC
, where h = K(h)
.
Note that an operator KEk is an infinite matrix K = ((kjk)), acting on a sequence of h=f2 by matrix multiplication. All matrices with only finitely many non-zero entries are necessarily compact, and every compact matrix is a limit in norm convergence of a sequence of such matrices of finite rank. Correspondingly, it can be shown that every operator of kNL(H) is an infinite matrix e where Ajk E L(H)
A = ((Ajk))j,k=1,2,...
(3.3)
Vice versa, a given such matrix (3.3) is in kNL(H) if and only if it is a limit, in norm convergence of L(H '), of a sequence
of matrices (3.3) each having only finitely many non-zero We have K(H) = k9K(H) C koL(H) C L(H-) entries Ajk all with ,
.
proper inclusions, and K(Hf) is a two-sided ideal of kk(H) We get A E C if and only if all entries AjkEC , j,k=1,25 .... and, again, A is limit of a norm convergent sequence of such
,
matrices with only finitely many non-zero entries. For such operator A E C we define a symbol a by setting (3.4)
Cr
A
(m) = ((a
Ajk
(m))) .
,k=1,2,...
Now it follows easily that aA:M in norm topology of k = K(h)
.
.
k is a continuous function
We have the result, below, where
303
X.3. Systems and Vectorbundles
304
E = K(H) is assumed for the remainder of sec.3, although an extension to the 2-link-case would meet no obstacles.
Theorem 3.1. Let (a0),(a1),(d1) hold, and let CCL(H) be a comparison algebra with compact commutator is Fredholm if
(a) Then an operator B = 1+A, with A E C
and only if the function 1+aA(m) is invertible in L(h), for every mEj (b) There exists a homomorphism u from the group HWI) of .
GL(l+k), under homotopy classes [a] of continuous maps a: operator multiplication as group operation, into Z, such that -
ffi
(3.5)
ind A = u[aA]
,
for all A E C' n V H')
.
thm.7 and thm.l0. For detailed proofs we refer to [BC21 There we also discuss the precise definition of the topological ,
tensor products ho-H Q C L(H)
and PkQ , for CC subalgebras P C L(h), and
.
The most important applications of thm.3.1 aim at (finite dimensional) matrices of operators with entries in a comparison and, more generally, at operators acting on crossections of a (finite dimensional) vector bundle over B Note that, algebra C
,
.
the homomorphism u of thm.3.1 in case of a compact manifold n essentially coincides with the relation between topological and ,
analytical index expressed by the Atiah-Singer-Bott index theorem. Finding u explicitly, i.e., obtaining an explicit index formula for a comparison algebra C' on a noncompact mainfold, will be a task of algebraic topology. We are not equipped to discuss this problem here, but, perhaps should make a few comments. Remark 3.2. A homotopy A:[0,11xffi 3 GL(l+k)
A(t,m)=a(t,m) + K(t,m)
(3.6)
,
,
in detail,
0
always can be deformed into a homotopy U:[ 0,11
,
mEffi
,
-* U(l+k)
, where
U(l+k) denotes the group of unitary operators in the algebra l+k C L(h) This follows, because a polar decomposition A = BU is .
possible within the algebra l+k , where B=(AA*)112
U=A(AAC)-1/2.
,
Here U is unitary, while the invertible self-adjoint positive definite operator B may be deformed into the identity, by the deformation titB+(1-t) E GL(l+k)
.
Remark 3.3. The problem of discussing the index homomorphism u may be reduced to a more conventional form in homotopy theory by
X.3. Systems and Vector bundles inducing the Bott group U(°°)
, defined as follows. For N=1,2,...,
let hN C h = 12 be the subspace of all sequences u=(ul,u2,... with uN+l=uN+2= ... =0
,
and let U(N) denote the subgroup of
U(l+h) consisting of all unitary matrices equal to 1 in h1 Regarding U(N) as a topological group with (Euclidean) topology .
one defines U(°) as inductive limit of the sequence U(N)
, with
inductive limit topology. Concretely, we thus have (3.7)
U(oo)
= U.- U(N) 3 -1
A subset MW(-) is closed if and only if MrU(N) is closed, for all N=1,2,... . A compact subset of U(oo) can have a non-void intersection with at most a finite number of U(N) Note that a continuous map A: X - U(l+k) for any compact .
,
space X can be written uniquely in the form (3.8)
A(x) = a(x)(1+k(x)), xEX , with aEC(X,S1) , kEC(X,K)
with the unit circle S1 = {cEc:Ic1=1} in T
.
,
On the other hand,
every continuous map F:X+U(oo) is continuous from X to U(l+h), and
vice versa, a continuous map G:X + U(l+h) of the form (3.8), with can be continuously deformed into a map F:X-U(oo) a(x) = 1 (Essentially, to construct F from G one uses the fact that U(oo) ,
.
is dense in the class of all unitary operators on It of the form 1+K
,
KEk, in norm topology of L(h)
,
and the polar decomposition
of rem.3.2.)
This reduces the discussion of the homotopy classes of thm. 3.1 to that of homotopy classes of maps 1 ; UN , with the unitary group UN of large dimension N.
Note that the theory of NxN-matrices A=((A.7k)).,k=1,...,N of operators AjkEC may be treated as a special case of thm.3.1.
Indeed, for such a matrix we define an operator A- E C' by (3.9) A-=((A-jk))j,k=1,2,..' A-ik=Ajk, for j,k
) if and only if A is a Fredholm operator of CN = L(fNW , and A, A" then will have the same index. The symbol of A will of course be defined as
(3.10)
aA(m) = aA-(m)IhN
noting that aA(m) is reduced by the decomposition h=hN ® hN . It is just as easy, of course, and, in fact, simpler, to
305
X.3. Systems and Vector bundles
redo the above theory with h replaced by the finite dimensional space hN = (EN .
In the following let Cm =
L(Cm')ffiC
, where C denotes any
comparison algebra. Let us come back to the comparison principle and A = DA , with of (2.1), and again set Q = A = ((Adjk)) ,
(3.11)
D =
((D.
7k
,k=l,...,m AA-1
Again we get a realization D = expression(s) D
k
D. E D
,
)).
of the (mxm-matrix of)
That realization always is a restriction of the closure of the minimal operator D0 , regardless of any condition .
(w), or (sk) In the results below we skip looking at the Fredholm index. An abstract index formula always follows in a similar .
way as in thm.1.9(A), thm.3.l,etc. Obtaining a concrete index formula should be easiest, in the present cases, but will not be a subject of our concern. Theorem 3.4. Assume cdn's (a0), (a1), (d0), and that C has compact commutator. The above realization D is a closed (unbounded) Fredholm operator of L(Hm), with HRL-TmNH, if and only if
the mxm-matrix aA(m)=((aD
(3.12)
A(m))) is invertible for mEI jl
Note that no cdn.(w) or (s) is required in thm.3.4. The proof is an immediate consequence of thm.3.1. Let us try to discuss the result. Again, we first consider There condition (3.12) restricted to the wave front space Wcffi .
it just means that the first order mxm-system D=((Djk)) is uniformly elliptic, with respect to the tensor hjk (3.13)
D = -iB1(x)3
.
.
That is, if
+ P(x)
X3
locally, with mxm-matrix-valued functions Bj(x)
,
P(x)
,
then
the function D(x,O) = B3(x)E. (globally defined as a function in C_(T*O,L(Mm)) ) is invertible for every (x,E)40, and satisfies (3.14)
HD-1(x,i;)II
with the matrix inverse
< X
< - , as xEQ, CE]kn, h3k(x)EiEk=1
at x,C
.
For a compact manifold 0 this is the only condition, since then I = W . On the other hand, if 0 is non-compact then there still may be comparison algebras with Ms = 0
.
For example, this
306
X.3. Systems and vector bundles is true under the assumptions of VII,thm.4.7, if 04 0 is compact.
On the other hand, our examples of IC11, and those of Sohrab Then we can apply thm.3.4 only if
show that we may have It s# 0
.
IDs and aA"ffis are accessible.
Note that thm.3.4 at least gives a sufficient criterion for the Fredholm property, if we have the assumptions of VII,thm.3.6 (or thm.3.6", for a suitable {St",dp",H"} (c> {S2,dp,H}). Then the
symbol is a restriction of the formal symbol of VII,(3.25). For a compact commutator algebra C the operator D of (3.11) then is Fredholm if (though not necessarily only if) the formal symbol se
TDA is invertible at each point of 88 0
Finally we will use our above theory to also consider operators acting on crossections of a vector bundle over Q. We have avoided vector bundles, so far, because our C -algebra approach to symbol and symbol space is developed best for complex-valued symbols, while its extension to vector bundles is a simple abstract matter, to be discussed now.
Let V be a vector bundle over Q with fiber an inner product space Vx of dimension m, with norm IuIx
,
u E Vx
.
We only con-
sider (C'-)vector bundles of the form V=VA#IS2 = 0-1(Q), where VA#
is a (continuous) vector bundle over the compactification M # of A We assume 0 induced by A# (cf.VII,l), with bundle projection 0 .
the euclidean metric 1.1x of the fiber defined for xC M # nuous in x
,
,
conti-
and C' over Q.
For the above we assume given a comparison triple on 2 , and generating classes A# D# satisfying (a0),(a1),(d0) , and then ,
will introduce operators acting on HV = L2(Q,V,dp), the space of crossections u of V satisfying Ilull2
=
J0Iu(x)l2dp
<
Let MA#=U1U...UUN be a finite atlas of the (compact) bundle VA#, and 1=V)1+...+PN a subordinate partition of unity, V .IOE C'(0)
Under the above assumptions we will construct an isomorphic imbedding of the Hilbert space HV into our above space HR=hRNHCH°° C = C(A#,D#). For with orthogonal projection PV:H-+HV in CR C C two such vector bundles V and W one then may consider operators of the form A"=PWA`PV , where A^'E C" . We get A"E C'; the restric-
307
X.3. Systems and vector bundles tion A^IHV = A is a map in L(HV,HW), and the set of such maps will be denoted by CVW Fredholm results for AE CVW will be obtained. .
1=1,...,p, denote the standard base Let e ( 6 1 ) 1 p 9P of MP. Define a linear map Uj from the space of local crossections ,
.
over Uj into the fNNTm-valued functions over U. by u(x) + eou(x), in the bundle coordinates of the chart U.. Here ij(x)u(x) + eNTiP.(x)u(x), where both sides may be extended 0 outside U. to obtainJglobal crossections of V and the product bundle x(gNN(m)=Z
respectively. A (global) crossection u of V may be written in the form u =
and a map O:TV + TZ, between the global crossec-
(where each iju
tions of f V and Z is defined by Ou = is mapped in the coordinates of Uj
).
The above construction defines a bundle homomorphism V+Z and an isomorphism: For an xE Uk U
k
,
in the bundle coordinates of
, we get Ou(x) = JN ejOR. ( ju), with the transition-matrices J=1 N Jk
Rjk. Thus Ou(x)=0 if and only if ipju=0, j=1,...,N , i.e., u=0 In particular, if ujl denotes the local crossection in U.
with ujl(x)=em in the coordinates of U., then ijuj1= vjl defines a global crossection, and fk(x)=IN=1e3mvj1() , xED global m-frame over M
,
C' over Q
.
,
defines a
Write V(x)=span{fl'" .,fm}.
We may identify the space MNNCm with the space hR of rem.3.3, writing eN0em = eN(0-1)+k
,
R =mN
Then the map 0 identifies
.
the space HV with the closed subspace of all measurable functions over 0 with value u(x)(= V(x), such that IIuIIV of (3.15) is finite: (3.15)
IIuIIV=
JO lu(x)Ix2du
Iu12={1mjk()ujuk}1/2,
,
u E HV
,
with a positive definite matrix ((mjk(x))), mjkE C # . Moreover, A the global frame fk is C # as well. A It follows that the orthogonal projection PV(x):h+V(x) (with the inner product of h) has matrix coefficients in C # and hence A is a (multiplication) operator in CRC C ,
For two vector bundles V,W, satisfying the above assumptions
308
X.3. Systems and vector bundles consider the class CV W above. It is natural to introduce a symbol GA (mod EV W=PWE°°IV, where E'=hnE) by setting OA(m) = PW(1(m))oA_(m)IV(1(m))
(3.16)
,
m E M
Clearly aA may be regarded as a homomorphism Vffi } WN , with the liftings V., Wffi of V and W onto M (by means of the map i of VII,l) Again consider only the case of E=K(H)
Theorem 3.5. An operator AE CVW is Fredholm in L(HV,HW) if and only if its symbol aA E hom(VM,WM) is an isomorphism onto WM Proof. Let m,m' be the fiber dimensions of V,W. Consider the case m<m'
;
in the other case we may take adjoints. The above glo-
bal frame fl.... fm for V(x) (and gj for W(x)) allows us to define a global isomorphism V(x)-W(x) by assigning fj,gj. Using the cor-
responding isomorphism HV}HW we may get restricted to the case iff V(x)CW(x). For m=m' get V=W and note that A°=l-PV+APVEC, is
A is , reducing thm.3.5 to thm.3.1. For m<m' aA cannot be surjective, and A cannot be The projection P onto span{gm,} is non0 hence POAPVEK(H-), a contradiction, if compact, but aP AP =0, :
0
A is
.
V
Q.E.D.
It is easy to derive an abstract index formula of the form (3.5) again. Details are left to the reader. Also, most of the results of this section can be generalized to the case of a multi-link ideal chain, with no or very little change in proof.
4. Discussion of algebras with a two-link ideal chain. In this section we want to look at case (B) of sec.l. That is, in detail, we assume that we have C D E 3 K(H)
(4.1)
,
where the commutator ideal E of the comparison algebra C properly contains K(H) , and where (4.2)
E/K(H) = C(ME,K(h))
with a locally compact space ffiE
,
,
called the symbol space of E
and the compact ideal K(h) of a (separable) Hilbert space h We of course also get
309
X.4. 2-link ideal chains A/E = C(N)
(4.3)
,
with another (compact) space N _ ffiC called the symbol space of since by design C/E is a commutative C'S-algebra with unit, C ,
,
so that the Gel'fand-Naimark theorem applies.
We have discussed two examples of the present form in VIII, 1, 2, and a third more complicated example in VIII,4. Another example, corresponding to the regular elliptic boundary-value problem of a half-space, was discussed in [C1],V Again it should be mentioned, that we know examples with .
longer ideal chains, i.e., E/K(H) is not a function algebra, but E D F D G D ... D K(H)
(4.4)
,
with each successive quotient of the form C(X,K(h)). (cf.Dynin [Dyl]). For most examples the space It has a concrete meaning; it occurs in some tensor product decomposition H=hNH- of H=L2(D,dp) On the abstract side of this discussion, Kaplanski [Kpl]
has introduced a class of C -algebras, called GCR-algebras, having a similar ideal chain, perhaps of infinite length (cf.Dixmier [Dx1],p.87).(There the type of algebra is called 'algebre postliminaire'.) Sakai [Skl] has shown that a C*-algebra is 'GCR' if and only if it is of type 1, in the sense that every representation is of type 1). It is easy to formulate an abstract result extending thm. 4.1, below, to longer chains. However, we will not discuss this, since all our examples are of type (A) or (B). Presently, for a two-link-chain, we have the following result.
Theorem 4.1.(a) An operator A E C is invertible modulo E if and only if its (C-)symbol aA, a certain continuous complex-valued function over ffi = ffiC does never vanish; (b) If an E-inverse P of AEC exist, so that we have (4.5)
AP = 1+E , PA = 1+F , with E,F E E
,
then 1+E and 1+F will be Fredholmif and only if the operator-valued functions 1+TE l+TF are invertible at all points of ME, and the inverses are bounded on NE , where TE TF denote the E-symbols ,
,
of E and F, respectively; (c) If the E-inverse P exists, and G, H are K(H)-inverses of 1+E and 1+F, then PG and HP are K(H)-inverses
310
X.4. 2-link ideal chains of A, and A is Fredholm; (d) Vice versa, if AEC is Fredholm, then there exists a K(H)-inverse P of A, which also is an E-inverse, so that (4.5) holds with E,F E K(H) c E We then have oA g 0 for all mEM and TE = TF = 0 on ffic so that 1+TE = 1+TF = 1 are invertible. .
,
,
The proof is a standard application of theory of commutative C -algebras.
Let us practice once more the comparison technique, by applying thm.4.l to the example of VIII,4, involving a complete Riemannian manifold with cylindrical ends. First, in that respect, recall that the secondary symbol space of that algebra C is void,
by VII,thm.4.2. Second, we note that we have cdn's (s),(ml1),(m6 (m.), (1j), j=1,2,3,4,5, and the assumption of VII,thm.3.6, all true. Accordingly we have the result, below.
Theorem 4.2. Let C be the comparison algebra of VIII,4. Then we have A = LAN E C
,
for every LELN . Moreover, the operator A=LAN
admits an E-inverse if and only if L is uniformly elliptic on S1
with respect to the tensor h;k
.
Indeed, IX,prop.3.2 is valid, and the algebra C has no secondary symbol space, so that the theorem is evident.
As a consequence of thm.4.2 we must look at the E-symbol of E = 1-AP and F=1-PA
,
for a suitable E-inverse P of A, in order
to decide whether or not A is Fredholm. Clearly, again, the realization L=AA-N of (2.4) of the expression L will be a closed Fredholm operator if and only if A is Fredholm.
First let us investigate the class LN (of IX,3) in the preD#, is found in
sent case. The description of the classes A#
VIII,4(i.e, they are determined by VIII,(2.2)
,
,
at each end of 0).
D# of VIII,4 were 'minimal', in the sense that we were interested only in supplying enough geneNote that the classes A#
,
rators for the algebra we had in mind, and to satisfy our basic conditions. Presently we should enlarge these classes (without changing the comparison algebra C ), in order to have 'large' classes
LN of differential expressions.
Clearly we may admit into A# any function aE A without enlarging the algebra. Since D# is a left A#-module, and D# a Lie-algebra we must watch about properties of the derivatives
in order to keep our cdn's (mj),(lj) preserved. A more careful investigation shows that we can reach the following classes LlpLN
311
X.4. 2-link ideal chains
of N-th order differential expressions. Definition 4.3. The class LN consists precisely of all expressions of order N with Co-coefficients, defined on 0 , such that
at each end, in the coordinates (x,t) of VIII, (2.1),L has the form (4.6)
k>O
limt;-Li(t)=L (eo), limt-.(a kLi)(t)=O
L=
Remark 4.4. Recall, that, at each end, we have a 'chart'
U=
(4.7)
{(x,t): xE BJ .
, O
In local coordinates (x1,...,xn-i) of the compact manifold B. formula (4.6) means that L=jLj(t)3t , where
L.(t) _
a.,a(x,t)Dxa , limt
I
aj,e(x,t)=aj, (x,oo),
lal<.N i
(4.8)
limt
0O
Sta. a(x,t) = Js
0
,
k > 0
A quick calculation shows that the classes LN are generated in the sense of IX,3 by the function algebra A!=L# and Lie-alge= L# , while the Stone-Weierstrass theorem may be used to show that A# and D#A are dense in A! and D!A , respectively, in so that C(A#,D#) = C(A!,D!) norm convergence of L(H)
bra D!
,
.
Let us go somewhat more into the detail of construction of an E-inverse P , for an operator A=LAN , since it is evident, that thm.4.1 is of little use for Fredholm theory, unless P is known, since the symbols
a(l-AP) , (,_PA)
otherwise cannot be obtained.
O HI N Let L E LN be uniformly elliptic, so that aA #Oon
,
for
We focus on one of the cylindrical ends Std=B1.x1, which Correspondingly we have H° C°, A!° = BxR D!° ,A°, all defined as the corresponding concept on the straight cylinder O°, with only 2 ends, which has the common domain U=Bxl+ A=LA
.
will be called S2°
,
.
with 0 , onto which we apply algebra surgery.
One may construct a uniformly elliptic expression L°EL°N coinciding with L on U
(4.9)
,
and then observe that
XLANX - .XL°A'NX E K(HU) , HU = L2(U)
.
This shows that construction of the E-inverse P , and even the later construction of a Fredholm inverse can be accomplished
312
X.4. 2-link ideal chains separately, at each end. Once we construct a Fredholm inverse P°
= Pj for A°
= L°A°N , at each end, a Fredholm inverse P is
given by (4.10)
P
M= number of ends,
JjM
with cut-off functions *j making
xMp 2=1 .
,
j=1,...,M
and a suitable
,
Also, P0 is just any operator in JO such that
aAP=1 over a compact neighbourhood of supp 0 . Working on the straight cylinder Bxl = S24 now, let us drop the superscript"' again, keeping in mind, that L=L° only is determined on the right end, t--, and is quite arbitrary for small negative t. According to (4.6) the limits Lj(oo)=limt+,L.(t) exist, and we may define the 't-translation invariant' expression L_ = IL (-)a3
(4.11)
Since L is arbitrary for negative t we may assume that L,, also is the limit of L, as t-+--
Then one concludes that L,ELN , and
.
A-A,, = LAN - LMAN E JO
(4.12)
.
The theorem, below, is self-explanatory. Theorem 4.5. If P,, is an E-inverse of A_ , then we have
AP. = J + 1 + E
(4.13)
,
P,,A = J'+ 1 + E'
, J,J'EJO , E,E'E E
Then, if we find K(H)with the minimal comparison algebra JO inverses 1+F, l+F', F,F'E E of 1+E and 1+E' , respectively, we .
,
get
(4.14) A(P,,(1+F))=l+K+C, ((1+F')P,)A=1+K'+C', K,K'EJO, C,C'EK(H). Indeed, we get (A-A,)P,, E JO (4.13)
,
,
P,,(A-A,,) E JO
, explaining
and then (4.14) is evident.
It is clear now that the right hand sides of the relations (4.14) are contained in the compact commutator algebra J=1+0 ' so that existence of K(H)- inverses for 1+K and l+K' is a matter of checking on the symbols. In addition, a proper choice of the cut-off function Xj in (4.10) will allows us to use any operator G. G' E 1+J0 satisfying aGa(l+K)=1, aG,a(1+K,)=1 near t=-. Thus we have proven the result, below.
313
X.4. 2-link ideal chains
Theorem 4.6. Assume L of thm.4.2 uniformly elliptic. At each end let L°
= L(co) be the translation invariant limit operator of L,
and let A° _ (1-A°)-1/2 be the (translation invariant) inverse
comparison operator square root. Then A= LAN is Fredholm if and only if, at each end, the translation invariant limits AA=L°A°N are Fredholm operators of L(H°) One will be tempted, of course, to conjugate AA with the t-Fourier transform Ft of VIII,2. Note that this takes A° onto the operator (4.15)
_ ik=OAk(TAx)kTk(t), Ak=ikLk(-)AX
AA
with (4.16)
Ax=(l-Ax)-1/2
,
(1+T2Ax2)-1/2
T(T) _
Recall also that the commutator ideal here is represented by the class CO(1,k)+K(H°) cf.VIII,2. k=K(L2(B)) A simple left multiplication of A^ by a Fredholm inverse P° of ,
,
the elliptic operator A0 on B will convert (4.15) into the form (4.17)
PA^ = 1+K(T)
,
K(T) E CB(R,k)
This, however, is no inversion mod E
.
.
Still the problem is
reduced to an equation involving compact operators over the compact manifold B (For more facts of. [CPo7.) .
Remark 4.7. It is clear that the operator K(T) of (4.17) is not only continuous but even analytic in the parameter T result of Gramsch [G2]
.
Using a
it therefore follows that the Fredholm ope-
tor-valued analytic operator function 1+K(T) has a meromorphic inverse if only it is invertible at a single T=T0 . Existence of such TO indeed can be established, using the Sobolev estimate of VI, thm.4.3, by a simple estimate, due to Agmon and Nirenberg, of. [AN1]. Then, of course, 1+K(T) must be invertible at all T E T
near the real axis, except at most countably many without
finite cluster. In fact, it follows that there are at most finitely many such points in some strip IIm Xl
,
n>0
.
We will get a Fredholm operator if none of these exceptional points lies on the real axis. While in general this may or may it always will be true not occur for a general expression L eatLe-5L for the expression for all but finitely many 161< n, 6E]k. ,
314
X.5. Sobolev spaces 5. Fredhoim theory and comparison technique in Sobolev spaces.
First let us point to a serious restriction of applicability of the present discussion. We generally assume here cdn.(s), which is never satisfied for a comparison triple with any nonvoid nonvoid subset of 2M # is A an n-l-dimensional manifold 30 such that dp and H extend to OL0, regular boundary, - i.e., if any open
satisfying all the conditions they otherwise satisfy on o Thus, in particular, while much of our L2-theory applies to the regular elliptic boundary problem, with Lopatinskij-Shapiro .
type boundary conditions, this is not so for theory in Sobolev spaces.
In fact, even our definition of Sobolev spaces Hs does not supply the commonly used Sobolev spaces if (s) is violated. In that case there is a choice of self-adjoint extensions of H0m for some integers m. One will get the common Sobolev space Hm only if Am is replaced by the inverse square root of the extension giving the largest domain for the corresponding form (u,Hmv)= (u,v)m , which then is the Sobolev space Hs This normally does not correspond to the Friedrichs extension (which gives the smal.
lest domain for the form). In case of a regular boundary the Friedrichs extension corresponds to the Dirichlet boundary condition, while the Sobolev space corresponds to the Neumann condition. In particular also, the As to be substituted for As in IX,(l.l) no longer will be the power of a fixed operator A
.
Thus we assume cdn.(s), for the remainder of sec.5. Also we require the same conditions as in IX,4, i.e., (1.), j=1,2, (m5), D# for a suitable P. within general reach of A# and (15)(P.) ,
,
of principal symbol type. (We always have (aj), j=0,1, and (d0), of course. ) Under these assumptions IX,thm.4.3 is valid in its strong form. A comparison algebra Cs is defined in Hs, sEa, from the generators IX,(1.10). Any two such comparison algebras Cs, C. are
isometrically isomorphic, under the natural isomorphism Ar:Hs-Ht and vice versa. The commur=t-s. For AECs we have A-rAAr E Ct tator ideals also map onto each other, under this conjugation. ,
The symbol spaces are homeomorphic under the associate dual map of the isomorphism Cs/Fs - Ct/Et induced. Also, for A E CO and even for A finitely generated by LAN
,
ANL
,
LC-QN=PN+LN '
315
X.5. Sobolev spaces
we have A-rAAr-A E E
,
316
so that the symbol of A in Hs coincides
with the symbol of A in Ht, if the associate dual map is used to identify the two symbol spaces. With that interpretation we obtain a common symbol space 1 for all algebras Cs , and the symbol aA is independent of s, whenever AEC, , or A is finitely generated
from LAN ,
ANL
for LE
,
, and
`PI
CO
Taking closure in the topological algebra 0(0) we obtain the Frechet algebra C, Since convergence Aj-A in C. implies convergence A.-A in every Cs , it follows that every AEC, has its .
symbol independent of s . Accordingly we define a symbol for the Frechet algebra C. by setting aA equal to the symbol of AECcCCS in Cs
for any fixed s, this definition being independent of s. Et/K(Ht) , s,tEg, We also get any two algebras Es/K(Hs) ,
,
isometrically isomorphic, under the same conjugation with Ar , r=t-s Thus, if the algebra E=E0 has a symbol of the second .
kind, i.e., if E/K(H) is isometrically isomorphic to C(E,K(h)), as in the examples of VIII,l, 2, and 4, then the same holds true for all the algebras Es/K(H5), sEE If y:E -
.
We get Es/K(H5)=C(E,K(h)).
K(H) denotes the symbol homomorphism of E , then the
symbol yE of an operator E (5.1)
YE = YE
Cs-
Cs is given by ,
Es = A-SEAS E C
S
where we know from IX,(4.4), here valid in its strong form, that ESE E so that yE is defined. ,
S
It now is natural to focus on the commutator ideal EO of the Note that Es is the closure of EO finitely generated algebra C2 .
in L(Hs), s E E and in 0(0) for s=- . For an operator EEEO we again have yE defined for all sEE , and the same is true for EEE_, ,
by continuous extension in 0(0) . While we found the Cs-symbol of an AEC. independent of s , in all cases examined, the corresponding property for the Ea-symbols would depend on the relation (5.2)
A_r EAr - E E K(H) for all EE E0, rE E
A check on our examples in sec.VIII,1,2,4
.
and [C1],V reveals
,
that (5.2) normally is not satisfied. As a substitute for 'yE=const.' , using V,thm.6.8, it follows that Es is an entire analytic function of s , defined for all for all s . All derireal and complex s, and taking values in E ,
vatives of Es (for s) again are in E
.
X.S. Sobolev spaces Then, using that the symbol homomorphism y: E/K(H)-+C(E,K(h))
is an entire function
is an isometry, one concludes that also yE s
of s (5.3)
.
Indeed, we get, with Fs=dEs/ds supPYE _YE s
-(s-s')YF s'
II
,
< IIEs-Es,-(s-s')Fs,u = o(Is-s'I).
s'
showing that the complex derivative of yE
for s exists and is s
a continuous function over E , with values in K(h) again.
Now, for a general operator E E E , the Frechet closure of , we obtain E as limit of a sequence E E E0 , for which i the sequence of symbols yE converges uniformly in s on compact E0 in 0(0)
,
is subsets of I1:
.
is entire again. We have proven:
Therefore yE s
Proposition 5.1. Under the assumptions of sec.5 the symbol ys(e), eEE , is an entire function of s with values in K(h), for EEE. ,
continuous in s and e , and CO(E) , for fixed s. Moreover, YE even is entire as a function E-+CO(E,K(h)). For fixed eEE and EEE' the
(entire Fredholm-operator-valued) function 1+yE(e), either has a nontrivial kernel for every (real and complex) s, or else has a meromorphic inverse, singular only at discrete points of M We have the following analogue of thm.4.l:
Theorem 5.2. Under the above assumptions, an operator A E Cs has an Es-inverse if its (primary) symbol aA does never vanish on the space ffi.
In case of AEC. the primary symbol is independent of s
so that an E -inverse exists either for no s or for all s E E An 'Es-inverted' operator K=1+E , EEEs is Fredholm if and .
only if its Es-symbol 1+ys is nonsingular for all eEE, and its inverse is in 1+CO(E,K(h)).
A Fredholm operator of Cs must have an Es-inverse, hence a non-vanishing primary symbol. Moreover, it must have an Esinverse B such that AB and BA have a non-singular Es-symbol on all of E
.
Among the examples offered one finds that the Hilbert space h of IX,(4.8) always has a concrete meaning as a factor in a topological tensor decomposition (5.4)
H = H0= h»h' ,
where then the E-symbol yE of a generator EEE0 may be easily cal-
317
X.S. Sobolev spaces
culated from that decomposition. Generally the operators As do not separate relative to (5.4). For example, in the algebra of VIII,2 (with generators VIII,(2.2)), setting n'=1, we have It = hx , h'= ht and ,
As = (1 -
(5.5)
a2r _
Ax)-s/2
but not As acting on the factor spaces. After an Ax with at Ft-conjugation, as in VIII,2, we get the operator function ,
(5.6)
,
(1+'r2Ax)-1/2
MS(T) = AxsTs(T) E K(h), with T(T) _
Here acting as multiplication operator on the factor h' it may be observed that Ms(T) provides an isomorphism It -+ .
(although not an isometry), depending on T
,
hs
for each fixed T
where, for a moment, hs denotes the L2-Sobolev space of order s of the compact manifold B , with respect to the metric on hand. Thus Hs , in this case, corresponds to the class of all hs-valued functions u(T) over I with f-
(IM-s(T)u(T)II2dT < -
One confirms easily that, in this case the the assumes values in the algebra CB ted algebra within the unique Laplace comparison Thus 1+yE(e) , for fixed eEE , compact space B an EEE0
,
,
.
of 0(0) , relative to B
,
.
symbol 1E for finitely genera,
algebra CB of the is an operator
and is K-invariant under H-conjugation,
in the sense of I,def.6.5, in view of IX,(4.4), and (5.6). Accordingly, I,thm.6.6 applies, and it follows that ker(l+ys) and ker (1+yE) have dimension independent of s case, although yE is not independent of s
.
,
Accordingly, in this it still follows that
l+yE(e) is invertible for one s=s0 if and only if it is invertible for all s .
To apply the last observation for construction of Fredholm inverses of an AE C_ we would require an E--inverse B of A We .
will not attempt its construction here, and leave further investigations to the reader.
Instead we now get restricted to the case E=K(H) by IX,thm.4.2. that then also Es = K(Hs), for sEl
.
We know
,
As in VI,3 two operators A, B E 0W are called Greeninverses of each other, if we have
(5.7)
AB - 1 E F.
,
BA - 1 E F.
,
318
X.5. Sobolev spaces where F_ denotes the ideal of operators of finite rank in 0(-°), as introduced in I, problem 6.8. There we stated that the 0(0)closure FW coincides with the class of K_ of operators AE0(0) with AEK(H ) for all sEl
.
s
Starting with any comparison algebra C. = C.(A!,D!) , under the assumptions of the present section (with C of compact commutator), we define the classes (subspaces of 0(s)) (5.8)
'PCs = {AA s
AEC_}
:
,
'PC. = U{'PCs: sEg}
.
It then follows that 'PC. is a graded algebra containing the
(differential operators of the) expressions in 9
within general
reach. In details, we have (5.9)
s,t,El
'PCs.TCt C 'PCs+t
Also TC0D K_ D Fw
QN C TCN
,
N=0,1,2,...
.
. K=AA-s
For an operator
a
E 'PCs the symbol aA of A is called
the symbol quotient of order s. Theorem 5.3. An operator KE'PCs admits a Green inverse in 0(-s)
if and only if its symbol quotient of order s does never vanish on the symbol space M . Moreover, any such Green inverse is contained in TC-s
.
The essential point of the proof is that operators in TCO=Cm are K-invariant under H-conjugation, in the sense of I,def.6.5, so that I,thm.6.6 applies for all Fredholm operators. This in essence implies the statement for s=0, while the general case is easily reduced to s=0. For details cf. [C1],IV,thm.4.1.
Theorem 5.4. Under the present assumptions, with a compact commutator algebra C, the algebra C. is a i -subalgebra of L(H) Proof. Suppose A G C_ admits an inverse B E L(H) Then it is Fredholm in C , hence aA90 on M By thm.5.2 it then admits a .
.
Green inverse BO E C
.
In fact, the proof of I,thm.6.6 shows that
the distinguished Green inverse B0:(ker A)1 im A of VI,3, with the orthogonal complement in H. is in C. im A=H , ker A =
0
.
But that must be B, since
, A being invertible, q.e.d.
319
App.A. Functions on manifolds Appendix A. Auxiliary results, concerning functions on manifolds. Let the manifold 0 satisfy the general assumptions of III,1. In particular we assume the existence of a countable locally fini0jclos j=1,2,...} , where each te atlas {Qj is compactly contai:
ned in some U. , where {U.) is another locally finite atlas of 0
Let 0` be an open subdomain of 0 (where 0 _ 0- is permitted.) Suppose f(x) and g(x) > 0 are functions over U- and 0 , respectively. We will use the Landau symbols in the following sense: Write f=O(g) (in c') if f(x)/g(x) is bounded over c-; write f=o0(g) (in D-) if f=O(g) (in 0-) and limx}.(f(x)/g(x))=0 (in U). (That is, for e>0 there exists a compact set K C 0 such that lf(x)/g(x)l
f=o(g)
,
< c for all x E 0-\K .) We shall write f=O(g)
(without "(in 0`)"
,
,
and
etc.) if no confusion can arise.
Lemma A.1. Let f,g be as above, and let g be continuous over U If f = o0(g)
,
then there exists a positive C-(U)-function ip
f = 0(*) (in R') , and V = o0(y) Proof. Let (x) = f(x)/g(x) , so that we have q(x) bounded over Consider U` and limx ¢(x) = 0 extended to 0 by setting fi(x) = 0 outside 0, then the limit still is zero. With our partition w. define nj = x E supp wj } Observe that nj>0, and 0 For there exists a compact set Ke C 0 such that Ifl<e outside K6 for every e>O Only finitely many of the sets 0. = supp wj can have points in common with Ke Else there exists X. E U nK e with a limit point xE 0 Ke Due to localy finite covesuch that
.
.
.
.
.
.
.
3k
ik
rage x0 has a neighbourhood N contained in only finitely many U. so cannot be a limit point of the x.
.
Hence, for e>0 there
k
exists NE such that all supp wj
,
Accordingly, nj =
j>Ne are completely outside Ke. supp wj} < e
.
Now we just define the function
where
the sum is locally finite, hence represents a C'(0-function. Since y is only continuous, the function $ thus defined is not yet C_(U)
but is at least continuous. It also satisfies the other
conditions: For any x E U we get = p(x)/g(x)
.
This implies Jf(x)I
(x)I =
Jnjwj(x)
=
p(x)
,
i.e.,
320
App.A. Functions on manifolds
. Also we note that limx4Injwj(x) =
f = 0(p) (in SZ-)
= o0(g)
.
Indeed, let KN =
:
j>N+l}
-
0
1j=N+lnjwj(x) < -
as N
,
so that
,
Clearly KN is compact,
S21U....US2N
while x E S2\KN implies that sup{nj
0
To fully establish the lemma we
.
now must make a C"-correction of 4 which does not disturb the other conditions already established. This is accomplished in lemma A.2, below. Lemma A.2. Let f,g E C(Q)
C%1)-functions y
,
,
g >
0
.
Then there exist positive
such that
6
f(x) < y(x) < f(x) + 6(x) , x E S2
(A.1)
,
6
= 0Q(g)
.
x E .jclos} Clearly ej> 0 , since Proof. Let ej = 2-JMin{g(x) UJclos are compact, and g(x) > 0 Using the coordinate transform QjclosC of Uj we may regard S2jC Uj as subsets of in Let wj > 0 .
:
.
.
w. E C0(S2j)
unity clos 0
.
'
jj=lwj(x) = 1 in a
,
i.e., {wj} is a partition of
Using regularizing techniques and the compactness of such that
it is possible to find fjE C0'(S2
wj(x)f(x) < f.(x) < wj(x)f(x) + ej , x E S2j
(A.2)
For each j let Xj(x) E CD(Uj)
,
0<Xj<1
Xj(x) =
1
,
x(x) , x E S2
.
,
.
as x E S2.
Then (A.2) implies that f(x) < If.(x) < f(x) + I
(A.3)
Here y(x) = Ifj(x)
, and 6(x) _ I ejXj(x) are well defined
C"(c2)-functions, since the sums are locally finite. Moreover, 0 < 6(x)/g(x) <
(A.4)
LxEsuPP(Xj)
2-j
.
For every N=1,2,..., a compact set KN may be found such that n KNmpl = 0 for all j < N due to local finiteness of
supp XJ .
,
,
{UTherefore the right hand side of (A.4) is < as x E Kcompl. We get lima->.,6(x)/g(x) =
0
,
JI+12-J=21-N
or, 6 = wS2(g), q.e.d.
321
App.B. Covarient derivatives Appendix B. Covariant derivatives, and curvature. By 'covariant derivatives' we mean the Riemannian covariant derivatives, with respect to the connection defined by the Christoffel symbols of the metric tensor hjk , where (B.1)
((h 7.k )).,k=l,...,n _ ((hjk))-l
.
Let bj and cj be covariant and contravariant tensors, (i.e., tangent or cotangent vector fields, respectively, depending smoothly on xES2
(B.2)
.
Then we have
bj lk = bj
x
k - r.kbl ,
c] lk = Cl
lxk
+ rmk cR1,
where the Christoffel symbols (B.3)
rjk = hilrjk,l
rjk,l =2(hjllxk + hlklxj - hjklxl)
,
are symmetric in j and k, and are not tensors. Both derivatives are tensors with two variable indices.
covariant
Covariant derivatives of tensors with multiple indices are defined similarly. For example, aijlk - aijlxk - aihrjk - ahjrik
aIk =
i
ah hi
a
xIx
k + ahjr hk
+
In general, r1r2 rI r2 as1s2... i = aS152, (B.6)
l
S
a-1 a+l rl as1...r jr
a
.
+
Ix i
ar1...
a
...
rr1
...
r1
s1...ss-llss+l... ssi Higher covariant derivatives are defined recursively. For
The covariant derivativative of the example alljk = (a'lj)lk tensor a'lj , of one covariant and one contravariant index. Etc. That is, the For sake of completeness one defines alj = a :
.
.
Ix
first covariant derivative of a (scalar) function a(x) is its gradient. Covariant derivatives do not necessarily commute, even if the coefficients are smooth. We have
322
App.B. Covarient derivatives (B.7)
aIik - alki
for a scalar a
.
323
'
However , for a tensor ai and b1 we get
ailjk - ailkj _ a1 ijk
(B.9)
biIjk - b1lki = -b11lljk '
with the Riemann curvature tensor Rijk (B.10)
Rlijk= r±klXj- rijlXk+
and Ri1jk Fikrmj-
.
Here we have
rijrmk' Rilkj =
hirRrlk..
Similarly (B.11)
aijlkl - aijllk - aihRjk l + ahjRikl
etc. It is known that the metric tensor hjk
as well as hjk behave like constants under covariant differentiation, i.e., (B.12)
hjkll =
0
, hjkll =
0
,
.
Also, covariant derivatives satisfy the usual differentiation rules. For example, (B.13)
(ajkbJk)I1 = ajkllbjk + ajkbjk11
,
etc. Finally we note the symmetry relations of the Riemann tensor: (B.14)
Rijkl
-Rjikl
-Rijlk ' Rijkl+ Riklj+ Riljk= 0
Also the Ricci tensor will be used: (B.15)
R. .
= Rijk
as well as the divergence formula (B.16)
hjka,jk =
.
j
x
IXk
(Notice that, as for all tensors, a covariant derivative may be 'raised', giving a contravariant derivative: (B.17)
ajlk
=
hJlajll , etc.
App.C. Summary of cdns(x)
Appendix C. Summary of the conditions
used.
In the following we give a list of the various conditions used to describe features of comparison algebras and their generating sets, in alphabetical order, with the section listed, where they were introduced. (a0) (V,l): a E A# implies a E A#
and A# D C'(c) (a1) (V,1): A# contains the constant functions. ,
(a1') (V,1): For every D E D# there exists an a E A# such that Va has compact support, and aD = Da = D .
(d0) (V,1): D=b33xi +pE D
implies I5=F3axj+pE D#, and D
con-
tains all Co-coefficient folpde's with compact support. (11) (IX,1): A# is an algebra under the pointwise product of functions, and D# is a Lie-algebra, under the commutator product of folpde's. Also, D# is a two-sided A module, as under (m2), and A#, interpreted as a set of zero-order expressions, is a subset of D# , Also, for a E A# , D E D# we have [a,D] E A# (12) (IX,1): DED# implies that D*ED (13) (IX,1): Cdn.(m3) holds, and, in addition, for every a E A# with a#0 at a closed set Q C M# there exists
a function b E A# with ab=l near Q (14) (IX,1): The expression H can be written as a finite sum of terms aDF, with a E A# , D,F E D# (15) (IX,1): For every a E A# D E D we have [a,H] E D E,FED# and [D,H] is a finite sum of terms bEF , bEA ,
(15') (IX,3): We have [ H,a] E L1,0
, [ H,D] E L22,0 , D E D! (l5)(P.) (IX,3): Cdn.(15) with L, replaced by Qj=Lj+P] for a graded algebra P_= UPN within reach.
for all a E A!
(I5')(P_) (IX,3): Cdn.(15') with Lj'O replaced by Qj'O P1.+L7,0 , with an algebra P.=UPN within reach and of principal symbol type.
(m1) (VII,2): For each D E D# the expression {D,D}(x) = h]k(x)5J(x)bk(x) + p(x)p(x)/q(x) defines a function in A# .
(ml)^ (VII,4): Same as (m1), but with {D,D} replaced by
324
App.C. Summary of cdns(x) {D,D}" , formed with the coefficients of some triple ",dU",H"}
(C> {c,du,H} (ml') (VII,3): In addition to {D,D} of (mI) also {D,D}0(x) {Sj
Ip(x)I2/q(x) defines a function in A#
_
.
(VII,4): Same as (ml') , but with respect to a
triple {Y",du",H"} (cf.(ml)" ). )
(ml')x" (VII,4): The correspon-
(ml)"x
(mlt
(mlx ,
x ,
'
ding conditions, (ml),..., stated only for the given x. (m2) (VII,2): A# and D# are complex vector spaces, and D# is a left A#-module, i.e., aDED# for all aEA# (m3) (VII,3): For each x0E M
A#
DED# and open neighbourhood N of ,
x0 there exists a function aEA# with 0
(m3)x (VII,4): Condition (m3) imposed only for x=x0 necessarily for any other xOEM # A
(m3) U (VIII,3): For every aE A# , DE D# with a=0
3U we have XUa E A#
,
, not
.
D=0 near XUD E D#, with the characteri,
stic function XU of U C 2. (m4) (VII,3): There exists an e>0 such that
A-1-e[a,A]E C, and A-EID,A1= A-'(DA-(AD)**) E C, for all aE A#, DE D (m5) (VII,3): We have A E C .
(m5') (IX,1): D# contains the 0-order folpde 1 (m6)x (VII,3): The dimension dx of S# = D#/D# , with Dx = {DED# {D,D}(x)=0} is maximal (i.e., 6x = n+l) (m6) (VII,3): Means (m6)x for all x E M :
A#
(m6)" (VII,4): Cdn. (m6), but with respect to {D,D}^ formed with the coefficients of some triple (m
6)x" :
{P",du",H"} (c> {c,du,H) Cf. (m6)" .
(mi)x (VII,3): There exists a DOE D#, such that {D,D}(x)90, but {D,D}1(x)=0
.
(m7) (VII,3): Means cdn.(m7)x for all x E M A
#
(m7)x" (VII,4): Means cdn. (m7)x ,but with {D,D} replaced by {D,D}"
,
as n (m6)"
(m7)" (VII,4): See (mi)x"
.
(V,6): (Same as 'H-compatible, V,def.6.4). (pl) ( I X , 3 ) : [H,L] E PN+l for all L E PN , N=0,1,... (p)
.
325
App.C. Summary of cdns(x) (p2) (IX,3): Every L E pN is within reach of L(H)
.
(ql) (VII,4): We have q=q^ in some neighbourhood of 3o (q2) (IX,6): The Riemann curvature tensor and all its coriant derivatives are bounded over 0 (q3) (IX,6): The potential q satisfies Vkq=O(ql+k/2), k=1,.. .
(s) (V,1): For all m=1,2,... the minimal operator (Hm)0 H0m of the m-th power HM is essentially self-adjoint. = (sj) (V,1): (where j=1,2,...,oo) (H3)0= H0J is essentially self-adjoint.
(w) (V,1): The minimal operator H0 is essentially selfadjoint.
326
List of symbols used
AA 14
E
ACB, BDA 2
e(x,y)
A# 126,129
E(A) 15
AC
213
graph A
A# c En
144
mn
199
198
Hd
78
219,236,248 62/63
2
B(x,q) 86
Hs, H. 30,157,160,251 H°° , Hm 303
B
i
blj
blb
,
322f
78
10 129
,
76
K(H)
24
cdn.(x3.) 324f
L(H)
2
CB
,
CB(X), CB(X,X) 229
ffi
CO
,
131
CO(X), C0(X,X) 230
ffis
C
129
M #
CO
129
C
257
S C
271
C0
257
C A#
187
, fp
A
0(g)
,
P(X,Y)
2(X,V)
213
Rs(A)
D#
144
dom A D
x
dx
2
Sx
,
Q(X) 2
13
194
T 0 , T 0x 161 161
194
w
194
(x.) 324f
126
P(X) 2
12
Sp(A) 86
{D,F} 193 E
,
187,206 E*0 115
D c
o0(g) 320
115
D# 126,129
d(x,y)
189
187
( x)
146
References
[AN1] S.Agmon and L.Nirenberg, Properties of solutions of ordinary differential equations in Banach space; Commun. Pure Appl. Math. 16 (1963) 121-239.
[Al1] W.Allegretto, On the equivalence of two types of oscillations for elliptic operators; Pac.J.Math. 55(1974) 319-328. [Al2]
,
Spectral estimates and oscillation of singular differential operators; Proc.Amer.Math.Soc.73(1979)51.
, Positive solutions and spectral properties of
[Al3]
second order elliptic operators;Pac.J.Math.92(1981) 15-25.
[APS] M.Atiyah, V.Patodi, and I.Singer, Spectral assymmetry and Riemannian geometry; Math. Proc. Phil. Cambridge Phil. Soc. 77 (1975) 43-69. [AS] M.Atiyah and I.Singer, The index of elliptic operators; I: Ann. of Math. 87 (1968) 484-530; III: Ann. of Math. 87 (1968) 546-604; IV: Ann. of Math. 92 (1970) 119138; V: Ann of Math. 92 (1970) 139-149, [BMSW] B.Barnes, G.Murphy, R.Smyth, T.West, Riesz- and Fredholm theory in Banach algebras; Pitman, Boston 1982. [BDT] P.Baum, R.Douglas, M.Taylor, Cycles and relative cycles in analytic K-homology; to appear.
[B1] R.Beals, A general calculus of pseudo-differential operators; Duke Math.J.42 (1975) 1-42. [B2]
,
Characterization of pseudodifferential operators and applications; Duke Math.J. 44 (1977) ,45-57 cor;
rection, Duke Math.J.46 (1979),215 [B31
,
.
On the boundedness of pseudodifferential.operators;
Comm.Partial Diff.Equ. 2(10) (1977) 1063-1070. [Bz11 I.M.Berezanski, Expansions in eigenfunctions of self-adjoint
operators; AMS transl. of math. monographs, vol.17, Providence 1968. [BW1] B.Booss and K Wojciechowski, Desuspension of splitting
elliptic symbols I,II; to appear. [BdM1] L.Boutet de Monvel, Boundary problems for pseudo-differential operators; Acta Math. 126 (1971) 11-51. [BS1] J.Bruening and R.Seeley, Regular singular asymptotics; Adv. [BS2]
Math. 58 (1985) 133-148. The resolvent expansion for second order regular ,
References singular operators; Preprint Augsburg 1985. [BS3]
, An index theorem for first order regular singular operators; Preprint Augsburg 1985.
[BC1] M.Breuer and H.O.Cordes,On Banach algebras with a-symbol; J.Math.Mech. 13 (1964) 313-324 .
[BC2] part 2 ; J.Math.Mech.14 (1965) 299-314 [Ca] A.Calderon, Intermediate spaces and interpolation,the complex ,
method; Studia Math.24 (1964) 113-190. [Ca1] T.Carleman, Sur la theorie mathematique de 1'equation de
Schroedinger; Arkiv f. Mat., Astr.,og Fys. 24B Nil (1934).
[Ct1] M.Cantor, Some problems of global analysis on asymptotically simple manifolds; Compos. Math. 38 (1979) 3-35. [CV1] A.Calderon and R.Vaillancourt,On the boundedness of pseudodifferential operators; J.Math.Soc.Japan 23 (1971) 374-378. [CV2]
, A class of bounded pseudodifferential operators; Proc.Nat.Acad.Sci.USA 69 (1972) 1185-1187.
[Che1]J.Cheeger, Spectral geometry of spaces with cone-like singularities;Proc.Nat.Acad.Sci. USA (1979) 2103. , M.Gromov, and M.Taylor, Finite propagation speed, ker[CGT] nel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds; J. Diff. Geom. 17 (1982) 15-53. [Ch] P.Chernoff, Essential selfadjointness of powers of generators of hyperbolic equations 12 (1973) 402-414
;
J.Functional Analysis
.
[CBC] Y.Choquet-Bruhat,and D.Christadolou, Elliptic systems in Ns 6-spaces on manifolds, elliptic at infinity; Acta Math.146 (1981) 129-150. [Cb1,2 1
L.Coburn, The C -algebra generated by an isometry, I:Bull. Amer.Math.Soc.73 (1967) 722-726; II: Transactions Amer.Math.Soc.137 (1969) 211-217.
[CL1]
L.Coburn and A.Lebow, Algebraic theory of Fredfholm opera-
tors; J.Math.Mech.15,577-584. [CM] R.Coifman and Y.Meyer, Au dela des operateurs pseudodifferentielles; Asterisque 57 (1979) 1-184.
[CC1] P.Colella and H.O.Cordes,The C -algebra of the elliptic boundary problem;Rocky Mtn.J.Math. 10 (1980) 217-238
.
329
References
[Cd1] E.A.Coddington,The spectral representation of ordinary selfadjoint differential operators;Ann.of Math.60 [CdL1]
(1954) 192-211. and N.Levinson, Theory of ordinary differential
equations,McGraw Hill,New York 1955.
[C1] H.O.Cordes,Elliptic pseudo-differential operators,an abstract theory.Springer Lecture Notes in Math.Vol.756, Berlin,Heidelberg,New York 1979. [C3]
,Techniques in pseudodifferential operators;mono-
[Cu]
,
[CQ]
, A matrix inequality, Proc.Amer.Math.Soc.ll (1960)
[CC]
,
graph,to appear.
On some C -algebras and Frechet-*-algebras of pseudodifferential operators; Proc.Sympos.P.Appl. Math.43 (1985) 79-104. 206-210.
[CD]
[CE]
On compactness of commutators of multiplications and convolutions,and boundedness of pseudodiffe-
rential operators; J.Functional Analysis,l8 (1975) 115-131. , A pseudo-algebra of observables for the Dirac equation; Manuscripta Math.45 (1983) 77-105. , A version of Egorov's theorem for systems of
hyperbolic pseudo-differential equations;J.of Functional Analysis 48 (1982) 285-300. [CEd]
, Ueber die Eigenwertdichte von Sturm-Liouville Problemen am unteren Rande des Spectrums; Math. Nachr.l9(1958), 64-71.
[CF]
, A pseudodifferential Foldy-Wouthuysen transform; Comm.in Partial Differential Equations, 8(13) (1983) 1475-1485.
[CFo]
,
[CG]
,
[CI]
,
[CL]
,
On maximal first order partial differential operators; Amer.J.Math.82 (1960) 63-91. On geometrical optics;lecture notes,Berkeley,1982 The algebra of singular integral operators in in; J.Math.Mech.14 (1965) 1007-1032.
On pseudodifferential operators and smoothness of special Lie-group representations;Manuscripta math. 28 (1979) 51-69
[CP]
.
, A global parametrix for pseudodifferential operators over In ;Preprint SFB 72 (Bonn) (1976)
330
References
(available as preprint from the author). [CS]
, Banach algebras,singular integral operators and
partial differential equations;Lecture Notes Lund,1971.(Available from the author). [CTJ
, An algebra of singular integral operators with two symbol homomorphisms;Bulletin AMS ,75 (1969) 37-42
[CWS]
,
.
Selfadjointness of powers of elliptic operators on noncompact manifolds;Math.Ann. 195 (1972) 257-272
[CPo]
,
On the two-fold symbol chain of a C -algebra of singular integral operators on a polycylinder; Revista Mat.Iberoamericana 1986, to appear.
[CDgl]
, and S.H.Doong, The Laplace comparison algebra of
a space with conical and cylindrical ends; To appear; Proceedings, Topics in pseudodifferential operators; Oberwolfach 1986; Springer LN.
[CEr] _,and A.Erkip, The N-th order elliptic boundary problem for non-compact boundaries;Rocky Mtn.J.of Math. 10 (1980) 7-24 .
[CHel]
,and E.Herman , Gelfand theory of pseudo-differential
operators; American J.of Math. 90 (1968)681-717. [CHe2]
[CLa]
[CM1]
Singular integral operators on a half-line; Proceedings Nat. Acad. Sci. (1966) 1668-1673. , and J.Labrousse, The invariance of the index in the metric space of closed operators;J.Math.Mech. 15 (1963) 693-720. and R.McOwen The C -algebra of a singular elliptic ,
problem on a non-compact Riemannian manifold; Math.Zeitschrift 153 (1977) 101-116 .
[CM2J
Remarks on singular elliptic theory for complete Riemannian manifolds; Pacific J.Math. 70 (1977) 133-141 , and S.Melo, An algebra of singular integral opera.
[CMe1]
tors with coefficients of bounded osecillation; To appear. [CSch]
and E.Schrohe, On the symbol homomorphism of a certain Frechet algebra of singular integral operators; Integral equ. and operator theory 120-133.
(1985)
331
References
[CW] _,and D.Williams, An algebra of pseudo-differential operators with non-smooth symbol ;Pacific J.Math. 78 (1978) 279-290
.
[CH] R.Courant and D.Hilbert,Methods of mathematical physics,I: Interscience Pub. ltd. New York 1955; II: Interscience Pub. New York 1966.
[Dd1] J.Dieudonne, Foundations of modern analysis,Acad.Press New York 1964
.
[Do1] H.Donnelly, Stability theorems for the continuous spectrum of a negatively curved manifold; Transactions AMS 264 (1981) 431-450. [Dgl] R.Douglas, Banach algebra techniques in operator theory; Academic Press, New York 1972. [Dg2]
,
Banach algebra techniques in the theory of Toeplitz operators, Expository Lec. CBMS Regional Conf. held at U. of Georgia, June 1972, AMS, Providence, 1973.
[Dg3]
,
C -algebra extensions and K-homology; Princeton Univ.Press and Tokyo Univ.Press, Princeton 1980.
[DG1] J.Duistermat and V.Guillemin, The spectrum of positive ellip tic operators and periodic bicharacteristics; [DS1,2,3 1
Inventiones Math. 29 (1975) 39-79. N.Dunford and J.Schwarz, Linear operators, Vol.I,II,III,
Wiley Interscience,New York 1958,1963,1971. [Dx1] Dixmier, Les C-algebres et leurs representations; GauthierVillars, Paris 1964. Sur une inegalite de E.Heinz, Math.Ann.l25 (1952)
[Dx2]
75-78.
[Dv1] A.Devinatz, Essential self-adjointness of Schroedinger operators; J.Functional Analysis 25 (1977) 51-69. [Du1] R.Duduchava, On integral equations of convolutions with dis[Du2]
continuous coefficients; Math. Nachr. 79 (1977) 75-98. , An application of singular integral equations to some problems of elasticity; Integral Equ. and Operator theory 5 (1982) 475-489.
(Du3]
,
On multi-dimensional singular integral operators,I:
The half-space case; J.Operator theory 11 (1984) 41-76
II: The case of a compact manifold; J. Operator theory 11 (1984) 199-214. [Du4]
,
;
Integral equations with fixed singularities; Teub-
332
References
ner Texte zur Math.,Teubner,Leizig,1979. [D11 J.Dunau ,Fonctions d'un operateur elliptique sur une variete compacte; J.Math.pures et appl. 56 (1977) 367-391
[Dyl] Dynin, Multivariable Wiener-Hopf operators; I: Integ.eq. and operator theory 9 (1986), 937-556; II: To appear. [Ei11 L.P.Eisenhart, Riemannian geometry; Princeton 1960. [Er1] A.Erkip, The elliptic boundary problem on the half-space; Comm. PDE 4(5) (1979) 537-554. [Er2]
, Normal solvability of boundary value problems in
half-space; to appear. [Es1] G.Eskin, Boundary problems for elliptic pseudo-differen-
tial equations; AMS Transl. Math. Monogr. 52 Providence 1981 (Russian edition 1973). [Fe I] C.Fefferman, The multiplier problem for the ball; Ann. Math. 94 (1971) 330-336.
[F11 J.Frehse,Essential selfadjointness of singular elliptic operators;Bol.Soc.Bras.Mat. 8.2(1977),87-107 [Fr1] K.O.Friedrichs, Symmetric positive linear differential equations; Commun.P.Appl.Math.11 (1958) 333-418. [Fr21
Spectraltheorie halbbeschraenkter Operatoren Math.Ann.109 (1934) 465-487 ,685-713 and 110 (1935) 777-779
.
[FL] K.O.Friedrichs and P.D.Lax,Proc.Sypos.Pure Math.Chicago,1966.
[Fh11 I.Fredholm, Sur une classe d'equations fonctionelles; Acta Math.27 (1903) 365-390. [Ga1] L.Garding,Linear hyperbolic partial differential equations with constant coefficients;Acta Math.85 (1-62) 1950 [Gf1] M.P.Gaffney, A special Stoke's theorem for complete Riemannian manifolds; Annals of Math. 60 (1954) 140-145 [GT1] D.Gilbarg,and N.Trudinger, Elliptic partial differential
equations of second order; 2-nd edition; Springer New York 1983. [GS] I.Gelfand and G.E.Silov, Generalized functions; Vol.1,Acad. Press, New York 1964.
[Gl1] I.M.Glazman, Direct methods of qualitative spectral analysis of singular differential operators; translated by Israel Prog. f. scientific transl. Davey, New York 1965.
[Gb] I.Gohberg, On the theory of multi-dimensional singular inte-
333
References
gral operators; Soviet Math. 1 (1960) 960-963. [GK] I.Gohberg and N.Krupnik, Einfuehrung in die Theorie der eindimensionalen singulaeren Integraloperatoren; Birkhaeuser, Basel 1979 (Russian ed. 1973).
[G1] B.Gramsch, Relative inversion in der Stoerungstheorie von Operatoren and ip-Algebren; Math. Ann. 269 (1984) 27-71. [G2J
, Meromorphie in der Theorie der Fredholm Operatoren mit Anwendungen auf elliptische Differentialopera-
toren; Math. Ann. 188 (1970) 97-112. [Gr1] G.Grubb, Problemes aux limites pseudo-differentiels depen-
dent d'un parametre; C.R.Acad.Sci.Paris 292 (1981) 581-583.
[Guil] V.Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues; Adv. Math. 55 (1985) 131-160. [GP1] V.Guillemin and A.Pollack, Differential topology; Prentice Hall, 1974. [GK] I.Gohberg and N.Krupnik,Einfuehrung in die Theorie der ein-
dimensionalen singulaeren Integrale; Birkhaeuser, Basel,Boston,Stuttgart 1979. [GLS] A.Grossmann,G.Loupias,and E.Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space; Ann.Inst. Fourier,Grenoble 18 (1968) 343-368.
[Hgl] S.Helgason, Differential geometry, Lie groups, and symmetric spaces; Acad. Press, New York 1978.
[Hz1] E.Heinz, Beitraege zur Stoerungstheorie der Spektralzerlegung; Math.Ann.123 (1951) 415-438.
[H1] E.Herman, The symbol of the algebra of singular integral operators; J.Math.Mech. 15 (1966) 147-156 .
[H11] E.Hilb, Ueber Integraldarstellungen willkuerlicher Funktionen; Math.Ann. 66 (1908) 1-66. [HP] E.Hille,and R.S.Phillips, Functional analysis and semi-groups
Amer.Math.Soc.Coll.Publ. Providence, 1957. [Hz1] F.Hirzebruch, Topological methods in algebraic geometry; Springer, Grundlehren, Vol. 131, 1966, Berlin Heidelberg, New York.
[Ho1] L.Hoermander,Linear partial differential operators;Springer, Berlin, Heidelberg, NewYork 1963.
334
References
[Ho] L.Hoermander, Pseudo-differential operators and hypoelliptic equations; Proceedings Symposia in pure and appl. Math. Vol 10 (1966) 138-183. [Ho3]
, The analysis of linear partial differential operators; Springer New York I: distribution theory and Fourier analysis, 1983; II: diff. operators pseudo-
with constant coefficients, 1983; III:
differential operators, 1985; IV, Fourier inteoperators, 1985. [I1] R.Illner, On algebras of pseudodifferential operators in LP (,n); Comm.Part.Diff.Equ. 2 (1977) 133-141. [Kpl] I.Kaplanski, The structure of certain operator algebras; Transactions Amer.Math.Soc.70 (1951) 219-255. [K1] T.Kato, Perturbation theory for linear operators; Springer New York 1966. [K2]
, Perturbation theory for nullity, deficiency and other quantities of linear operators; Journal
[K3,
, Notes on some inequalities for linear operators;
d'Analyse Math. 6 (1958) 261-322. Math.Ann.125 (1952) 208-212. [Ke1] J.L.Kelley, General topology; Van Nostrand, Princeton 1955.
[KG1] W.Klingenberg, D.Grommol, and W.Meyer, Riemannsche Geometrie in Grossen; Springer Lecture Notes Math. Vol. 55 New York 1968. [Ko1] K.Kodaira, On ordinary differential equations of any even
order and the corresponding eigenfunction expansions; Amer.J.Math. 72 (1950) 502-544. [Kr1] M.G.Krein,On hermitian operators with direction functionals; Sbornik Praz. Inst. Mat. Akad. Nauk Ukr. SSR 10 (1948) 83-105. [Ku1] H.Kumano-go, Pseudodifferential operators; MIT-Press 1982.
[Le1] N.Levinson, The expansion theorem for singular self-adjoint linear differential operators; Ann.o.f Math.ser.2 59 (1954) 300-315 [Lk] R.Lockhart, Fredholm properties of a class of elliptic ope.
rators on noncompact manifolds; Duke Math.J. 48 (1981) 289-312. [LMgl] J.Lions and E.Magenes, Non-homogeneous boundary value problems and applications; I, II, Springer,
335
References New York 1972.
[LM] R.Lockhart and R.McOwen, Elliptic differential operators on noncompact manifolds; Acta Math.151 (1984) 123-234.
[Lw1] K.Loewner, Ueber monotone Matrixfunktionen; Math.Zeitschr. 38 (1934) 177-216. [MaM] R.Mazzeo and R.Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature; preprint Dept.Math.MIT, 1986.
[M0] R.McOwen, Fredholm theory of partial differential equations on complete Riemannian manifolds; Ph.D. Thesis, [M1] R.McOwen
Berkeley 1978. Fredholm theory of partial differential equations
on complete Riemannian manifolds;Pacific J.Math. 87 (1980) 169-185 .
[M2] R.McOwen
On elliptic operators in In ;Comm. Partial Diff.
equations, 5(9) (1980) 913-933. [MS1] E.Meister and F.O.Speck, The Moore-Penrose inverse of Wiener
Hopf operators on the half-axis and the quarter plane; J. Integral equations 9 (1985) 45-61. [Me1] R.Melrose, Transformation of boundary problems, Acta Math. 147 (149-236).
[MM1] R.Melrose and G.Mendoza,Elliptic boundary problems on spaces with conic points; Journees des equations differentielles, St. Jean-de-Monts, 1981. , Elliptic operators of totally characteristic type;
[MM2]
MSRI-preprint, Berkeley, 1982.
[Mi1] J.Milnor, Topology from a different viewpoint; Univ.of Virginia press, 1965. [Ms1] J.Moser,
On Harnack's theorem for elliptic differential equations; Comm.P.Appl.Math.l4 (1961) 577-591.
[MP1] W.Moss and J.Piepenbrinck, Positive solutions of elliptic equations; Pac.J.Math.55(1978) 219-226. [NSz1] B.v.Sz.Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes; Springer, New York 1967. [Nb1,2 1
G.Neubauer, Index abgeschlossener Operatoren in Banachraeumen I: Math.Ann.160 (1965) 93-130; II:Math.
Ann. 162 (1965) 92-119. [vN1] J.v.Neumann,Allgemeine Eigenwerttheorie Hermitescher Funk-
336
References
tionaloperatoren; Math.Ann.102 (1929) 49-131. [Ne1] M.A.Neumark, On the defect index of linear differential operators; Doklady Akad.Nauk SSSR 82 (1952) 517-520. [Ne2]
,
Lineare Differentialoperatoren; Akademie Verlag Berlin 1960.
[Nt1] F.Noether, Ueber eine Klasse singulaerer Integralgleichungen; Math. Ann. 82 (1921) 42-63. [NW1] L.Nirenberg, and H.F.Walker, The null spaces of elliptic partial differential operators in Rn J. Math. analysis and Appl. 42 (1973) 271-301. ;
[Ppl] J.Piepenbrink, Nonoscillatory elliptic equations; J.Diff.Eq. 15(1974) 541-550. [Pp2]
, A conjecture of Glazman; J.Diff.Eq.24(1977) 173-176
[P1] S.C.Power, Fredholm theory of piecewise continuous Fourier integral operators on Hilbert space; J. Operator theory 7 (1982) 52-60
.
[PW1] M.Protter, and H.Weinberger, Maximum principles in differential equations; Prentice Hall, Englewood Cliffs 1967. [RS] M.Reed and B.Simon, Methods of modern mathematical physics; Academic Press, New York. I: Functional analysis,
1975; II: self-adjointness and Fourier analysis,
1975; III: Scattering theory, 1978; IV: Spectral theory, 1978; V: Perturbation theory, 1979 [Re1] F.Rellich, Halbbeschraenkte Differentialoperatoren hoeherer Ordnung; Proceedings International Kongr. Math. Amsterdam 1954 , vol.3, 243-250. [RS1] S.Rempel and B.W.Schulze, Parametrices and boundary symbo-
lic calculus for elliptic boundary problems without the transmission property; Math.Nachr. 105 (1982) 45-149. [R1] C.E.Rickart, General theory of Banach algebras; v.Nostrand Princeton 1960. [Ril] F.Riesz, Ueber die linearen Transformationen des komplexen
Hilbertschen Raumes;Acta Sci.Math.Szeged 5 (1930) 23-54.
[RN] F.Riesz and B.Sz.-Nagy, Functional analysis; Frederic Ungar, New York 1955.
[Rol] D.Robert, Proprietes spectrales d'operateurs pseudo-differentiels; Comm. PDE 3(9) (1978) 755-826.
337
References
[Sa1] D.Sarason, Toeplitz-operators with semi-almost periodic symbols; Duke Math.J. 44 (1977) 357-364.
[Schu1] W.Schulze, Ellipticity and continuous conormal asymptotics on manifolds with conical singularities; to appear; Math. Nachrichten. , Mellin expansions of pseudo-differential opera-
[Schu2]
tors and conormal asymptotics of solutions; Proceedings, Oberwolfach conference on Topics in pseudodifferential operators 1986; to appear. [Sk1] S.Sakai, C -algebras and W -algebras; Springer, Berlin New York 1971. [Ro1l W.Roelcke, Ueber Laplaceoperatoren auf Riemannschen Mannigfaltigkeiten mit diskontintuierlichen Gruppen; Math.Nachr.21 (1960) 132-149
.
[Schr1] E.Schrohe, The symbol of an algebra of pseudo-differential operators; to appear. , Potenzen elliptischer Pseudodifferentialoperato-
[Schr2]
ren; Thesis, Mainz 1986.
[Se1] R.T.Seeley, Topics in pseudo-differential operators, CIME Conference at Stresa 1968.
Integro-differential operators on vector bundles; Transactions AMS 117 (1965) 167-204 Ca [S1] H.Sohrab, The -algebra of the n-dimensional harmonic [Se2]
.
oscillator; Manuscripta.Math.34 (1981),45-70
.
C -algebras of singular integral operators on the
[S21
line related to singular Sturm-Liouville problem; Manuscripta Math. 41 (1983) 109-138. [S3]
,
Pseudodifferential C -algebras related to Schroe-
dinger operators with radially symmetric potential; Quarterly J.Math.Oxford 37 (1986) 105-115. [So] A.Sommerfeld, Atombau and Spektrallinien; Vieweg,Braunschweig 1957
[Sp1] F.O.Speck, General Wiener-Hopf-factorization methods; Pitman London 1985. [St1] M.H.Stone, Linear transformations in Hilbert space; Amer.
Math. Soc. Coll. Publ. New York 1932
.
[T1] M.Taylor , Pseudo-differential operators; Springer Lecture Notes in Math. Vol. 416, New York 1974 .
[T2]
, Pseudodifferential operators;Princeton Univ.Press, Princeton 1981
.
338
References
[Ti1] E.C.Titchmarsh, Eigenfunction excpansions associated with second order differential equations;part I,2nded. Clarendon Oxford 1062; part II, ibid. 1958. [Tr1] F.Treves,
Introduction to pseudodifferential operators and Fourier integral operators; Plenum Press,New York and London 1980
[Up1] H.Upmeier, Toeplitz,C -algebras on bounded symmetric domains; Ann. of Math.(2) 119 (1984) 549-576. [W1] A.Weinstein, A symbol class for some Schroedinger equations on
In
; Amer.J.Math.107 (1985) 1-21.
[We1] H.Weyl, Ueber gewoehnliche Differentialgleichungen mit Singularitaeten, and die zugehoerigen Entwicklungen willkuerlicher Funktionen; Math.Ann.68 (1910) 220-269. [We2]
, The method of orthogonal projection in potential theory; Duke Math.J. 7(1940) 411-444.
[Wd1] H.Widom, Asymptotic expansions for pseudodifferential operators on bounded domains; Springer Lecture Notes No 1152, 1985. [Wi1] E.Wienholtz, Halbbeschraenkte partielle Differentialoperatoren vom elliptischen Typus; Math.Ann.135 (1958) 50-80.
[Z1] S.Zelditch, Reconstruction of singularities for solutions of Schroedinger equations;Ph.D.Thesis,Univ.of Calif. Berkeley (1981).
339
Index
condition (xj),x=a,d,... 324
A-bounded 294 A-compact 294
covariant derivatives 134f, 322 curvature (Riemannian) 279f, 323
adjoint operator 3 adjoint relation 5
defect index 7
algebra surgery 239
defect spaces 6,38 definite case 41
boundary, regular 67 boundary condition 41,68,315 ,Dirichlet 53,68 ,Neumann 315
Dirichlet condition 53,68 operator 60,68 problem 87
realization 59
boundary space 44 bounded operator 1
domain (of an
Carleman alternatives 41
E-symbol 218f
unbounded operator) 2
change of dependent variable 93f
for periodic
Christoffel symbols 134,322 closed operator 2
for polycylinder
closing 39
closure of an operator 2
coefficients 227f algebra 235f elliptic differential expression 37
co-sphere bundle 170 covariant derivative 134,322
equivalent triples 93
commutator of differential expressions 133f
essential self-adjointness 123 of H0 103, 111
, compactness 137f
commutator ideal E 126,131 commuting unbounded operators 17 compactness criteria 71f of commutators 144
essentially self-adjoint 5
of H0m
105f,lll
expresssion (differential) 36
within reach 127,155,262 extension (of an operator) 2
comparison operator 61,127f comparison algebra 127f,157 , minimal 129,161f
4 (property) 293
comparison triple 128
formal adjoint 36 Fredholm domain 302
complete spaces,
with cylindrical ends 246
folpde 126,129
index 33,293
341
operator 33,180,293 inverse 180,295
maximum principle 91 minimal operator 37 minimal comparison algebra 129
distinguished 34,180 ,
multiplication operator 134
special 295
Frehse's theorem 118f
noncompact commutators,
Friedrichs extension 11,71f
algebras with 218f normal forms 93,94
fundamental solution 42,78
Sturm-Liouville 95,151 graph (of an unbounded operator) 2 Green inverse 161,181,318
order classes 30,267,318 ordinary diff.expression 37
, distinguished 181
Green's function 78f
periodic coefficients 154f,221 poly-cylinder algebra 228f
H-compatible expresssion 157,160
positive operator 14
Hs-comparison algebra 257,267f,315f
positivity of the Green's function 100,130
Harnack inequality 87f Heinz, lemma of 24
preclosed operator 3
hermitian operator 5 HS-chain 29
principal part 36 symbol type 266 symbol space 193
hull 191
projection 14
hypo-elliptic expression 37
positive square root 12
isometry, betw.Hs-spaces 31,268
reach, expression
indefinite case 41 irreducible C -algebra 130 K-invariant under H-conjugation 33,181 Landau symbols 320 Laplace comparison operator 93
within 127,155f,186,262f algebra within 266 general 266 real operator 10
realization,of a differential expression 35,37 regular endpoint 45 regularity, boundary 67f
limit circle case 41,53
Rellich's criterion 75,77,179
limit point case 41,53
resolvent 12
Loewner, lemma of 24
compactness 71f expansion of
maximal null space 47
of H-compat. expr.158
maximal operator 37
formula for As 139,163
342
set 12
square root of an operator 12
Riesz-v.Neumann extens.thm.8
strongly hypo-elliptic 38 Sturm-Liouville problem 51,53
second order express.on 0 132 secondary symbol space 193 rel.to formal symbol 207 local charac.195,211,214
subextending triple 76 summation convention 60 surgery, algebra 239f symbol of an operator 191,137
for Schroedinger opera-
of an algebra 191
tors on In
function 170
215f
self-adjoint operator 4 semi-bounded operator 10 separated boundary cdn's 48 singular endpoint 45,49 Sobolev space 61,157,159,251f first 128 on compact Q 157 on noncompact 0 251f
of integral order 279f local 272 Sobolev estimate 181f,184 imbedding 182,217,251 ,
comparison
algebra in 253,257,315 spectral density matrix 52 spectral family 15
spectral theorem 12f spectrum 13 essential 265
formal algebra 187,206 symbol space ffi
131,189,191,145
, principal 193
, secondary 193
systems of differential expressions 303f tensor 132,322 tensor products,
topological 226,229,303 two-link ideal chains 218f,309 unbounded operator 1 Vector bundles, comparison algebras on 303f
wave front space 161f,169f Weyl's lemma 39,42,60
