C*-ALGEBRAS
VOLUME 4: HILBERT SPACES
C*-ALGEBRAS
VOLUME 4: HILBERT SPACES
North-Holland Mathematical Library Board...
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C*-ALGEBRAS
VOLUME 4: HILBERT SPACES
C*-ALGEBRAS
VOLUME 4: HILBERT SPACES
North-Holland Mathematical Library Board of Honorary Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, W.A.J.Luxemburg, F. Peterson, I.M. Singer and A.C. Zaanen Board of Advisory Editors: A. Bjomer, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sjostrand, J.H.M. Steenbrink, F. Takens and J. van Mill
VOLUME 61
ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo
C*-Algebras Volume 4: Hilbert Spaces
Corneliu Constantinescu Departement Mathematik, ETH Zurich CH-8092 Zurich Switzerland
ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 2 1 1, 1000 A E Amsterdam, The Netherlands
0 2001
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Preface Funct,ional analysis plays an important role in the program of studies a t the Swiss Federal Instit,ute of Technology. At present, courses entitled F~lnctional .4nalysis I and 11 arc taken during the fifth and sixth semester, rcspectivcly. I have taught these courses several times and after a while typewritten lecture notes resulted that were distributed to the students. During the academic year 1987/88: I was fortunate enough to have an eager enthusiastic group of students that I had already erlcoul~teredpreviously in other lecture courses. These stutlt>rltswanted to learn rrlore in the area and asked rrie to design a cont,inuation of' the courses. Accordingly. I proceeded during the academic year, following, wit,h a series of special lectures, Functional Analysis I11 and IV, for which I again distrih~~tcd typewritten Icct,~lrcnotes. At the end I found that t,hcrc had accum~llatcda mass of text,ual material, and I asked myself if I should not publish it in the form of a book. Cnfortunately, I realized that the two special lecture series (they had been given only once) had been badly organized and contained ~rlaterialthat should have been included in the first two portions. And so I came to the conclusion that I should write everything anew - and if a t all then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost impcrccptihly and at t,hc end it had more than do~lhlcd.Also: the English language turned o ~ to~ be t a st~~mhling 1)lock for me; I would likc to take this opport~lnityto thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aqnilino, who wrote, first a b \ ~ o r d ~ 4 A R Cand ' ~ ~ then , a Lq$J'r'M v e r s i o ~with ~ great co~r~petence, a ~ ~ g e l patience, ic a11d utter devotion, in spite of illness. My thanks also go to the Swiss Federal Inst,it,utcof Tc,chnology that. genero~lslyprovided the infrast,ructure for this extensive enterprise and to my c~ollcagncswho showed their understanding for i t -
Corneliu Constantinescu
This Page Intentionally Left Blank
Table of Contents of Volume 4
Introduction 5
...............................
Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Hilbert sums . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Orthogonal Projections of Hilbert Space . . . . . . . . . . . . . 5.2.1 Projections onto Convex Sets . . . . . . . . . . . . . . . 5.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . 5.2.4 Mean Ergodic Theorems . . . . . . . . . . . . . . . . . . 5.2.5 The Frechet-Riesz Theorem . . . . . . . . . . . . . . . . 5.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 5.3.3 Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . 5.3.4 Normal Operators . . . . . . . . . . . . . . . . . . . . . . 5.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cyclic R.epresentation . . . . . . . . . . . . . . . . . . . 5.4.2 General Representations . . . . . . . . . . . . . . . . . . 5.4.3 Example of Representations . . . . . . . . . . . . . . . . 5.5 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hilbert Dimension . . . . . . . . . . . . . . . . . . . . . 5.5.3 Standard Examples . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Fourier-Plancherel Operator . . . . . . . . . . . . . 5.5.5 Operators and Orthonormal Bases . . . . . . . . . . . . 5.5.6 Self-.normal Compact Operators . . . . . . . . . . . . . . 5.5.7 Examples of R.eal C*-Algebras . . . . . . . . . . . . . .
xix
3 3 3 14 19 24 24 29 33 54 63 72 72 86 108 123 130 130 146 156 166 166 191 206 218 223 243 258
...
v1l1
Table of Contents
5.6 Hilbert right C*-Modules . . . . . . . . 5.6.1 Some General Results . . . . . . 5.6.2 Self-duality . . . . . . . . . . . . 5.6.3 Von Neumann right W*-modules 5.6.4 Examples . . . . . . . . . . . . . 5.6.5 K E . . . . . . . . . . . . . . . . . 5.6.6 Matrices over C'-algebras . . . . 5.6.7 Type I W'-algebras . . . . . . . Name Index Subject Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
286 286 310 341 373 430 477 515
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
Contents of All Volumes
Table of Contents of Volume 1
Some Notation and Terminology
. . . . . . . . . . . . . . . . . . . .
1 Banact1 Spaces . . . . . . . . . . . . . . . . . . 1.1 Xormed Spaces . . . . . . . . . . . . . . . . 1.1.1 Gencral Results . . . . . . . . . . . . 1.1.2 Somc Standard Examples . . . . . . 1.1.3 A.linkowski's Thcorcm . . . . . . . . . 1.1.4 Locallv Compact Norn~edSpaces . . 1.1.5 Prodncts of Normed Spaccs . . . . . 1.1.6 S n m n ~ a l ~Families le . . . . . . . . . . Exercises. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Oprrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Resnlts . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Standard Exarrlples . . . . . . . . . . . . . . . . . . . . . 1.2.3 Infinit.e Matricrs . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Qnotient Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Cornplernented Sut~spaces . . . . . . . . . . . . . . . . . 1.2.6 Tht! 'Topology of Pointwise Convergence . . . . . . . . . 1.2.7 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . .
1
7 7 7 12 31 33 37 40 58 61 61 74 92 113 120 134 138 148 150 153 139 159 171
The Alaoglu Bor1rt)aki Theorcrn . . . . . . . . . . . . . . Rilir~carMaps . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thc llahn-Banad1 Theorem . . . . . . . . . . . . . . . . . . . . 1.3.1 Thc Banach Theorern . . . . . . . . . . . . . . . . . . . . 1.0.2 Exarnples in Measure Theory . . . . . . . . . . . . . . . I .3.3 'I'llc Hahn Ranach Thcorcrn . . . . . . . . . . . . . . . . 180 1.3.4 'The Transposc of an Operator . . . . . . . . . . . . . . . 191 1.2.8 1.2.9
1
Table of Contents
Polar Sets . . . . . . . . . . . . The Bidual . . . . . . . . . . . The Krein-~mulian Theorem . Reflexive Spaces . . . . . . . . . Completion of Normed Spaces . Analytic Functions . . . . . . . Exercises . . . . . . . . . . . . . 1.4 Applications of Baire's Theorem . . . . 1.4.1 The Banach-Steinhaus Theorem 1.4.2 Open Mapping Principle . . . . Exercises . . . . . . . . . . . . . 1.5 Banach Categories . . . . . . . . . . . 1.3.5 1.3.6 1.3.7 1.3.8 1.3.9 1.3.10
1.5.1 Definitions . . . 1.5.2 Functors . . . . 1.6 Nuclear Maps . . . . . 1.6.1 General Results 1.6.2 Examples . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
1.7 Ordered Banach spaces . . . . . . 1.7.1 Ordered normed spaces . . 1.7.2 Order Continuity . . . . . Name Index Subject Index
. . . . . . . . . . . . . . 199 . . . . . . . . . . . . . . 211 . . . . . . . . . . . . . . 228
. . . . . . . . . . . . . . 240 . . . . . . . . . . . . . . 245 . . . . . . . . . . . . . . 246 . . . . . . . . . . . . . . 254 . . . . . . . . . . . . . . 256 . . . . . . . . . . . . . . 256 . . . . . . . . . . . . . . 264 . . . . . . . . . . . . . . 280 . . . . . . . . . . . . . . 281 . . . . . . . . . . . . . . . . . 281 . . . . . . . . . . . . . . . . . 288 . . . . . . . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . 322 . . . . . . . . . . . . . . . . . 334 . . . . . . . . . . . . . . . . . 334 . . . . . . . . . . . . . . . . . 340
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Table of Contents of Volume 2
Introduction 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Bartach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.1.1 2.1.2 2.1.3 2.1.4 2.1.5
2.2
2.3
2.4
3 3
General Results . . . . . . . . . . . . . . . . . . . . . . . 3 Invert.iblcElcmcnts . . . . . . . . . . . . . . . . . . . . . 13 Tho Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 17 Standard Examples . . . . . . . . . . . . . . . . . . . . . 32
Con~~~lexification of Algebras . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . Norrnod Algcl, ras . . . . . . . . . . . . . . . . . . . . . . . . . .
51 65 69
. . . . . . . . . . . . . . . . . . . . . . . 2.2.1 GortoralRes~~lts 2.2.2 The Standard Examples . . . . . . . . . . . . . . . . . . 2.2.3 'The Kxponcntial Function and the Neumann Series . . . 2.2:1 Invertible Elements of IJnit.al Ranach Algc.11ras . . . . . . 2.2.5 Thc Theorems of Ricsz and Gclfand . . . . . . . . . . . . 2.2.6 Poles of Rosolvtrnts . . . . . . . . . . . . . . . . . . . . . 2.2.7 Mod~llcs . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
69 82 114 125 153 161 174
197 Invol~itiveBanach .i\lgrbras . . . . . . . . . . . . . . . . . . . . 201 2.3.1 Involntivc Algebras . . . . . . . . . . . . . . . . . . . . . 201 2.3.2 Invo11it.ivc~Banach Algebra. . . . . . . . . . . . . . . . . 241 2.3.3 Scsql~iliriearForms . . . . . . . . . . . . . . . . . . . . . 275 2.3.4 I'ositive I,irtear Forms . . . . . . . . . . . . . . . . . . . 287
2.3.5 2.3.6
The State Space . . lnvol~itivcMotiules Exercises . . . . . . Gelfarltl Algebras . . . . .
. . . . . . . . . . . . . . . . . . . . . 305 . . . . . . . . . . . . . . . . . . . . . 322
2.4.1 2.4.2
. . . . . . . . . . . . . . . . . . . . . 328 . . . . . . . . . . . . . . . . . . . . . 331 The Gclfand Transform . . . . . . . . . . . . . . . . . . . 331 Invol~~tive Gelfand Algebras . . . . . . . . . . . . . . . . 343
Table of Contents
xii
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . Locally Compact Additive Groups . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . The Fourier Transform . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
358 365 378 390 396
3 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Fredholm Operators . . . . . . . . . . . . . . . . . . . . 3.1.4 Point Spectrum . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Spectrum of a Compact Operator . . . . . . . . . . . . . 3.1.6 Integral Operators . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . 3.2.1 Boundary Value Problems for Differential Equations . . . 3.2.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 3.2.3 Linear Partial Differential Equations . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
399 399 399 419 437 468 477 489 517 518 518 530 549 563
2.4.3 2.4.4 2.4.5 2.4.6
Name Index Subject Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
...
Xlll
Table of Contents of Volume 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
C* -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 3
Introduction 4
4.1
Tlic General Thcory 4.1.1 4.1.2
General Rc?sults . . . . . . . . . . . . . . . . . . . . . . . The Symnlctry of C*-Algebra . . . . . . . . . . . . . . .
Functional c a l c ~ ~ l uins C*-Algebras . . . The Theorem of Fuglede-P~itnam . . . . T h e Order Relation . . . . . . . . . . . . . . . . 4.2.1 Definition and General Propcrt.ies . . . . 4.2.3 Exa~nples . . . . . . . . . . . . . . . . . 4.1.3 4.1.4
4.2
. . . . . . . . . . . . . . . . . . . . . . . .
4 30
. . . . . . . . . 56 . . . . . . . . .
75
. . . . . . . . .
92
. . . . . . . . .
92 . . . . . . . . . 116 4.2.4 Powers of Positive Elen~cnts . . . . . . . . . . . . . . . . 123 4.2.5 Thc Mod1111ls . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2.6 Ideals and Quotients of C*-Algebras . . . . . . . . . . . 150 4.2.7 T h e Ordered Set of Orthogonal Projectior~s . . . . . . . 162 4.2.8 Approximate Unit . . . . . . . . . . . . . . . . . . . . . 178 4.3 Snpplemcntary Rcsnlt.s on C* Algebras . . . . . . . . . . . . . . 208 4.3.1 Tho Exterior Mllltiplication . . . . . . . . . . . . . . . . 208 4.3.2 Order Complet.e C*--Algebra5 . . . . . . . . . . . . . . . 215 4.3.3 The Carrier . . . . . . . . . . . . . . . . . . . . . . . . . 243 4.3.4 Heretiitilry C*-S11ba1gebra.s . . . . . . . . . . . . . . . . 263 4.3.5 Simple C*-algebras . . . . . . . . . . . . . . . . . . . . . 276 4.3.6 Sl~pplement. ary Res~iltsConcerning Complcxifiratior~ . . 286 4.4 IV*-A1get)ras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.4.1 General Properties . . . . . . . . . . . . . . . . . . . . . 297 4.4.2 F as arl E s u b m o d ~ ~of l e E' . . . . . . . . . . . . . . . 309 4.4.3 Polar Rrpresent ation . . . . . . . . . . . . . . . . . . . . 335 4.4.4 W*-Homorr~orphisms. . . . . . . . . . . . . . . . . . . . 361
xiv
Table of Contents
Subject Index
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Symbol Index . . . . .
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. . 388
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411
Table of Contents of Volume 4
Introduction 5
...............................
Hilbcrt Spaces . . . . . . . 5.1 Pre-Hilbert Spaces . . 5.1.1 General Results 5.1.2 Examples . . . 5.1.3 Hilbert sums . 5.2
5.3
5.4
5.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Projections of Hilbert Space . . . . . . . . . . . . . 5.2.1 Projections onto Convex Sets . . . . . . . . . . . . . . . 5.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . 5.2.4 Mean Ergodic Theorems . . . . . . . . . . . . . . . . . . 5.2.5 Tho Frbchet.. Riesz Theorem . . . . . . . . . . . . . . . . Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General Res~llts . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 5.3.3 Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . 5.3.4 Normal Operators . . . . . . . . . . . . . . . . . . . . . . R.epresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cyclic: Representation . . . . . . . . . . . . . . . . . . . 5.4.2 General Representations . . . . . . . . . . . . . . . . . . 5.4.3 Example of Representations . . . . . . . . . . . . . . . . Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hilhert Dimension . . . . . . . . . . . . . . . . . . . . . 5.5.3 Standard Examples . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Fourier-Plancherel Operator . . . . . . . . . . . . . 5.5.5 Operators and Orthonormal Bases . . . . . . . . . . . . 5.5.6 Self -normal Cornpac:t Operators . . . . . . . . . . . . . . 5.5.7 Examples of Real C*-Algebras . . . . . . . . . . . . . .
xix
3 3 3 14 19 24 24 29 33 54 63 72 72 86 108 123 130 130 146 156 166 166 191 206 218 223 243 258
xvi
Table of Contents
5.6 Hilbert right C* -Modules . . . . . . . . 5.6.1 Some General Results . . . . . . 5.6.2 Self-duality . . . . . . . . . . . . . 5.6.3 Von Neumann right W* -modules 5.6.4 Examples . . . . . . . . . . . . . 5.6.5 ICE . . . . . . . . . . . . . . . . . 5.6.6 Matrices over C*-algebras . . . . 5.6.7 Type I Lit*-algebras . . . . . . . Name Index Subject Index
. . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
286 286 310 341 373 430 477 515
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
xvii
Table of Contents of Volume 5
...............................
xix
6 Selected Chapters of C*-Algebras . . . . . . . . . . . . . . . . . . . 6.1 LP-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Characteristic Families of Eigenvalues . . . . . . . . . . . 6.1.2 Characteristic Sequences . . . . . . . . . . . . . . . . . . 6.1.3 Properties of the LP-spaces . . . . . . . . . . . . . . . . 6.1.4 Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . 6.1.5 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 Duals of LP-spaces . . . . . . . . . . . . . . . . . . . . . 6.1.7 Exterior Mi~ltiplicationand LP-Spaces . . . . . . . . . . 6.1.8 The Canonical Projection of the Tridual of K . . . . . . 6.1.9 Integral Operators on FIilbert Spaces . . . . . . . . . . . 6.2 Selfadjoint Linear Differential Equations . . . . . . . . . . . . . 6.2.1 Selfadjoint Boundary Value Problems . . . . . . . . . . . 6.2.2 The Reglilar Sturm-Liouville Theory . . . . . . . . . . . 6.2.3 Selfadjoint Linear Differential Equations on T . . . . . . 6.2.4 Associated Parabolic and Hyperbolic Evolution Equations 6.2.5 Selfadjoint Linear Partial Differential Equations . . . . . 6.2.6 Associated Parabolic arid Hyperbolic Evolution Equations 6.3 Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Strong Topology . . . . . . . . . . . . . . . . . . . . 6.3.2 Bidual of a C*-algebra . . . . . . . . . . . . . . . . . . . 6.3.3 Extensiorl of the Functional Calculus . . . . . . . . . . . 6.3.4 Von Neumann-Algebras . . . . . . . . . . . . . . . . . . 6.3.5 The Commlrt.ants . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Irredllcible Representations . . . . . . . . . . . . . . . . 6.3.7 Commutative von Neumann Algebras . . . . . . . . . . . Algebras . . . . . . . . . . . . . 6.3.8 Representations of W*-. 6.3.9 Finite-dimensional C*-algebras . . . . . . . . . . . . . .
3 3 3 10 21 46 56 72 79 102 116 124 125 139 150 153 184 192 202 203 218 263 283 293 299 320 325 334
Introduction
xviii
Table of Contents
6.3.10 A generalization . . . . . . . . . . . . . . . . . . . . . . . 355 7
C' algebras Generated by Groups . . . . 7.1 Projective Representations of Groups 7.1.1 Schur functions . . . . . . . . 7.1.2 Projective Representations . . 7.1.3 Supplementary Results . . . . 7.1.4 Examples . . . . . . . . . . . 7.2 Clifford Algebras . . . . . . . . . . . 7.2.1 General Clifford Algebras . . . . . . . . . . . . . . . . 7.2.2 7.2.3 CL(IN) . . . . . . . . . . . . . Namc Index S~lbjcctIndex
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369 369 369 404 431 466 492 492 518 538
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . 563
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
xix
Introduction This book has evolved from the I(~c:tnrccourso on F~~rrctiorral Analysis I had givc,rr sc~vt,raltirrres at thc! ETH. 'fht, text has a strict logical order, in the style of "D(+irriton Theorern I'roof - Example Excrcisrs". Thc proofs arc9rathcr -
thorough and there
-
many rxanrplcs.
The, first part of the t~ook(the first three chapters. resp. the first two vol~l-
rnes) is devotc.d to the theory of Banach spaccs in th(>most general sense of the t,errrr. The pllrpose of the first chaptcr (resp. first v o l ~ ~ n ris e )to introduce those rcs~rltso n Banach spaccs which are used later or which are closely conrrc,ctcd with the book. It therefore only contains a small part of the theory, and several rcsults ilte stated (and proved) in ;I dilutod form. The second chaptvr (which t,ogethc,r with Chapter 3 rriakes thc second volume) deals with Barrach algcbras (anti invol~itivcnanach algebras). which constitute t h main ~ topic of the first part of the t~ook.T l ~ ethird c:lrapter deals with conrpact operators 011 Banach spaces i111tllirrear (ordinary and partial) differential eq~lations- applications of tht, thc~)l.yof Rarrach algct)ras. The sc~c,ondpart of the t~ook(the last four chapters. resp. the last three, vol~~rrres) is d(~vot.c!tlto the theory of Hilhcrt sl)ac:t!s. oncc again in tht. gc!r~c!riil scrrsr of the, ttrrr~.It 1)egirrswith a chapter (Chapter 4 , rrsp. Volnrr~e3) on the theor.v of C* iilgct)ras and I,I''- algebras which arc essentially the focns of thc I)ook. Chapter 5 (resp. Volurne 4) t.rt!ats Hilbc'rt spacvs for which wc had no ~rccdcarlic,r. It contains t h rcprosentation ~ theorc:ms, i.e. the theorems on isomclt rifs hetnrclen a1)strac.t C*-;rlgcl)ras and the c,oncrc'tc (i" -algebras of operators on Hilbert spaccs. Clraptcr G (which together with Chapter 7 rnakes Vol~irr~o 5) presents tht! tlic,ory of CP spaccs of operators, its application to the self achoirrt linear (ordinary and partial) diftercrrtial c'cl~~atiorrs,and the von N(:~imannillgcbras. Firrallj~:Chapter 7 present,s examples of C* algc1)ras tl(f~nc!dwith the, aid of gl.o~lps,in partic~rlarthe Clifford algebras. Many important domains of C* algr1)ras arc igrrored in the present 1)ook. It stro~lltlt ~ ecmpha.sized that t . h ~ whole theory is constr~lc~ted in parallcl for the real and for the c o n ~ l ~ l en~~rnt)rrs. x i.c. t,hc. C* algel~rasare real or complex.
In addition to the a1)ovt: (vcrtic:al) str~lcturcof t,ho t~ook:thcrc, is also i1 second (horizontal) division. It c,onsists of a main st.rand, eight l)ri~nc.h(,s. alltl ad(litionii1 material. The rcsnlts belonging to the main strand arc rnarked wit11 ( 0 ) . 1,ogically sprakirrg. a reader coultl rcst.rict hirnself/ht!rsclf t o thcsc anti igrrort, the rest. R c s ~ ~ lon t s the eight sl~bsitliarybranches are rnarkecl with ( 1 ) , (2). (3) . (4) . (5) . ( 6 ) . (7) , and (8) . The key is
Infinite Matrices Banach Categories Nl~rlcarMaps Locally Compact Groups Differential Equations Laurent Series Clifford Algebras Hilbert C*-Modules These arc (logically) independent of each other, hut all depend on the main strand. Finally, the results which belong to the additional material have no marking and from a logical perspective may he ignored. So the reader can shorten for himselflherself this very long book using the above marks. Also, si~lcethe proofs are given with almost all refererices, it is possible to get into the book a t any level and not to read it linearly. U'r assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point set topology, and integration t,hcory. The hook addrcssrs itself mainly to mathematicians, or to physicists interested in C* algebras. I would like to apologize for any omissions in citations occasioned by the fact that mv acquaintance with the history of functional analysis is, unfortunately, very ~estricted.For this history we recorrirr~endthe following texts. 1 . BIRKHOFF: G. and KREYSZIG, E.: The Establishment of Functional Analysis, Historia Mathcmatica 11 (1984), 238 321.
2. BOURBAKI, N., Elements of the History of Mathematics, (21. Topological Vector Spaces). Springer-Verlag (1994). 3. DIEUDONYB, .I.. History of Functional Analysis, North Holland (1981). 4. DIEUDONNE, J.: A Panorama of Pure Mathematics (Chapter C 111: Spectral Theory of Operators), Academic Press (1982).
5. HEUSER., H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die wcrdcndc Rlnctionalanalysis), Tcubner (1986); 3. Auflagc (1992). G . KADISON, R.V., Operator Algebras, the First Forty Years, in: Proceedings of Syrriposia in Pure Mathe~natics38 1 (1982). 1-18.
7. MONNA, A.F., Fi~nctionalAnalysis in Historical Pcrspcctivc, .John WhiIcy k Sons (1973).
8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359 382. Thcrc is no shortagc of cxcellcnt books on C* algebras. Neverthclcss, we hope that this book will be also of some utility to the mathematics community.
This Page Intentionally Left Blank
VOLUME 4: HILBERT SPACES
This Page Intentionally Left Blank
5 . Hilbert Spaces
Most books on functional analysis treat the theory of Hilbert spaces before the theory of C*-algebras if they treat the latter at all. This is didactically justifiable since the former is simpler and is the main source of examples for the latter. But in a strictly logical sense, the theory of C*-algebras has priority. The development of this theory does not require even the definition of a Hilbert space. Conversely the theory of Hilbert spaces benefits from the theory of C*-algebras because the set of operators on a Hilbert space is a C*-algebra in a natural way. There is a didactic principle which says that in a book on mathematics the degree of difficulty should be roughly speaking an increasing function of the page number. While we do not wish to underestimate the importance of this principle, we have chosen in this instance to let logical priority be our guide and to rely on the reader not to completely ignore Hilbert spaces. This inversion of the customary order does have substantial benefits, for when dealing with the algebra of operators on a Hilbert space we may avail ourselves of all the results from the theory of C'-algebras.
5.1 Pre-Hilbert Spaces A number of general results, constructions, and examples from the theory of Hilbert spaces are pesented i r ~this first section.
5.1.1 General R e s u l t s Definition 5.1.1.1 ( 0 ) (von Neumann, 1928) Let E be a vector space. A scalar product on E is a positive sesquilinear form f on E (Propositzon 2.3.3.3 e)), such that for every x E E ,
4
5. Hilbert Spaces
A pre-Halbert space is a vector space endowed with a scalar product. A preHilbert space is called real (complex) if the ground field i n R (C). W e shall usually denote a scalar product i n a pre-Hzlbert space E by
Ex E
+M ,
( x ,y ) H
(XI
y)
(scalar product of x and y )
and for x E E , we define
J. von Neumann defined only spearable Hilbert spaces. Non separable Hilbert spaces wrre considered by H. Lowing (1934) and F. Rellich (1935). Proposition 5.1.1.2
(0)
If E is a pre-Hilbert space, then the m a p
zs a n o r m and
for all x , y E E . T h e above n o r m is called the canonical norm on the preHilbert space E . Unless otherwise stated, we always regard a preHzlbert space as endowed with zts canonical norm. A Halbert space is a complete pre-Hilbert space. A real (complex) Hilbert space is a complete real (complex) pre-Hilbert space. The assertion follo~vsimmediately from Proposition 2.3.3.9.
Corollary 5.1.1.3
(0)
If E is a pre-Hilbert space, then the m a p
is contznuous By Schwarz's Inequality!
for all ( x , y ) 6 E x E and the assertion now follows frorn Proposition 1.2.9.2 c + a.
5.1 Pre-Hzlbert Spaces
5
Corollary 5.1.1.4 ( 0 ) Let E , F be pre-Hilbert spaces and u : E -t F a bijective lznear map. u is a n isometry of the n o n e d spaces E and F iff
( ~ x I ~= Y( 5) 1 ~ ) for all x, y E E
.
The necessity follows from Propositiori 2.3.3.7 a =. b and Proposition 2.3.3.8
a
+ c & d . The sufficiency follows from ll~x11~ = (ux1ux) =
(215) =
llx112
for every x E E .
Corollary 5.1.1.5 Let E # (0) be a pre-Hilbert space. Take u E C ( E ) , a E K , and x E E # . If
then x
E
Ker (a1 - u ) and
CY
E up(u)
< IaI2 - 2 a h + llu112 = 0 (Corollary 2.3.3.4). Hence
x E Ker (a1 - u )
If u
# 0 , the11 a # 0 and x # 0 . Thus a E u p ( u ) .If u = 0 , then CY
= 0 E up(u).
Theorem 5.1.1.6 ( 0 ) (Jordan, von Neumann) Let E be a normed space. T h e following are equivalent: a ) There is a scalar product o n E generating the n o r m of E
5. Hilbert Spaces
6
If these conditions are fi~lfillled, then the scalar product i n a) is unique. The ~lniquencssof the scalar product follows from Proposition 2.3.3.7 a + b and Proposition 2.3.3.8 a + c & d . a + b & c follows from the parallelogram law (Proposition 2.3.3.2 b)). b + c (resp. c + b). Put
Then
1
2 (RSP. 5 ) j ( l l a + bl12 + lla - bl12) = 2(llxl12+ IIYII~). b&c+a. Case1
IK=R
Given x , y E E : put
I
=
I
1
+l 2
-
Ilx
-
Y112).
Then, for every x , y E E ,
( 4 ~= )( ~ 1 2 ) and
(ZIT)
= 11x1I2.
Thus only linearity in the first variable needs further proof. Step 1 Now
x,y, z
E
E
+ ( x + ylz) = ( x ( z )+ ( y ( z )
5.1 Pre-Hilbert Spaces
Step2
nEINU{O},x,y~E+(nx~y)=n(x~y)
The assertion follows from Step 1, by complete induction. Step 3
X,
yE E
* (-xly)
= -(xly)
By Step 1, 0 = (01~)= (x
-
~ I Y=) ( 4 ~ + ) (-xIy).
Hence (-XIY) = -(XI?/).
Step 4
(Y
E Q , s,y E E
There are nl E Z , n E IX with
* (crxly) = cr(x1y)
(1 =
F . By Steps 2 arid 3,
TL(CYXIY) = (nosly) = (mxly) = m(x1y).
Hencr 711
(QXIY)= -(zly) n
-(xly) = (-xlv)
=4x1~).
5 II - xll llvll = llxll llvll
by the triangle inequality and Step 3. Thus
5. Hzlbert Spaces
Step 6 Let
R ,x, y E E 3 (axly) = a(xly)
cr E
be a sequence in
& converging to a . Then
I ( a x l ~) (~nxIy)l= I ( @ x l ~+) (-anxly) = I((@- an)xly)I I
I II(Q - an)xlI l l ~ l l= la - a n 1 IIxII IIYII by Steps 3, 1, and 5. Thus
(axJy) = lim (ansly) = lim an(xly) = cr(xly) n+m
n-cc
by Step 4.
Case 2
IK = C
Giver1 x, y E E , define 1
re(xlv) := 4 (llx + y1I2 - llx
Step 1
xEE
-
* (xlx) = 1 1 ~ 1 1 ~
Now
re(xlx) = l l ~ l l ~ ,
yIl2)
>
5.1 Pre-Halbert Spaces
=
1 4
(/I
-
-
ix
+ Y112
-
llix
+ Y112)
=
-rc(ylix) = -im(?/lx) .
Hencc ( ~ I Y= ) (~14.
a , P E a , X, y , z E E * ( a x + P y l z )
Step 3
= ~ ( x l +P(YIz) ~ )
By Case 1, (ax + Pylz) = re(ax
Step 4
+ PyIz) + i r e ( a x + P Y ~ ~=z )
x, y E E =+ (ixly) = i(x1y)
By Steps 2 and 3,
irn(ix1y) = -irn(yliz) = -re(yl
-
x) = -re(-x(y) = re(x1y).
Hencc (ixly) = re(ix1y)
Step 5
+ iirn(ix1y) = -irn(xly) + zrc(x1y) =
a , 0 E c , x, y, z E E 3 (xlay
+ Pz) = 6(xIy) + D(x)z).
By Steps 2: 3, and 4, (slay + 02) = (NY+ PzIx) = ~ ( Y I x+) P(zIx) = 6 (~1s+ )
=
p ( 2 1 ~ )= ~ ( X I Y+)~ ( x I Y ) .
Rerr~nrk. This theorem makes it possible to define pre-Hilbert (Hilbert) spaces as norrncd (Banach) spaces satisfying the parallelogram law.
5. Halbert Spaces
10
-
Corollary 5.1.1.7 ( 0 ) Let E be a pre-Hilbert space and E the completion of the associated normed space. There is a unique scalar product o n E generatzng the n o r m of E . E with this scalar product is called the completion of E . The scalar product of E is the restrzction to E of the scalar product of
E.
-
Take x , y E E . There are sequences ( x , ) , , ~ , ( Y ~ in ) E ~converging ~ ~ to x and y , respectively. By the parallelogram law (Proposition 2.3.3.2 b ) ) ,
+
) ) x y)I2 + ) ) x- y ) J 2= lim J J x n+ yn(I2+ lim IJxn - YnI12 = n+w
n+co
By Theorem 5.1.1.6, there is a unique scalar product on E generating its norm. The restriction to E of this scalar product generates the norm of E . Hence it coincides with the original scalar product of E (Theorem 5.1.1.6). Corollary 5.1.1.8 Let E , F be pre-Hilbert spaces and take u E L ( E ,F ) . T h e n the following are equivalent:
a ) lluxll = llxll for every x E E
b) ( u z l u y ) = ( x J y )for all x , y E E . a
* b.
The map E+ImE,
x-us
is an isometry and the assertion follows from Theorem 5.1.1.6. b a is trivial.
*
rn
Proposition 5.1.1.9 ( 0 ) Let E be a real (complez) pre-Hilbert space, f a sesquzlinear form o n E , and take a E IR+ . If
for every x E E , then
(IS(X,Y)I
for all x , y E E .
+ I ~ ( Y , X ) 1I 2 4 1 x 1 1I I Y I I )
5.1 Pre-Hilbert Spaces
We may assume without loss of generality that x
11
# 0 and y # 0 . Consider
By Proposition 2.3.3.2 b),c),
5
1
( I+ I
+I
( - Y)1)
1 . Let s , t be two distinct elements of T . Then
By the parallelogram law (Proposition 2.3.3.2 b)), the norm of P ( T ) is not generated by a scalar product.
Example 5.1.2.7 Let n E IN and take a E ReIK,,, . The map
is a scalar product iff u(a) c 10, m[ and e v e y scalar product on IKn is of this form. By Example 2.3.3.6, f is a positive sesquilinear form iff u(a) C Assume that 0 E ~ ( ( 1 )Then . there is an x E IKn\{O) with
R+.
and so
Hence f is not a scalar product. Now assume that 0 $ o ( a ) . Take x E IKn with
(1 . 11 is the Euclidean norm. Then, by Lagrange's Theorem, there is an a E IR with
where
for every i E N n . Thus a E u(a) and
It follows that
5.1 Pre-Hilbert Spaces
for every y E IKn\{O). Hence f is a scalar product. Let g be a scalar product on IKn . Given i , j E K, , put
4, := g(e,, e,) Then
for all x, y E IKn , i.e. g is of the above form.
Example 5.1.2.8 Every infinite-dimensional vector space E admits two scalar products, which generate non-equivalent norms. Let ( x , ) , be ~ ~an algebraic basis of E and f a strictly positive real function on I such that inf f ( ~ =) 0 , LEI
sup f ( ~ =) oc LEI
Then
are two scalar products on E which generate non-equivalent norms.
Proposition 5.1.2.9 ( 0 ) Let E be a vector space and f a positive sesquilinear form on E . Define
b)
F is a vector subspace of E . W e write u : E + E I F for the quotient map.
c) There is a unique map
such that
for every x , y E E
5. Halbert Spaces
18
d ) g is a scalar product, called the scalar product associated to f a) is a consequence of Schwarz's Inequality (Proposition 2.3.3.9). b) follows from a ) and Corollary 2.3.3.4. c) Take X , Y E E I F and x 1 , x 2 X~ , y , , y , ~ Y . T h e n
so that
by a ) . This proves the existence of g . The uniqueness is trivial. d ) is easy to check.
Example 5.1.2.10
The map
is a n injective homomorphism of unital real algebras. Identifyzng G with its image wzth respect to the above map, M becomes a two-dimensional complex vector space and the map
((aP , , y , 6 ) , (a',P', r ' , 6 ' ) )
* ( a + Pi)(crt - P'i) + (Y + 6 i ) ( ~-' b'i)
is a scalar product generating the euclidean n o r m o n M The proof is a straightforward verification.
5.1 Pre-Hilbert Spaces
5.1.3 Hilbert sums
(0)
Proposition 5.1.3.1 and put
O E := ~
LEI
Let ( E L ) r EbeI a family of pre-Hilbert spaces
I I X
E L E~ I
a) @EL is a vector subspace of LEI
, ~L El I l
E
1
xLl12<m
fl E,
.
rEI
b ) The map
(
) ( x
LEI
)
, ( x . Y ) H~
+
(
X
~
I
Y
~
)
LEI
is a scalar product with associated norm @EL + R+
x ++
1
LEI
(x
1
llxLl12)5
LEI
a)E, with this scalar product is called the Hilbert
sum of the family
LEI
(ELILEI.
c) If each
EL ( L
I) is complete then @EL is also complete.
E
LEI
d ) If each EL ( L
I) is separable and if I is countable, then a)E, is also
E
LEI
separable.
a) Take x , y E @EL and a, P E IK . Then LEI
(
(cIx
+
Py!Ll12)
LEI
Hence ax + fly
E
=
(x
lax
+
. Y L 2
LEI
@ E L ,i.e. a)E, is a vector subspace of LEI
LEI
n EL
LEI
5. Hilbert Spaces
b) Take
x ,
y , 2 E @EL and a , /J E IK . By Schwarz's Inequality, LEI
C
5
I(X~IY~)I
LEI
x
I I X ~ II II
Y ~ III
LEI
LE I
LEI
We have
+
= C ( a ( x L 1 z L ) /J(y,lzL))= a x ( x L l z L + ) LEI
LEI
LEI
LEI
x ( x , l x L )= LEI
PC(Y~I~L), LEI
LEI
E1 1 ~ ~ 1 1IR+~ E
I
LEI
and the assertion follows C) Let ( x , ) , , ~ be a Cauchy sequence in
@EL.For every
E
> 0 , there is
LEI
an n, E IN such that
>
for all m, n E N with m 2 n, , n n, . It follows that for every L E I , ( X , ( L ) ) , ~ N is a Cauchy sequence, and hence a convergent sequence. Define
Then
for every
E
> 0 and n
E IN, n
2 n, . Hence
5.1 Pre-Hilbert Spaces
and lim x , = x .
n+m
d) Given
L
E I , let A, be a countable dense set of El with 0 6 A , . Define
A
:=
in
A, ( { L
x 6
E I ( x , # 0) is finite}
'€1
It is easy to see that A is a countable dense set of @EL. LEI
Remark. If EL= IK for every
L
E I , then
Corollary 5.1.3.2 If (EL)LEI is a finite family ofpreeHilbert spaces, then the EL is generated by the scalar product Euclidean n o r m on
n
LEI
If each EL ( L E I ) is a Hilbert space, then
n E, is also a Hilbert space.
LEI
.
Proposition 5.1.3.3 ( 0 ) Let (El),,, , (F,),,, be two families of preHilbert spaces and take ( z L , ) ,E~ I C(EL,FL)wzth
n
LEI
W e define
5. Hzlbert Spaces
which proves a). Since O u , is linear we deduce further that LEI
Take X E I and let
XA E
E,# . Define
Then x E a)El and LEI
so that
Since
XA
is arbitrary,
Since X is also arbitrary,
and
Proposition 5.1.3.4 ( 0 ) Let ( E l ) , E I (FL)lEI, , and ( G L ) l Ebe~ families of pre-Hilbert spaces. Take a , E IK ,
5.1 Pre-Hilbert Spaces
(wL)LEI E
n
C(FL,G L )
LEI
with
sup l l ~ l l l< m, LEI
SUP I I v L I I LEI
<m
7
SUP llwcll LEI
Then
o .L, LEI
These relations are easy to check
= (ow,) LEI
0
(mu.) LEI
< m.
24
5. Halbert Spaces
5.2 Orthogonal Projections of Hilbert Space If F is a closed vector subspace of a Hilbert space, then we can define an orthogonal projection onto F , just as in the case of Euclidean spaces. This projection is used to prove the Frkchet-Riesz Theorem that every continuoris linear form on a Hilbert space can be written as (.lx) , with Il(.lx)ll = llxll . Thus the dual of a Hilbert space may be identified with the Hilbert space itself, whereby the isometry in question is a conjugate linear one.
5.2.1 Projections onto Convex Sets Proposition 5.2.1.1 convex set of E . a) y, 2 E A
* Ily
(0)
- 2112
Let E be a pre-Hilbert space, x E E , and A a
< 2\12 - ~ 1 1 ' + 2115 -
211' -
4d~(~)'
b) Every sequence ( x , ) , ~i n~ A with lim JIx - x,)) = dA(x)
n+m
is a Cauchy sequence. If it converges,then
a) i(y
+ 2)
belongs to A . By the parallelogram law (Proposition 2.3.3.2
b)),
5 2llx - y1I2 + 2112 - z\I2- 4 d ~ ( x ) ' b) follows from a ) .
Theorem 5.2.1.2 ( 0 ) (0.Nikodyrn, 1931; F. Riesz, 1934) Let E be a pre-Hilbert space. Take x E E and let A be a nonempty convex set of E . If
25
5.2 Orthogonal Projections of Hilbert Space
A is conlplete with respect to the induced metric, then there is a unique y
E
A
such that
y is characterized by the property z EA
r e ( x - yly - z ) 1 0
W e define
Step 1
Uniqueness
Take y, z E A with 112
- yll = llx - zll = d a ( x ) .
B y Proposition 5.2.1.1 a ) , Ily
-
z1I2
< 2112 - yll + 2112 - zll
-
4da(z) = 0
Hence
Step 2
Existence
Let ( x , ) , , ~ b e a sequence in A with lim Ilx, - xi1 = d ~ ( x ) .
n+m
By Proposition 5.2.1.1 b ) , ( x , ) , ~is~ a Cauchy sequence in A . Since A is ~ to some y E A . By Proposition 5.2.1.1 b ) , complete, ( x , ) , ~converges
Step 3
y, z E A , Ilx - yll = d A ( x ) + re(x - yly - z )
+ ( 1 - o ) y E A and so; by Corollary 2.3.3.4, YI12 = d ~ ( x0
5. Hilbert Spaces
Since a is arbitrary, re(x - yly - 2 ) Step 4
y E A,
(2
E
A
>0
+ re(x - yly - 2 ) > 0 ) 3 113: - yll
=~ A ( x )
By Corollary 2.3.3.4,
I"
for every
2
- 2112 = II(z
-
Y) + (Y - 2)112 =
E A . Hence -
yll = ~
A ( x )
Remark. The above theorem does not hold for ge,~eralBanach spaces. Let A be the closed convex hull of {%en In E I N ) in t " . Then d A ( 0 )= 1 but
11x11 > 1 for every x 6 A .
Proposition 5.2.1.3 ( 0 ) Let E be a pre-Hilbert space and A a nonempty convex set of E which is complete with respect to the induced metric.
b)
KA
is uniformly contznuous.
a ) By Theorem 5.2.1.2,
5.2 Orthogonal Projections of Hzlbert Space
re(y - ~
> 0,
IKA(Y)
-~
I~A(x)
-~ A ( Y ) )
A ( Y )
A ( X ) )
It follows t h a t ~ ~ ( T A (Y )Y
2 0,
and so
b) follows from a ) . C) T A ( X )
E A,
so that K A o K A ( x ) = T A ( K A ( 5 ) )= K A ( x )
for every x E E . Hence 71.4 0 T A
=7 r ~ .
Proposition 5.2.1.4 Let E be a pre-Hilbert space, Q a nonempty downward (upward) directed set of nonempty convex sets of E which are complete i n the induced metric, 5 the lower (upper) section filter of Q , and put
If B is nonempty and complete i n the induced metric, then lim r A( x ) = K B ( x ) A,$
for every x E E .
5. Hilbert Spaces
We first remark that B is convex. Consider
(p:%--+E,
A++TA(X).
We have llv(A) - cp(~')Il' = IITA(x)- ~ A ~ ( x ) III '
for all A, A' E 2l with A' c A (Proposition 5.2.1.1 a)). Sirice (dA(~))AEll is an increasing bounded (decreasing) family in R + ,it follows that p(3) is a Cauchy filter. Hence l i r n a ~ ( x )exists and belongs to B . Now A,3
~ llux11'
5.2 Orthogonal Projections of Hilbert Space
e from
+ f.
Assume that u
# 0 . Then F # ( 0 ) . Take x
I I X I I=
E F\{O). It follows
11~x11 I I I U I I11x11,
that
u(x-ux+(Yy)=ux-u2x+(Yuy=(Yy,
so that J C X ) ~ ) )= ~ ) ll0yJ1~ ) ~ =
JIu(x - ux
+ ( Y Y ) ) 0 .
-
Since /3 is arbitrary, we deduce that (x - uxlv) = 0 , and so
UX
(Proposition 5.2.3.1 b
3
= KFX
a). Hence u = a~ .
I
(YY))~
5. Halbert Spaces
38
Corollary 5.2.3.6 ( 0 ) Let E be a pre- Hilbert space and F, G, H complete subspaces of E . If XFXC is a projection of E onto H , then
By Corollary 5.2.3.2,
and the assertion follows now from Proposition 5.2.3.5 e
Proposition 5.2.3.7 subspace of E , then
(0)
+ a.
H
Let E be a pre-Hilbert space. If F is a complete
and the map
is an isometry. If E is also complete then
By Corollary 5.2.3.2, 5.2.3.5 a + d ,
KF
is a projection of E onto F and by Proposition
Ker T,V = F' By Murray's Theorem (Theorem 1.2.5.8 b
+ a),
E = F ~ F ~ . By Pythagoras' Theorem, the map FQF'+E, is an isometry. \?ie have F c F" (y, z ) E F x F L with
(X,Y)MX+Y
(Proposition 5.2.2.2 b)). Take x E F L L . There is a pair
z=y+z.
From
5.2 Orthogonal Projections of Hilbert Space
(Proposition 5.2.2.2 a)) we deduce that z = 0 . (Proposition 5.2.2.2 e)), and
Hence F L L c F , F" = F . By Murray's Theorem (Theorem 1.2.5.8) and the above considerations, l E - T,V is a projection of E onto F' with Ker ( I E - xF) = F Thus, by Proposition 5.2.3.5 d
=F ~ L
+a,
Hence 1~ = TF + K P L
Corollary 5.2.3.8 ( 0 a complemented subspace.
)
Every closed vector subspace of a Hilbert space is
Corollary 5.2.3.9 ( 0 ) Let E be a pre-Hilbert space and A a subset of E . If E is complete or if A is finite, then -4'' is the closed vector subspace of E generated by A . Let F be the closed vector subspace of E generated by A . Since A L L is a closed vector subspacc of E containing A (Proposition 5.2.2.2 a),b)), F cA~L. If E is complete, then F is complete as well. If A is finite, then F is finitedimensional and therefore complete. Take x E A L L . By Proposition 5.2.3.7, there is a pair ( y , z ) E F x F L with
By the above considerations, y E A". deduce further
By Proposition 5.2.2.2 a),b),e), we
5. Hilbert Spaces
Hence
Remark. The hypothesis " E complete or A finite" cannot be relinquished, as the following example shows. Put
and let E denote the vector subspace of C2 generated by A U {x} . Then A L = (0) , A L L = E , and x does not belong to the closed vector subspace of E generated by A . Corollary 5.2.3.10 ( 0 ) Let E be a Hilbert space and F a vector subspace of E . T h e n the following are equivalent: a) F = E .
From FLL=
F
(Corollary 5.2.3.9) we deduce that
E=FICBF (Proposition 5.2.3.7) and the assertion now follows. Definition 5.2.3.11 ( 0 ) Let E be a Hilbert space and take u E L ( E ) . A closed vector subspace F of E is said to reduce u if F and F L are U znvariant. Corollary 5.2.3.12 ( 0 ) Let E be a Hilbert space, F a closed vector subspace of E , and take u E L ( E ) .
5.2 Orthogonal P~ojectzonsof Hzlbert Space
a ) F is u-invariant iff
b) F reduces u iff
a) is easy to see. b) If F reduces u , then by a ) and Proposition 5.2.3.7,
Suppose that RFU
= UXF .
Take x E F and y E F' . Then
Ill;uy = u7rf.y = 0 (Proposition 5.2.3.5 a =+ d) , Uy E KernF = F' (Proposition 5.2.3.5 a =+ d ) , so that u(F)c F,
u ( F L ) C F1
Thus F reduces u . Proposition 5.2.3.13 ( 0 ) Let F, G be closed vector subspaces of the Hilbert space E . Then the following are equivalent:
5. Hilbert Spaces
42
f)
KG
- KF = 7 T c " p l .
g)
KG
- AF is a projection in E .
a @ b is trivial. a&b+c.Byb),
i.e. KFAG is a projection in E . By a), it is a projection onto F and so the assertion follows from Corollary 5.2.3.6. c + d . By Corollary 5.2.3.5 a + e ,
d
@
e . By Proposition 5.2.3.1,
= (ACXIX). I I ~ F X =~ ~( ~~ F x ~ x )llrcxll2 ,
d
=+
a . Take x E F . By Proposition 5.2.3.1,
1 1 ~ 1 =1 ~ l l ~ ~ xIl IlI ~~ G x I I ~ = 1 1 ~ 1 -1 ~ 115 - ~cx1I25 1 1 ~ 1 1 ~ Thus
and
FCC. a & b & c & e + f . Wehave (KG
i.e.
KG - K F
= K$
-
- AGKF
- AFKG
+
A;
= AG
- KF
is a projection in E . Take x E E . Then X - A G X E G ~ C
(Proposition 5.2.3.1 a
3
F ~ X , -AFXEFI
b). Hence
X G X - KFX
= (x
(Proposition 5.2.2.2 a)). Thus
-
rFx) -
(5 - KGX)
E
F'
,
5.2 Orthogonal Projections of Hilbert Space
Im(Tc-TF)
cGnF1
Since
whenever x E G n F L (Corollary 5.2.3.5 a =. d), it follows that projection of E onto G n F L . By e),
for every x E E . By Corollary 5.2.3.5 b
TG - x p
is a
+a,
f + g is trivial. g + a . Sirice
it follows that 1 T ~ T+ G r G n F ). 2
TF = - (
Take x E F . Then
IIxII
=
1
+ TGTFXI I
I I K ~ ~ l =l -I~TFTCX 2 I
(Corollary 5.2.3.5 a
3
1
1
I -2 (IITFRGXII+ IIT~XII)I
(II~TG~II + II~cxII)= ll~GxI1
e). Thus
(Proposition 5.2.3.1)
FcG. Proposition 5.2.3.14 ( 0 ) Let F, G be closed vector subspaces of a Hilbert space E and H := F n G . T h e n the following are equivalent:
44
5. Hzlbert Spaces
C)
7iGTF
= TH.
d ) (Fn H L ) 1 ( G n H L ) .
e) ( F n H L ) 1 G .
f) F = H @ ( F n G L ) . If these conditzons hold, then F + G is closed and
a
+b
b
3
TCTF
is trivial. c . Since
is a projection in E . Since X E
E===+K~T~X=KF ET FG nT G F= XH , XE
TGTF
H==+T~T~x=x,
is a projection onto H . By Corollary 5.2.3.6, TGTF
c
+d.
Take
x E
(Proposition 3.2.3.5 a
F n H I . Then
+ d), so that X E
(Proposition 5.2.3.5 a
= TH
KerrccGL c (GnHL)L
+ c). Hence (Fn H L ) 1 ( Gn H I ) .
d
+ e.
Take x E G . There is a (y, z ) E H x H L with x=y+z
(Proposition 5.2.3.7). By d ) ,
5.2 Orthogonal Projections of Halbert Space
Z = X - ~ E G ~ H ' C ( F ~ H ~ ) ~ . From F n H L c HL it follows that H C H L L c( F ~ H ' ) ' , x = ~ + z E ( F ~ H ~ ) ' (Proposition 5.2.2.2 a),b),c)) and so (Fn H L ) 1 G . e + f . Bye): FnH'c
F~G'.
Since
G' c
HI,
it follows that FnHL=FnGL. Take x E F . There is a unique (y, z ) E H x H L with x=y+z (Proposition 5.2.3.7). Now Z = X - Y E
F~H'=F~G',
so that
(Corollary 1.4.2.6). f + d . Take x E F n H I . By f), there is a pair (y,z) E H x (F n GL) with
46
5. Hilbert Spaces
From
it follows that y = 0 and
i.e.
( F n H1)1(GnH I ) . f
3
a . Take x E E . By f ) , there is a pair ( y , z ) E H x ( F n G L ) with T F X = Y + Z .
Then KGTFX
=Y=THTFX =THX
(Proposition 5.2.3.5 a =. d, Proposition 5.2.3.13 a TGTF = KH
+ c ) , i.e.
.
By d o f and by the symmetry of d), we deduce from the above equality that
TFTG
=T
~ K F .
We now prove the final assertion. Put := K F + K G - K f j .
U
By a),c), and Proposition 5.2.3.13 a 3 b & c, UTF =TF
,
UKc
= TG
,
UTH
= Tfj
.
Thus u2=u,
i.e. u is a projection in E . Take x r F
+ G . There is a
( y , z ) E F x G with
5.2 Orthogonal Projections of Halbert Space
so that
since the reverse inclusion is obvious, u is a projection of E onto F + G . Hence F + G is closed. Given x E E ,
(Proposition 5.2.3.13 a
+ e), so 71
(Proposition 5.2.3.3 b
= TF+G
+ a).
Corollary 5.2.3.15 ( 0 ) Let F , G be closed vector subspaces of the Hilbert space E . Then the following ale equivalent: a) F I G
b)
7 r F 7 r ~=
c) F
+G
0.
is closed and
a e b follows from Proposition 5.2.3.5 a b + c . Clearly, T ~ K F T G=
By Proposition 5.2.3.14, F
+d
0 =7l~Tc.
+ G is closed and
5. Hilbert Spaces
Thus
(Proposition 5.2.3.5 a
+ c) and FIG.
Corollary 5.2.3.16 Let space and
K
F be afinite set of orthogonal projections i n a Hilbert
is an orthogonal projectzon iff
whenever u , v E 3 are dzstinct. In this case,
The sufficiency of the condition and the last assertion follow from Corollary 5.2.3.15 b + c by complete induction. We now assume that K is an orthogonal projection. Let u , v be two distinct elements of 3 and x E 11nu . Then
1 1 ~ 1 >1 ~ (KxIx) = ~
>
+ 1 1 " J ~ 1 1 ~= 1 1 ~ 1 +1 ~ l l v ~ 1 1 ~
l l ~ X 1 1 ~ llu~11~
( W X I X= )
we3
we3
(Proposition 5.2.3.1), so
It follows vu=o. Proposition 5.2.3.17 Let F be a closed vector subspace of the Hilbert space E and take u E C ( E ). T h e n the following are equivalent:
a ) F L c K e r v and I m u c F .
5.2 Orthogonal Projections of Hilbert Space
a
+ b . It follows immediately from Imuc F,
that 7 r ~ U= 21.
Take x E E . Then
x - 7rpx E F J (Proposition 5.2.3.1 a
b
+a
* b), so that
is obvious.
Proposition 5.2.3.18 Let E be a Hilbert space, F a vector subspace of E , and G a Banach space. Take u L(F, G ) . T h e n there is a v E L ( E : G) such that
Since u is uniformly continuous, we may assume that F is closed. By Corollary 5.2.3.2, the map
has the required properties. Proposition 5.2.3.19 Let
(0)
0 := inf LEI
and take
< E t 2 ( I ) such that
Let ( x , ) , ~ , be a family i n the Hilbert space E .
llxlll,
P := sup llxLll< 00, LEI
5. Hzlbert Spaces
50
for distinct i n E and
1,
XE
I .
Then, given 77 E e 2 ( I ) , the family
Let J be a finite subset of
I .
( ~ ( L ) X , ) , , is ~
summable
Then
It follows t h a t
112v(l)xL1l'x 5
LEJ
Hence
( ~ ( L ) X , ) is , ~ summable ~
+
x
5
~ ~ ( 1 ) ~ 2 ~ ~ l x( ~~ ( ~ ) ~x t2 I v ( ~ ) ~ A ) l llall ,
then
for every
3:
E E
Tho uniqueness follows from the fact that { X ~ LE I)" is the closed vector subspace of E generated by { x , I L E I ) (Corollary 5.2.3.9). Sow we prove the existence and the last assertion. By Proposition 5.2.3.19, the map
are well-defined and
Hence I m v is the closed vector subspace of E generated by { x , I L E I ) (Proposition 1.2.1.18 c)). Define
52
5. Hilbert Spaces
Then, by Proposition 1.2.1.18 b),
IIuxI12 = ll~v-'nIrnvx11~ I (rC2 + ~ ~ v J ~ ~ ) I I ~ -5~ ~ I T T ~ ~ x I ~ ~
for every x E E , so that
X E
If {x, 1 L E I)' E,
= (0)
, then Im v = E and by Proposition 1.2.1.18 b), given
Proposition 5.2.3.21 ( 0 ) Let E , F be infinite-dimensional Hilbert spaces. Take u E L ( E , F ) , and let u be the equivalence class of u i n L ( E , F ) / K ( E , F) . Then there is a sequence ( x , ) ~ , i~n E such that
for all m, n E IN and
llull =
II~xnll.
We construct the sequence ( x , ) , ~ inductively ~ such that for every n E IN
Take n E IIV and assume the sequence has been constructed up to n - 1 . Let G be the vector subspace of E generated by {x, ( m E INn-,) . Then u 0 n c E K ( E , F) , so that
Hence there is an x
E
E# such that
llull
-
, 1
< Il(u - u O .rrc)xll
5.2 Orthogonal Projections of Hilbert Space
By Proposition 5.2.3.1, r c l x E E# and
Hence
I uIG'I
> IIuT~LxII
=
I~UX
- ~ r c x l l=
II(u
1
.
- U 0 ~c)x11> llull
-
-n.
Then there is an xn E G I with
Proposition 5.2.3.22 ( 0 ) Let E be a Hilbert space and finite-dimensional vector subspaces of E . Then
5
the set of
for everg u E C ( E ). By Proposition 5.2.3.5 a
+ e,
r
sup I I ~ F U K F I I I I ~ I I . FE3
Take cu E ]0,11ull[. There is an x E E# such that
11ux11>0 . Let F be the vector subspace of E generated by { x ,u x ) . Then F E 5 and
I I ~ F ~ ~ F >I I ( I ~ F u ~ F x ~ =~ 1 1 ~ ~ ~ = x 1llu~ll 1 > 0. Since cu is arbitrary, it follows
54
5. Hilbert Spaces
5.2.4 Mean Ergodic Theorems
The main ideas of these results originate in a paper of J. von U I eumann.
Let E be a n unital Banach algebra. Take x E E with ( X " ) , ~ N bounded. Let A be the convex hull of {xn-' n E IN) i n E and let ( a n p ) n , p Ebe~ a famzly i n { x ) ' , such that is absolutely summable for every n E IN and
Proposition 5.2.4.1
Then for every n E I N , (anpxP-')pENis absolutely summable and for every V EA
,
Put
Then
Thus (anpxP-')p,, is absolutely summable for every n E IN Given n E I N , put
Then
5.2 Orthogonal Projections of Hzlbert Space
for every
7~
E
N and s o lim (x - l)x, = 0
n+w
Since
lim (xP- l)xn = 0
n+w
for any p E IN. There is a finite family
in IR+ with I C N ,
I t frdlows t h a t
for every n E IN. Hcncc lirn (y - l)x, = 0
n+cc
Lemma 5.2.4.2
There is a y €
for every n E N and t E For every p E N, put
Then
[&,$1
n+ with
5. Hilbert Spaces
56
p>tn+l*crp
n+l- 1, 1 t 1 ($''(I - t)" n+ 1
'
such t h a t Ikn - tn1 < 1 and
is an increasing function for p 5 kn and a decreasing function for p Hence this function takes its supremum a t k := kn E I N n - , , such t h a t
> k,.
By the Stirling inequalities (for n > 1 ),
=2
(;)
t k ( l - t1n-k =
2n!tk(l- t)n-k k !- k)!
&nneheken-ktk(~
-
t)n-k
-
- k)n-k
2 e n m k k J-(n
where
for every n E IN\{l). Given n E N \ { l ) , 2eh k + l hn5z(T)k+'(-)
n-k+l n-k
n-k+d
,
so that y := sup 6 < ntN
-
2eh
-s u p
6n E N
<m.
.
5.2 Orthogonal Projections of Halbert Space
Proposition 5.2.4.3
Let E be a Hilbert space. Take u E C ( E ) # . Put
F and let
be a family i n IR,
((Y,,),,,~N
for every n E
:= Ker
(1 - u )
, such that
W and
Then
and
for every x E E . In particular, 1
"
lirn - ):u p - ' x = R F X n+m n p=l lim ( t n l
n+m
+ (1
(Mean Ergodic Theorem),
-
t , ) ~ ) ~=xT F X
for every x E E and every sequence (tn)nEm i n ]0,1[ wath lim n t n ( l - t,) = GO.
n+m
We have
for every n E W . In particular, (anpuP-l)pENis absolutely surnmable and
Take z E E . Let A be the convex hull of {up-'x ( p E N ) and take y E A . There is a finite family (Pp)pEIin IR, such that I c N ,
5. Hilbert Spaces
Put
v belongs to the convex hull of
=
x
x
I p E Ih')
{up-'
Q ~ ~ u -~ - ~a x n
p ~ p=- l) ~:QnpUp-' ~ (1
PC r-4
PEN
and
[PEN
)
- V)X
for every n E IN . By Proposition 5.2.4.1,
Given n E N ,
Since
it follows
F ) . Then u* and u* o u are Fredholm operators and Dim Ker U * = Dim Coker u , Ind u* = -1nd u
Dirt1 Ker u = Dirn Coker u'
,
, Ind (u o u * ) = 0 ,
If u has index 0 , then Dim Ker u = Dim Ker U * = D i ~ nCoker u = Dim Coker u' < co In particular if E = F then ue(u*)=
la E ue(u)}.
5. Hilbert Spaces
Since Dim Ker u* = Dim Ker u' = Dim Cokcr u (Theorem 5.3.1.4, Proposition 3.1.3.4), we deduce that Dim Ker u = Dim Ker u*' = Dim Coker u* (Proposition 5.3.1.5 a)). In particular, u* is Fredholm and Ind u* = -1nd u .
By Proposition 3.1.3.7, u o u* is Fredholm and Ind(u o u') = 0 . Since I m u and Im (u o u*) are closed, Im u = Im ( u o u*) = (Ker u')' by Proposition 5.3.2.4. The assertion for Ind u = 0 follows. In order to prove the final assertion take a E IK. By Proposition 3.1.3.25 a),
Corollary 5.3.2.7 Let E be a Hilbert space. Take u E K ( E ) and a E IK . If E i s finite-dzmensional or if a # 0 , then Dim Ker ( a 1 - U) = Dim Ker (Gl - u*) =
= Dim Coker ( a 1 - u) = Dim Coker (81 - u*)
Im ( a 1 - U) = (Ker (51 - u*))' ,
< cc ,
5.3 Adjoint Operators
By the Corollaries 3.1.3.12 and 3.1.3.13,
a1 - u
E
3 ( E ) , I n d ( a 1 - u) = 0 .
By Proposition 5.3.2.6,
Dim Ker ( a 1 - u) = Dim Ker(G1 - u*) = = Dim Coker ( a 1 - u) = Dim Coker(a1 - u*) < oo ,
Im(a1 - u )
=
(Ker ( ~ - lu*))'
It follows that a E up(u) e CY E u p ( u * ) . Proposition 5.3.2.8 ( 0 ) Let u be a projection i n the Hilbert space E . Then the following are equivalent: a)
u is a n orthogonal projection (in the sense of Corollary 5.2.3.2).
b) u is selfadjoint, i.e. u is a n orthogonal projection of the C*-algebra C(E) (Theorem 5.3.1.13 and Definition 4.1.2.18). c) u is positive.
a a b . Take x , y ~ E . T h e n (uxly-uy) = o = (ux -x1uy) (Proposition 5.2.3.1 a (uxly) = (uxly - uy)
+ b), so that
+ (uxIuy) = (uxIuy) = (ux
i.e. u is selfadjoint (Theorem 5.3.1.4) b + a . Since
-
xluy)
+ (xluy) = ( ~ I U Y ) ,
92
5. Hilbert Spaces
By Proposition 5.2.3.5 e + a , u is an orthogonal projection. b 3 c follows from Proposition 4.2.1.22. c 3 d is trivial. d 3 a . Xow
i.e. u'u is a projection. Since u'u is selfadjoint, we deduce from b is an orthogonal projection. Hence
* a , that
U*U
llu1I2 =
l l ~ * ~ lIl 1
(Theorem 5.3.1.4, Proposition 5.2.3.5 a
* e), so that
.
llull 5 1 and u is an orthogonal projection (Proposition 5.2.3.5 e
+ a).
Proposition 5.3.2.9 ( 0 ) Let F be a closed vector subspace of the Hilbert space E . Take u E L ( E ) . T h e n F is u-invariant iff F L is u* invariant. In particular, F reduces u iff F is u-invarzant and u* -invariant. In this case,
for every x i F , where
If 3 is a n involutzve set of C ( H ) , then F is 3 - i n v a n a n t iff
T.D
E
3 '
The first assertion follows from x, Y E E
* (uxly) = ( x l u * ~ )
.
and F = F1' (Corollary 5.2.3.9). Then second assertion follows from
The final assertion follows from the first one and Corollary 5.2.3.12 b).
Let E be a Hzlbert space. Take u E L ( E ) . Let (F,),,I be a famzly of closed vector subspaces of E such that each F, ( L E I ) reduces u . Then both F, and the closed vector s~ibspaceof E generated by U F, Corollary 5.3.2.10
reduce u .
LEI
LEI
¤
5.3 Adjoint Operators
99
Corollary 5.3.2.11 Let E be a Hilbert space, u a normal (selfadjoint, unitary) operator on E , F a closed vector subspace of E which reduces u and
W
Then v is normal (selfadjoint, unitary).
Definition 5.3.2.12 ( 0 ) Let E be a pre-Hilbert space and F a vector space. Given ( x ,y) E E x F , define
I f F is normed, then (.lx)y E L ( E ,F ) and II(.Ix)yII = llxll llyll for every ( x ,y )
E
E x F.
Proposition 5.3.2.13
(0)
Let E , F, G be Hilbert spaces.
a ) The map
is sesquzlinear. b) ( x ,Y ) E E x F C)
* ((.lx)y)' = ( . I y ) x .
( x ,y) E E x F , v
E
C ( F ,G ) 3 v
o
( ( . l x ) y )= ( . l x ) u y .
d ) u E L ( E , F ) , ( y ,2 ) E F x G 3 ( ( . I Y ) z 0) u = (.lu*y)z e)
(X,YI)EEXF,(YL,Z)EFXG* 3
f)
If
((.l742)2)0 ( ( . I x ) Y I= ) (Y~IY~)(.Ix)z.
( X I , yl), ( x 2 ,y
then
where
~E)
E x F such that
5. Halbert Spaces
a ) is easy to see. b) Given (a, b) E E x F , (((.lx)y)alb) = ((alx)ylb) = (alx)(ylb) = (al(bly)x) = (al((.ly)x)b) . c) Given a E E ,
u
0
((.Ix)y)a = u(a1x)y = (a1x)uy = ((.[x)uy)a.
d ) Given a E E ,
e) By c) and a) ( ( ' 1 ~ 2 )O~ (('Ix)YI) ) = ('Ix)((YIIY~)z) = (YI~Y~)(.~x)z. f) We have
and
It follows
Remark. If we put
for every n E N (Example 5.1.2.3),then (x;x,),,~ (x,x;),~N is not. Indeed, by b) and e),
is sumrnable in C(e2) , but
5.3 Adjoint Operators
for every n E N . Take A E !Q,(lr\J). For
E E e2,
(Pythagoras' Theorem) and
so that
By Proposition 1.1.6.6, ( X ; X , ) , ~ ~is summable and (X,X:),~~ is not sumrnable.
Corollary 5.3.2.14
(0)
If E
zs
a Hilbert space, then
Take u E {(.lx)x 1 x E
ElC.
Define
@N if x # O f:E+K,
x-
11~112
O
if x = O .
If x E E , then by Proposition 5.3.2.13 c),d),f),
It follows
= (f(x+y))(x+y) = f ( x + y ) x + f ( x + ~ ) ~ 7
f(x) = f ( x + y ) = f
(~)
for all x, y E E . Hcncc f is constant and so u E E ~ E .
5. Halbert Spaces
96
Proposition 5.3.2.15 ( 0 ) Let E be a Hilbert space, F a Banach space, @ a71 upward directed set of closed subspaces of E the union of which is dense i n E , and 5 the upper section filter of 6 . If u E K ( E , F ) , then
Let v be the isometry of real Banach spaces
(Corollary 5.2.5.3). We have u' E K ( F t ,E ' ) (Theorem 3.1.1.22 a is compact (Proposition 3.1.1.1 I ) ,
(Corollary 1.3.4.5, Theorem 5.3.1.4, Proposition 5.3.2.8 a
+ b ) , v-'
out
+ b ) , and
for every G E 6 (Theorem 1.3.4.2 t))). By Proposition 5.2.3.3, lim IIu c,3
0
KG -
ull = lim llrGo (v-I G.3
0
u t ) - (v-I
0
ul)ll = 0 ,
Corollary 5.3.2.16 ( 0 ) Let E be a Hilbert space, 5 art upward directed set of finite-dimenszonal vector subspaces of E , the union of which is dense i n E , 6 the upper section filter of 5 , and define
Then ~ ( 6 zs) an approximate unzt of K ( E ) (Proposition 5.3.1.17). The assertion follows immediately from Propositions 5.2.3.3 arid 5.3.2.15 (and Corollary 5.2.3.2).
Corollary 5.3.2.17 ( 0 ) Let E, F be Hilbert spaces, E (resp. 5 ) an upward directed set offinite-dzmensional vector subspaces of E (resp. F ) the union of which is dense i n E (resp. i n F ) , and take u E K ( E , F ) . Then for every E > 0 , there are Eo E E and Fo E 5 such that llrHo u o KG - u I I < E
forall G E E and HE^ with Eo c G , I;bc H
5.3 Adjoint Operators
By Corollary 5.3.2.15, there are Eo E @ and Fo E
5 such t h a t
for all G E @ and H E 5 with Eo c G , F o c H . Take G E @ and H E 5 with Eo C G , Fo C H . Then (Theorem 5.3.1.4, Proposition 5.3.1.7, Corollary 5.2.3.2)
< ~ ~ ~ * - u * o n , , ~ ~ + ~< ~-2+u- =o2E n. ~ - u ~ ~ E
E
Example 5.3.2.18 If u,, up denote the right an left shift of and q denotes the quotient map
e 2 , respectively,
then qu, , qup are unit(l,ry and (qur)' = qur, but there is no v E L ( t 2 ) / K ( t 2 )with
By the last assertion of Proposition 5.3.2.2, q is a homomorphism of involutive unital algebras (Proposition 5.3.1.17) and by Example 5.3.1.6,
By Example 3.1.2.16,
Hence qu, and qup arc unitary. Assume there is a v E L ( t 2 ) / K ( e 2 )with ev = qur
Take w E C(e2) with
98
5. Hilbert Spaces
Then
Hence, by Proposition 3.1.3.21 c), Ind u ,
=
Ind ew = 0
since eW is invertible. This is a contradiction, since Indu,
=
¤
-1.
Definition 5.3.2.19 ( 0 ) Given the Hilbert space E , take x E E and 3 C L ( E ) . T h e n 3 is sazd to act irreducibly on E if 3\{0) # 0 and if 0 and E are the only 3-invariant closed vector subspaces of E . 3 zs said to act non-degenemtely on E if
x is called cyclic for 3 , if
x is called separating for 3 if
It is obvious that if there is a cyclic element for degenerately on E .
Proposition 5.3.2.20 a ) If every x E E\{O)
(0)
F ,then 3 acts non-
Let E be a Hzlbert space and take 3 C L ( E ) .
is cyclic for 3 , then 3 acts irreducibly o n E .
b) If 3 is a subalgebra of L ( E ) acting irreducibly o n E , then every
x E E \{O) is cyclic for 3 . If p is a n orthogonal projection of E belongzng to 3 such that p 3 p is one-dimensional, then there is a n x E E such that
5.3 Adjoint Operators
99
a) It is obvious that 3 \ { O ) # 0. Let F be an 3-invariant closed vector subspace of E . Assume that F # (0) and take x E F\{O). By Corollary 5.2.3.9,
E = {ux 1 u
E 3)"
c F"
=F
Hence F = E and 3 acts irreducibly on E . b) We put
Then F is an F-invariant vector subspace of E and so F is also 3-invariant. Assume F = {O). Then IKx is 3-invariant, so that IKx = E . Take u E F \ { O ) . Then ux # 0 and this contradicts the assumption F = ( 0 ) . Hence F = E . By Corollary 5.2.3.9,
x is thus cyclic for 3. Assume Im p is not one-dimensional. Then there are x,y E I m p such that
Since x is cyclic for 3 , there is a u E 3 such that
Since p 3 p is one-dimensional, there is an cu E IK such that pup = a p . Thus
which is a contradiction. Hence I m p is one-dimensional. Take x E I m p with 11x11 = 1 . Then Ker p = (Imp)' = Ker (.lx)x (Proposition 5.2.3.5 a
+ d ) and
Hence p = (.Ix)x.
5. Halbert Spaces
100
Proposition 5.3.2.21 ( 0 ) Let E be a Hilbert space and take 3 c C ( E ). Then the following are equivalent:
a ) 3 acts non-degenerately on E b ) Given x E E\{O} , there is a u E 3 with u'x # 0 . If these conditions are fulfilled, 3 is a subalgebra of L ( H ) , and 5 is an approximate unit of 3 , then limux = x u,3
for every z E E
a
+ b.
.
Assume that
u w x= 0 for every u E 3 . Then
(xluy) = (u*xIy)= 0 for every
(11,
y) E 3 x E . Hence x E {uy I ( u ,Y ) E 3 x E l i = { O ) , x=o.
b
+ a.
Take
x E
{UY
I ( u ,y) E 3 x
ElL.
Then
11u*x11' = ( ? ~ ' X ~ U * X=) (xIuu*x)= 0 for every u 6 3 ,so that x = 0 . Thus {UY
I ( u ,Y ) E 3 x El'
1
{ I L ~ (u,y )
E
T x E)"
= (01,
=E
5.3 Adjoint Operators
Hence 3 acts non~-deneratelyon E . We nowr prove the last assertion. Given
(11,
x) E 3x E ,
lirn u u x = u x u,3
Hence if F is the vector s~ibspaceof E , generated by { v x 1 ( v ,x ) E 3 x E ) , then lirn u x = x u,3
for every x E F . By a ) and Corollary 5.2.3.9, F is dense in E . Hence lirn u x = x u,3
for every x E E
Proposition 5.3.2.22
(0)
Let E be a Hilbert space and take 3 C L ( E ) .
a) If x is cyclic for 3 , then x separates 3" b) If x separates P r F and 3 is a quaszunital involutive subtr1,qebra of L ( E ) acting non-degenerately o n E , then x is cyclic for 3 . a) Take u E
FCwith
u v x = vux = 0 for every v E F .Hence if F denotes the vector subspace of E generated by { v x I u E 3 1 , then
F c Keru. By Corollary 5.2.3.9,
Hence x separates 3' b) Define
5. Halbert Spaces
102
F is the closed vector space generated by {ux 1 u E 3 ) (Corollary 5.2.3.9).It is therefore 3-invariant. By Corollary 5.3.2.9, T F E 3 ' . Let 5 be an approximate unit of 3 . By Proposition 5.3.2.21,
Hence
(1
-
rF)x = 0.
Since 1 - K F E Pr 3' and x separates Pr F C , it follows that
F=E Hence x is cyclic for 3 . Proposition 5.3.2.23 ( 0 ) Let E be a separable Hilbert space and 3 a commutative involutive subalgebra of L ( E ) . If 3 acts non-degenerately o n E , then 3 has a separatzng vector. Let '2 be the set of subsets A of E such that
for every x E
A and
for distinct x ,y E A . By Zorn's Lemma, ZI contains a maximal element A . Take y E { U X ~ ( U , XE ) 3 x A)'
Take u,,v E 3 and x E A . Then
5.3 Adjoint Operators
(ux Ivy) = (v'ux I y) = 0 Hence
Since A is maximal in U,
Hence
= U ( ~ e r u ) ' c {uxI(u,x) E 3 x
A)"
~ € 3
(Proposition 5.3.2.4),
Since E is separable, A is countable. Let cp : A put
+ IN
be an injective map and
Take v E 3 with
Since the elements of the fanlily ( V X ) , , ~are pairwisc orthogonal,
for every x E A (Pythagoras' Theorem). Hence vux = UVX = 0 for every (u, x) E 3 x A . It follows that
(Corollary 3.2.3.9). Thus y separates 3 .
5. Hilbert Spaces
104
Proposition 5.3.2.24
(0)
Let E be a Hilbert space, F a closed vector subspace o j E , and 3 a subalgebra of L ( E ) acting irreducibly or1 E . Given ~ € 3 define ,
T h e n {ii I u E 3 ) acts irreducibly o n F Take x , y E F with x # 0 . By Proposition 5.3.2.20 b), x is cyclic. Hence ~ that ~ there is a sequence ( u ~ in ) 3~such lim u,x = y
n+m
(Corollary 5.2.3.9). Thus lim ZLnx = y .
n+cc
Hence x is cyclic for {ii 1 u E 3). By Proposition 5.3.2.20 a), {ii 1 u E 3) acts irreducibly or1 F . Definition 5.3.2.25 Let E , F be Hilbert spaces. A n operator u called a partial isometry if
1121x11=
:
E
-t
F zs
I I X I I.
for all x E (Ker u)' . Proposition 5.3.2.26 Let E, F be Halbert spaces and take u E L ( E , F ) . T h e n the jollowzng are equivalent:
a ) u is a partial isometry.
b)
U*
zs a partial isometry.
c) u' o u zs a n orthogonal projection
d ) u o u* is a n orthogonal projection. If these conditzons are fulfilled, then (Ker u)' = Irn u* ,
(Ker u')'
= Im u
,
and u* 0 u (resp. u 0 v* ) is the orthogonal projection of E on Im u' (of F on Im u ).
5.3 Adjoint Operators
a
+ b & c, and the last assertion. By the hypothesis, the map
is an isometry. Hence I m u is closed and by Proposition 5.3.2.4, Im u = (Ker u')'
.
Take x, y E (Ker u)' . Then (ux I UY)= ( ~ I Y ) (Corollary 5.1.1.8 a
b), so that
Hence the map Im IL + (Ker u ) ~ , z
++ U*Z
is the inverse of u . It follows that u' is also a partial isometry and therefore Imu* = ( ~ eu)' r Moreover, u* o u (resp. u o U* ) is the orthogonal projection of E on Irn u* (of F or1 I m u ) . c + a . We have Ker u = Ker (u* o u ) , ( ~ e11)'r
= (Ker (u'ou))'
=
Im (U*OIL)
(Proposition 5.3.2.4) and = llu*112 = llu* 0 uI1
I1
(Theorem 5.3.1.4). Take x E (Ker u)' . Then 11x11 = Ilu'uxll I IluxII
I IIxII
3
so that IIuxll = llxll and u is a partial isometry. b+a&dandd+b followfroma+b&candc~a.
5. Halbert Spaces
106
Proposition 5.3.2.27 Let (H,),,, , (K,),,, be two families of Hilbert spaces ~ and E the C*-direct s u m of the family (K(H,, K L ) ) L .EPut
and for each u E E , define
Then
and the m a p
is un injective homomorphism of znvolutive vector spaces (algebras i f (HL)lE,= (K')Gl ). The first assertion follows from Proposition 3.1.2.17. It is obvious that the map
is injective. By Proposition 5.1.3.4 and Proposition 5.3.2.3, this map is a homomorphism of involrltive algebras. Proposition 5.3.2.28 E:
T h e following are equivalent for every Hilbert space
a ) E is finite-dzmensional.
c) K ( E ) is unital. a + b is trivial. b + c follows from Theorem 4.4.1.8 h). c + a . Let p be the unit of K(E) and put F := I m p . Then p E Pr K(E) so that p = T F (Proposition 5.3.2.8 b + a). Take x E F 1 . By Proposition 5.3.2.13 c),
5.3 Adjoint Operators
so that
Hence F'
= (0)
,
and
By Proposition 3.1.1.10, E is finite-dimensional.
Proposition 5.3.2.29 Let E be a n infinite-dimensional Hilbert space, 3 a closed ideal of L ( E ) , and G a C*-subalgebra of L ( E ) . T h e n 3 G is the C' -subalgebra of L ( E ) generated by 3 u G and the C * - algebras (3 G ) / F and G / ( F n G) are canonically isometric. If i n addition F n G = (0) then
+
for all ( u ,v) E 3 x
+
6
By Theorem 5.3.1.13, L ( E ) is a C'palgebra and the assertion follows from W Corollary 4.2.6.7.
5. Hilbert Spaces
108
5.3.3 Selfadjoint O p e r a t o r s P r o p o s i t i o n 5.3.3.1 ( 0 ) Let E , F be Hilbert spaces and S the involutive vector space of continuous sesqilinear forms on E (Proposition 2.3.3.3 c)). Given u E L ( E ) , define
a) ii E S for e v e y u E L ( E )
b) The map L ( E ) -+ S , u
-
-
.ii
zs an isomorphism of involutzve vector spaces. c) u* o u is positive for e v e y u E & ( E , F ) . a ) Now
and so, by Proposition 1.2.9.2c b) Consider
+ a , .ii is continuous.
It is obvious that cp is linear. By Corollary 5.3.1.2 and Theorern 5.3.1.4, cp is bijective. Take u E L ( E ) . Then
-
- u * ( x ,y ) = (u'xly) = ( x l u * * y )= ( x l u y ) = ( u y l x ) = .ii(y,x) = & * ( x ,y )
for all x , y E E (Theorem 5.3.1.4, Proposition 5.3.1.5 a ) ) . Thus
and 9 is involutive. c) Given x E E ,
-
Thus by b), u* o u is positive. Corollary 5.3.3.2 An operator u on a real Hilbert space E is selfadjoint ifl (ux~?,)=
for all x , y E E .
I -
4
((u(x
+ Y ) lz + Y ) - ( 4 5
-
Y ) Ix
-
Y))
5.3 Adjoint Operators
109
This follows immediately from Proposition 5.3.3.1 b) and Proposition 2.3.3.7. H Corollary 5.3.3.3 selfadjoint iff
(0)
A n operator u o n a cornplez Hilbert space E is
for every x E E This follows immediately from Proposition 5.3.3.1 b) arid Proposition 2.3.3.8 a ~ b . Corollary 5.3.3.4 that
for every z
E
(0)
Let u be an operator o n a Hilbert space E such
E . If u is selfadjoznt or zf IK = C , then u = 0
By Proposition 5.3.3.1 b) and Proposition 5.1.1.9,
for every x E E and so u
=0
Remark. Let u be the operator on R 2 ,defined by the matrix
Then
for every x E IR'. This example shows that the above Corollary is false without some hypothesis such as the selfadjointness of u or IK = C . P r o p o s i t i o n 5.3.3.5 space E , then
(0 )
If u is a selfadjoint operator o n the Hilbert
11u1( = sup I(uxIx)I = inf { a E r€E#
R+ ( x E E ==+ )(uxlx)l 5
all~11~) .
5.3 Adjoint Operators
Theorem 5.3.3.6 E # 10) and put
(0)
111
Let u be a selfadjoint operator o n the Hilbert space
/3 := sup{(uxlx)
IX
E E , llxll = 1 1 ,
(Corollary 5.3.3.3). T h e n c-u, P E a ( u ) C [a,Dl.
Take y
€1
-
co,a [ .Then
0 < cu - Y I (UXIZ) - Y(XIX) = ((u - Y ~ ) X I I X )I I ( U -Y~)XII for every x E E with
((XI(
= 1 . By Corollary 5.3.2.5, y
4
$ a ( u ) . Hence
~ c )[ a ,
I t follows that
arid
4.1 c [a,PI . Put y := inf a ( u ) .
Then " ( u - y l ) = U(U)- 7 C I R + ,
i.e. u - y l is positive. Thus (uxlx) = ((u - y1)xlx)
+ y = ((u
for every x E E , llxll = 1 , and
and
SO
-
yl)1'2x I (u - y1)1'2x)
a 2 y . Hencc
+y > y
112
5. Hilbert Spaces
Corollary 5.3.3.7 ( iff zt is seljadjoznt and
0)
T h e operator u o n the Hzlbert space E is positive
for every x E E . I n the complex case, we m a y ornzt the adjective "selfadjoint". In particular, (.lx)x and the map
are positive for every x E E The first assertion follows from Theorem 5.3.3.6 and the second from Corollary 5.3.3.3. The final assertion follows from the first one, Corollary 4.2.2.10, and Proposition 5.3.2.13 b). Remark. The example given in the Remark to Corollary 5.3.3.4shows that we cannot drop "selfadjoint" in the real case. Corollary 5.3.3.8 ( 0 ) If F and G are closed vector subspaces of a Hilhert xc:. In particular, if H is the closed vector space E , then F c G iff n~ subspace of E generated by F u G (by F n G ) , then x~ is the supremum (znfimum) of { n ~xG} , zn Pr L ( E ) .
0 .
(0)
Let E , F be Hilbert spaces. Take
a) Ker (u*o u)" = Ker u b) u compact e (u* o u ) zs~ compact.
E L ( E ,F ) ,
5.3 Adjoant Operators
By Corollary 5.3.3.9, u' o u is positive. Thus (u'u)" is well-defined.
a)
Step1
vERe~(E),n~K~Kerv~"=Kerv
By Proposition 5.3.2.4,
Ker v2 = Ker v , and so the assertion follows by complete induction. Step 2
v
E L(E)+ ,
0 < 4 < y + ~ ev4r c Kerv*
The assertion follows frorn v* = .*-4,p.
Step 3
v E L ( E ) + + Kerv" = Ker v
Take n E N such that 2-" < a < 2" . By Steps 1 and 2,
Ker v C Ker vQ2n= Ker ( u " ) ~ '= Ker vQ c ~ ev2" r = Ker v , and so Ker v" = Kcr v Step 4
Ker ( u * ~ ) = " Ker u
By Proposition 5.3.2.4 and Step 3:
Ker (u* o u)" = Ker (u' o U) = Ker u
b)
Stcp 1
v E R e L ( E ) , n E IN
+ (v
compact H v2" compact)
By Proposition 5.3.2.2 a H c ,
v compact u v2 compact. The assertion thus follows by complete induction. Step 2
v E .C(E)+ , 0 < /3 < 7 ,va conipact
+ v*
compact
5. Halbert Spaces
114
The assertion follows from u1
= u7-4v4
and Proposition 3.1.1.11. Step 3
u E t ( E ) ++ ( v compact
@
ua compact)
Let n E IN with
2-" < ff By Steps 1 and 2, v compact ==+ uo2" compact
* ua compact
< 2".
-
(va)'" compact
*
==+ u2' compact ti u compact.
Thus
u compact eva compact. Step 4
u compact H (u*o u ) compact ~
By Propositioii 5.3.2.2 a H c and Step 3, u compact
Corollary 5.3.3.11 then, for every n E IN
u* o u compact
(0 )
(u*o u)O compact.
If u is a normal operator o n a Hilbert space,
Ker un = Ker u ,
u compact
un compact
Since
( u * u )=~ ( U * ) ~ U " = (un)*un, it follows by Proposition 5.3.3.10, that
-
~ Ker ((un)'un)= Ker un , Ker u = Ker ( u ' u ) = u compact
( u * ~compact ) ~ M (un)'un compact e
+=+
11"
compact.
5.3 Adjoint Operators
115
Remark. W e cannot drop the adjective "norrrial" in the above corollary, as the following example shows:
Then u is not compact but u2 = 0 Corollary 5.3.3.12 define
(0)
Let E be a Hilbert space. Take u E C ( E ) and
a) f is Hermitian iff u is selfadjoint.
b ) f is positive i f f u is positive. I n this case { x E El f ( x , x ) = 0 ) = Keru. c ) f zs a scalar product iff u is injective and posztive. a) follows from Proposition 5.3.3.1 a),b). b ) By a) and Corollary 5.3.3.7, f is positive iff u is positive. Take x E E . Then f ( x , x ) = (uxIx) =
(,tX
Iutx)
= luf212
Thus, by Corollary 5.3.3.11, f ( z , x ) = 0 ex
E
~erui
x E Keru
c ) follows from b ) . Proposition 5.3.3.13 ( 0 ) Let E , F,G be Hzlbert spaces. Take u E C ( E ,F ) and let p (resp. q ) be the orthogonal projection of E onto (Keru)' (of F 071 G).
a ) p (resp. q ) is the smallest orthogonal projection r in E (in F ) such that
b) Given v E L ( G , E ) with u o II = 0 , we have that p o v
=0
c ) Given v E C ( F ,G ) with v o u = 0 , we have that v o q = 0
5. Hilbert Spaces
116
a ) 1 - p (resp. 1 - q ) is the orthogonal projection of E onto Keru (of F on Im u' ) and so
uop =u,
(qou=u).
Now uo(1-r)=O,
((1-r)ou=O),
so that Kerr=Im(l-r)cKeru,
(ImucKer(1-r)=Irnr),
(Corollary 5.3.3.8) b) Since Irnv
c Keru = Kerp,
we have that
c) Since 1 r n q = ~ ~ K e r v , we have that voq=o Remark. If E = F = G , then the above proposition says that p and q are the right and left carrier of u , respectively. Theorem 5.3.3.14 ( 0 ) Let E be a Hilhert space, F a nonenpty upward directed set of L ( E ) , and 5 its upper section filter. Then the following are equivalent:
5.3 Adjoint Operators
a ) 3 is bounded above. b) 7 possesses a supremum i n C ( E ) If these conditions are fulfilled, then:
I n particular, C(E) and K ( E ) are C-order complete (Corollary 5.3.1.12).
>
a + b & c . Replacing F by {u - uo I u E 7, u uo) for some uo E 7 ,if necessary, we may assume that 7 c L ( E ) . Let v be an upper bound for F . Then +
for every x E E (Corollary 5.3.3.7). Define f :E x E
---t
IK ,
(x, y)
H
lim(ux1y) u,3
*
(Propositions 5.3.3.1 a),,), 2.3.3.7 a + b , and 2.3.3.8 a c & d). Then f is a positive sesquilincar form on E (Proposition 5.3.3.1 a),b)) and
for all z, y 6 E (Schwarz's Inequality). Hence f is continuous (Proposition 1.2.9.2 c + a). By Corollary 5.3.1.2, there is a U I E L ( E ) such that
for all x, y E E . Thus w E L ( E ) + (Corollary 5.3.3.12 b)) and
whenever u E F and x E E . Hence
for every u E F (Corollary 5.3.3.7). w is thus thc supremum of F in C ( E ) . Hence L ( E ) is C order cornpletc. By Proposition 5.3.1.17, K ( E ) is also C order complete. b + a is trivial.
5. Halbert Spaces
118
Corollary 5.3.3.15
If E, F are complex Hzlbert spaces, then
{u E L(E, F ) 1 u is an isomorphism (isometry))
is path- connected. Define
U V
:= {u
:= { UE C ( E ) I u is invertible (unitary)}
L ( E ,F ) 1 u is an isomorphism (isometry)) .
Take u, v E V . Then 11-' o v E U (Corollary 5.3.1.22). By Corollary 4.3.2.9 (Corollary 4.3.2.7). and Theorem 5.3.3.14, there is a continuous map
with
Then
[O,ll+V,
@c--1~f(~)
is a continuous path connecting u with v .
Remark. The corollary does not hold in the real case. In the finite-dimensional case, the set of isomorphisms (isometries) has two connected components. Corollary 5.3.3.16 (Atkinsorl, 1931) If E , F are cornplex Halbert spaces, then for each n E Z, {U E F ( E , F ) I Ind u = n )
is path-connected
Case 1
n=O
Take u, v E 3 ( E ,F) with Indu = Indv = 0
5.3 Adjoint Operators
119
By Corollary 3.1.3.15, there are isomorphisms uo, vo : E -t F and continuous paths (, 7 in F ( E , F ) connecting u and uo and v and vo, respectively. By Corollary 5.3.3.15, there is a continuous path in F ( E , F ) connecting uo and vo . Then v-'Y)) = (P((x, ~))tkl and N,, = F . b)
The jactonzation v of the inclusion m a p E znjective.
c ) If X , Y E E / F and x
E
-+
L 2 ( p ) through E / F is
X , y E Y , then
d ) v m a y be extended uniquely to a n isometry of Hilbert spaces
-
e) If we identify E / F with L 2 ( p ) using the isometry i n d), then
Z ( Y ) = XY
1
llZll
=
llxlSllO0
for every ( x ,y ) E E x L 2 ( p ) . f)
T h e jollowzng assertions are equivalent: fl)
S is a one-poznt set.
f2)
the representation of E associated to
f3)
the Hilbert space associated to x' is one-dimensional.
Example 5.4.3.2 M b ( T ) +with
XI
is irreducible.
Let T be a Hausdorfl topological space. Take p, u E
5.4 Representations
T h e n the following are equivalent:
a ) The representations of C ( T ) associated to x' and y' are equivalent. b) There are positive real Bore1 functions f and g on T such that
f E L2(d>
v=f2./1,
g E L 2 ( v ) , p = g2.v.
Given x E C ( T ), define 'PX : L 2 ( p ) -+
L2(p),
t 4xF
By Example 5.4.2.1, ( L 2 ( p jcp) , . and ( L 2 ( v )$) , are the representations of C ( T ) associated t o x' and y', respectively. a + b . Let u : L 2 ( v )+ L 2 ( p ) be an isometry of Hilbert spaces such that
for every x E C ( T ). Let K be a compact set of T and q E L 2 ( v ) with q=7)e~.
Then for every x E C ( T ) with
Hence
5. Halbert Spaces
158
T\K. I f K is a p-null set, then K is a v-null set as well. Hence v is pabsolutely continuous. By tho Radon-Nikodym Theorem, there is a positive real Borel function f on T such that f€L2(p), v=f2.p. It follows that there is a positive real Borel function g on T such that
b
+a.
Define
u, v are operators preserving the norms and
Hence u and v are isometries and
Let x E C ( T ) and
7)
E
L 2 ( v ) .We have
so that
'px = u'
0
(*x) 0 u
Hence the representations associated to x' and y' arc cquivalent.
Example 5.4.3.3
Let H be a Halbert space, A a nonempty set of pairwise orthogonal elements of H\{O) , such that
and E a C*-subalgebra of L ( H ) . Put
x': E - + I K ,
x++x(x E'
= lim(uxtn luxk,) = lim lluxkn n ,3
n,3
Hence 12 # 0 . d ) It is obvious that the map is a unital algebra homomorphism. Take u E L ( E ) and x, y, E 3.Then
and the factorization referred to in d) is a unital representation. By c), it is faithful.
164
5. Halbert Spaces
e) Take x, y E that
F\g. There is a decreasing sequence (An),En in 5 such 1
Ilx(i)ll > illxll for every i E A1 and
11412
1 l ( x ( ~ ) l x ( j ) ) l3 < , l(~(i)l~(j))l 2,
then the topologzcal cardinality of L ( E , F ) is equal to
C.
c ) If both E and F are dzfferent from (0) then L ( E , F ) is separable z f f one of these spaces zs finzte dimenszonal and the other is separable. a) \$'e have
b) The map
5.5 Orthonormal Bases
is an isometry of real Banach spaces, so we may assume a 5 b . Then, by a), bO
< bb = 2b < ab ,
so that
By Example 1.1.2.5 and Theorem 5.5.2.5, there is a dense set B of F with Card B = b Let A be an orthonormal basis of E . Then
F := {U
= C(E, F)lu(A) c B }
is a dense set of L ( E , F ) and Card F 5 Card BA = bn = c Now Irt A and B be orthonormal bases of E ant1 F , respectively. Put
G For f E G let
UJ
:= { f E B" (f is injective}
.
be the element of L ( E , F ) with "J" = f (")
for every x E A (Proposition 5.5.1.22). Then
1 1 , -~ ~"911 > JZ for all distinct f , g E G . Hence, if 31 is a dense set of C(E, F ) , then Card 31
> Card G = bO = c .
c) If E is finite-dimensional, then CardL(E, F ) = Card Fa and so L ( E , F ) is separable iff F is separable. We use again the isometry L ( E , F) --+ C(F, E ) ,
u c,u * ,
to conclude that if F is finite-dimensional, then C(E, F ) is separable iff E is separable. If both E and F arc: infinite-dimensional, then, by b), L(E, F ) is not separable.
198
5. Hilbert Spaces
Proposition 5.5.2.12 ( 0 ) Let E , F be Hilbert spaces different from ( 0 ) . Let N ( E ) and N ( F ) be the Hilbert dimensions of E and F , respectively, and
put
a) A n y dense set of K ( E , F ) has cardinality at least N b) If N is infinite, then K ( E , F ) has topological cardinality N c) K ( E , F ) is separable iff E and F are separable. d ) The C--algebras K ( E ) and K ( F ) are isomorphic i f f the Hilbert spaces E and F are isomorphic. e) If E is infinzte-dimensional, then the topological cardinality of C ( E ) / K( E ) is 2 H ( E.) a ) Let A and B be orthonormal bases of E and F , respectively. Then for all X I , 2 2 E A , x1 # 5 2 and y l , y2 E B , yl # y2 we have
Hence there is a set of pairwise disjoint open sets of K ( E , F ) of cardinality N . Thus any dense set of K ( E , F ) has cardinality at least N . b) Let A and B be dense sets of E and F , respectively, of cardinality a t most N and let ( x ,y ) E E x F . Then for all ( a ,b) E A x B ,
5 Ila - xll llbll + llxll Ilb - Yll It follows that ( . l x ) y belongs to the closure of
{(.I.)bl(a,b) in K ( E , F) . Since
E
A
x
B)
5.5 Orthonormal Bases
Card {(.la)bl(a,b) E A x B )
O
n+m
By Corollary 5.5.1.9 (and Corollary 1.4.1.6 a + c) , ( U X , ) , ~converges ~ weakly ~ to 0 . Hence we can construct inductively an increasing sequence ( k n ) n E in IN such that
for every n E IN. We may assume G to be infinite-dimensional. Let (y,,),,~ be an orthonormal basis of G (Corollary 5.5.2.7 a + b). By Proposition 5.5.1.22, there is a u E C(E) such that
5. Hilbert Spaces
200
vYn = X k , for every n E W and v = 0 on G I . We have
C 32"m2 = a28 < m24 -
so that
cp : G / F 4K ( H ) is an isometry. Take < , q E K , < # O , a n d ~ > O . T a k ex ~ G \ F . T h e r e i s a< o € K such that (*x) 0 .
¤
Hence Il.llG and 11.11 are equivalent.
Proposition 5.6.2.6 ( 8 ) (W. Paschke, 1973) Let E be a C*-algebra, G, H inner-product right E-modules, and u E Z E ( ~H, ) Then (uSIu0 I llu112(O ) ( z ~ . y ~ ) = II
By the above, lim q(r,,yl)(t> 0) = ( es
~
O)(rl.vt) 1
*
c4,
5.6 Hzlbert right C*-Modules
Proposition 5.6.2.10
(8)
327
Let H be an inner-product right E-module
0
0
and H its complexification (Proposition 5.6.1.6). W e identify L ( H , E ) and 0
0
-
L ( H , E ) using the isomorphism of complex vector spaces
(
HE)
-
( h)
( u ,v ) +-+ ( u ,v )
defined in Proposition 2.1.5.6 (Proposition 4.3.6.1 f), Proposition 5.6.1.6)
0
b) H is self-dual 2ff H is self-dual. 0
a) Take ( u ,v ) E
i?.For every
( -70))
Hence
0
and H is self-dual. is self-dual. Take u E Now assume
k
0
fi
-
and put
w := (u, 0) . 0
By a), w E H . Since H is self dual, there is a ( 0 . By b), there is a J E p f ( I ) such that
for every K E
for every K
p f (I\J)
c I\J.
. By continuity,
Let K l ,K Z E p ( I ) with
5. Hzlbert Spaces
366
Then
Hence the map
is continuous. Since a is arbitrary, the map
is continuous. It follows that
is a compact set of E E . By Banach-Steinhaus Theorem (Theorem 1.4.1.2), it is a bounded set of E .
Corollary 5.6.3.18
Let E be a W*-algebra, (x,),,,
a summable family in
E E , and
(Proposition 5.6.3.17 e)). For every a E E and J C I put - ( a , J ) := sup
{
(gX.,
.)I
K
c I\.J)
5 ~llall.
5.6 Hilbert right C*-Modules
a ) If a E E , J
cI,
K E q f ( I \ J ) , and a E l m ( I ) then
b) For every a E E and
E
> 0 , there i s a J
for every a E e m ( I ) and K E C)
For every
CY
E
pf(I)such
that
(I\J) .
E e m ( I ) , ( a , ~ , ) ,i ,s ~summable i n E, and
d) T h e maps
are continuous and
as a compact set of E,.
a ) follows from Proposition 1.1.1.5. b) By Proposition 5.6.3.17 a b , there is a J E y j ( I ) such that
5. Hilbert Spaces
368
for every (Y E em(I) arid K E p f ( I \ J ) . c) By b) and Proposition 5.6.3.17 b =. a , ( ~ , x , ) , ~isIsummable in EE for every a E !"(I). The inequality follows from a ) . d) By c) , the map em(I) + E , is continuous. Let a E E and
E
a
-
E
a,xL LEI
> 0 . By b) and c ) , there is a J E p f ( I ) such that
for every a E (IK#)' and K
c I\J.
Let a ,0 E (IK')' E
1% - PtI < 3 ( 1 + 87) for every
L
E J . By c) ,
Hence the map
is continuous. Since a is arbitrary, the map
is continuous. It follows that
is a compact set of E,
such that
5.6 Hzlbert right C*-Modules
36.9
Let E be a W*-al.qebra and let (x,),,~, ( Y X ) A E L be sum~nablefamilies i n E E .
Proposition 5.6.3.19
5
a)
XEL
( ( eL XE IL ) Y X )
(..
= E (5 ")) (5") ( f Y X ) . LEI
=
XEL
XEL
LEI
b) T h e map
(Proposition 5.6.3.17 e)) is continuou.~,where p ( I ) and p ( L ) are identified with { O , l ) ' and {0,1)', , respectzvely. a ) For a E E ,
( j (eYA), (yA.atxL) ((ezL) = ( eX yE LX , a = xL ELI )
=
LEI
A €1,
XEL
LEI
= L E I (z.,
(5
X E L .A)
= XEL
a) =
XEL
LEI ( 2 .
=
yX,a)
(2
,A€ t, y X )
,
.a)
and t,he assertion follows from the fact that a is arbitrary. b) By Proposition 5.6.3.17 e), ( X ~ ) ~ ~and . J ( Y , ) , , ~ are summable in E, for all .I C I and K C L . The assertion follows from a ) and C. Constantinescu, Spaces of Measures, Theorem 3.5.2 c). Proposition 5.6.3.20
Let ( x , ) , ~be~ a fa~nily of the W * algebra E such
that
for all distinct
L,
X E I . Then the following are equivalent:
5. Hzlbert Spaces
3 70
a) (x,),,~ zs summable zn E, b ) (x:x,),,I
.
zs summable zn E,
If these condztzons are fulfilled then c)
LEI
LEI
a
LEI
* b & c . By Propositions 5.6.3.17 c),d) and 5.6.3.19 a),
( ) ( ) (5.:)(5 5 5 LEI
LEI
=
LEI
E
= ):
LEI
E
xX) =
€ 1
(x:
AEI
xA) =
E
x:xA
=
X~X,
LEI
rE1 XEI
*
b a . Let a E E and let (y, l a [ ) be its polar decomposition (Theorem 4.4.3.5 a)). For every J E T J ( I ) , b y Proposition 2.3.4.6 c ) ,
5 (Y*
(p)(5
XL)
By Proposition 5.6.3.17 a =+ b , for every that
E
Y? l a , ) ( l , , a l ) =
> 0 , there is a J
E T J ( I ) such
5.6 Hilbert right C ' M o d u l e s
for every K E g f ( I \ J ) . By the above,
for every K E p , ( I \ J ) . By Proposition 5.6.3.17 b in E,.
+a,
( x , ) , ~ Iis summable
Remark. The above conditions imply that ( ~ ~ , l is ~ sumrnable ) ~ , ~ in E, for every cr 2 2 , since
, ~ ~summable in is a cofinite subset of I . But it may happen that ( I X , ~ ~is) not EE for every (Y E ]0,2[. A counterexample in this sense is given by the sequence (x,),,~ in L(e2),where
for every n
N
Corollary 5.6.3.21 such that
for all distznct
L, X
Let E be a W*-algebra and (p,),El a family in Pr E
E I . Then (p,),,,
is summable i n E, and
The set
is upward directed and bounded (by 1) so that (P,),,~ is summable in E, (Theorem 4.4.1.8 b)). By Proposition 5.6.3.17 c) and Proposition 5.6.3.20 a =+ b&c,
5. Halbert Spaces
372
Proposition 5.6.3.22 such that
for all distinct
L,
Let E be a W*-algebra and ( P . ) [ ~ a~ famzly zn Pr E
X E I and
Then the map
LEI
(Corollary 5.6.3.18 d)) is a unztal injective W * - h o m o m o r p h i s m with pretranspose
For a , B E e r n ( [ ) ,by Propositions 5.6.3.19 a ) and 5.6.3.17 c),
Hcnce the map is an ir~volutivealgebra homomorphism. It is obviously unital and injective. Moreover, for a E t m ( I ) and a E E ,
so that the map is a W-homomorphism with pretranspose
5.6 Hilbert right G' -Modules
5.6.4 Examples Proposition 5.6.4.1 ( 0 ) Let ( H l ) , , I be a family of (weak) semi-innerproduct right E-modules and
If
:=
@ H,
:= {
fE
LEI
n
the family ((
a ) e2(1,F ) is a vector subspace of F1 and e 2 ( I ,F ) endowed with the maps
t 2 ( 1 ,F ) x e 2 v ,F )
4E
,
( I ,7 )
-
7;E' iEl
is a Hilbert right E module.
C)
If F is an ideal of E and ( x , ) , , ~is a bounded family in E then for every E E e 2 ( I ,F ) ,
In partzcular, t 2 ( 1 ,F ) endowed with the map E x e2(1,F )
('(1,F ) ,
--t
( x ,E )
(xEL)LEI
is a (unital) Hilbert E-module. d) If F is an ideal of E and
I* :=
(
Let
<E H
for every
L
srich that
E
I . Then for every
L
E
I,
t;
:b,xp,lx
0 = ( H,)
and ii* = i*. In particular, cp is involutive. It is easy
LE l
to see that cp is an injective unital algebra homomorphism. We want to show that cp is surjective. Let u E LE( H,) and let 3 be an ultrafilter on v f ( I ) finer than 3 1 .
a>
LEI
Then
for every ( E H and J E ?,(I). By a) and Proposition 5.6.3.3 a define
* b , we may
where the limits are considered in HH . Take