Limit Operators and Their Applications in Operator Theory

403 Pages·2004·31.117 MB·English

by  Vladimir Rabinovich, Bernd Silbermann, Steffen Roch (auth.)

OperatorTheory: Advances and
Applications
Vol. 150
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I. Gohberg
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Limit Operators
and Their Applications
in Operator Theory
Vladimir Rabinovich
Steffen Roch
Bernd Silbermann
Springer Basel AG
Authors:
Vladimir Rabinovich Steffen Roch
Instituto Politecnico Nacional Department of Mathematics
ESIME Zacatenco Technical University of Darmstadt
Avenida IPN Schlossgartenstrasse 7
Mexico, D,F. 07738 64289 Darmstadt
Mexico Oermany
v [e-mail] [e-mail]
Bernd Silbermann
Department of Mathematics
Technical University of Chemnitz
09107 Chemnitz
Oermany
[e-mail]
2000 Mathematics Subject Classification 47L80; 35S05, 47A53, 47030, 65R20
ISBN 978-3-0348-9619-1 ISBN 978-3-0348-7911-8 (eBook)
DOI 10.1007/978-3-0348-7911-8
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Library of Congress, Washington D.C., USA
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© Springer Basel AG 2004
Originally published by Birkhăuser Verlag AG 2004
Softcover reprint of the hardcover 1 st edition 2004
Part of Springer Science+Business Media
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Cover design: Heinz Hiltbrunner, Basel
ISBN 3-7643-7081-5
987654321 www.birkhăuser-science.com
Contents
Preface . Xl
1 Limit Operators
1.1 Generalized compactness, generalized convergence 1
1.1.1 Compactness, strong convergence, Fredholmness 1
1.1.2 P-compactness .... 4
1.1.3 P-Fredholmness 10
1.1.4 P-strong convergence 11
1.1.5 Invertibility ofP-strong limits 15
1.2 Limit operators . . . . . . . 17
1.2.1 Limit operators and the operator spectrum 17
1.2.2 Operators with rich operator spectrum. 19
1.3 Algebraization . 23
1.3.1 Algebraization by restriction 24
1.3.2 Symbol calculus . 25
1.4 Comments and references . . . . . . 29
2 Fredholmness ofBand-dominated Operators
2.1 Band-dominated operators . 31
2.1.1 Function spaces on TLN . 31
2.1.2 Matrix representation . 32
2.1.3 Operators ofmultiplication 33
2.1.4 Band and band-dominated operators 35
2.1.5 Limit operators ofband-dominated operators 40
2.1.6 Rich band-dominated operators . 43
2.2 P-Fredholmness ofrich band-dominated operators . 45
2.2.1 The main theorem on P-Fredholmness .... 45
2.2.2 Weakly sufficient families of homomorphisms 51
2.2.3 Symbol calculus for rich band-dominated operators. 53
2.2.4 Appendix A: Second version ofa symbol calculus 56
2.2.5 Appendix B: Commutative Banach algebras ..... 59
vi Contents
2.3 Local P-Fredholmness: elementary theory . . . . . . 61
2.3.1 Local operator spectra and local invertibility 61
2.3.2 PR-compactness, PR-Fredholmness . . . . . 62
2.3.3 Local P-Fredholmness of band-dominated operators 64
2.3.4 Allan's local principle . . . . . . . . . . . . . . . . . 65
2.3.5 Local P-Fredholmness of band-dominated operators
in the sense ofthe local principle . . . 69
2.3.6 Operators with continuous coefficients 72
2.4 Local P-Fredholmness: advanced theory . . 74
2.4.1 Slowly oscillating functions . . . . . 74
2.4.2 The maximal ideal space of SO(ZN) 79
2.4.3 Preliminaries on nets . . . . . . . . . 82
2.4.4 Limit operators with respect to nets 87
2.4.5 Local invertibility at points in Moo(SO(ZN)) 89
2.4.6 Fredholmness of band-dominated operators
with slowly oscillating coefficients. . . . . . . 93
2.4.7 Nets vs. sequences . . . . . . . . . . . . . . . 94
2.4.8 Appendix A: A second proofofTheorem 2.4.27 . 95
2.4.9 Appendix B: A third proofofTheorem 2.4.27 . 100
2.5 Operators in the discrete Wiener algebra. . . . . . . . . 103
2.5.1 The Wiener algebra . . . . . . . . . . . . . . . . 103
2.5.2 Fredholmness ofoperators in the Wiener algebra 107
2.6 Band-dominated operators with special coefficients 111
2.6.1 Band-dominated operators with almost
periodic coefficients 111
2.6.2 Operators on half-spaces. . . . . . . . . 113
2.6.3 Operators on polyhedral convex cones 119
2.6.4 Composed band-dominated operators on Z2 124
2.6.5 Difference operators ofsecond order . . 128
2.6.6 Discrete Schrodinger operators . . . . . 131
2.7 Indices ofFredholm band-dominated operators 135
2.7.1 Main results. . . . . . . . . . . . . . . . 136
2.7.2 The algebra A(Z) as a crossed product. 138
2.7.3 The K1-group ofA(Z) . 139
2.7.4 The K1-group ofA± . . . . . . . . 142
2.7.5 ProofofTheorem 2.7.1 144
2.7.6 Unitary band-dominated operators 147
2.8 Comments and references . . . . . . . . . 150
3 Convolution Type Operators on lRN
3.1 Band-dominated operators on LP(lRN) . 153
3.1.1 Approximate identities and P-Fredholmness . 153
3.1.2 Shifts and limit operators . 155
Contents vii
3.1.3 Discretization . 155
3.1.4 Band-dominated operators on LP(JRN) 157
3.2 Operators ofconvolution . 159
3.2.1 Compactness ofsemi-commutators . 159
3.2.2 Compactness ofcommutators .... 164
3.3 Fredholmness ofconvolution type operators 169
3.3.1 Operators ofconvolution type . 169
3.3.2 Fredholmness..... . 172
3.4 Compressions ofconvolution type operators 179
3.4.1 Compressions ofoperators in A(BUC(JRN), Cp) 180
3.4.2 Compressions to a half-space .... 181
3.4.3 Compressions to curved half-spaces. 182
3.4.4 Compressions to curved layers 184
3.4.5 Compressions to curved cylinders .. 184
3.4.6 Compressions to cones with smooth cross section 185
3.4.7 Compressions to cones with edges .... 190
3.4.8 Compressions to epigraphs offunctions 193
3.5 A Wiener algebraofconvolution-type operators . 194
3.5.1 Fredholmness ofoperators in the Wiener algebra 194
3.5.2 The essential spectrum ofSchrodinger operators 195
3.6 Comments and references . 199
4 Pseudodifferential Operators
4.1 Generalities and notation . . . . . . . . . . . . 201
4.1.1 Function spaces and Fourier transform. 201
4.1.2 Oscillatory integrals . . . . . 203
4.1.3 Pseudodifferentialoperators....... 204
4.1.4 Formal symbols . . . . . . . . . . . . . . 205
4.1.5 Pseudodifferential operators with double symbols 206
4.1.6 Boundedness on L2(JRN) . . . . . . . . . . . . . . 207
4.1.7 Consequences ofthe Calderon-Vaillancourt theorem 210
2
4.2 Bi-discretizationofoperators on L (JRN) . . 211
4.2.1 Bi-discretization . . . . . . . . . . . 211
4.2.2 Bi-discretization and Fredholmness . 213
4.2.3 Bi-discretization and limit operators 215
4.3 Fredholmness ofpseudodifferential operators. 218
4.3.1 A Wiener algebraon L2(JRN) . . . . . 218
4.3.2 Fredholmness ofoperators in W$(L2(JRN)) 222
4.3.3 Fredholm properties ofpseudodifferential
operators in 0P88,0 . . . . . . . . . . . . . . . . . . . . . . 224
viii Contents
4.4 Applications. 228
4.4.1 Operators with slowly oscillating symbols 228
4.4.2 Operators with almost periodic symbols 230
4.4.3 Operators with semi-almost periodic symbols 233
4.4.4 Pseudodifferential operators ofnonzero order 234
4.4.5 Differential operators. 236
4.4.6 Schrodinger operators 239
4.4.7 Partial differential-difference operators 242
4.5 Mellin pseudodifferential operators 243
4.5.1 Pseudodifferential operators with analytic symbols 243
4.5.2 Mellin pseudodifferential operators 247
4.5.3 Mellin pseudodifferential operators with
analytic symbols 250
4.5.4 Local invertibility ofMellin pseudodifferential
operators 251
4.6 Singular integrals over Carleson curves
with Muckenhoupt weights 254
4.6.1 Carleson curves and Muckenhoupt weights . 254
4.6.2 Logarithmic spirals and power weights . 255
4.6.3 Curves and weights with slowly oscillating data . 257
4.6.4 Local representatives and local spectra of
singular integral operators . 258
4.6.5 Singular integral operators on composed curves 262
4.7 Comments and references 265
5 Pseudodifference Operators
5.1 Pseudodifference operators. . . . . . . . . . . . . . 267
5.2 Fredholmness ofpseudodifference operators .... 273
5.3 Fredholm properties of pseudodifference operators
on weighted spaces . . . . . . . . . . . . 276
5.3.1 Boundedness on weighted spaces 276
5.3.2 Fredholmness on weighted spaces 278
5.3.3 The Phragmen-Lindelofprinciple 279
5.4 Slowly oscillating pseudodifference operators. 280
5.4.1 Fredholmness on lP-spaces ..... 280
5.4.2 Fredholmness on weighted spaces,
Phragmen-LindelOfprinciple 284
5.4.3 Fredholm index for operators in OPSO 287
5.5 Almost periodic pseudodifference operators 288
5.6 Periodic pseudodifference operators . 289
5.6.1 The one-dimensional case . 290
5.6.2 The multi-dimensional case . 292
Contents ix
5.7 Semi-periodic pseudodifference operators .. 293
5.7.1 Fredholmness on unweighted spaces 293
5.7.2 Fredholmness on weighted spaces 296
5.7.3 Fredholm index . 297
5.8 Discrete Schrodinger operators . 297
5.8.1 Slowly oscillating potentials . 298
5.8.2 Exponential decay ofeigenfunctions 299
5.8.3 Semi-periodic Schrodinger operators 301
5.9 Comments and references . 302
6 Finite Sections ofBand-dominated Operators
6.1 Stability ofthe finite section method 304
6.1.1 Approximation sequences . 304
6.1.2 Stability vs. invertibility ... 306
6.1.3 Stability vs. Fredholmness .. 307
6.2 Finite sections ofband-dominated operators on Zl and Z2 312
6.2.1 Band-dominated operators on Zl: the general case 313
6.2.2 Band-dominated operators on Zl: slowly
oscillating coefficients . . . . . . . . . . . . . 315
6.2.3 Band-dominated operators on Z2 . . . .... 318
6.2.4 Finite sections ofconvolution type operators 320
6.3 Spectral approximation . 321
6.3.1 Weakly sufficient families and spectra . 322
6.3.2 Interlude: Spectra ofband-dominated operators
on Hilbert spaces . . . . . . . . 326
6.3.3 Asymptotic behavior of norms 327
6.3.4 Asymptotic behavior ofspectra 328
6.4 Fractality ofapproximation methods . 332
6.4.1 Fractal approximation sequences 333
6.4.2 Fractality and norms . 335
6.4.3 Fractality and spectra . 336
6.4.4 Fractality ofthe finite section method for a class
ofband-dominated operators 339
6.5 Comments and references . 342
7 Axiomatization ofthe Limit Operators Approach
7.1 An axiomatic approach to the limit operators method 345
7.2 Operators on homogeneous groups 361
7.2.1 Homogeneous groups .. 361
7.2.2 Multiplication operators 362
7.2.3 Partition of unity ... 363
7.2.4 Convolution operators 364
7.2.5 Shift operators .... 365
x Contents
7.3 Fredholm criteria for convolution type operators with shift. 368
7.3.1 Operators on homogeneous groups 368
7.3.2 Operators on discrete subgroups 372
7.4 Comments and references . . . . . . . . 373
Bibliography 375
Index .... 387