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Bibliographic Details
Main Authors:	Rabinovich, Vladimir (Author), Roch, Steffen 1958- (Author), Silbermann, Bernd 1941- (Author)
Format:	Book
Language:	German
Published:	Basel Birkhäuser 2004
Series:	Operator theory 150
Subjects:	Differential operators Fredholm operators Operatortheorie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012892973&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	392 S.
ISBN:	3764370815

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MARC


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adam_text	LIT OPERATORS AND THEIR APPLICATIONS IN OPERATOR THEORY VLADIMIR RABINOVICH STEFFEN ROCH BERND SILBERMANN BIRKHAUSER VERLAG BASEL * BOSTON * BERLIN CONTENTS PREFACE XI 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 OF P-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 OF BAND-DOMINATED OPERATORS 2.1 BAND-DOMINATED OPERATORS 31 2.1.1 FUNCTION SPACES ON Z N 31 2.1.2 MATRIX REPRESENTATION 32 2.1.3 OPERATORS OF MULTIPLICATION 33 2.1.4 BAND AND BAND-DOMINATED OPERATORS 35 2.1.5 LIMIT OPERATORS OF BAND-DOMINATED OPERATORS 40 2.1.6 RICH BAND-DOMINATED OPERATORS 43 2.2 V- FREDHOLMNESS OF RICH BAND-DOMINATED OPERATORS 45 2.2.1 THE MAIN THEOREM ON V- 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 OF A SYMBOL CALCULUS 56 2.2.5 APPENDIX B: COMMUTATIVE BANACH ALGEBRAS 59 CONTENTS 2.3 LOCAL P-FREDHOLMNESS: ELEMENTARY THEORY 61 2.3.1 LOCAL OPERATOR SPECTRA AND LOCAL INVERTIBILITY 61 2.3.2 P7-COMPACTNESS, P7-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 OF THE 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(Z N ) 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 M(SO(Z N )) 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 PROOF OF THEOREM 2.4.27 95 2.4.9 APPENDIX B: A THIRD PROOF OF THEOREM 2.4.27 100 2.5 OPERATORS IN THE DISCRETE WIENER ALGEBRA 103 2.5.1 THE WIENER ALGEBRA 103 2.5.2 FREDHOLMNESS OF OPERATORS IN THE WIENER ALGEBRA 107 2.6 BAND-DOMINATED OPERATORS WITH SPECIAL COEFFICIENTS ILL 2.6.1 BAND-DOMINATED OPERATORS WITH ALMOST PERIODIC COEFFICIENTS ILL 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 1? 124 2.6.5 DIFFERENCE OPERATORS OF SECOND ORDER 128 2.6.6 DISCRETE SCHRODINGER OPERATORS 131 2.7 INDICES OF FREDHOLM 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 IFI-GROUP OF A(Z) 139 2.7.4 THE I^I-GROUP OF A 142 2.7.5 PROOF OF THEOREM 2.7.1 144 2.7.6 UNITARY BAND-DOMINATED OPERATORS 147 2.8 COMMENTS AND REFERENCES 150 CONVOLUTION TYPE OPERATORS ON R N 3.1 BAND-DOMINATED OPERATORS ON L P (R N ) 153 3.1.1 APPROXIMATE IDENTITIES AND P-FREDHOLMNESS 153 3.1.2 SHIFTS AND LIMIT OPERATORS 155 CONTENTS 3.1.3 DISCRETIZATION 155 3.1.4 BAND-DOMINATED OPERATORS ON L P (R N ) 157 3.2 OPERATORS OF CONVOLUTION 159 3.2.1 COMPACTNESS OF SEMI-COMMUTATORS 159 3.2.2 COMPACTNESS OF COMMUTATORS 164 3.3 FREDHOLMNESS OF CONVOLUTION TYPE OPERATORS 169 3.3.1 OPERATORS OF CONVOLUTION TYPE 169 3.3.2 FREDHOLMNESS 172 3.4 COMPRESSIONS OF CONVOLUTION TYPE OPERATORS 179 3.4.1 COMPRESSIONS OF OPERATORS IN A{BUC(R N ), C P ) 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 OF FUNCTIONS 193 3.5 A WIENER ALGEBRA OF CONVOLUTION-TYPE OPERATORS 194 3.5.1 FREDHOLMNESS OF OPERATORS IN THE WIENER ALGEBRA 194 3.5.2 THE ESSENTIAL SPECTRUM OF SCHRODINGER OPERATORS 195 3.6 COMMENTS AND REFERENCES 199 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 PSEUDODIFFERENTIAL OPERATORS 204 4.1.4 FORMAL SYMBOLS 205 4.1.5 PSEUDODIFFERENTIAL OPERATORS WITH DOUBLE SYMBOLS 206 4.1.6 BOUNDEDNESS ON L 2 (R N ) 207 4.1.7 CONSEQUENCES OF THE CALDERON-VAILLANCOURT THEOREM .... 210 4.2 BI-DISCRETIZATION OF OPERATORS ON L 2 (M. N ) 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 OF PSEUDODIFFERENTIAL OPERATORS 218 4.3.1 A WIENER ALGEBRA ON L 2 (R N ) 218 4.3.2 FREDHOLMNESS OF OPERATORS IN W S (L 2 (R N )) 222 4.3.3 FREDHOLM PROPERTIES OF PSEUDODIFFERENTIAL OPERATORS IN OPS$ 0 224 II 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 OF NONZERO 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 OF MELLIN 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 PSEUDODIFFERENCE OPERATORS 5.1 PSEUDODIFFERENCE OPERATORS 267 5.2 FREDHOLMNESS OF PSEUDODIFFERENCE 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-LINDELOF PRINCIPLE 279 5.4 SLOWLY OSCILLATING PSEUDODIFFERENCE OPERATORS 280 5.4.1 FREDHOLMNESS ON Z P -SPACES 280 5.4.2 FREDHOLMNESS ON WEIGHTED SPACES, PHRAGMEN-LINDELOF PRINCIPLE 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 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 OF EIGENFUNCTIONS 299 5.8.3 SEMI-PERIODIC SCHRODINGER OPERATORS 301 5.9 COMMENTS AND REFERENCES 302 FINITE SECTIONS OF BAND-DOMINATED OPERATORS 6.1 STABILITY OF THE 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 OF BAND-DOMINATED OPERATORS ON Z 1 AND Z 2 312 6.2.1 BAND-DOMINATED OPERATORS ON Z 1 : THE GENERAL CASE 313 6.2.2 BAND-DOMINATED OPERATORS ON Z 1 : SLOWLY OSCILLATING COEFFICIENTS 315 6.2.3 BAND-DOMINATED OPERATORS ON Z 2 318 6.2.4 FINITE SECTIONS OF CONVOLUTION TYPE OPERATORS 320 6.3 SPECTRAL APPROXIMATION 321 6.3.1 WEAKLY SUFFICIENT FAMILIES AND SPECTRA 322 6.3.2 INTERLUDE: SPECTRA OF BAND-DOMINATED OPERATORS ON HILBERT SPACES 326 6.3.3 ASYMPTOTIC BEHAVIOR OF NORMS 327 6.3.4 ASYMPTOTIC BEHAVIOR OF SPECTRA 328 6.4 FRACTALITY OF APPROXIMATION 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 OF THE FINITE SECTION METHOD FOR A CLASS OF BAND-DOMINATED OPERATORS 339 6.5 COMMENTS AND REFERENCES 342 AXIOMATIZATION OF THE 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
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series	Operator theory
series2	Operator theory
spellingShingle	Rabinovich, Vladimir Roch, Steffen 1958- Silbermann, Bernd 1941- Limit operators and their applications in operator theory Operator theory Differential operators Fredholm operators Operatortheorie (DE-588)4075665-8 gnd
subject_GND	(DE-588)4075665-8
title	Limit operators and their applications in operator theory
title_auth	Limit operators and their applications in operator theory
title_exact_search	Limit operators and their applications in operator theory
title_full	Limit operators and their applications in operator theory Vladimir Rabinovich ; Steffen Roch ; Bernd Silbermann
title_fullStr	Limit operators and their applications in operator theory Vladimir Rabinovich ; Steffen Roch ; Bernd Silbermann
title_full_unstemmed	Limit operators and their applications in operator theory Vladimir Rabinovich ; Steffen Roch ; Bernd Silbermann
title_short	Limit operators and their applications in operator theory
title_sort	limit operators and their applications in operator theory
topic	Differential operators Fredholm operators Operatortheorie (DE-588)4075665-8 gnd
topic_facet	Differential operators Fredholm operators Operatortheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012892973&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000970
work_keys_str_mv	AT rabinovichvladimir limitoperatorsandtheirapplicationsinoperatortheory AT rochsteffen limitoperatorsandtheirapplicationsinoperatortheory AT silbermannbernd limitoperatorsandtheirapplicationsinoperatortheory

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