Verfügbarkeit: Holomorphic operator functions of one variable and applications

Holomorphic operator functions of one variable and applications: methods from complex analysis in several variables

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Bibliographische Detailangaben
Hauptverfasser:	Gohberg, Yiśrāʿēl Z. 1928-2009 (VerfasserIn), Leiterer, Jürgen (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Basel [u.a.] Birkhäuser 2009
Schriftenreihe:	Operator theory 192
Schlagworte:	Holomorphic functions Operator theory Holomorphe Funktion Operatorfunktion
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XX, 422 S. 235 mm x 165 mm
ISBN:	9783034601252

Internformat

MARC


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Datensatz im Suchindex

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adam_text	CONTENTS PREFACE XI INTRODUCTION XIII NOTATION XIX 1 ELEMENTARY PROPERTIES OF HOLOMORPHIC FUNCTIONS 1 1.1 DEFINITION AND FIRST PROPERTIES . . . . . . . . . . . . . . . . . . . . . 1 1.2 THE MAXIMUM PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 CONTOUR INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 THE CAUCHY INTEGRAL THEOREM . . . . . . . . . . . . . . . . . . . . . 11 1.5 THE CAUCHY FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 THE HAHN-BANACH CRITERION . . . . . . . . . . . . . . . . . . . . . . 15 1.7 A CRITERION FOR THE HOLOMORPHY OF OPERATOR FUNCTIONS . . . . . . . . 18 1.8 POWER SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.9 LAURENT SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.10 ISOLATED SINGULARITIES . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.11 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 SOLUTION OF U = F AND APPLICATIONS 29 2.1 THE POMPEIJU FORMULA FOR SOLUTIONS OF U = F ON COMPACT SETS . . 29 2.2 RUNGE APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 SOLUTION OF U = F ON OPEN SETS . . . . . . . . . . . . . . . . . . . . 41 2.4 O E -COCYCLES AND THE MITTAG-LEFFLER THEOREM . . . . . . . . . . . . . 43 2.5 RUNGE APPROXIMATION FOR INVERTIBLE SCALAR FUNCTIONS AND THE WEIERSTRASS PRODUCT THEOREM . . . . . . . . . . . . . . . . . . . . . 44 2.6 O E -COCYCLES WITH PRESCRIBED ZEROS AND A STRONGER VERSION OF THE MITTAG-LEFFLER THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7 GENERALIZATION OF THE WEIERSTRASS PRODUCT THEOREM . . . . . . . . . 54 2.8 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 VIII CONTENTS 3 SPLITTING AND FACTORIZATION WITH RESPECT TO A CONTOUR 59 3.1 SPLITTING WITH RESPECT TO A CONTOUR . . . . . . . . . . . . . . . . . . 59 3.2 SPLITTING AND THE CAUCHY INTEGRAL . . . . . . . . . . . . . . . . . . . 61 3.3 H¨OLDER CONTINUOUS FUNCTIONS SPLIT . . . . . . . . . . . . . . . . . . . 65 3.4 THE SPLITTING BEHAVIOR OF DIFFERENTIABLE FUNCTIONS . . . . . . . . . . 72 3.5 APPROXIMATION OF H¨OLDER CONTINUOUS FUNCTIONS . . . . . . . . . . . . 75 3.6 EXAMPLE: A NON-SPLITTING CONTINUOUS FUNCTION . . . . . . . . . . . . 77 3.7 THE ADDITIVE LOCAL PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . 82 3.8 FACTORIZATION OF SCALAR FUNCTIONS WITH RESPECT TO A CONTOUR. FIRST REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.9 FACTORIZATION OF H¨OLDER FUNCTIONS . . . . . . . . . . . . . . . . . . . 89 3.10 FACTORIZATION OF WIENER FUNCTIONS . . . . . . . . . . . . . . . . . . . 91 3.11 THE MULTIPLICATIVE LOCAL PRINCIPLE . . . . . . . . . . . . . . . . . . . 93 3.12 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4 THE ROUCH´E THEOREM FOR OPERATOR FUNCTIONS 97 4.1 FINITE MEROMORPHIC FREDHOLM FUNCTIONS . . . . . . . . . . . . . . . 97 4.2 INVERTIBLE FINITE MEROMORPHIC FREDHOLM FUNCTIONS . . . . . . . . . . 101 4.3 SMITH FACTORIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4 THE ROUCH´E THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 MULTIPLICATIVE COCYCLES ( O G -COCYCLES) 113 5.1 TOPOLOGICAL PROPERTIES OF GL ( E ) . . . . . . . . . . . . . . . . . . . 114 5.2 TWO FACTORIZATION LEMMAS . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 O E -COCYCLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 RUNGE APPROXIMATION OF G -VALUED FUNCTIONS. FIRST STEPS . . . . . . 125 5.5 THE CARTAN LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.6 O G -COCYCLES. DEFINITIONS AND STATEMENT OF THE MAIN RESULT . . . . . 131 5.7 REFINEMENT OF THE COVERING . . . . . . . . . . . . . . . . . . . . . . . 133 5.8 EXHAUSTING BY COMPACT SETS . . . . . . . . . . . . . . . . . . . . . . 137 5.9 PROOF OF THE MAIN THEOREM IN THE CASE OF SIMPLY CONNECTED OPEN SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.10 RUNGE APPROXIMATION OF G -VALUED FUNCTIONS. GENERAL CASE . . . . . 143 5.11 PROOF OF THE MAIN THEOREM IN THE GENERAL CASE . . . . . . . . . . . . 147 5.12 O G ( E ) -COCYCLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.13 WEIERSTRASS THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.14 WEIERSTRASS THEOREMS FOR G * ( E ) AND G * ( E )-VALUED FUNCTIONS . . . 156 5.15 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6 FAMILIES OF SUBSPACES 159 6.1 THE GAP METRIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 KERNEL AND IMAGE OF OPERATOR FUNCTIONS . . . . . . . . . . . . . . . 172 6.3 HOLOMORPHIC SECTIONS OF CONTINUOUS FAMILIES OF SUBSPACES . . . . . . 184 CONTENTS IX 6.4 HOLOMORPHIC FAMILIES OF SUBSPACES . . . . . . . . . . . . . . . . . . 185 6.5 EXAMPLE: A HOLOMORPHIC FAMILY OF SUBSPACES WITH JUMPING ISOMORPHISM TYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.6 INJECTIVE FAMILIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.7 SHUBIN FAMILIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.8 COMPLEMENTED FAMILIES . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.9 FINITE DIMENSIONAL AND FINITE CODIMENSIONAL FAMILIES . . . . . . . . 209 6.10 ONE-SIDED AND GENERALIZED INVERTIBLE HOLOMORPHIC OPERATOR FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.11 EXAMPLE: A GLOBALLY NON-TRIVIAL COMPLEMENTED HOLOMORPHIC FAMILY OF SUBSPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.12 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7 PLEMELJ-MUSCHELISHVILI FACTORIZATION 219 7.1 DEFINITIONS AND FIRST REMARKS ABOUT FACTORIZATION . . . . . . . . . . 220 7.2 THE ALGEBRA OF WIENER FUNCTIONS AND OTHER SPLITTING R -ALGEBRAS . . 222 7.3 H¨OLDER CONTINUOUS AND DIFFERENTIABLE FUNCTIONS . . . . . . . . . . . . 230 7.4 REDUCTION OF THE FACTORIZATION PROBLEM TO FUNCTIONS, HOLOMORPHIC AND INVERTIBLE ON C * . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.5 FACTORIZATION OF HOLOMORPHIC FUNCTIONS CLOSE TO THE UNIT . . . . . . 240 7.6 REDUCTION OF THE FACTORIZATION PROBLEM TO POLYNOMIALS IN Z AND 1 /Z 240 7.7 THE FINITE DIMENSIONAL CASE . . . . . . . . . . . . . . . . . . . . . . 242 7.8 FACTORIZATION OF G * ( E )-VALUED FUNCTIONS . . . . . . . . . . . . . . . 245 7.9 THE FILTRATION OF AN OPEATOR FUNCTION WITH RESPECT TO A CONTOUR . . 251 7.10 A GENERAL CRITERION FOR THE EXISTENCE OF FACTORIZATIONS . . . . . . . . 259 7.11 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8 WIENER-HOPF OPERATORS, TOEPLITZ OPERATORS AND FACTORIZATION 269 8.1 HOLOMORPHIC OPERATOR FUNCTIONS . . . . . . . . . . . . . . . . . . . . 269 8.2 FACTORIZATION OF G * ( E )-VALUED FUNCTIONS . . . . . . . . . . . . . . . 273 8.3 THE SPACE L 2 (* , H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 8.4 OPERATOR FUNCTIONS WITH VALUES ACTING IN A HILBERT SPACE . . . . . . 287 8.5 FUNCTIONS CLOSE TO THE UNIT OPERATOR OR WITH POSITIVE REAL PART . . . 291 8.6 BLOCK T¨OPLITZ OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . 297 8.7 THE FOURIER TRANSFORM OF L 1 ( R , E ) . . . . . . . . . . . . . . . . . . 305 8.8 THE FOURIER ISOMETRY U OF L 2 * R , H * . . . . . . . . . . . . . . . . . 313 8.9 THE ISOMETRY V FROM L 2 ( T , H ) ONTO L 2 ( R , H ) . . . . . . . . . . . . 317 8.10 THE ALGEBRA OF OPERATOR FUNCTIONS L ( H ) * L 1 * R , L ( H ) * . . . . . . 320 8.11 FACTORIZATION WITH RESPECT TO THE REAL LINE . . . . . . . . . . . . . . 324 8.12 WIENER-HOPF INTEGRAL OPERATORS IN L 2 * [0 , * [ , H * . . . . . . . . . . . 325 8.13 AN EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 8.14 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 X CONTENTS 9 MULTIPLICATIVE COCYCLES WITH RESTRICTIONS ( F -COCYCLES) 343 9.1 F -COCYCLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 9.2 THE MAIN RESULTS ON COCYCLES WITH RESTRICTIONS. FORMULATION AND REDUCTION TO O D,Z,M . . . . . . . . . . . . . . . . . . . . . . . . . . 346 9.3 THE CARTAN LEMMA WITH RESTRICTIONS . . . . . . . . . . . . . . . . . 347 9.4 SPLITTING OVER SIMPLY CONNECTED OPEN SETS AFTER SHRINKING . . . . . . 352 9.5 RUNGE APPROXIMATION ON SIMPLY CONNECTED OPEN SETS . . . . . . . . 354 9.6 SPLITTING OVER SIMPLY CONNECTED OPEN SETS WITHOUT SHRINKING . . . . 358 9.7 RUNGE APPROXIMATION. THE GENERAL CASE . . . . . . . . . . . . . . . 362 9.8 THE OKA-GRAUERT PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . 365 9.9 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 10 GENERALIZED INTERPOLATION PROBLEMS 369 10.1 WEIERSTRASS THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . . 369 10.2 RIGHT- AND TWO-SIDED WEIERSTRASS THEOREMS . . . . . . . . . . . . . . 371 10.3 WEIERSTRASS THEOREMS FOR G * ( E )- AND G * ( E )-VALUED FUNCTIONS . . . 374 10.4 HOLOMORPHIC G * ( E )-VALUED FUNCTIONS WITH GIVEN PRINCIPAL PARTS OF THE INVERSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 10.5 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 11 HOLOMORPHIC EQUIVALENCE, LINEARIZATION AND DIAGONALIZATION 379 11.1 INTRODUCTORY REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . 379 11.2 LINEARIZATION BY EXTENSION AND EQUIVALENCE . . . . . . . . . . . . . 380 11.3 LOCAL EQUIVALENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 11.4 A THEOREM ON LOCAL AND GLOBAL EQUIVALENCE . . . . . . . . . . . . . . 392 11.5 THE FINITE DIMENSIONAL CASE . . . . . . . . . . . . . . . . . . . . . . 394 11.6 LOCAL AND GLOBAL EQUIVALENCE FOR FINITE MEROMORPHIC FREDHOLM FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 11.7 GLOBAL DIAGONALIZATION OF FINITE MEROMORPHIC FREDHOLM FUNCTIONS . 406 11.8 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 BIBLIOGRAPHY 413 INDEX 419
any_adam_object	1
author	Gohberg, Yiśrāʿēl Z. 1928-2009 Leiterer, Jürgen
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author_facet	Gohberg, Yiśrāʿēl Z. 1928-2009 Leiterer, Jürgen
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dewey-ones	515 - Analysis
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dewey-tens	510 - Mathematics
discipline	Mathematik
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series	Operator theory
series2	Operator theory
spelling	Gohberg, Yiśrāʿēl Z. 1928-2009 Verfasser (DE-588)118915878 aut Holomorphic operator functions of one variable and applications methods from complex analysis in several variables Israel Gohberg ; Jürgen Leiterer Basel [u.a.] Birkhäuser 2009 XX, 422 S. 235 mm x 165 mm txt rdacontent n rdamedia nc rdacarrier Operator theory 192 Holomorphic functions Operator theory Holomorphe Funktion (DE-588)4025645-5 gnd rswk-swf Operatorfunktion (DE-588)4202830-9 gnd rswk-swf Operatorfunktion (DE-588)4202830-9 s Holomorphe Funktion (DE-588)4025645-5 s DE-604 Leiterer, Jürgen Verfasser aut Erscheint auch als Online-Ausgabe 978-3-0346-0126-9 Operator theory 192 (DE-604)BV000000970 192 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017726587&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Gohberg, Yiśrāʿēl Z. 1928-2009 Leiterer, Jürgen Holomorphic operator functions of one variable and applications methods from complex analysis in several variables Operator theory Holomorphic functions Operator theory Holomorphe Funktion (DE-588)4025645-5 gnd Operatorfunktion (DE-588)4202830-9 gnd
subject_GND	(DE-588)4025645-5 (DE-588)4202830-9
title	Holomorphic operator functions of one variable and applications methods from complex analysis in several variables
title_auth	Holomorphic operator functions of one variable and applications methods from complex analysis in several variables
title_exact_search	Holomorphic operator functions of one variable and applications methods from complex analysis in several variables
title_full	Holomorphic operator functions of one variable and applications methods from complex analysis in several variables Israel Gohberg ; Jürgen Leiterer
title_fullStr	Holomorphic operator functions of one variable and applications methods from complex analysis in several variables Israel Gohberg ; Jürgen Leiterer
title_full_unstemmed	Holomorphic operator functions of one variable and applications methods from complex analysis in several variables Israel Gohberg ; Jürgen Leiterer
title_short	Holomorphic operator functions of one variable and applications
title_sort	holomorphic operator functions of one variable and applications methods from complex analysis in several variables
title_sub	methods from complex analysis in several variables
topic	Holomorphic functions Operator theory Holomorphe Funktion (DE-588)4025645-5 gnd Operatorfunktion (DE-588)4202830-9 gnd
topic_facet	Holomorphic functions Operator theory Holomorphe Funktion Operatorfunktion
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017726587&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000970
work_keys_str_mv	AT gohbergyisraʿelz holomorphicoperatorfunctionsofonevariableandapplicationsmethodsfromcomplexanalysisinseveralvariables AT leitererjurgen holomorphicoperatorfunctionsofonevariableandapplicationsmethodsfromcomplexanalysisinseveralvariables

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