Hankel Operators and Their Applications

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Springer Science & Business Media, Jan 14, 2003 - Mathematics - 784 pages
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The purpose of this book is to describe the theory of Hankel operators, one of the most important classes of operators on spaces of analytic func tions. Hankel operators can be defined as operators having infinite Hankel matrices (i. e. , matrices with entries depending only on the sum of the co ordinates) with respect to some orthonormal basis. Finite matrices with this property were introduced by Hankel, who found interesting algebraic properties of their determinants. One of the first results on infinite Han kel matrices was obtained by Kronecker, who characterized Hankel matri ces of finite rank as those whose entries are Taylor coefficients of rational functions. Since then Hankel operators (or matrices) have found numerous applications in classical problems of analysis, such as moment problems, orthogonal polynomials, etc. Hankel operators admit various useful realizations, such as operators on spaces of analytic functions, integral operators on function spaces on (0,00), operators on sequence spaces. In 1957 Nehari described the bounded Hankel operators on the sequence space 2. This description turned out to be very important and started the contemporary period of the study of Hankel operators. We begin the book with introductory Chapter 1, which defines Hankel operators and presents their basic properties. We consider different realiza tions of Hankel operators and important connections of Hankel operators with the spaces BMa and V MO, Sz. -Nagy-Foais functional model, re producing kernels of the Hardy class H2, moment problems, and Carleson imbedding operators.
 

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Contents

I
1
II
2
III
13
IV
19
V
23
VI
25
VII
37
VIII
39
LXXIII
417
LXXIV
420
LXXV
424
LXXVI
431
LXXVII
433
LXXVIII
440
LXXIX
443
LXXX
453

IX
46
X
61
XI
62
XII
66
XIII
71
XIV
74
XV
76
XVI
81
XVII
84
XVIII
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XIX
88
XX
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XXI
97
XXII
103
XXIII
109
XXIV
119
XXV
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XXVI
125
XXVII
126
XXVIII
131
XXIX
135
XXX
142
XXXI
145
XXXII
147
XXXIII
148
XXXIV
170
XXXV
173
XXXVI
187
XXXVII
212
XXXVIII
228
XXXIX
231
XL
232
XLI
239
XLII
243
XLIII
253
XLIV
257
XLV
267
XLVI
275
XLVII
281
XLVIII
292
XLIX
296
L
303
LI
306
LII
312
LIII
316
LIV
318
LV
338
LVI
341
LVII
344
LVIII
355
LIX
356
LX
359
LXI
362
LXII
371
LXIII
376
LXIV
379
LXV
380
LXVI
384
LXVII
390
LXVIII
402
LXIX
405
LXX
408
LXXI
415
LXXXI
454
LXXXII
457
LXXXIII
466
LXXXIV
472
LXXXV
473
LXXXVI
476
LXXXVII
479
LXXXVIII
485
LXXXIX
489
XC
493
XCI
497
XCII
499
XCIII
503
XCIV
509
XCV
516
XCVI
518
XCVII
520
XCVIII
522
XCIX
526
C
529
CI
531
CII
536
CIII
542
CIV
548
CV
553
CVI
554
CVII
555
CVIII
557
CIX
560
CX
561
CXI
562
CXII
565
CXIII
568
CXIV
574
CXV
578
CXVI
580
CXVII
586
CXVIII
592
CXIX
594
CXX
600
CXXI
602
CXXII
607
CXXIII
618
CXXIV
623
CXXV
636
CXXVI
643
CXXVII
661
CXXVIII
662
CXXIX
672
CXXX
681
CXXXI
682
CXXXII
685
CXXXIII
687
CXXXIV
693
CXXXV
694
CXXXVI
703
CXXXVII
705
CXXXVIII
717
CXXXIX
739
CXL
776
CXLI
780
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