Harmonic Analysis of Operators on Hilbert Space

Title Page......Page 1
Copyright......Page 2
Foreword to The Original Edition......Page 3
Foreword to the English Edition......Page 7
CONTENTS......Page 9
1. Unilateral shifts. Wold decomposition......Page 13
2. Bilateral shifts......Page 16
3. Contractions. Invariant vectors and canonical decomposition......Page 18
4. Isometric and unitary dilations of a contraction......Page 22
5. Matrixonstruction of the unitary dilation......Page 28
6. Commutative systems of isometrics and contractions......Page 31
7. Positive definite functions on a group......Page 36
8. Some applications......Page 40
9. Regular unitary dilations of commutative systems of contractions......Page 44
10. Another method to construct isometric dilations. An example concerning systems of differential equations......Page 51
11. Unitary p-dilations......Page 57
Notes......Page 63
1. Structure of the space of the minimal unitary dilation......Page 69
2. Structure of the space of the minimal isometric dilation. Dilation of commutants......Page 73
3. The residual part of the minimal unitary dilation. Quasi-affinities and quasi-similarities......Page 80
4. A classification of contractions. Canonical triangulations......Page 84
5. Invariant subspaces and their behaviour with respect to quasi-similarity......Page 88
6. Spectral relations......Page 93
7. Spectral multiplicity......Page 98
S. Similarity of the operators of class rI' to contractions......Page 104
Notes......Page 107
1. Hardy classes. Inner and outer functions......Page 111
2. Functional calculus : the classes H- and Hz......Page 121
3. The role of the outer functions in the functional calculus......Page 130
4. Contractions of class Co and their minimal functions......Page 134
5. Relations between minimal function and spectrum......Page 138
6. Relations between minimal function and invariant subspaces......Page 141
7. Characteristic vectors of a contraction of class Co and unicellularity......Page 146
8. Continuous one parameter semi-groups of contractions; their generators and co-generators......Page 152
9. Unitary dilation of semi-groups. Semi-groups of isometrics......Page 157
Notes......Page 164
1. Calculation rules......Page 167
2. Representation of q(T) as a limit of p,(T)......Page 173
3. Functions limited by a sector......Page 175
4. Accretive and dissipative operators......Page 179
5. Fractional powers of maximal accretive operators......Page 185
Notes......Page 192
1. The spaces L'(%) and H'(41)......Page 195
2. Operator valued analytic functions. Inner and outer functions......Page 198
3. Lemmas on Fourier representations. Invariant subspaces for unilateral shifts.......Page 204
4. Factorizations......Page 212
5. Existence of non-trivial factorizations for contractive analytic functions.......Page 222
6. Contractile analytic functions with scalar multiples......Page 228
7. Factorization theorems for functions with scalar multiple......Page 239
8. Analytic kernels......Page 243
Notes......Page 245
1. Characteristic functions......Page 249
2. Functional models for a given contraction......Page 253
3. Functional models for given contractive analytic functions......Page 255
4. The characteristic function and the spectrum......Page 271
5. The characteristic function and the minimal function......Page 277
6. Spectral type of the minimal unitary dilation......Page 283
Notes......Page 288
1. The fundamental theorem......Page 293
2. Some additional propositions......Page 301
3. Regular factorizations......Page 306
4. Arithmetic of regular divisors......Page 315
5. Invariant subspaces for contractions of class C......Page 322
6. Spectral decomposition of contractions of class C whose characteristic function admits a scalar multiple......Page 328
Notes......Page 331
1. Existence of a scalar multiple for the characteristic function......Page 335
2. Decomposition Cc---C11......Page 339
3. Spectral decomposition of weak contractions......Page 346
Notes......Page 352
1. Contractions which are similar to unitary operators......Page 353
2. Quasi-similarity of contractions......Page 357
3. Cri teria of unicellularity......Page 362
4. Disaipalive operators. Class (610')......Page 368
5. Dissipalive operators similar to self-adjoint oats......Page 376
Notes......Page 378
REFERENCES......Page 385
INDEX......Page 397
NOTATIONS......Page 401