INDEX

Abel means 348 bounded semigroup 409 abstract Cauchy problem 199 Bram–Halmos criterion for subnormality 76 algebra of compact operators 293 Burnside theorem 290 algebraic operator 254, 256 ∗ almost algebraic operator 255, 257 C -algebra 29, 385 almost periodic group 403 Calkin algebra 304 almost periodic operator 403 Carleman transform 421 analytic semigroup 252 Carleson theorem 281 antisymmetric tensor product 146 Cauchy inequality 261 anti-commutator 387 Cauchy integral formula 70, 248 approximate eigenvalue 416 Cauchy problem 325, 414 approximate point spectrum 416 Cauchy–Schwarz inequality 65, 166, 356 Araki–Lieb–Thirring inequality 120 Cayley transform 390 arithmetic operator mean 157 Ces`aroaverages 250 ascent 310 Ces`aromeans 112, 113, 308, 348, 351 asymptotically periodic sequence 409 common boundary point of Gerschgorin circles Atiyah–Singer index 398 431 attractor 329 commutativity of self-adjoint operators 54 Aupetit–Zem´anektheorem 295 commutator, quasinilpotent 293 automorphic function 318 compact commutative semigroup 218 compact group 219 Banach–Steinhaus argument 64, 67 complementary triangular forms 444 Berezin transform 364, 398 complete isometry 228 Berger characterization of subnormality 77 completely bounded mapping 228 Bergman operator 386 completely bounded norm 228 Bergman shift 316 complex K¨ahlergeometry 385 Bergman space 284, 363, 374 composition operator 366 Beurling–Lax theorem 321 compound matrix 146 Binet–Cauchy theorem 147 connected component 438 Bishop property 266, 282 consecutive differences of powers 300 Bishop theorem 270 boundedness 301, 302 boundary projection 218, 219, 222, 404 convergence 302, 308 boundary spectrum splitting-off theorem 218 convergence to zero 10, 251, 304, 305, 381, boundary subspace 217, 404 406 bounded approximate identity 13 power dominated 301 bounded radial function 376 relative compactness 306, 307, 308, 381

[453] 454 INDEX consecutive differences of powers, Gelfand–Naimark majorization theorem 131 sublinear decay 250, 262 generalized eigenvector 267 constant norm of resolvent 39 generalized numerical radius 340, 348 continuous semigroup 194 generalized scalar operator 268, 274 contraction 224 generalized singular values 126 convex function 123 generalized Wold decomposition 316 convex hull of Gerschgorin centres 431 generator 331 convex open cone 388 geometric criteria of rationality of the bound- convex structure 222 ary spectrum 223 corona problem 281 geometric operator mean 157 Gerschgorin centres 430, 436 Dahlquist–Lozinski˘ıformula 430 Gerschgorin discs 428 decomposable operator 265, 271 Gerschgorin radii 430 decomposition of the identity 198 Gerschgorin region 428 decreasing rearrangement 123 Gerschgorin theorem 428 descent 310 Gleason part 320 determinant inequalities 175 Golden–Thompson inequality 120 diagonalizable matrix 194, 195 strengthened 150 direct Cauchy theorem 320 Green function 320 discrete Hilbert transform 110 group of operators 289, 290 double power bounded 9, 217, 219, 310 group of unitary operators 427 doubly stochastic matrix 124 growth function 247, 255, 259, 261 growth of entire function 249 eigenfunction expansion 265 eigenvalue 416, 430 Haagerup norm 229 eigenvector 267, 430 Haagerup tensor product 229 elementary rational matrix functions 446 Haar measure 418 empathy relation 327 Hankel matrix 79, 80 Engel theorem 288 Hankel operator, compactness 379 ergodic semigroup 412 Hankel rank 86 ergodic spectrum 402 Hardy space 27, 316, 317, 362 essential norm 126 Hardy type operator 316, 320 Esterle theorem 10 harmonic function 60, 371 evolution problem 205 harmonic majorant 317 exponential type 10 harmonic measure 319 finite difference system 339 harmonic operator mean 157 Fock space 363, 392 Hausdorff dimension 321 Fourier transform 197 Helson set 422 Fredholm operator 28, 30, 280 Hilbert module 232 Frobenius norm 430, 436 Hilbert transform 112 Fuglede theorem 54 Hilbert–Schmidt class 138 functional calculus 194, 297 Hille theorem 10 Furuta inequality 163, 170 Hille–Yosida theorem 248, 249, 335, 412 holomorphic cross-section 317 Gaussian measure 363 H¨olderinequality 139, 152, 156, 170 Gelfand theorem 10, 310, 312 Horn majorization 131 Gelfand transform 30, 36, 417, 420 hyponormal operator 75, 422 INDEX 455 idempotent 56 Lomonosov theorem 293 inner boundary 321 L¨owner–Heinzinequality 143, 148 integrated semigroup 205, 326 Lumer formula 430 interior spectrum 217 Lyubich extension theorem 417 interior subspace 217 Lyubich theorem on nonemptieness of spec- invariant subspace 277, 293, 420, 422, 433, 449 trum 417 irreducible matrix 430 irreducible semigroup of operators 292 M-hyponormal operator 274 isometry 17, 404 Macaev ideal 138 iterative methods 256 majorization 119 Markov operator 408 Jordan algebra 387 McCoy theorem 443 Jordan form 266, 312, 428 mild solution 414 Jordan homomorphism 55 minimal polynomial 254 Jordan triple 385 Mlak absolute continuity theorem 315 M¨obiusbounded operator 70, 71, 345 Kaplansky theorem 288 M¨obius-invariance 370 Katznelson–Tzafriri theorem 11, 407 M¨obiusmapping 249 Kittaneh inequality 173 moments of a sequence 76, 77 Kolchin theorem 288 multiplication operator 321, 373, 385 Kreiss matrix theorem 247, 250, 259, 312, 339, von Neumann algebra 231 352 von Neumann condition 353 Ky Fan majorization 128 nilpotent matrix 266 Laguerre polynomials 14 nilpotent operator 50, 291 Laplace–Beltrami operator 399 nonnegative inverse eigenvalue problem 187 Laplace operator 185 nonnegative matrix 187, 290 Laplace–Stieltjes transform 198 nonnegative operator 222 norm, majorization 429 Laplace transform 201, 326, 394, 445 normal extension 316 Lasota–Li–Yorke theorem 408 normal matrix 429, 435 de Leeuw–Glicksberg decomposition 218, 403, normal operator 54, 75 411, 416, 420 numerical radius 71 347, 435 left operator module 230 numerical range 430, 436, 437 LeVeque–Trefethen–Spijker technique 262 Levitzki theorem 288, 291 operator mean 157 Lidskii theorem 176 operator monotone function 143 Lidskii–Wielandt majorization theorem 119, operator of Hardy type 316 128 orbit 298 Lie bracket 288 orthonormal basis 429 limit isometric semigroup 402 linear preserver 49 Parseval formula 362 Liouville theorem 52, 252, 256, 274 Parseval identity 258 Littlewood–Paley theorem 116 partial isometry 390 local principle of Allan and Douglas 29 peripheral spectrum 217, 297 locally compact abelian group 105 permutable function 287 logarithmic growth 352 permutable spectral radius 292 log-majorization 120, 125 permutable spectrum 290 456 INDEX permutable trace 293 relatively compact orbit 299, 307, 311 permutation matrix 124, 430 reproducing functional 361 Perron–Frobenius theorem 188 reproducing kernel 361 Phragm´en–Lindel¨oftechnique 253 resolvent 249, 274, 297, 334, 402 Plancherel theorem 65 resolvent condition 69, 247, 259, 272, 344, 345, Planck constant 385 348, 351, 353 Poincar´e’sminmax principle 184 Riemann mapping theorem 366 point spectrum 402 Riesz operator 255, 260, 310 Poisson kernel 345 right operator module 230 polar decomposition 130, 131 Rota theorem 423 pole of resolvent 254, 297, 308 Rouch´etheorem 431 polyhedral normed space 224 polynomially hyponormal operator 76 Sarason example 375 positive Hankel extension 87 scalar operator 198 positive Hankel matrix 80 scalar-type spectral operator 106, 117 positive Toeplitz matrix 89 Schatten p-class 138, 293 Post–Widder theorem 331 Schur triangularization theorem 188, 343, 353, potent operator 56 427, 431, 443 power bounded element 9, 61, 71, 298, 302, Segal–Bargmann space 26, 44, 363, 392, 396 305, 308 self-adjoint idempotent 55 power bounded operator 59, 114, 217, 225, 247, self-adjoint matrix 429 249, 251, 260, 342, 345, 381, 402, 422 self-adjoint operator on Banach space 105 power bounded orbit 311 semigroup of compact quasinilpotent operators power dominated 13, 298, 301,305 293 power inequality 348 semigroup of idempotents 289 power operator mean 157 semigroup of nilpotent operators 288 Powers–Størmer inequality 143 semigroup of operators 287 projection 434 C-semigroup 326 projection-valued 197 Shilov boundary 389, 419 projective operator tensor product 230 Shilov idempotent theorem 418, 420 pseudospectrum 25 similar operators 15 pseudo-hyperbolic distance 374 similarity 267, 428 pseudo-unitary group 387 similarity to contraction 346, 422 similarity to self-adjoint operator 66 quadratic equation 429 similarity to unitary operator 59, 63 quadratic operator 433, 434 simple nest property 451 quasi-nilpotent operator 253, 298, 449 simple pole 381 quasi-power-bounded 13 single-valued extension property 421 quasi-similar operators 280 singular values 427, 435 spectral continuity 183 Radon–Nikodym property 200, 335 spectral decomposition 265 rank 50 spectral family 107 Rayleigh quotient 184 spectral idempotent 297 reducible set of operators 287, 290 spectral inclusion theorem 316 reducing subspace 434 spectral mapping theorem 252, 417 regularized semigroup 326 spectral operator 268 relative entropy 154 spectral projection 195, 197, 252 INDEX 457 spectral radius 9, 50, 288, 291, 292, 297, 340, trigonometrically well-bounded operator 109 347 Trotter–Kato product formula 120, 149 spectrum 9, 36, 50, 68, 69, 106, 109, 188, 249, truncated complex moment problem 90 266, 273, 287, 290, 297, 315, 340, 381, 382, truncated Hamburger moment problem 80, 87, 402, 419, 427 88 Spijker lemma 262, 263 truncated Stieltjes moment problem 81 square bounded in average 59, 61, 63, 65 stable subspace 404 uniform boundedness principle 327 Stirling approximation 249, 252 uniform linear growth 344 Stone theorem 67 uniformly convergent means 414 uniformly radical Banach algebra 14 strongly constrictive C0-semigroup 415 strongly constrictive operator 407 unilateral weighted shift 76 strongly continuous semigroup 66, 199 unipotent operator 288, 290 strongly stable operator 402 unitarily invariant norm 135 strongly stable semigroup 409 unitary equivalence 316, 397 Sturm–Liouville operator 183 unitary extension 17 subharmonic function 52 unitary group 427, 433 submultiplicativity of spectral radius 288, 292 unitary matrix 241, 342, 353, 429, 431, 437 submultiplicativity of spectrum 288, 292, 293 unitary operator 19, 55, 108, 317 subnormal completion problem 79 unitary similarity orbit 427, 431 subnormal operator 75, 316 unit sphere of operators 428 Sushkevich kernel 218 universal covering space 317 symmetric gauge function 134 Vesentini theorem 52 symmetrically normed ideal 137 Volterra algebra 14 Szeg˝omeasure 320 Volterra operator 253, 449 Sz.-Nagy–Foia¸stheorem 423 Weyl majorization theorem 147 Tauberian theorem 377 weak log-majorization 125 Taylor coefficients 11, 249, 344 weak majorization 123 Toeplitz matrix 27 weighted shift 116, 257 Toeplitz operator 27, 184, 317, 368, 373, 396 well-bounded operator 108, 198, 202 compactness 376, 378, 379 Widder inversion theorem 200, 335 trace 189, 287, 288, 436 Widder operator 330 trace class 138, 287 Widom formula 35 trace norm 128, 435 Wiener algebra 405 transference 114 winding number 28, 30 translation operator 105 triangularizable algebra of compact operators Yosida approximation 249, 346 293 Young inequality 142 triangularizable set of operators 287, 288, 290 trigonometric formulas 433 Zelazko˙ theorem 419