57. T. Ichinose, Spectral properties of linear operators I,Trans.Amer.Math.Soc. 235 (1978),

Transcription

1 References 1. L.V. Ahlfors, Complex Analysis, 3rd edn. McGraw-Hill, New York, P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space Volume I, Pitman, London, 1981; reprinted: Dover, New York, N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space Volume II, Pitman, London, 1981; reprinted: Dover, New York, W. Arveson, An Invitation to C*-Algebras, Springer, New York, W. Arveson, A Short Course in Spectral Theory, Springer, New York, F.V. Atkinson, The normal solvability of linear equations in normed spaces, Mat. Sbornik 28 (1951), G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, 1966; reprinted: Dover, Mineola, B.A. Barnes, RieszpointsandWeyl stheorem, Integral Equations Operator Theory 34 (1999), B.A. Barnes, G.J. Murphy, M.R.F. Smyth, and T.T. West, Riesz and Fredholm Theory in Banach Algebras, Pitman, London, R.G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley, New York, 1995; enlarged 2nd edn. of The Elements of Integration, Wiley, New York, R. Beals, Topics in Operator Theory, The University of Chicago Press, Chicago, R. Beals, Advanced Mathematical Analysis, Springer, New York, S.K. Berberian, Notes on Spectral Theory, Van Nostrand, New York, S.K. Berberian, An extension of Weyl s theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), C.S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, 187 DOI / , Springer Science+Business Media, LLC 2012

2 188 References 16. S.K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1971), S.K. Berberian, Lectures in Functional Analysis and Operator Theory, Springer, New York, S.K. Berberian, Introduction to Hilbert Space, 2nd edn. Chelsea, New York, J. Bram, Subnormal operators, Duke Math. J. 22 (1955), A. Brown and C. Pearcy, Spectra of tensor products of operators, Proc. Amer. Math. Soc. 17 (1966), A. Brown and C. Pearcy, Introduction to Operator Theory I Elements of Functional Analysis, Springer, New York, A. Brown and C. Pearcy, An Introduction to Analysis, Springer, New York, S.R. Caradus, W.E. Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 9. Marcel Dekker, New York, L.A. Coburn, Weyl s theorem for nonnormal operators, Michigan Math. J. 13 (1966), J.B. Conway, Every spectral picture is possible, Notices Amer. Math. Soc. 24 (1977), A J.B. Conway, Functions of One Complex Variable, Springer, New York, J.B. Conway, A Course in Functional Analysis, 2nd edn. Springer, New York, J.B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, Vol. 36, Amer. Math. Soc., Providence, J.B. Conway, A Course in Operator Theory, Graduate Studies in Mathematics, Vol. 21, Amer. Math. Soc., Providence, K.R. Davidson, C*-Algebras by Example, Fields Institute Monographs, Vol. 6, Amer. Math. Soc., Providence, J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, D.S. Djordjević, Semi-Browder essential spectra of quasisimilar operators, Novi Sad J. Math. 31 (2001), R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972; 2nd edn. Springer, New York, H.R. Dowson, Spectral Theory of Linear Operators, Academic Press, New York, B.P. Duggal and S.V. Djordjević, Generalized Weyl s theorem for a class of operators satisfying a norm condition, Math.Proc.RoyalIrishAcad.104 (2004),

3 References B.P. Duggal, S.V. Djordjević, and C.S. Kubrusly, Hereditarily normaloid contractions, Acta Sci. Math. (Szeged) 71 (2005), B.P. Duggal and C.S. Kubrusly, Weyl s theorem for direct sums, Studia Sci. Math. Hungar. 44 (2007), N. Dunford and J.T. Schwartz, Linear Operators Part I: General Theory, Interscience, New York, N. Dunford and J.T. Schwartz, Linear Operators Part II: Spectral Theory Self Adjoint Operators in Hilbert Space, Interscience, New York, N. Dunford and J.T. Schwartz, Linear Operators Part III: Spectral Operators, Interscience, New York, P.A. Fillmore, Notes on Operator Theory, Van Nostrand, New York, P.A. Fillmore, A User s Guide to Operator Algebras, Wiley, New York, K. Gustafson, Necessary and sufficient conditions for Weyl s theorem, Michigan Math. J. 19 (1972), K. Gustafson and D.K.M. Rao, Numerical Range, Springer, New York, P.R. Halmos, Measure Theory, Van Nostrand, New York, 1950; reprinted: Springer, New York, P.R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, 2nd edn. Chelsea, New York, 1957; reprinted: AMS Chelsea, Providence, P.R. Halmos, Finite-Dimensional Vector Spaces, Van Nostrand, New York, 1958; reprinted: Springer, New York, P.R. Halmos, Shifts on Hilbert spaces, J. ReineAngew. Math. 208 (1961), P.R. Halmos, A Hilbert Space Problem Book, Van Nostrand, New York, 1967; 2nd edn. Springer, New York, P.R. Halmos and V.S. Sunder, Bounded Integral Operators on L 2 Spaces, Springer, Berlin, R. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker, New York, R. Harte and W.Y. Lee, Another note on Weyl s theorem, Trans. Amer. Math. Soc. 349 (1997), G. Helmberg, Introduction to Spectral Theory in Hilbert Space, North-Holland, Amsterdam, D. Herrero, Approximation of Hilbert Space Operators Volume 1, 2nd edn. Longman, Harlow, E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, Colloquium Publications Vol. 31, Amer. Math. Soc., Providence, 1957; reprinted: 1974.

4 190 References 57. T. Ichinose, Spectral properties of linear operators I,Trans.Amer.Math.Soc. 235 (1978), V.I. Istrǎţescu, Introduction to Linear Operator Theory, MarcelDekker, New York, M.A. Kaashoek and D.C. Lay, Ascent, descent, and commuting perturbations, Trans. Amer. Math. Soc. 169 (1972), T. Kato, Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin, 1980; reprinted: D. Kitson, R. Harte, and C. Hernandez, Weyl s theorem and tensor products: a counterexample, J. Math. Anal. Appl. 378 (2011), C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, C.S. Kubrusly, Hilbert Space Operators, Birkhäuser, Boston, C.S. Kubrusly, A concise introduction to tensor product, FarEastJ.Math.Sci. 22 (2006), C.S. Kubrusly, Measure Theory, Academic Press/Elsevier, San Diego, C.S. Kubrusly, The Elements of Operator Theory, Birkhäuser/Springer, New York, 2011; enlarged 2nd edn. of Elements of Operator Theory, Birkhäuser, Boston, C.S. Kubrusly and B.P. Duggal, On Weyl and Browder spectra of tensor products, Glasgow Math. J. 50 (2008), C.S. Kubrusly and B.P. Duggal, On Weyl s theorem of tensor products, toappear. 69. D.C. Lay, Characterizations of the essential spectrum of F.E. Browder, Bull. Amer. Math. Soc. 74 (1968), W.Y. Lee, Weyl s theorem for operator matrices, Integral Equations Operator Theory 32 (1998), W.Y. Lee, Weyl spectrum of operator matrices, Proc. Amer. Math. Soc. 129 (2001), J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), V. Müller, Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras, 2nd edn. Birkhäuser, Basel, G. Murphy, C*-Algebras and Operator Theory, Academic Press, San Diego, C.M. Pearcy, Some Recent Developments in Operator Theory, CBMSRegional Conference Series in Mathematics No. 36, Amer. Math. Soc., Providence, H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer, Berlin, 1973; 2nd edn. Dover, New York, F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar, New York, 1955; reprinted: Dover, New York, 1990.

5 78. H.L. Royden, Real Analysis, 3rd edn. Macmillan, New York, References W. Rudin, Real and Complex Analysis, 3rd edn. McGraw-Hill, New York, W. Rudin, Functional Analysis, 2nd edn. McGraw-Hill, New York, M. Schechter, On the essential spectrum of an arbitrary operator. I,J.Math. Anal. Appl. 13 (1966), M. Schechter, Principles of Functional Analysis, Academic Press, New York, 1971; 2nd edn. Graduate Studies in Mathematics, Vol. 36, Amer. Math. Soc., Providence, J. Schwartz, Some results on the spectra and spectral resolutions of a class of singular operators, Comm. Pure Appl. Math. 15 (1962), Y.-M. Song and A.-H. Kim, Weyl s theorem for tensor products, Glasgow Math. J. 46 (2004), V.S. Sunder, Functional Analysis Spectral Theory, Birkhäuser, Basel, B. Sz.-Nagy, C. Foiaş, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Springer, New York, 2010; enlarged 2nd edn. of B. Sz.-Nagy and C. Foiaş, North-Holland, Amsterdam, A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, Wiley, New York, 1980; reprinted: Krieger, Melbourne, 1986; enlarged 2nd edn. of A.E. Taylor, Wiley, New York, A. Uchiyama, On the isolated points of the spectrum of paranormal operators, Integral Equations Operator Theory 55 (2006), J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo27 (1909),

6 Index absolutely continuous, 96 adjoint, 9 algebra homomorphism, 51, 91 algebra isomorphism, 91 algebra with identity, 91 algebraic complements, 3, 166 algebraic multiplicity, 129, 157 analytic function, 29 analytic function on neighborhoods, 110 analytic function on spectra, 110, 111 approximate eigenvalue, 32 approximate point spectrum, 32 approximation spectrum, 32 arc, 103 ascent of an operator, 163 Atkinson Theorem, 145 Axiom of Choice, 31, 74 backward bilateral shift, 49 backward unilateral shift, 49 Banach algebra, 91 Banach Steinhaus Theorem, 2, 41 bilateralshift,49 bilateral weighted shift, 50 Bolzano Weierstrass Property, 57 Borel measure, 64 Borel sets, 64 Borel σ-algebra, 64 bounded below, 1 bounded component, 51 bounded inverse, 3 Bounded Inverse Theorem, 3 bounded linear transformation, 1 bounded measurable function, 65, 95 bounded sequence, 2 bounded variation, 103 Browder operator, 165 Browder spectrum, 170 Browder s Theorem, 173, 179 Calkin algebra, 143 Calkin spectrum, 144 canonical bilateral shift, 49 canonical orthonormal basis, 85 canonical unilateral shift, 49 Cartesian decomposition, 24, 93 Cartesian Decomposition Theorem, 25 Cauchy domain, 106 Cauchy Integral Formula, 107 Cauchy Theorem, 109 clockwise oriented curve, 106 clopen set, 115 closed curve, 103 closed region, 105 cohyponormal operator, 12 coisometry, 14 C.S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, DOI / , Springer Science+Business Media, LLC

7 194 Index commutant, 126, 175 commutator, 185 commuting operators, 12, 51, 80 83, 101, 114, 116 compact linear transformation, 18 compact operator, 18 compact perturbation, 147, 152, 157, 177, 185 compact set, 17 Compactness Theorem, 17 complementary linear manifolds, 3, 166 complementary projection, 7 complementary spectral sets, 122 complementary subspaces, 3, 166 complemented Banach space, 134 completely continuous, 18 completely normal, 88 complex algebra, 91 component of a set, 51, 105, 147 compression spectrum, 32 connected, 105 continuous linear transformation, 1 continuous spectrum, 30 continuously differentiable, 103, 104 contraction, 2 counterclockwise oriented curve, 106 C*-algebra, 92 curve, 103 cyclic operator, 73, 74 cyclic vector, 73, 74 densely intertwined, 23 derivative, 29 descent of an operator, 163 diagonal operator, 85 diagonalizable operator, 49, 61, 62, 84, 85, 88 direct sum of operators, 7, 182 direct sum of spaces, 3, 6, 7 direct summand, 181 directed pair, 103 disconnected, 105, 115 domain, 105 dominant operator, 181 double commutant, 126 Double Commutant Theorem, 126 eigenspace, 16, 30 eigenvalue, 30 eigenvector, 30 entire function, 29 equivalent measures, 96 equivalent operators, 23 essential range, 86 essential singularity, 127 essential spectrum, 144 essentially bounded functions, 85, 95 essentially invertible, 145 extension by continuity, 25 finite algebraic multiplicity, 129, 157 finite geometric multiplicity, 129, 157 finite measure, 64 finite-rank operators, 45 finite-rank transformation, 24 Fourier series expansion, 84, 85 Fredholm Alternative, 21, 44, 135, 136, 140, 152, 161, 185 Fredholm index, 134 Fredholm operator, Fredholm spectrum, 145 Fuglede Theorem, 80 Fuglede Putnam Theorem, 84 full spectrum, 52, 87 functional calculus, 91, 94, 100 Functional Calculus Theorems, 95, 97, 99, 100, 111 Gelfand Beurling formula, 38, 51, 93 geometric multiplicity, 129, 157 Hahn Banach Theorem, 29, 30, 109 Heine Borel Theorem, 18 Hermitian element, 92 Hermitian operator, 11 hole, 51, 147, 151, 153 holomorphic function, 29 homomorphism, 51, 91 hyperinvariant subspace, 23, 81, 121 hyponormal operator, 12 idempotent function, 7 index stability, 139 inside of a path, 106 integral over a curve, 105 intertwine, 23, 87 invariant linear manifold, 2 invariant subspace, 2, 81, 181

8 Index 195 Inverse Mapping Theorem, 3 invertible element, 91, 141 involution, 91 involutive algebra, 92 irreducible operator, 11 isolated eigenvalues, 157 isolated points of spectra, isolated singularity, 127 isoloid operator, 181 isometric isomorphism, 94 isometrically isomorphic algebras, 94 isometry, 14 isomorphic algebras, 91 Jordan closed region, 106 Jordan curve, 103 Jordan Curve Theorem, 106 Jordan domain, 107 kernel, 2 Laurent expansion, 38, 127 Laventriev Theorem, 76 left essential spectrum, 144 left inverse, 141 left semi-fredholm operator, 131, 132 left spectrum, 142 length of a curve, 103 linear span, 6 Liouville Theorem, 29 logarithmic additivity, 136 measurable set, 64 multiplication operator, 67, 85, 86 multiplicity of a point in an arc, 103 multiplicity of a shift, 49 multiplicity of an arc, 103 multiplicity of eigenvalue, 30, 60, 129 mutually orthogonal projections, 8 natural map, 143 natural quotient map, 143 negatively oriented curve, 106 negatively oriented path, 106 Neumann expansion, 4 nilpotent operator, 38 nonnegative measure, 64 nonnegative operator, 12 nonscalar operator, 81 nontrivial projection, 81 nontrivial subspace, 2, 81 normal eigenvalues, 157 normal element, 92 normal operator, 12, 141, 162, 184 normaloid operator, 15, 43 normed algebra, 91 nowhere dense, 88 null space, 2 numerical radius, 42 numerical range, 41 open mapping, 3 Open Mapping Theorem, 3 operator, 1 operator norm property, 1 opposite oriented curve, 105 ordinary sum of subspaces, 3 oriented curve, 103 orthogonal complement, 5 orthogonal complementary subspaces, 6 orthogonal direct sum, 5 orthogonal direct sum of operators, 7 orthogonal direct sum of subspaces, 5, 7 orthogonal family of projections, 8 orthogonal projection, 8, 13 orthogonal projection on M, 9 orthogonal sequence of projections, 8 orthogonal sets, 5 Orthogonal Structure Theorem, 6, 8 orthogonal subspaces, 5 orthogonal vectors, 5 outside of a path, 106 parallelogram law, 22, 43 parameterization, 103 paranormal operator, 182 part of an operator, 181 partial isometry, 24 partition of an interval, 103 path, 106 path about a set, 110 piecewise continuously differentiable, 104 piecewise smooth curve, 104 piecewise smooth function, 104 point spectrum, 30 polar decomposition, 24 Polar Decomposition Theorem, 25 polarization identity, 21, 42, 77

9 196 Index pole, 127 positive measure, 64 positive operator, 12 positively oriented curve, 106 positively oriented path, 106 power bounded operator, 38 power inequality, 42 power series, 4, 38, 95, 112 power set, 64 projection, 7 Projection Theorem, 6, 8 proper contraction, 2 pseudohole, 147, 151, 153 Pythagorean Theorem, 8, 56 quasiaffine transform, 23 quasiaffinity, 23 quasiinvertible transformation, 23 quasinilpotent operator, 38 quasisimilar operators, 23 Radon Nikodým derivative, 67, 102 range, 2 rectifiable curve, 103 reducible operator, 11, 81 reducing subspace, 11, 81, 181 reductive operator, 11, 88 region, 105 residual spectrum, 30 resolution of the identity, 8 resolvent function, 28 resolvent identity, 28 resolvent set, 27 reverse oriented curve, 105 Riemann Stieltjes integral, 104 Riesz Decomposition Theorem, 121 Riesz Functional Calculus, 111 Riesz idempotent, 116, 154 Riesz point, 155, 156 Riesz Representation Theorems, 65, 69 right essential spectrum, 144 right inverse, 141 right semi-fredholm operator, 131, 132 right spectrum, 142 Rosenblum Corollary, 52 scalar operator, 16, 81 scalar spectral measure, 97 scalar type operator, 128 Schechter Theorem, 159 self-adjoint element, 92 self-adjoint operator, 11 semi-fredholm operator, seminormal operator, 12 separating vector, 98 sequentially compact set, 17 σ-finite measure, 64 similar operators, 23, 88 similarity to a strict contraction, 49 simple curve, 103 simple function, 128 singularity, 127 smooth curve, 104 smooth function, 103 span of a set, 6 spectral decomposition, 59, 79 Spectral Mapping for polynomials, 35, 37 Spectral Mapping Theorems, 35, 37, 101, 102, 114 spectral measure, 64, 128 spectral operator, 128 spectral partition, 31 spectral picture, 152 spectral radius, 38, 91 spectral set, 115 Spectral Theorem: compact case, 58 Spectral Theorem: general case, 67, 76 spectraloid operator, 43, 49 spectrum, 27, 91 square-integrable functions, 85 square root, 23 Square Root Theorem, 25 square-summable family, 6 stability, 40 -algebra, 91 star-cyclic operator, 73, 74 star-cyclic vector, 67, 73, 74 -homomorphism, 92 -isomorphism, 92 Stone Weierstrass Theorem, 68, 76, 77, 100 strict contraction, 2 strictly positive operator, 12 strong convergence, 2 strongly stable operator, 40 subspace, 2 summable family, 6 support of a measure, 64, 86

10 Index 197 tensor product of operators, 183 topological sum of subspaces, 6 total variation, 103 totally bounded set, 17 totally hereditarily normaloid, 182 uniform convergence, 2 uniform stability, 40 uniformly stable operator, 40, 49 unilateral shift, 49, 184 unilateral weighted shift, 50 unitalalgebra,1,91 unital complex Banach algebra, 2, 27, 51, 91, 174 unital homomorphism, 91, 143 unitarily equivalent operators, 23, 85 unitary element, 92 unitary operator, 14 unitary transformation, 14, 23 von Neumann algebra, 126 weak convergence, 2 weakly stable operator, 40 Weierstrass Approximation Theorem, 76 Weierstrass Theorem, 29 weighted sum of projections, 9 Weyl operator, 140 Weyl spectrum, 157 Weyl spectrum identity, 183 Weyl s Theorem, 162, winding number, 106 Zorn s Lemma, 37, 71, 177