Banach Algebra Techniques in Operator TheorySpringer Science & Business Media, 6 בדצמ׳ 2012 - 198 עמודים Operator theory is a diverse area of mathematics which derives its impetus and motivation from several sources. It began with the study of integral equations and now includes the study of operators and collections of operators arising in various branches of physics and mechanics. The intention of this book is to discuss certain advanced topics in operator theory and to provide the necessary background for them assuming only the standard senior-first year graduate courses in general topology, measure theory, and algebra. At the end of each chapter there are source notes which suggest additional reading along with giving some comments on who proved what and when. In addition, following each chapter is a large number of problems of varying difficulty. This new edition will appeal to a new generation of students seeking an introduction to operator theory. |
תוכן
Banach Algebras | 30 |
Abstract Index in a Banach Algebra | 33 |
The Space of Multiplicative Linear Functions | 36 |
The Gelfand Transform | 37 |
The GelfandMazur Theorem | 39 |
The Gelfand Theorem for Commutative Banach Algebras | 41 |
The Spectral Radius Formula | 42 |
The StoneWeierstrass Theorem | 43 |
The Polar Decomposition | 88 |
Weak and Strong Operator Topologies | 91 |
WAlgebras | 92 |
Isomorphisms of LSpaces | 94 |
Normal Operators with Cyclic Vectors | 95 |
Maximal Abelian WAlgebras | 98 |
Homomorphisms of CAlgebras | 100 |
The Extended Functional Calculus | 102 |
The Generalized StoneWeierstrass Theorem | 46 |
The Disk Algebra | 47 |
The Algebra of Functions with Absolutely Convergent Fourier Series | 50 |
The Algebra of Bounded Measurable Functions | 52 |
Notes | 53 |
Geometry of Hilbert Space | 58 |
The CauchySchwarz Inequality | 59 |
The Pythagorean Theorem | 60 |
Hilbert Spaces | 61 |
The Riesz Representation Theorem | 66 |
The Existence of Orthonormal Bases | 69 |
The Dimension of Hilbert Spaces | 70 |
Notes | 71 |
Operators on Hilbert Space and CAlgebras | 74 |
Normal and Selfadjoint Operators | 77 |
Projections and Subspaces | 78 |
Multiplication Operators and Maximal Abelian Algebras | 80 |
The Bilateral Shift Operator | 82 |
CAlgebras | 83 |
The GelfandNaimark Theorem | 84 |
The Spectral Theorem | 85 |
The Square Root of Positive Operators | 86 |
The Unilateral Shift Operator | 87 |
The Fuglede Theorem | 103 |
Notes | 104 |
Compact Operators Fredholm Operators and Index Theory | 108 |
Approximation of Compact Operators | 110 |
Integral Operators | 112 |
The Calkin Algebra and Fredholm Operators | 113 |
Atkinsons Theorem | 114 |
The Index of Fredholm Operators | 115 |
The Fredholm Alternative | 116 |
Volterra Integral Operators | 118 |
Connectedness of the Unitary Group in a WAlgebra | 119 |
Quotient CAlgebras | 124 |
Representations of the CAlgebra of Compact Operators | 126 |
Notes | 129 |
The Hardy Spaces | 133 |
Reducing Subspaces of Unitary Operators | 135 |
Beurlings Theorem | 137 |
The Maximal Ideal Space of H | 138 |
The InnerOuter Factorization of Functions in H2 | 141 |
The Conjugates of H¹ and LH | 144 |
The Closedness of H +C | 145 |
The GleasonWhitney Theorem | 149 |
מהדורות אחרות - הצג הכל
Banach Algebra Techniques in Operator Theory <span dir=ltr>Ronald G. Douglas</span> תצוגה מקדימה מוגבלת - 1998 |
מונחים וביטויים נפוצים
abelian Banach space Borel measure bounded linear functional bounded variation C*-algebra closed subspace closure commutative Banach algebra compact Hausdorff space compact operator completes the proof contained continuous function converges Corollary defined Definition denote dense element exists follows Fredholm operator function ƒ function in H function in L∞(T functional calculus ƒ and g ƒ in H Gelfand transform Hausdorff space Hilbert space Hint homomorphism implies inner function inner product invariant subspace isometric isomorphism LC(H Lemma linear space M₁ mapping Math maximal ideal maximal ideal space Moreover multiplicative linear functional nonzero norm normal operator obtain operator on H orthogonal orthonormal basis outer function partial isometry preceding proof is complete Proposition result Riesz self-adjoint subalgebra space H spectrum subset Theorem Theory Toeplitz operators unit ball v₁ w*-topology weak operator topology