Operator and Spectral Theory
نویسنده
چکیده
This lecture is a complete introduction to the general theory of operators on Hilbert spaces. We particularly focus on those tools that are essentials in Quantum Mechanics: unbounded operators, multiplication operators, self-adjointness, spectrum, functional calculus, spectral measures and von Neumann’s Spectral Theorem. 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Operators, Domains, Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Bounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Closed and Closable Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Definitions, Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Adjoint and Closability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 The Case of Multiplication Operators . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Orthogonal Projectors, Unitaries, Isometries . . . . . . . . . . . . . . . . . . 12 1.3.1 Orthogonal Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Unitaries and Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 Partial Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Symmetric and Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.2 Basic Criterions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.3 Normal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.1 Functional Analysis Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.3 Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.6 Spectral Theorem for Bounded Normal Operators . . . . . . . . . . . . . 27 1.6.1 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.6.3 Bounded Borel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.4 Multiplication Operator Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Stéphane ATTAL Institut Camille Jordan, Université Lyon 1, France e-mail: [email protected]
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تاریخ انتشار 2013