A Short Course on Spectral Theory
This book presents the basic tools of modern analysis within the context of what might be called the fundamental problem of operator theory: to c- culate spectra of speci?c operators on in?nite-dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more re?ned methods that allow one to approach problems that go well beyond the computation of spectra; the mathematical foundations of quantum physics, noncommutative K-theory, and the classi?cation of sim- ? ple C -algebras being three areas of current research activity that require mastery of the material presented here. The notion of spectrum of an operator is based on the more abstract notion of the spectrum of an element of a complex Banach algebra. - ter working out these fundamentals we turn to more concrete problems of computing spectra of operators of various types. For normal operators, this amounts to a treatment of the spectral theorem. Integral operators require 2 the development of the Riesz theory of compact operators and the ideal L of Hilbert–Schmidt operators. Toeplitz operators require several important tools; in order to calculate the spectra of Toeplitz operators with continuous symbol one needs to know the theory of Fredholm operators and index, the ? structure of the Toeplitz C -algebra and its connection with the topology of curves, and the index theorem for continuous symbols.
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adjoint assertion Banach space Borel function bounded linear bounded operator C∗-algebra closed subspace commutative compact Hausdorff space compact operator complex algebra complex numbers consider continuous function converges Corollary Deduce deﬁned denote diagonalizable direct sum example Exercises ﬁnd ﬁnite ﬁrst follows formula Fredholm operator function f functional calculus Gelfand map Hausdorff space hence Hilbert space Hilbert–Schmidt operators homomorphism implies infinite-dimensional integral invertible isometric isomorphism kernel Lemma linear functional linear map linear span matrix maximal ideal measure space multiplication operator Neumann algebra nonzero norm normal operator operator topology operators acting orthonormal basis PROOF quotient Riesz satisﬁes satisfying self-adjoint element sequence sp(A space H spectral measure spectral theorem spectrum subalgebra subset theory Toeplitz operators Tr(a unilateral shift unique unit ball unital Banach algebra unitarily equivalent unitary operator vector space von Neumann algebra zero