Let H be an infinite-dimensional separable Hilbert space, and let K: HH be a compact self- adjoint operator. 1. Spectral Decomposition: Show that K can be expressed as K = -Ση(,en)ens n=1 where {e} is an orthonormal basis of H consisting of eigenvectors of K, and {A,,} is a sequence of real numbers converging to zero. 2. Singular Spectral Measures: Prove that the spectral measure E associated with K is purely atomic, and each atom corresponds to an eigenvalue >,, with multiplicity equal to the dimension of the corresponding eigenspace. 3. Measure-Theoretic Implications: Suppose K is trace-class. Show that the trace of K can be expressed as Τι(Κ) = Σλη, n=1 and discuss the measure-theoretic conditions under which this equality holds, particularly in relation to the convergence of the series and the properties of the spectral measure E.

Let H be an infinite-dimensional separable Hilbert space, and let K: HH be a compact self- adjoint operator. 1. Spectral Decomposition: Show that K can be expressed as K = -Ση(,en)ens n=1 where {e} is an orthonormal basis of H consisting of eigenvectors of K, and {A,,} is a sequence of real numbers converging to zero. 2. Singular Spectral Measures: Prove that the spectral measure E associated with K is purely atomic, and each atom corresponds to an eigenvalue >,, with multiplicity equal to the dimension of the corresponding eigenspace. 3. Measure-Theoretic Implications: Suppose K is trace-class. Show that the trace of K can be expressed as Τι(Κ) = Σλη, n=1 and discuss the measure-theoretic conditions under which this equality holds, particularly in relation to the convergence of the series and the properties of the spectral measure E.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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