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Problem: Let A be a densely defined self-adjoint operator on a Hilbert space H with domain D(A). Prove that the operator A has a spectral representation in terms of a spectral measure E on the Borel subsets of the spectrum σ(A), where: A = = SOLAY AdE(X). σ(A) Your solution should address the following points: • The domain of A and the closure of the operator. • The concept of the spectral theorem for unbounded operators. • The role of the resolvent operator and how it connects to the spectral measure. ⚫ The existence of the spectral decomposition and the necessary conditions on the operator. Solution Outline: 1. Self-Adjoint Operators: Discuss the self-adjointness of the operator A and its domain D(A). 2. Spectral Theorem for Unbounded Operators: Present the spectral theorem for unbounded self- adjoint operators and the conditions required for the spectral measure. 3. Resolvent Operator: Show how the resolvent operator is related to the spectral measure and how it allows us to construct the spectral decomposition. 4. Spectral Representation: Using the spectral theorem, prove that A can be written as an integral over its spectrum using a projection-valued measure.

Problem: Let A be a densely defined self-adjoint operator on a Hilbert space H with domain D(A). Prove that the operator A has a spectral representation in terms of a spectral measure E on the Borel subsets of the spectrum σ(A), where: A = = SOLAY AdE(X). σ(A) Your solution should address the following points: • The domain of A and the closure of the operator. • The concept of the spectral theorem for unbounded operators. • The role of the resolvent operator and how it connects to the spectral measure. ⚫ The existence of the spectral decomposition and the necessary conditions on the operator. Solution Outline: 1. Self-Adjoint Operators: Discuss the self-adjointness of the operator A and its domain D(A). 2. Spectral Theorem for Unbounded Operators: Present the spectral theorem for unbounded self- adjoint operators and the conditions required for the spectral measure. 3. Resolvent Operator: Show how the resolvent operator is related to the spectral measure and how it allows us to construct the spectral decomposition. 4. Spectral Representation: Using the spectral theorem, prove that A can be written as an integral over its spectrum using a projection-valued measure.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.1: The Geometry And Algebra Of Vectors
Problem 24EQ
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Problem: Let A be a densely defined self-adjoint operator on a Hilbert space H with domain D(A). Prove that the operator A has a spectral representation in terms of a spectral measure E on the Borel subsets of the spectrum σ(A), where: A = = SOLAY AdE(X). σ(A) Your solution should address the following points: • The domain of A and the closure of the operator. • The concept of the spectral theorem for unbounded operators. • The role of the resolvent operator and how it connects to the spectral measure. ⚫ The existence of the spectral decomposition and the necessary conditions on the operator. Solution Outline: 1. Self-Adjoint Operators: Discuss the self-adjointness of the operator A and its domain D(A). 2. Spectral Theorem for Unbounded Operators: Present the spectral theorem for unbounded self- adjoint operators and the conditions required for the spectral measure. 3. Resolvent Operator: Show how the resolvent operator is related to the spectral measure and how it allows us to construct the spectral decomposition. 4. Spectral Representation: Using the spectral theorem, prove that A can be written as an integral over its spectrum using a projection-valued measure.
Transcribed Image Text:Problem: Let A be a densely defined self-adjoint operator on a Hilbert space H with domain D(A). Prove that the operator A has a spectral representation in terms of a spectral measure E on the Borel subsets of the spectrum σ(A), where: A = = SOLAY AdE(X). σ(A) Your solution should address the following points: • The domain of A and the closure of the operator. • The concept of the spectral theorem for unbounded operators. • The role of the resolvent operator and how it connects to the spectral measure. ⚫ The existence of the spectral decomposition and the necessary conditions on the operator. Solution Outline: 1. Self-Adjoint Operators: Discuss the self-adjointness of the operator A and its domain D(A). 2. Spectral Theorem for Unbounded Operators: Present the spectral theorem for unbounded self- adjoint operators and the conditions required for the spectral measure. 3. Resolvent Operator: Show how the resolvent operator is related to the spectral measure and how it allows us to construct the spectral decomposition. 4. Spectral Representation: Using the spectral theorem, prove that A can be written as an integral over its spectrum using a projection-valued measure.
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