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Problem: Let T be a bounded linear operator on a Hilbert space H such that T is self-adjoint. Prove that the operator T admits a spectral decomposition. Specifically, show that there exists a projection-valued measure E on the Borel sets of the spectrum o(T) of T such that: T = AE(A). Outline all the necessary steps in the proof, including: ⚫ The concepts of spectral measures and the spectral theorem. • The existence of the spectral measure. • How the operator T is represented as an integral over its spectrum. • The connection between the spectral theorem and functional calculus. Solution Outline: 1. Self-Adjointness and Spectral Theorem: Begin by stating the spectral theorem for bounded self-adjoint operators. Recall that T is diagonalizable in the sense of its spectral measure. 2. Definition of Spectral Measure: Define the spectral measure E, which is a projection-valued measure. 3. Functional Calculus: Use functional calculus to construct the spectral representation of T. 4. Integration over Spectrum: Show that I can be written as an integral over the spectrum using E. 5. Projection-valued Measure: Prove that the spectral measure satisfies the properties of a projection-valued measure, ensuring that T is appropriately represented by the spectral integral.

Problem: Let T be a bounded linear operator on a Hilbert space H such that T is self-adjoint. Prove that the operator T admits a spectral decomposition. Specifically, show that there exists a projection-valued measure E on the Borel sets of the spectrum o(T) of T such that: T = AE(A). Outline all the necessary steps in the proof, including: ⚫ The concepts of spectral measures and the spectral theorem. • The existence of the spectral measure. • How the operator T is represented as an integral over its spectrum. • The connection between the spectral theorem and functional calculus. Solution Outline: 1. Self-Adjointness and Spectral Theorem: Begin by stating the spectral theorem for bounded self-adjoint operators. Recall that T is diagonalizable in the sense of its spectral measure. 2. Definition of Spectral Measure: Define the spectral measure E, which is a projection-valued measure. 3. Functional Calculus: Use functional calculus to construct the spectral representation of T. 4. Integration over Spectrum: Show that I can be written as an integral over the spectrum using E. 5. Projection-valued Measure: Prove that the spectral measure satisfies the properties of a projection-valued measure, ensuring that T is appropriately represented by the spectral integral.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 11EQ
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Problem: Let T be a bounded linear operator on a Hilbert space H such that T is self-adjoint. Prove that the operator T admits a spectral decomposition. Specifically, show that there exists a projection-valued measure E on the Borel sets of the spectrum o(T) of T such that: T = AE(A). Outline all the necessary steps in the proof, including: ⚫ The concepts of spectral measures and the spectral theorem. • The existence of the spectral measure. • How the operator T is represented as an integral over its spectrum. • The connection between the spectral theorem and functional calculus. Solution Outline: 1. Self-Adjointness and Spectral Theorem: Begin by stating the spectral theorem for bounded self-adjoint operators. Recall that T is diagonalizable in the sense of its spectral measure. 2. Definition of Spectral Measure: Define the spectral measure E, which is a projection-valued measure. 3. Functional Calculus: Use functional calculus to construct the spectral representation of T. 4. Integration over Spectrum: Show that I can be written as an integral over the spectrum using E. 5. Projection-valued Measure: Prove that the spectral measure satisfies the properties of a projection-valued measure, ensuring that T is appropriately represented by the spectral integral.
Transcribed Image Text:Problem: Let T be a bounded linear operator on a Hilbert space H such that T is self-adjoint. Prove that the operator T admits a spectral decomposition. Specifically, show that there exists a projection-valued measure E on the Borel sets of the spectrum o(T) of T such that: T = AE(A). Outline all the necessary steps in the proof, including: ⚫ The concepts of spectral measures and the spectral theorem. • The existence of the spectral measure. • How the operator T is represented as an integral over its spectrum. • The connection between the spectral theorem and functional calculus. Solution Outline: 1. Self-Adjointness and Spectral Theorem: Begin by stating the spectral theorem for bounded self-adjoint operators. Recall that T is diagonalizable in the sense of its spectral measure. 2. Definition of Spectral Measure: Define the spectral measure E, which is a projection-valued measure. 3. Functional Calculus: Use functional calculus to construct the spectral representation of T. 4. Integration over Spectrum: Show that I can be written as an integral over the spectrum using E. 5. Projection-valued Measure: Prove that the spectral measure satisfies the properties of a projection-valued measure, ensuring that T is appropriately represented by the spectral integral.
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