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Let H be a separable Hilbert space, and let T: HH be a bounded self-adjoint operator. 1. Spectral Measure Construction: Prove that there exists a unique projection-valued measure E: B(R) B(H) (where B(R) is the Borel σ-algebra on R) such that T= AdE(A). 2. Functional Calculus Extension: Using the spectral measure E, extend the functional calculus to all bounded Borel measurable functions f: RC and prove that the map f f(A) dE(A) is a *-homomorphism. 3. Measure-Theoretic Properties: Show that for any vector EH, the map Ex: B(R) → [0, ∞) defined by Exx (B) (E(B), x) is a regular Borel measure on R. Requirements: • • • = Utilize measure-theoretic concepts to construct and analyze the spectral measure. Apply the Riesz Representation Theorem where appropriate. Demonstrate the uniqueness and properties of the spectral measure in the context of bounded self-adjoint operators.

Let H be a separable Hilbert space, and let T: HH be a bounded self-adjoint operator. 1. Spectral Measure Construction: Prove that there exists a unique projection-valued measure E: B(R) B(H) (where B(R) is the Borel σ-algebra on R) such that T= AdE(A). 2. Functional Calculus Extension: Using the spectral measure E, extend the functional calculus to all bounded Borel measurable functions f: RC and prove that the map f f(A) dE(A) is a *-homomorphism. 3. Measure-Theoretic Properties: Show that for any vector EH, the map Ex: B(R) → [0, ∞) defined by Exx (B) (E(B), x) is a regular Borel measure on R. Requirements: • • • = Utilize measure-theoretic concepts to construct and analyze the spectral measure. Apply the Riesz Representation Theorem where appropriate. Demonstrate the uniqueness and properties of the spectral measure in the context of bounded self-adjoint operators.

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 15EQ
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Let H be a separable Hilbert space, and let T: HH be a bounded self-adjoint operator. 1. Spectral Measure Construction: Prove that there exists a unique projection-valued measure E: B(R) B(H) (where B(R) is the Borel σ-algebra on R) such that T= AdE(A). 2. Functional Calculus Extension: Using the spectral measure E, extend the functional calculus to all bounded Borel measurable functions f: RC and prove that the map f f(A) dE(A) is a *-homomorphism. 3. Measure-Theoretic Properties: Show that for any vector EH, the map Ex: B(R) → [0, ∞) defined by Exx (B) (E(B), x) is a regular Borel measure on R. Requirements: • • • = Utilize measure-theoretic concepts to construct and analyze the spectral measure. Apply the Riesz Representation Theorem where appropriate. Demonstrate the uniqueness and properties of the spectral measure in the context of bounded self-adjoint operators.
Transcribed Image Text:Let H be a separable Hilbert space, and let T: HH be a bounded self-adjoint operator. 1. Spectral Measure Construction: Prove that there exists a unique projection-valued measure E: B(R) B(H) (where B(R) is the Borel σ-algebra on R) such that T= AdE(A). 2. Functional Calculus Extension: Using the spectral measure E, extend the functional calculus to all bounded Borel measurable functions f: RC and prove that the map f f(A) dE(A) is a *-homomorphism. 3. Measure-Theoretic Properties: Show that for any vector EH, the map Ex: B(R) → [0, ∞) defined by Exx (B) (E(B), x) is a regular Borel measure on R. Requirements: • • • = Utilize measure-theoretic concepts to construct and analyze the spectral measure. Apply the Riesz Representation Theorem where appropriate. Demonstrate the uniqueness and properties of the spectral measure in the context of bounded self-adjoint operators.
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