Problem Statement: Let H be a Hilbert space, and let T : D(T) CHH be an unbounded self-adjoint operator. 1. Resolvent Set and Spectral Measure: Define the resolvent set p(T) and the spectrum σ(T) of T. Construct the spectral measure E associated with T and prove that I can be represented as T = (T) AdE(X). 2. Measurability of the Spectral Measure: Prove that for each x, y EH, the map A (E(d)x, y) is a complex measure, and demonstrate its measurability with respect to the Borel σ-algebra on σ(T). 3. Spectral Theorem for Unbounded Operators: Extend the spectral theorem to unbounded self- adjoint operators by verifying that the integral representation of T satisfies D(T) = {x = H | √(T) |A|² d (E(A)x, x) < ∞0}. E Jo(T) AdE(X)x. Requirements: ° • • and that T acts on x = D(T) as Tx = Address the complexities introduced by unbounded operators in the spectral theorem. Employ measure-theoretic integration for unbounded measurable functions. Carefully handle domain issues associated with unbounded operators in Hilbert spaces.
Problem Statement: Let H be a Hilbert space, and let T : D(T) CHH be an unbounded self-adjoint operator. 1. Resolvent Set and Spectral Measure: Define the resolvent set p(T) and the spectrum σ(T) of T. Construct the spectral measure E associated with T and prove that I can be represented as T = (T) AdE(X). 2. Measurability of the Spectral Measure: Prove that for each x, y EH, the map A (E(d)x, y) is a complex measure, and demonstrate its measurability with respect to the Borel σ-algebra on σ(T). 3. Spectral Theorem for Unbounded Operators: Extend the spectral theorem to unbounded self- adjoint operators by verifying that the integral representation of T satisfies D(T) = {x = H | √(T) |A|² d (E(A)x, x) < ∞0}. E Jo(T) AdE(X)x. Requirements: ° • • and that T acts on x = D(T) as Tx = Address the complexities introduced by unbounded operators in the spectral theorem. Employ measure-theoretic integration for unbounded measurable functions. Carefully handle domain issues associated with unbounded operators in Hilbert spaces.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 52E
Related questions
Question
Transcribed Image Text:Problem Statement:
Let H be a Hilbert space, and let T : D(T) CHH be an unbounded self-adjoint operator.
1. Resolvent Set and Spectral Measure: Define the resolvent set p(T) and the spectrum σ(T) of
T. Construct the spectral measure E associated with T and prove that I can be represented as
T =
(T)
AdE(X).
2. Measurability of the Spectral Measure: Prove that for each x, y EH, the map A
(E(d)x, y) is a complex measure, and demonstrate its measurability with respect to the Borel
σ-algebra on σ(T).
3. Spectral Theorem for Unbounded Operators: Extend the spectral theorem to unbounded self-
adjoint operators by verifying that the integral representation of T satisfies D(T) =
{x = H | √(T) |A|² d (E(A)x, x) < ∞0}.
E
Jo(T) AdE(X)x.
Requirements:
°
•
•
and that T acts on x = D(T) as Tx =
Address the complexities introduced by unbounded operators in the spectral theorem.
Employ measure-theoretic integration for unbounded measurable functions.
Carefully handle domain issues associated with unbounded operators in Hilbert spaces.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 6 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage