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Question 9: Spectral Theory of Weighted Shift Operators and Measure- Theoretic Dynamics Problem Statement: Consider the Hilbert space l² (N) and define the weighted shift operator S₁, by Sw(en) = Wnen+1, where {e} is the standard orthonormal basis and {w} is a bounded sequence of positive weights. 1. Spectrum Determination: Determine the spectrum σ(S) of Sw in terms of the weight sequence {wn}. Provide a proof that relates the spectral properties to the limit superior and inferior of the weights. 2. Spectral Measure Analysis: Construct the spectral measure E associated with S₁, and analyze its support. Show how the properties of {wn} influence the nature of E. 3. Measure-Theoretic Dynamical Systems: Relate the spectral properties of S to measure- theoretic dynamical systems by interpreting Sw as an operator encoding a shift dynamical system. Prove results concerning the ergodicity or mixing properties of the system based on the spectral measure E.

Question 9: Spectral Theory of Weighted Shift Operators and Measure- Theoretic Dynamics Problem Statement: Consider the Hilbert space l² (N) and define the weighted shift operator S₁, by Sw(en) = Wnen+1, where {e} is the standard orthonormal basis and {w} is a bounded sequence of positive weights. 1. Spectrum Determination: Determine the spectrum σ(S) of Sw in terms of the weight sequence {wn}. Provide a proof that relates the spectral properties to the limit superior and inferior of the weights. 2. Spectral Measure Analysis: Construct the spectral measure E associated with S₁, and analyze its support. Show how the properties of {wn} influence the nature of E. 3. Measure-Theoretic Dynamical Systems: Relate the spectral properties of S to measure- theoretic dynamical systems by interpreting Sw as an operator encoding a shift dynamical system. Prove results concerning the ergodicity or mixing properties of the system based on the spectral measure E.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 46E
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Question 9: Spectral Theory of Weighted Shift Operators and Measure- Theoretic Dynamics Problem Statement: Consider the Hilbert space l² (N) and define the weighted shift operator S₁, by Sw(en) = Wnen+1, where {e} is the standard orthonormal basis and {w} is a bounded sequence of positive weights. 1. Spectrum Determination: Determine the spectrum σ(S) of Sw in terms of the weight sequence {wn}. Provide a proof that relates the spectral properties to the limit superior and inferior of the weights. 2. Spectral Measure Analysis: Construct the spectral measure E associated with S₁, and analyze its support. Show how the properties of {wn} influence the nature of E. 3. Measure-Theoretic Dynamical Systems: Relate the spectral properties of S to measure- theoretic dynamical systems by interpreting Sw as an operator encoding a shift dynamical system. Prove results concerning the ergodicity or mixing properties of the system based on the spectral measure E.
Transcribed Image Text:Question 9: Spectral Theory of Weighted Shift Operators and Measure- Theoretic Dynamics Problem Statement: Consider the Hilbert space l² (N) and define the weighted shift operator S₁, by Sw(en) = Wnen+1, where {e} is the standard orthonormal basis and {w} is a bounded sequence of positive weights. 1. Spectrum Determination: Determine the spectrum σ(S) of Sw in terms of the weight sequence {wn}. Provide a proof that relates the spectral properties to the limit superior and inferior of the weights. 2. Spectral Measure Analysis: Construct the spectral measure E associated with S₁, and analyze its support. Show how the properties of {wn} influence the nature of E. 3. Measure-Theoretic Dynamical Systems: Relate the spectral properties of S to measure- theoretic dynamical systems by interpreting Sw as an operator encoding a shift dynamical system. Prove results concerning the ergodicity or mixing properties of the system based on the spectral measure E.
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