Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Holder inequality: ≤ (6)(C) j=1 1 where p > 1 and + P 1 1. Cauchy-Schwarz inequality: Σ Kk;% ≤ (²)* Σαρ j=1 Minkowski inequality: + k=1 m=1 (Ex)'s (Eur)² + (Σ~)' where p>1. k=1 m-1 Problem 8: Spectral Theory in Hilbert Spaces Problem Statement: Let H be a separable Hilbert space and let T: H→H be a compact, self-adjoint operator. Tasks: a) Spectral Theorem: State the Spectral Theorem for compact, self-adjoint operators on Hilbert spaces. Use it to describe the spectrum of T. b) Eigenvalues and Eigenvectors: Prove that the non-zero spectrum of T consists of eigenvalues with finite multiplicity, accumulating only at zero. c) Hilbert-Schmidt Operators: Define Hilbert-Schmidt operators and prove that every Hilbert- Schmidt operator is compact. d) Visualization: For H=2(N) and T represented by a diagonal matrix with entries {\,} converging to zero, visualize the action of T on basis vectors and illustrate the convergence properties of the operator.
Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Holder inequality: ≤ (6)(C) j=1 1 where p > 1 and + P 1 1. Cauchy-Schwarz inequality: Σ Kk;% ≤ (²)* Σαρ j=1 Minkowski inequality: + k=1 m=1 (Ex)'s (Eur)² + (Σ~)' where p>1. k=1 m-1 Problem 8: Spectral Theory in Hilbert Spaces Problem Statement: Let H be a separable Hilbert space and let T: H→H be a compact, self-adjoint operator. Tasks: a) Spectral Theorem: State the Spectral Theorem for compact, self-adjoint operators on Hilbert spaces. Use it to describe the spectrum of T. b) Eigenvalues and Eigenvectors: Prove that the non-zero spectrum of T consists of eigenvalues with finite multiplicity, accumulating only at zero. c) Hilbert-Schmidt Operators: Define Hilbert-Schmidt operators and prove that every Hilbert- Schmidt operator is compact. d) Visualization: For H=2(N) and T represented by a diagonal matrix with entries {\,} converging to zero, visualize the action of T on basis vectors and illustrate the convergence properties of the operator.
Chapter3: Graphs And Functions
Section3.4: Graph Linear Inequalities In Two Variables
Problem 3.82TI: Veronica works two part time jobs in order to earn enough money to meet her obligations of at least...
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Transcribed Image Text:Instructions:
*Do not Use AI. (Solve by yourself, hand written preferred)
* Give appropriate graphs and required codes.
* Make use of inequalities if you think that required.
* You are supposed to use kreszig for reference.
Holder inequality: ≤ (6)(C)
j=1
1
where p > 1 and +
P
1
1.
Cauchy-Schwarz inequality: Σ Kk;% ≤ (²)* Σαρ
j=1
Minkowski inequality: +
k=1
m=1
(Ex)'s (Eur)² + (Σ~)'
where p>1.
k=1
m-1
Problem 8: Spectral Theory in Hilbert Spaces
Problem Statement:
Let H be a separable Hilbert space and let T: H→H be a compact, self-adjoint operator.
Tasks:
a) Spectral Theorem: State the Spectral Theorem for compact, self-adjoint operators on Hilbert
spaces. Use it to describe the spectrum of T.
b) Eigenvalues and Eigenvectors: Prove that the non-zero spectrum of T consists of eigenvalues
with finite multiplicity, accumulating only at zero.
c) Hilbert-Schmidt Operators: Define Hilbert-Schmidt operators and prove that every Hilbert-
Schmidt operator is compact.
d) Visualization: For H=2(N) and T represented by a diagonal matrix with entries {\,}
converging to zero, visualize the action of T on basis vectors and illustrate the convergence
properties of the operator.
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