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2. Let T be a self-adjoint compact operator on a Hilbert space H. Prove that the spectrum of T consists of a sequence of real eigenvalues converging to zero. 3. Show that if I is a normal operator on a Hilbert space, then the spectral theorem implies that there exists a measure space (M, μ) such that I can be represented as a multiplication operator on L²(M, μ). 4. Illustrate the spectral properties of T with a graph of the spectrum on the complex plane, and show examples of compact operators with discrete spectra. Problem 5: Topological Manifolds and Covering Spaces Let M be a topological manifold of dimension 2 (a surface), and consider the universal covering space M. 1. Prove that if M is a connected and simply connected 2-dimensional surface, then M is homeomorphic to the plane R². 2. Find the universal covering spaces for the surfaces S² (2-sphere), 7² (torus), and the Klein bottle, and describe their properties. 3. Show that any covering space of a compact surface can be constructed using a tiling of the universal cover, and draw a graphical representation of the covering map for the torus and the Klein bottle. 4. Use the concept of deck transformations to describe the automorphism group of the covering space, and show its relation to the fundamental group of the base space.

2. Let T be a self-adjoint compact operator on a Hilbert space H. Prove that the spectrum of T consists of a sequence of real eigenvalues converging to zero. 3. Show that if I is a normal operator on a Hilbert space, then the spectral theorem implies that there exists a measure space (M, μ) such that I can be represented as a multiplication operator on L²(M, μ). 4. Illustrate the spectral properties of T with a graph of the spectrum on the complex plane, and show examples of compact operators with discrete spectra. Problem 5: Topological Manifolds and Covering Spaces Let M be a topological manifold of dimension 2 (a surface), and consider the universal covering space M. 1. Prove that if M is a connected and simply connected 2-dimensional surface, then M is homeomorphic to the plane R². 2. Find the universal covering spaces for the surfaces S² (2-sphere), 7² (torus), and the Klein bottle, and describe their properties. 3. Show that any covering space of a compact surface can be constructed using a tiling of the universal cover, and draw a graphical representation of the covering map for the torus and the Klein bottle. 4. Use the concept of deck transformations to describe the automorphism group of the covering space, and show its relation to the fundamental group of the base space.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 27EQ
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2. Let T be a self-adjoint compact operator on a Hilbert space H. Prove that the spectrum of T consists of a sequence of real eigenvalues converging to zero. 3. Show that if I is a normal operator on a Hilbert space, then the spectral theorem implies that there exists a measure space (M, μ) such that I can be represented as a multiplication operator on L²(M, μ). 4. Illustrate the spectral properties of T with a graph of the spectrum on the complex plane, and show examples of compact operators with discrete spectra. Problem 5: Topological Manifolds and Covering Spaces Let M be a topological manifold of dimension 2 (a surface), and consider the universal covering space M. 1. Prove that if M is a connected and simply connected 2-dimensional surface, then M is homeomorphic to the plane R². 2. Find the universal covering spaces for the surfaces S² (2-sphere), 7² (torus), and the Klein bottle, and describe their properties. 3. Show that any covering space of a compact surface can be constructed using a tiling of the universal cover, and draw a graphical representation of the covering map for the torus and the Klein bottle. 4. Use the concept of deck transformations to describe the automorphism group of the covering space, and show its relation to the fundamental group of the base space.
Transcribed Image Text:2. Let T be a self-adjoint compact operator on a Hilbert space H. Prove that the spectrum of T consists of a sequence of real eigenvalues converging to zero. 3. Show that if I is a normal operator on a Hilbert space, then the spectral theorem implies that there exists a measure space (M, μ) such that I can be represented as a multiplication operator on L²(M, μ). 4. Illustrate the spectral properties of T with a graph of the spectrum on the complex plane, and show examples of compact operators with discrete spectra. Problem 5: Topological Manifolds and Covering Spaces Let M be a topological manifold of dimension 2 (a surface), and consider the universal covering space M. 1. Prove that if M is a connected and simply connected 2-dimensional surface, then M is homeomorphic to the plane R². 2. Find the universal covering spaces for the surfaces S² (2-sphere), 7² (torus), and the Klein bottle, and describe their properties. 3. Show that any covering space of a compact surface can be constructed using a tiling of the universal cover, and draw a graphical representation of the covering map for the torus and the Klein bottle. 4. Use the concept of deck transformations to describe the automorphism group of the covering space, and show its relation to the fundamental group of the base space.
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