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Let X be a Banach space, and let T : X → X be a bounded linear operator. Consider the space L(X), the space of all bounded linear operators on X, and define the operator T as follows: Part 1: Norm and Spectrum 1. Operator Norm: Prove that the operator norm T| satisfies the following inequality for any x Є X: ||Tx|| ≤ ||T|| · ||x||. (Hint: Use the definition of operator norm.) 2. Spectrum: Define the spectrum o(T) of the operator T. Show that for any A € C, if AI - T is not invertible, then > € σ(T). Classify the spectrum into three disjoint sets: the point spectrum, continuous spectrum, and residual spectrum. Part 2: Compact Operators and Fredholm Operators - 3. Compact Operator: Let K: X → X be a compact operator. Prove that if ||T – K|| < || ||, then T + K is invertible. (Hint: Use the Neumann series for the inverse of operators.) 4. Fredholm Operator: Suppose that T is a Fredholm operator with index ind(T). Prove that: ind(T) = dim(ker(T)) - dim(coker(T)), where coker(T) = x/ran(T). Further, show that if K is compact, then T + K is also Fredholm with the same index. Part 3: Spectral Theorem for Compact Operators 5. Spectral Theorem: Suppose T is a compact self-adjoint operator on a Hilbert space H. Prove the following: The spectrum σ (T) of T consists of 0 and a sequence of eigenvalues {} such that An → 0 as n→ ∞. For each non-zero eigenvalue X, the eigenspace corresponding to \,, is finite- dimensional, and T can be diagonalized as: Tr=An(x, en)en, n=1 where {e} is an orthonormal basis of eigenvectors corresponding to the eigenvalues Ʌn.

Let X be a Banach space, and let T : X → X be a bounded linear operator. Consider the space L(X), the space of all bounded linear operators on X, and define the operator T as follows: Part 1: Norm and Spectrum 1. Operator Norm: Prove that the operator norm T| satisfies the following inequality for any x Є X: ||Tx|| ≤ ||T|| · ||x||. (Hint: Use the definition of operator norm.) 2. Spectrum: Define the spectrum o(T) of the operator T. Show that for any A € C, if AI - T is not invertible, then > € σ(T). Classify the spectrum into three disjoint sets: the point spectrum, continuous spectrum, and residual spectrum. Part 2: Compact Operators and Fredholm Operators - 3. Compact Operator: Let K: X → X be a compact operator. Prove that if ||T – K|| < || ||, then T + K is invertible. (Hint: Use the Neumann series for the inverse of operators.) 4. Fredholm Operator: Suppose that T is a Fredholm operator with index ind(T). Prove that: ind(T) = dim(ker(T)) - dim(coker(T)), where coker(T) = x/ran(T). Further, show that if K is compact, then T + K is also Fredholm with the same index. Part 3: Spectral Theorem for Compact Operators 5. Spectral Theorem: Suppose T is a compact self-adjoint operator on a Hilbert space H. Prove the following: The spectrum σ (T) of T consists of 0 and a sequence of eigenvalues {} such that An → 0 as n→ ∞. For each non-zero eigenvalue X, the eigenspace corresponding to \,, is finite- dimensional, and T can be diagonalized as: Tr=An(x, en)en, n=1 where {e} is an orthonormal basis of eigenvectors corresponding to the eigenvalues Ʌn.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let X be a Banach space, and let T : X → X be a bounded linear operator. Consider the space L(X), the space of all bounded linear operators on X, and define the operator T as follows: Part 1: Norm and Spectrum 1. Operator Norm: Prove that the operator norm T| satisfies the following inequality for any x Є X: ||Tx|| ≤ ||T|| · ||x||. (Hint: Use the definition of operator norm.) 2. Spectrum: Define the spectrum o(T) of the operator T. Show that for any A € C, if AI - T is not invertible, then > € σ(T). Classify the spectrum into three disjoint sets: the point spectrum, continuous spectrum, and residual spectrum. Part 2: Compact Operators and Fredholm Operators - 3. Compact Operator: Let K: X → X be a compact operator. Prove that if ||T – K|| < || ||, then T + K is invertible. (Hint: Use the Neumann series for the inverse of operators.) 4. Fredholm Operator: Suppose that T is a Fredholm operator with index ind(T). Prove that: ind(T) = dim(ker(T)) - dim(coker(T)), where coker(T) = x/ran(T). Further, show that if K is compact, then T + K is also Fredholm with the same index. Part 3: Spectral Theorem for Compact Operators 5. Spectral Theorem: Suppose T is a compact self-adjoint operator on a Hilbert space H. Prove the following: The spectrum σ (T) of T consists of 0 and a sequence of eigenvalues {} such that An → 0 as n→ ∞. For each non-zero eigenvalue X, the eigenspace corresponding to \,, is finite- dimensional, and T can be diagonalized as: Tr=An(x, en)en, n=1 where {e} is an orthonormal basis of eigenvectors corresponding to the eigenvalues Ʌn.
Transcribed Image Text:Let X be a Banach space, and let T : X → X be a bounded linear operator. Consider the space L(X), the space of all bounded linear operators on X, and define the operator T as follows: Part 1: Norm and Spectrum 1. Operator Norm: Prove that the operator norm T| satisfies the following inequality for any x Є X: ||Tx|| ≤ ||T|| · ||x||. (Hint: Use the definition of operator norm.) 2. Spectrum: Define the spectrum o(T) of the operator T. Show that for any A € C, if AI - T is not invertible, then > € σ(T). Classify the spectrum into three disjoint sets: the point spectrum, continuous spectrum, and residual spectrum. Part 2: Compact Operators and Fredholm Operators - 3. Compact Operator: Let K: X → X be a compact operator. Prove that if ||T – K|| < || ||, then T + K is invertible. (Hint: Use the Neumann series for the inverse of operators.) 4. Fredholm Operator: Suppose that T is a Fredholm operator with index ind(T). Prove that: ind(T) = dim(ker(T)) - dim(coker(T)), where coker(T) = x/ran(T). Further, show that if K is compact, then T + K is also Fredholm with the same index. Part 3: Spectral Theorem for Compact Operators 5. Spectral Theorem: Suppose T is a compact self-adjoint operator on a Hilbert space H. Prove the following: The spectrum σ (T) of T consists of 0 and a sequence of eigenvalues {} such that An → 0 as n→ ∞. For each non-zero eigenvalue X, the eigenspace corresponding to \,, is finite- dimensional, and T can be diagonalized as: Tr=An(x, en)en, n=1 where {e} is an orthonormal basis of eigenvectors corresponding to the eigenvalues Ʌn.
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