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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Need detailed answers without AI, use kreszig
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.
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 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
