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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2021 • Question 3

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2021 • Question 3

Let \(X\) be a Banach space and \(T : X \to X\) be a compact linear operator. Use the Fredholm alternative to show that if \(\lambda \notin \sigma _p(T)\) and \(\lambda \neq 0\) then \(\lambda \in \rho (T)\), or in other words: that all non-zero spectral values of a compact operator are eigenvalues.

Raw LaTeX

Let \(X\) be a Banach space and \(T : X \to X\) be a compact linear operator. Use the
Fredholm alternative to show that if \(\lambda \notin \sigma_p(T)\) and \(\lambda \neq 0\) then \(\lambda \in \rho(T)\), or in other
words: that all non-zero spectral values of a compact operator are eigenvalues.

Topic Tags

Banach and Normed Spaces Bounded Linear Operators Fredholm Alternative Spectral Theory of Operators