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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2025 • Question 3

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2025 • Question 3

Let \(T : X \to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \neq 0\) the null space \(N(T - \lambda I)\) is finite dimensional.

Raw LaTeX

Let \(T : X \to X\) be a compact linear operator on a normed space \(X\). Show that for every
\(\lambda \neq 0\) the null space \(N(T - \lambda I)\) is finite dimensional.

Topic Tags

Banach and Normed Spaces Bounded Linear Operators Compact Operators