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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2021 • Question 5

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2021 • Question 5

Let \(T : X \to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \(\{\lambda \in \mathbb {C} \mid |\lambda | \le \|T\|\}\).

Raw LaTeX

Let \(T : X \to X\) be a bounded linear operator on a complex Banach space \(X\).
Prove that \(\sigma(T)\) lies in the complex plane disk: \(\{\lambda \in \mathbb{C} \mid |\lambda| \le \|T\|\}\).

Topic Tags

Banach and Normed Spaces Bounded Linear Operators Dual Spaces and Bounded Linear Functionals