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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2019 • Question 2

Question

Math 6420 • PDE Spring 2019 • Question 2

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that the spectrum \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}:\lvert \lambda \rvert \le \lVert T\rVert \}. \]

Raw LaTeX

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that
the spectrum \(\sigma(T)\) lies in the complex plane disk:
\[
  \{\lambda\in\mathbb{C}:\lvert\lambda\rvert\le\lVert T\rVert\}.
\]

Topic Tags

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