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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Fall 2024 • Question 4

Question

Math 6710 • Applied Linear Operators and Spectral Methods Fall 2024 • Question 4

Let \(H\) be a Hilbert space and \(A : H \to H\) be a bounded self-adjoint linear operator.

  1. Prove that

    \[\|A\| = \sup _{x \neq 0} \frac {|\langle Ax, x\rangle |}{\|x\|^2}.\]
  2. Prove that at least one of \(\|A\|\) or \(-\|A\|\) is an element of \(\sigma (A)\).

Raw LaTeX

Let \(H\) be a Hilbert space and \(A : H \to H\) be a bounded self-adjoint linear operator.
\begin{enumerate}

\item
Prove that

\[\|A\| = \sup_{x \neq 0} \frac{|\langle Ax, x\rangle|}{\|x\|^2}.\]



\item
Prove that at least one of \(\|A\|\) or \(-\|A\|\) is an element of \(\sigma(A)\).

\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Hilbert Spaces and Orthogonality