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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2019 • Question 4

Question

Math 6420 • PDE Spring 2019 • Question 4

Let \(H\) be a Hilbert space. Recall that an operator \(T\in B(H)\) is Hilbert-Schmidt if there exits a complete orthonormal sequence, \((e_n)\in H\) such that \[ \sum _{n=1}^{\infty }\lVert Te_n\rVert ^2<\infty . \] Show that every Hilbert-Schmidt operator is compact.

Raw LaTeX

Let \(H\) be a Hilbert space. Recall that an operator \(T\in B(H)\) is Hilbert-Schmidt if
there exits a complete orthonormal sequence, \((e_n)\in H\) such that
\[
  \sum_{n=1}^{\infty}\lVert Te_n\rVert^2<\infty.
\]
Show that every Hilbert-Schmidt operator is compact.

Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Orthonormal Bases and Total Sets Sequence Spaces