Measure, Integration & Real Analysis · Problem 10C.6

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · compact-operators-hilbert-schmidt

Problem

Suppose

T

is a bounded operator on a Hilbert space

V

. Prove that if there exists an orthonormal basis

\{e_k\}_{k \in \Gamma}

V

such that

\sum_{k \in \Gamma} \|Te_k\|^2 < \infty,

then

T

is compact.