Measure, Integration & Real Analysis · Problem 10C.5

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · compact-diagonal-operators

Problem

Suppose

(b_1, b_2, \ldots) \in \ell^\infty

. Define

T: \ell^2 \to \ell^2

T(a_1, a_2, \ldots) = (a_1 b_1, a_2 b_2, \ldots).

Prove that

T

is compact if and only if

\lim_{n\to\infty} b_n = 0