Measure, Integration & Real Analysis · Problem 10C.2

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · compact-operators

Problem

Prove that if TT is a compact operator on L2([0,1])L^2([0,1]), then lim⁡n→∞n∥T(xn)∥2=0\lim_{n\to\infty} \sqrt{n}\|T(x^n)\|_2 = 0, where xnx^n means the element of L2([0,1])L^2([0,1]) defined by x↦xnx \mapsto x^n.

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