Measure, Integration & Real Analysis · Problem 10C.4

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · compact-multiplication-operators

Problem

Suppose h∈L∞(R)h \in L^\infty(\mathbf{R}). Define Mh∈B(L2(R))M_h \in \mathcal{B}\bigl(L^2(\mathbf{R})\bigr) by Mhf=fhM_h f = fh. Prove that if ∥h∥∞>0\|h\|_\infty > 0, then MhM_h is not compact.

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