Measure, Integration & Real Analysis · Problem 10A.10

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · multiplication-operator-properties

Problem

Suppose

h \in \mathcal{L}^\infty(\mathbf{R})

and

M_h: L^2(\mathbf{R}) \to L^2(\mathbf{R})

is the bounded operator defined by

M_h f = fh

. (a) Show that

M_h

is injective if and only if

|\{x \in \mathbf{R} : h(x) = 0\}| = 0

. (b) Find a necessary and sufficient condition (in terms of

h

) for

M_h

to have dense range. (c) Find a necessary and sufficient condition (in terms of

h

) for

M_h

to have closed range. (d) Find a necessary and sufficient condition (in terms of

h

) for

M_h

to be invertible.