Measure, Integration & Real Analysis · Problem 10B.24

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · partial-isometries-multiplication-operators

Problem

Suppose

(X, \mathcal{S}, \mu)

is a

\sigma

-finite measure space and

h \in L^\infty(\mu)

. As usual, let

M_h \in \mathcal{B}\bigl(L^2(\mu)\bigr)

denote the multiplication operator defined by

M_h f = fh

. Prove that

M_h

is a partial isometry if and only if there exists a set

E \in \mathcal{S}

such that

|h| = \chi_E

. [A bounded operator

T

on a Hilbert space is called a partial isometry if

\|Tf\| = \|f\|

for all

f \in (\mathrm{null}\, T)^\perp