Measure, Integration & Real Analysis · Problem 10B.26

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · partial-isometries-characterization

Problem

Suppose

T

is a bounded operator on a Hilbert space

V

. Prove that

T

is a partial isometry if and only if

T^*T = P_U

for some closed subspace

U

V

. [A bounded operator

T

on a Hilbert space is called a partial isometry if

\|Tf\| = \|f\|

for all

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