Measure, Integration & Real Analysis · Problem 10B.25

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · partial-isometries

Problem

Suppose

T

is an isometry on a Hilbert space. Prove that

T^*

is a partial isometry. [A bounded operator

T

on a Hilbert space is called a partial isometry if

\|Tf\| = \|f\|

for all

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