Measure, Integration & Real Analysis · Problem 10B.2

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · spectrum-bounded-operators

Problem

Suppose

T

is a bounded operator on a Hilbert space

V

. ((a)) Prove that

\mathrm{sp}(S^{-1}TS) = \mathrm{sp}(T)

for all bounded invertible operators

S

V

. ((b)) Prove that

\mathrm{sp}(T^*) = \{\bar{\alpha} : \alpha \in \mathrm{sp}(T)\}

. ((c)) Prove that if

T

is invertible, then

\mathrm{sp}(T^{-1}) = \left\{\frac{1}{\alpha} : \alpha \in \mathrm{sp}(T)\right\}