Measure, Integration & Real Analysis · Problem 10B.23

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · operator-exponential

Problem

For

T

a bounded operator on a Banach space, define

e^T

e^T = \sum_{k=0}^{\infty} \frac{T^k}{k!}.

((a)) Prove that if

T

is a bounded operator on a Banach space

V

, then the infinite sum above converges in

\mathcal{B}(V)

and

\|e^T\| \leq e^{\|T\|}

. ((b)) Prove that if

S, T

are bounded operators on a Banach space

V

such that

ST = TS

, then

e^S e^T = e^{S+T}

. ((c)) Prove that if

T

is a self-adjoint operator on a complex Hilbert space, then

e^{iT}

is unitary.