Measure, Integration & Real Analysis · Problem 10B.6

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · spectrum-analytic-functions

Problem

Suppose

T

is a bounded operator on a complex nonzero Banach space

V

. ((a)) Prove that the function

\alpha \mapsto \varphi\left((T - \alpha I)^{-1} f\right)

is analytic on

\mathbf{C} \setminus \mathrm{sp}(T)

for every

f \in V

and every

\varphi \in V'

. ((b)) Prove that

\mathrm{sp}(T) \neq \emptyset