Measure, Integration & Real Analysis · Problem 10B.16

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · normal-operators-eigenvectors

Problem

Suppose

\{e_k\}_{k \in \mathbf{Z}^+}

is an orthonormal basis of a Hilbert space

V

. Suppose also that

T

is a normal operator on

V

and

e_k

is an eigenvector of

T

for every

k \geq 2

. Prove that

e_1

is an eigenvector of

T