MathDB

Let

H

be a complex Hilbert space. Let

T:H\to H

be a bounded linear operator such that

|(Tx,x)|\le\lVert x\rVert^2

for each

x\in H

. Assume that

\mu\in\mathbb C

|\mu|=1

, is an eigenvalue with the corresponding eigenspace

E=\{\phi\in H:T\phi=\mu\phi\}

. Prove that the orthogonal complement

E^\perp=\{x\in H:\forall\phi\in E:(x,\phi)=0\}

E

T

-invariant, i.e.,

T(E^\perp)\subseteq E^\perp

Problem Statement