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1998 VJIMC, Problem 1

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 $E^\perp=\{x\in H:\forall\phi\in E:(x,\phi)=0\}$ of $E$ is $T$-invariant, i.e., $T(E^\perp)\subseteq E^\perp$.

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