91Ó°ÊÓ

Prove that if \(T\) is a compact operator on a Hilbert space \(V\) and \(e_{1}, e_{2}, \ldots\) is an orthonormal sequence in \(V\), then \(\lim _{n \rightarrow \infty} T e_{n}=0\).

Suppose \(T\) is a bounded operator on a Hilbert space \(V\). (a) Prove that sp \(\left(S^{-1} T S\right)=\operatorname{sp}(T)\) for all bounded invertible operators \(S\) on \(V\). (b) Prove that \(\operatorname{sp}\left(T^{*}\right)=\\{\bar{\alpha}: \alpha \in \operatorname{sp}(T)\\}\). (c) Prove that if \(T\) is invertible, then \(\operatorname{sp}\left(T^{-1}\right)=\left\\{\frac{1}{\alpha}: \alpha \in \operatorname{sp}(T)\right\\}\).

Suppose \(E\) is a bounded subset of \(\mathbf{F}\). Show that there exists a Hilbert space \(V\) and \(T \in \mathcal{B}(V)\) such that the set of eigenvalues of \(T\) equals \(E\).

(a) Prove or give a counterexample: If \(T\) is a bounded operator on a Hilbert space such that \(T\) and \(T^{*}\) are both injective, then \(T\) is invertible. (b) Prove or give a counterexample: If \(T\) is a bounded operator on a Hilbert space such that \(T\) and \(T^{*}\) are both surjective, then \(T\) is invertible.

Prove that if \(T\) is a normal operator on a Hilbert space, then \(\left\|T^{n}\right\|=\|T\|^{n}\) for every \(n \in \mathbf{Z}^{+}\).

Prove that every nonzero eigenvalue of a compact operator on a Hilbert space has finite algebraic multiplicity.

Find the singular values of the Volterra operator.