91Ó°ÊÓ

Show that if \(T, S\) are bounded linear operators on a Banach space \(X\) and one of them is compact, then \(T S\) and \(S T\) are compact.

Let \(X\) be an infinite-dimensional Banach space. Show that there is a bounded linear non-compact operator from \(X\) into \(c_{0}\).

Let \(K\) be a compact set in the scalar field. Show that there is an operator \(T \in \mathcal{B}\left(\ell_{2}\right)\) such that \(\sigma(T)=K\).

(i) Let \(L\) be a left shift operator in \(\ell_{2}, L\left(x_{1}, x_{2}, \ldots\right)=\left(x_{2}, x_{3}, \ldots\right)\). Show that the set of all eigenvalues of \(L\) is the open unit disk. (ii) Let \(R\) be a right shift operator on \(\ell_{2}, R\left(x_{1}, x_{2}, \ldots\right)=\left(0, x_{1}, x_{2}, \ldots\right)\). Show that the set of all eigenvalues of \(R\) is empty. (iii) Show that \(\sigma(L)\) and \(\sigma(R)\) are both equal to the closed unit disk.

Consider the right shift \(R\) on \(\ell_{2}\) and the diagonal operator \(D\) associated with \(d_{i}=2^{-i}\). Define a weighted shift operator \(T\) on the complex space \(\ell_{2}\) by \(T=R \circ D .\) Show that \(T\) is a compact operator with spectral radius 0 and \(T\) is one-to-one. Thus, \(T\) has no eigenvalues and \(\sigma(T)=\\{0\\}\).

Let \(H\) be a separable Hilbert space. An operator \(T \in \mathcal{B}(H)\) is called a Hilbert-Schmidt operator if there is an orthonormal basis \(\left\\{e_{i}\right\\}\) of \(H\) such that \(\sum\left\|T\left(e_{i}\right)\right\|^{2}<\infty .\) Show that if \(\left\\{f_{i}\right\\}\) is another orthonormal basis of \(H\), then \(\sum\left\|T\left(f_{i}\right)\right\|^{2}=\sum\left\|T\left(e_{i}\right)\right\|^{2}\) The number \(\|T\|_{H S}=\left(\sum\left\|T\left(e_{i}\right)\right\|^{2}\right)^{\frac{1}{2}}\) is called the Hilbert-Schmidt norm of \(T .\) Show that \(\|T\|_{H S} \geq\|T\|\).

Show that every Hilbert-Schmidt operator \(T\) on a Hilbert space \(H\) is compact. Find a compact operator that is not a Hilbert-Schmidt operator.

Let \(C\) be a closed convex bounded subset of a Banach space \(X\). Show that if \(T: C \rightarrow C\) is a nonexpansive map, then \(\inf \\{\|x-T(x)\| ; x \in C\\}=0\).