Problem 4
Let A be a continuous linear operator mapping a Banach space \(\mathrm{E}\) onto another Banach space \(E_{1}\). Prove that there is a constant \(\mathrm{a}>0\) such if \(\mathrm{B} \in \mathscr{L}\left(E, E_{1}\right)\) and \(\|A-B\|<\mathrm{a}\), then \(\mathrm{B}\) also maps \(\mathrm{E}\) onto (all of) \(E_{1}\).
Problem 5
Let A be an operator mapping a Hilbert space \(H\) into itself. Then a subspace \(M \subset H\) is said to be invariant under \(A\) if \(x \in M\) implies Ax \(\in M\). Prove that if \(M\) is invariant under \(A\), then its orthogonal complement \(M^{\prime}=H \ominus M\) is invariant under the adjoint operator \(A^{*}\) (in particular, under A itself if A is self-adjoint).
Problem 7
Give an example of an operator whose spectrum consists of a single point.
Problem 8
Given a bounded linear operator A mapping a Banach space E into itself, prove that the limit $$ r=\lim _{n \rightarrow \infty} \sqrt[n]{\left\|^{n}\right\|} $$ exists. Show that the spectrum of \(\mathrm{A}\) is contained in the disk of radius \(\mathrm{r}\) with center at the origin. Comment. The quantity \(r\) is called the spectral radius of the operator A. This result contains Theorem 8 as a special case, since \(\left\|A^{n}\right\|<\|A\|^{n}\).
Problem 9
Let \(R,=(\mathrm{A}-\lambda I)^{-1}\) and \(R,=(\mathrm{A}-\mu I)^{-1}\) be the resolvents corresponding to the points \(A\) and \(\mu\). Prove that \(R_{\lambda} R_{\mu}=R_{\mu} R_{\lambda}\) and $$ R_{\mu}-R_{\lambda}=(\mu-\lambda) R_{\mu} R_{\lambda} $$ Hint. Multiply both sides of (19) by \((\mathrm{A}-\lambda I)(A-\mu L)\). Comment. It follows from (19) that if \(\mathrm{A}\), is a regular point of \(\mathrm{A}\), then the derivative of \(R\), with respect to \(A\) at the point \(\lambda_{0}\), i.e., the limit. $$ \lim _{\Delta \lambda \rightarrow 0} \frac{R_{\lambda_{0}+\Delta \lambda}-R_{\lambda_{0}}}{\Delta \lambda} $$ (in the sense of convergence with respect to the operator norm) exists and equals \(R_{\lambda_{0}^{\circ}}^{2}\)
Problem 10
Let A be a bounded self-adjoint operator mapping a complex Hilbert space Hinto itself. Prove that the spectrum of A is a closed bounded subset of the real line.
Problem 11
Prove that every bounded linear operator defined on a complex Banach space with at least one nonzero element has a nonempty spectrum.
