Fundamentals of spectral theory of bounded operators in Banach spaces

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Using problem 9.39, prove that $\rho (A)$ is an open set (hence $\sigma (A)$ is a closed set) in $\mathbb {C}$.

Using problem 9.39, prove that the spectrum of a bounded operator $A$ lies inside the disk $\left\{ \lambda \in \mathbb {C}:\left|\lambda \right| \leqslant \left\| A\right\| \right\}$. Prove that for $\left|\lambda \right| > \left\| A\right\|$ the resolvent is given by the C. Neumann series:

Prove Theorem 12.1.

Theorem 12.1: Let $X$ be a Banach space. If $A \in \mathcal{B}(X)$, then the limit $\lim_{n \rightarrow \infty }\left\| A^{n}\right\|^{1 / n}$ exists and

where $r(A) = \max_{\lambda \in \sigma (A)}\left|\lambda \right|$ is the spectral radius of the operator $A$ (Definition 12.2).

© Let $\lambda , \mu \in \rho (A)$. Prove the Hilbert resolvent identity:

Using this identity, prove that

Let $\lambda_{0}, \lambda \in \rho (A)$. Prove that if $\left|\lambda -\lambda_{0}\right| < \frac{1}{\left\| R_{\lambda_{0}}(A)\right\| }$, then

Prove that $R_{\lambda }(A)$ is

Let $X$ be a Banach space, $A \in \mathcal{B}(X)(X \neq \left\{ 0\right\} )$. Prove that the spectrum of $A$ is nonempty.

Let $X$ be a Banach space, $A \in \mathcal{B}(X), \lambda \in \sigma (A)$, and suppose there exists $c > 0$ such that $\left\| A x-\lambda x\right\| > c\left\| x\right\|$ for all $x \in X$. Using problem 9.49, prove that $\lambda \in \sigma_{\mathrm{r}}(A)$.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Prove that if for a point $\lambda \in \mathbb {C}$ there exists a Weyl sequence, then $\lambda \in \sigma (A)$. Prove that if $\lambda \in \sigma_{\mathrm{p}}(A) \cup \sigma_{\mathrm{c}}(A)$, then a Weyl sequence exists for it (see also problem 12.23).

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$ an invertible operator. Prove that $\lambda \in \sigma (A)$ if and only if $1 / \lambda \in \sigma \left(A^{-1}\right)$. Prove this statement with $\sigma$ replaced by $\sigma_{\mathrm{p}}, \sigma_{\mathrm{c}}$ and $\sigma_{\mathrm{r}}$. Prove that the eigenvectors of the operators $A$ and $A^{-1}$ corresponding to the eigenvalues $\lambda$ and $1 / \lambda$ coincide.

Prove that the spectrum of an invertible isometric operator acting in a normed space lies on the unit circle. Prove that the spectrum of a unitary operator acting in a Hilbert space lies on the unit circle. Find the resolvent of these operators for $\lambda \in \mathbb {C}$, $\left|\lambda \right| \neq 1$.

Let $X = X_{0} \oplus X_{1}$ be a Banach space, represented as a direct sum of two of its closed subspaces, and let $P \in \mathcal{B}(X)$ be the projector onto $X_{0}$ along $X_{1}$. Find and classify its spectrum. Find its resolvent.

(R. S. Phillips, 1955). Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Prove that $\sigma (A) = \sigma \left(A^{\prime }\right)$.

(R. S. Phillips, 1955). Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that ¹ $\sigma (A) = \overline{\sigma \left(A^{*}\right)}$.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Prove that if $\lambda \in \sigma_{\mathrm{r}}(A)$, then $\lambda \in \sigma_{\mathrm{p}}\left(A^{\prime }\right)$; if $\lambda \in \sigma_{\mathrm{p}}(A)$, then $\lambda \in \sigma_{\mathrm{p}}\left(A^{\prime }\right) \cup \sigma_{\mathrm{r}}\left(A^{\prime }\right)$; if $\lambda \in \sigma_{\mathrm{c}}(A)$, then $\lambda \in \sigma_{\mathrm{c}}\left(A^{\prime }\right) \cup \sigma_{\mathrm{r}}\left(A^{\prime }\right)$.

Let $X$ be a reflexive Banach space, $A \in \mathcal{B}(X)$. Prove that $\sigma_{\mathrm{c}}(A) = \sigma_{\mathrm{c}}\left(A^{\prime }\right), \sigma_{\mathrm{r}}\left(A^{\prime }\right) \subset \sigma_{\mathrm{p}}(A)$.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that if $\lambda \in \sigma_{\mathrm{r}}(A)$, then $\bar{\lambda } \in \sigma_{\mathrm{p}}\left(A^{*}\right)$, if $\lambda \in \sigma_{\mathrm{p}}(A)$, then $\bar{\lambda } \in \sigma_{\mathrm{p}}\left(A^{*}\right) \cup \sigma_{\mathrm{r}}\left(A^{*}\right)$, if $\lambda \in \sigma_{\mathrm{c}}(A)$, then $\bar{\lambda } \in \sigma_{\mathrm{c}}\left(A^{*}\right)$.

Find the spectra of the following operators (with classification) and compute their resolvents:

Find the spectra of the operators $A$ and $A^{*}$, if $A$ acts in $l_{2}(\mathbb {Z})$ as follows: $(A x)_{n} = x_{n+1}$ for $n \geqslant 0,(A x)_{n} = 0$ for $n < 0$.

Find the spectrum, spectral radius, and resolvent of the operator $(A x)(t) = \int_{0}^{t} x(s) d s$ in the spaces:

Find the spectrum, spectral radius, and resolvent of the operator $A: l_{p} \rightarrow l_{p}, p \in [1, \infty )$:

Find the spectrum and resolvent of the following operators:

Give examples of bounded operators in Banach spaces showing that for a number $\lambda \in \mathbb {C}$ lying in the residual spectrum, a Weyl sequence may or may not exist (cf. problem 12.9).

Using problems 8.75, 8.77, prove that normal operators have no residual spectrum.

(H. Weyl, 1910). Let $A$ be a normal operator. Prove that $\lambda \in \sigma (A)$ if and only if a Weyl sequence exists.

Prove that the spectrum of a self-adjoint operator is real.

Prove that if the spectrum of a normal operator is real, then this operator is self-adjoint.

Prove that if the spectrum of a normal operator lies on the circle $\left\{ z \in \mathbb {C}:\left|z\right| = 1\right\}$, then this operator is unitary.

Let $A$ be a self-adjoint operator in a Hilbert space $H$. Prove that $\sigma_{\text{ess }}(A) \backslash \sigma_{\mathrm{p}}(A) = \sigma_{\mathrm{c}}(A)$.

Let $A$ be a bounded self-adjoint operator in a Hilbert space $H$. Prove that if there exists an orthonormal Weyl sequence for the operator $A-\lambda I$, then $\lambda \in \sigma_{\text{ess }}(A)$.

Find the essential spectrum for the following operators:

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$ and $A = A^{*}$. Define the numbers $m: = \inf_{\left\| x\right\| = 1}(A x, x)$ and $M: = \sup_{\left\| x\right\| = 1}(A x, x)$. Prove that $\sigma (A) \subset [m, M]$ and both points $m$ and $M$ lie in the spectrum.

Using problem 8.52, prove that for any bounded operator $A$ in a Hilbert space $\sigma \left(A^{*} A\right) \subset \left[0,\left\| A\right\|^{2}\right]$, and moreover the point $\left\| A\right\|^{2}$ lies in the spectrum.

Prove that eigenvectors corresponding to distinct eigenvalues are orthogonal for:

Let $H$ be a Hilbert space, $A$ a normal operator. Prove that $r(A) = \left\| A\right\|$ (see problem 8.78).

Let $X, Y$ be Banach spaces, $A \in \mathcal{B}(X), B \in \mathcal{B}(Y)$ similar operators (see Definition 9.4). Prove that $\sigma_{\mathrm{p}}(A) = \sigma_{\mathrm{p}}(B), \sigma_{\mathrm{c}}(A) = \sigma_{\mathrm{c}}(B)$ and $\sigma_{\mathrm{r}}(A) = \sigma_{\mathrm{r}}(B)$ (whence it follows that $\sigma (A) = \sigma (B))$.

Definition 9.4: Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X)$, $B \in \mathcal{B}(Y)$. The operators $A$ and $B$ are called similar if there exists a bounded operator $S$ with bounded inverse (that is, $S \in \mathcal{I B}(Y, X)$) such that $A S = S B$ (notation $A \sim B$).

Give an example of two operators $A$ and $B$ in a Banach space that are not similar but have the same spectra: $\sigma (A) = \sigma (B), \sigma_{\mathrm{p}}(A) = \sigma_{\mathrm{p}}(B), \sigma_{\mathrm{c}}(A) = \sigma_{\mathrm{c}}(B), \sigma_{\mathrm{r}}(A) = \sigma_{\mathrm{r}}(B)$ and $\sigma_{\mathrm{ess}}(A) = \sigma_{\mathrm{ess}}(B)$.

Using problem 12.36, find the spectrum of the following operators:

The operator $A$ in the space $L_{2}[0,1]$ is given by the formula

Classify the spectrum, determining $\sigma_{\mathrm{p}}, \sigma_{\mathrm{c}}, \sigma_{\mathrm{r}}$ and $\sigma_{\mathrm{ess}}$.

Let $X$ be a Banach space, $A, B \in \mathcal{B}(X)$. Prove that $\sigma (A B) \cup \left\{ 0\right\} = \sigma (B A) \cup \left\{ 0\right\}$. Prove this equality with $\sigma$ replaced by $\sigma_{\mathrm{p}}, \sigma_{\mathrm{c}}$ and $\sigma_{\mathrm{r}}$. Prove that if at least one of the operators $A$ or $B$ is invertible, then the operators $A B$ and $B A$ are similar and, hence, $\sigma (A B) = \sigma (B A)$.

Let $X$ be a Banach space, and suppose for operators $A \in \mathcal{L}(X)$ and $B \in \mathcal{L}(X)$ the relation $A B-B A = I$ holds. Using problem 12.40, prove that one of the operators is unbounded.

Prove that for every compact set $\Omega \subset \mathbb {C}$ there exists a normal operator $A$ in a Hilbert space such that $\sigma (A) = \Omega$. Prove that every compact subset of the real axis is the spectrum of some self-adjoint operator. Prove that every closed subset of the unit circle is the spectrum of some unitary operator.

Prove that for every compact set $\Omega \subset \mathbb {C}$ there exists an operator A in a Hilbert space such that $\sigma (A) = \sigma_{\mathrm{p}}(A) = \Omega$.

Let $H$ be a Hilbert space, $U \in \mathcal{B}(H)$. Prove that $U$ is an orthogonal reflection if and only if $U^{*} = U = U^{-1}$. Find $\sigma (U)$.

Let $X$ be a normed space, $A, B \in \mathcal{B}(X)$. Prove that if $A$ and $B$ commute, then $R_{\lambda }(A)$ commutes with $B$ for all $\lambda \in \rho (A)$. Prove that if $B$ and $R_{\lambda }(A)$ commute for some $\lambda \in \rho (A)$, then $A$ and $B$ commute.

Let $X$ be a Banach space, $A, B \in \mathcal{B}(X)$ and $A B = B A$. Prove that

Give an example of a sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(X)$ and an operator $A$ in a Banach space $X$ such that $A_{n} \rightrightarrows A, \sigma \left(A_{n}\right)$ is the unit circle, and $\sigma (A)$ is the unit disk.

Let $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(X)$ and $A_{n} \rightrightarrows A$.

Find the numerical range of the following operators:

Prove the first statement of Theorem 12.2: for any bounded operator $A$ in a complex Hilbert space $H$ the numerical range $W(A)$ is a convex bounded set.

Prove the second statement of Theorem 12.2: for any bounded operator $A$ in a complex Hilbert space $H$ the inclusions $\sigma (A) \subset \overline{W(A)} \subset \left\{ z \in \mathbb {C}:\left|z\right| \leqslant \left\| A\right\| \right\}$ hold.

Let $H$ be a complex Hilbert space, $A$ a self-adjoint operator. Prove that $\overline{W(A)} = [m, M]$, where $m = \inf \left\{ \lambda : \lambda \in \sigma (A)\right\}$, and $M = \sup \left\{ \lambda : \lambda \in \sigma (A)\right\}$. Must the set $W(A)$ be closed?

Let $H$ be a complex Hilbert space, $A$ a self-adjoint operator. Prove that $\overline{W(A)} = [m, M]$, where either $M = \left\| A\right\|$, or $m = -\left\| A\right\|$.

Let $H$ be a complex Hilbert space, $A \in \mathcal{B}(H)$, and suppose for every $x$ the number $(A x, x)$ is zero. Prove that $A = 0$. Give an example of a nonzero operator $A$ in a real Hilbert space with the property $(A x, x) = 0$ for every $x$.

Let $H$ be a complex Hilbert space, $A \in \mathcal{B}(H), w(A): = \sup \left\{ \left|z\right|: z \in W(A)\right\}$. Prove that $w(A) \leqslant \left\| A\right\|$.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that if $\left|\lambda \right| = \left\| A\right\|$ and $\lambda \in W(A)$, then $\lambda \in \sigma_{\mathrm{p}}(A)$.

Prove that for a normal operator $A \in \mathcal{B}(H)$ in a Hilbert space, the set $\overline{W(A)}$ coincides with the convex hull of the spectrum $\sigma (A)$.

Find the numerical range of the following operators:

Let $X$ be an infinite-dimensional Banach space, $A \in \mathcal{K}(X)$. Prove that $0 \in \sigma_{\text{ess }}(A)$.

Let $X$ be an infinite-dimensional Banach space, $A \in \mathcal{K}(X)$. Relying on the Fredholm alternative, prove that:

Prove that a nonzero eigenvalue of a compact operator has finite multiplicity.

Let $X$ be a Banach space, $A \in \mathcal{K}(X)$. Prove that for every $\delta > 0$ the set $\sigma_{\mathrm{p}}(A) \cap \left\{ z:\left|z\right| > \delta \right\}$ consists of a finite number of points, each of which is an eigenvalue of the operator $A$ of finite multiplicity.

Prove that the integral operator

is a Volterra operator in the space

Let $A$ be a Volterra operator and $\lambda \in \mathbb {C}$. Prove that the equation $(I-\lambda A) x = y$ in a Banach space $X$ is solvable for every right-hand side and every $\lambda$. Prove that the solution is representable as the series $x = \sum_{k = 0}^{\infty } \lambda^{k} A^{k} y$.

Solve the equation $x(t)+\lambda \int_{0}^{t} x(s) d s = y(t)$ (with respect to the function $x$, with the function $y$ known) in the following spaces:

Let $X$ be a complex infinite-dimensional Banach space, $A \in \mathcal{B}(X)$ and $\sigma (A) = \left\{ 0\right\}$. Is it true that $A$ is compact?

Let $X$ be an infinite-dimensional Banach space. Give examples of operators $A \in \mathcal{K}(X)$ for which

By investigating the eigenvalues of the Riemann-Liouville operator in the space $L_{2}[0,1]$ with arbitrary parameter $r > 1 / 2$ (see problem 6.35), prove the non-compactness of this operator.

Find the eigenvalues of the Hardy operator

in the spaces

Find the eigenvalues of the operator

in the space $L_{p}[0,1](1 \leqslant p < \infty )$.

In the space $C[0, \pi ]$, find the eigenvalues and eigenfunctions of the integral operator

in the case

In the space $L_{2}[-\pi , \pi ]$, find the eigenvalues and eigenfunctions of the integral operator

if $K(t, s) = \sum_{n \in \mathbb {Z}} 3^{-\left|n\right|} e^{-i n(t-s)}$.

In the space $L_{2}[0, \pi ]$, find the eigenvalues and eigenfunctions of the integral operator

if

In the space $L_{2}[0,1]$, find the eigenvalues and eigenfunctions of the integral operator

if

(H. Weyl, 1910). Let $X$ be a Banach space, $A \in \mathcal{B}(X), \lambda \in \sigma (A) \backslash \sigma_{\mathrm{p}}(A)$ and $B = A+K$, where $K \in \mathcal{K}(X)$. Prove that $\lambda \in \sigma (B)$.

Let $T_{r} \in \mathcal{B}\left(l_{p}(\mathbb {Z})\right), p \in [1, \infty )$, be the right shift operator, i.e., $\left(T_{r} x\right)_{n} = x_{n-1}, n \in \mathbb {Z}$. Prove that $\sigma \left(T_{r}\right)$ coincides with the unit circle, and construct a one-dimensional operator $K \in \mathcal{B}\left(l_{p}(\mathbb {Z})\right)$ such that $\sigma \left(T_{r}+K\right)$ coincides with the unit disk. (See Example 12.1 and cf. problem 12.47).