Problems to Solve

Let $A \in \mathcal{L}(X)$. Prove that for $\left|\lambda \right| > \left\| A\right\|$ the inequality $\left\| R_{\lambda }(A)\right\| \leq (\left|\lambda \right|- \left\| A\right\| )^{-1}$ holds.

Let $X$ be a Banach space, $A \in \mathcal{L}(X), \lambda \in \rho (A)$. Prove that $A$ and $R_{\lambda }(A)$ commute.

The operator $A \in \mathcal{L}\left(l_{\infty }\right)$ is defined by the formula $A x = \left(\alpha_{1} x_{1}, \alpha_{2} x_{2}, \ldots \right)$, $x \in l_{\infty }$, where $\left(\alpha_{1}, \alpha_{2}, \ldots \right) \in l_{\infty }$. Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$, the resolvent and the spectral radius of the operator $A$.

Find the spectrum, the set of eigenvalues, the spectral radius and the resolvent of an operator $A: l_{p} \rightarrow l_{p}, 1 \leq p \leq +\infty$.

Find the norm, the spectral radius, the set of eigenvalues and the spectrum of an operator $A: l_{2} \rightarrow l_{2}$.

Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$ and $R_{\lambda }(A)$.

Let $X = L_{p}(T, \mu ), 1 \leq p < +\infty$, where $\mu$ is a $\sigma$-finite measure on $T, a \in L_{\infty }(T, \mu ),(A x)(t) = a(t) x(t), t \in T, x \in X$. Prove that

(this set is called the essential range of the function $a$). Find the set of eigenvalues of the operator $A$.

Let $X$ be a Banach space, an operator $A \in \mathcal{L}(X)$ has a continuous inverse $A^{-1}$. Prove that

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

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$ and there exists a nonzero polynomial $P$ such that $P(A) = 0$. Prove that

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$ and $A^{2} = A$. Prove that $\sigma (A) \subset \left\{ 0,1\right\}$. Provide examples of operators $A$, for which $A^{2} = A$ and

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ and $A^{*} = A$. Prove that if $\sigma (A) \subset \left\{ 0,1\right\}$, then $A$ is an orthogonal projector.

Let $X = C(\left[-1,1\right])$. Find the spectra of the operators

Let $X = C(\left[-1,1\right])$. Find the eigenvalues, the eigenvectors, and the spectrum of the operator $A \in \mathcal{L}(X)$, which is defined by the formula $(A x)(t) = x(-t), t \in [-1,1], x \in X$.

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ be a normal operator (for example, self-adjoint or unitary). Prove that

Let $X = L_{2}(\left[-1,1\right]), a, b \in \mathbb {C}$. Find the eigenvalues, the spectrum, and the resolvent of the operator $A \in \mathcal{L}(X)$, which is defined by the formula $(A x)(t) = a x(t)+b x(-t), t \in [-1,1], x \in X$.

Prove that any nonempty compact subset of $\mathbb {C}$ is the spectrum of some continuous linear operator.

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$ be an operator such that $\left\| A x\right\| = \left\| x\right\| , x \in X$, and $0 \in \rho (A)$. Prove that

Let $X = C(\left[0,1\right])$ and $(A x)(t) = x\left(t^{2}\right), t \in [0,1]$. Prove that $\sigma (A) \subset \left\{ \lambda \in \mathbb {C} \mid \left|\lambda \right| = 1\right\}$.

Let $H$ be a Hilbert space, $P$ be an orthogonal projector in $H, P \neq 0$ and $P \neq I$. Prove that $\sigma (P) = \sigma_{p}(P) = \left\{ 0,1\right\}$, moreover

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

Let $\frac{1}{p}+\frac{1}{q} = 1, A x = \left(x_{2}, x_{3}, \ldots \right), x \in l_{p}, B x = \left(0, x_{1}, x_{2}, \ldots \right), x \in l_{q}$. Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$ and do the same for the operator $B$, if

Find $\sigma (A), \sigma_{p}(A), \sigma_{c}(A), \sigma_{r}(A)$, the resolvent, and the spectral radius of the operator $A: l_{p} \rightarrow l_{p}, 1 < p < +\infty$, if

Let $y, z$ be fixed elements of a Hilbert space $H$. The operator $A \in \mathcal{L}(H)$ is defined by the formula $A x = (x, y) z$. Find $\sigma_{p}(A), \sigma (A), R_{\lambda }(A), A^{n}$ for all $n \geq 1$, and $r(A)$.

An operator $A \in \mathcal{L}(X)$, where $X$ is a Banach space, is called quasinilpotent if $\lim_{n \rightarrow \infty }\left\| A^{n}\right\|^{\frac{1}{n}} = 0$. Prove that $A$ is quasinilpotent if and only if $\sigma (A) = \left\{ 0\right\}$.

Consider the operator $A \in \mathcal{L}\left(L_{2}(\left[0,1\right])\right)$, which is defined by the formula $(A x)(t) = \int_{0}^{t} x(s) d s, t \in [0,1]$.

Find the spectrum of the Volterra integral operator in $L_{2}(\left[0,1\right])$ with a continuous kernel $K$, i.e., the operator $(A x)(t) = \int_{0}^{t} K(t, s) x(s) d s, t \in [0,1], x \in L_{2}(\left[0,1\right])$, where $K \in C(\left\{ (t, s) \mid 0 \leq s \leq t \leq 1\right\} )$.

Find the spectrum of the operator $A \in \mathcal{L}\left(L_{2}(\left[-1,1\right])\right)$, which is defined by the formula $(A x)(t) = \int_{-t}^{t} x(s) d s, t \in [-1,1]$.

Let $X_{1}, X_{2}$ be Banach spaces, $A_{1} \in \mathcal{L}\left(X_{1}\right)$ and $A_{2} \in \mathcal{L}\left(X_{2}\right)$. Operators $A_{1}$ and $A_{2}$ are called similar, if there exists a continuously invertible operator $B$ : $X_{2} \rightarrow X_{1}$ such that $A_{2} = B^{-1} A_{1} B$. Let the operators $A_{1}$ and $A_{2}$ be similar. Prove that

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$. Prove that the sets $\sigma (A), \sigma_{p}(A)$, $\sigma_{c}(A), \sigma_{r}(A)$ remain the same, if the norm in $X$ is replaced with any equivalent norm.

Let $s \in \mathbb {R} \backslash \left\{ 0\right\}$ be a fixed number. The operator $A_{s}$ in $L_{2}(\mathbb {R})$ is defined by the formula $\left(A_{s} x\right)(t) = x(t+s), t \in \mathbb {R}$. Prove that $\sigma_{p}\left(A_{s}\right) = \sigma_{r}\left(A_{s}\right) = \varnothing$, and $\sigma \left(A_{s}\right) = \sigma_{c}\left(A_{s}\right) = \left\{ \lambda \in \mathbb {C} \mid \left|\lambda \right| = 1\right\}$.

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

Let $X = C\left(\left[0, \frac{1}{2}\right]\right)$ and $(A x)(t) = t x\left(t^{2}\right), t \in \left[0, \frac{1}{2}\right], x \in X$. Prove that $r(A) = 0$.

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$. Prove that if for $\lambda \in \mathbb {C}$ there exists a sequence $\left\{ x_{n}: n \geq 1\right\} \subset X$, such that $\left\| x_{n}\right\| = 1$ for $n \geq 1$ and

then $\lambda \in \sigma (A)$.

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H), A = A^{*}$. Prove that

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H), A = A^{*}$. Prove that if $R(A-\lambda I) = H$, then $\lambda \in \rho (A)$.

The operator $A$ in $C(\left[0,1\right])$ is defined by the formula

Prove that $r(A) = 1$.

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H), A = A^{*}, \lambda \in \rho (A) \cap \mathbb {R}$. Prove that the resolvent $(A-\lambda I)^{-1}$ is a self-adjoint operator.

Let $X$ be a Banach space, $A, B \in \mathcal{L}(X)$. Prove that for all $\lambda \in \rho (A) \cap \rho (B)$ the following identity holds:

Prove that $r(A B) = r(B A)$ for all $A, B \in \mathcal{L}(X)$.

Let $X$ be a Banach space, $A, B \in \mathcal{L}(X)$. Prove that $\sigma (A B) \backslash \left\{ 0\right\} = \sigma (B A) \backslash \left\{ 0\right\}$ and for all $\lambda \in \rho (A B) \backslash \left\{ 0\right\}$ the identity

holds.

Let $X$ be a Banach space, $A, B \in \mathcal{L}(X)$ and $A B-B A = c I$ for some $c \in \mathbb {C}$. Prove that $c = 0$.