Problems to Solve

Let $X_{1}, X_{2}$ be normed linear spaces, $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$. Prove the equalities

Prove that a continuous linear operator $A: X \rightarrow Y$ remains continuous if the norms in $X$ and $Y$ are replaced with equivalent ones.

Let $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ be two equivalent norms on a linear space $X$. Denote by $X_{i}$ the linear space $X$ endowed with the norm $\left\| \cdot \right\|_{i}, i = 1,2$. Prove that the norms in $\mathcal{L}\left(X_{1}\right)$ and $\mathcal{L}\left(X_{2}\right)$ are equivalent.

Let $X_{1}$ be a finite-dimensional NLS, $X_{2}$ be an arbitrary NLS. Prove that every linear operator $A: X_{1} \rightarrow X_{2}$ is continuous. In particular, any linear operator in a finite-dimensional NLS is continuous.

Find the general form of a linear operator $A: X \rightarrow Y$ (which is continuous according to problem $6.8^{\circ }$) and calculate its norm in the following cases:

Let $\left\{ \alpha_{n}: n \geq 1\right\}$ be a bounded sequence. Find the norm of the diagonal operator $A x = \left(\alpha_{1} x_{1}, \alpha_{2} x_{2}, \ldots \right)$ in $l_{p}, 1 \leq p \leq \infty$.

Let $\alpha = \alpha (t)$ be a fixed function from $C(\left[a, b\right]), A$ is the operator of multiplication by the function $\alpha :(A x)(t) = \alpha (t) x(t)$.

Let $A$ be the operator of multiplication by a Lebesgue measurable function $\alpha$ in $L_{p}(\left[a, b\right]), 1 \leq p \leq +\infty :(A x)(t) = \alpha (t) x(t)$. Prove that $A$ is bounded if and only if $\alpha \in L_{\infty }(\left[a, b\right])$. Find the norm of $A$.

Find the norm of the operator $A: \mathbb {R}^{m} \rightarrow l_{2}$, if

$x = \left(x_{1}, \ldots x_{m}\right) \in \mathbb {R}^{m}$, if $\mathbb {R}^{m}$ is endowed with the Euclidean norm.

Let $\left(\alpha_{j k}\right)_{j, k = 1}^{\infty }$ be an infinite matrix with real or complex entries such that $\sum_{j, k = 1}^{\infty }\left|\alpha_{j k}\right|^{2} < \infty$. Prove that the operator of multiplication by this matrix, defined by the equalities $(A x)_{j} = \sum_{k = 1}^{\infty } \alpha_{j k} x_{k}, j \geq 1$, is linear and continuous in the space $l_{2}$. Estimate the norm of $A$.

The integral operator $A: C(\left[a, b\right]) \rightarrow C(\left[a, b\right])$ with a continuous kernel $K \in C\left(\left[a, b\right]^{2}\right)$ is defined by the formula $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s, t \in [a, b]$, $x \in C(\left[a, b\right])$. Prove that $A$ is linear and continuous. Also prove that $\left\| A\right\| = \max_{t \in [a, b]} \int_{a}^{b}\left|K(t, s)\right| d s$.

The integral operator $A: L_{1}(\left[a, b\right]) \rightarrow L_{1}(\left[a, b\right])$ with a continuous kernel $K \in C\left(\left[a, b\right]^{2}\right)$ is defined by the formula $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s, t \in [a, b]$, $x \in L_{1}(\left[a, b\right])$. Prove that $A$ is linear and continuous. Also prove that $\left\| A\right\| = \max_{s \in [a, b]} \int_{a}^{b}\left|K(t, s)\right| d t$.

The integral operator $A: L_{2}(\left[a, b\right]) \rightarrow L_{2}(\left[a, b\right])$ with kernel $K \in L_{2}\left(\left[a, b\right]^{2}\right)$ is defined by the formula $(A x)(t) = \underset {\left[a, b\right]}{\int } K(t, s) x(s) d s, t \in [a, b]$. Prove that $A$ is linear and continuous. Also prove the estimate

The Volterra integral operator $A: C(\left[a, b\right]) \rightarrow C(\left[a, b\right])$ with continuous kernel $K$ is defined by the formula $(A x)(t) = \int_{a}^{t} K(t, s) x(s) d s, t \in [a, b]$, where $K \in C\left(\left\{ (t, s) \in \mathbb {R}^{2} \mid a \leq s \leq t \leq b\right\} \right)$. Prove that $A$ is linear and continuous. Find $\left\| A\right\|$.

Let $u$ be a continuous $2 \pi$-periodic function on $\mathbb {R}$. The integral operator $A: L_{1}(\left[-\pi , \pi \right]) \rightarrow L_{1}(\left[-\pi , \pi \right])$ is defined by the formula $(A x)(t) = \int_{-\pi }^{\pi } u(t- s) x(s) d s, t \in [-\pi , \pi ]$. Prove that $A$ is linear and continuous. Find $\left\| A\right\|$.

Let $u, v \in L_{2}(\left[a, b\right])$. Prove that the integral operator $A: L_{2}(\left[a, b\right]) \rightarrow L_{2}(\left[a, b\right])$, which is defined by the formula

is linear and continuous. Also prove that $\left\| A\right\| = \left\| u\right\| \cdot \left\| v\right\|$.

Find the norm of the operator defined as $A x = x$, which acts:

Let $K \in C\left(\left[a, b\right]^{2}\right), \alpha < 1$. Prove that the integral operator $A: C(\left[a, b\right]) \rightarrow C(\left[a, b\right])$, where $(A x)(t) = \int_{a}^{b} \frac{K(t, s)}{\left|t-s\right|^{\alpha }} x(s) d s, t \in [a, b]$, is linear and continuous.

Let the numbers $\alpha , \beta , \gamma$ be such that $0 \leq \gamma \leq \alpha < \beta$. Denote by $C_{\alpha }$ the Banach space of functions $x \in C([0,+\infty ))$ for which

with norm $\left\| x\right\|_{\alpha } = \sup_{0 \leq t < \infty } e^{\alpha t}\left|x(t)\right|$. Show that the operator $A: C_{\alpha } \rightarrow C_{\gamma }$, defined by the formula $(A x)(t) = \int_{0}^{t} e^{-\beta (t-s)} x(s) d s, t \geq 0$, is linear and continuous. Find $\left\| A\right\|$.

Let $H$ be a Hilbert space, $y, z \in H, A x = (x, y) z, x \in H$. Show that $A \in \mathcal{L}(H)$, and find $\left\| A\right\|$.

Let $u, v \in C(\left[a, b\right])$ and $(A x)(t) = \int_{a}^{b} u(t) v(s) x(s) d s$. Show that $A \in \mathcal{L}(X)$, and find the norm $\left\| A\right\|$. Consider the cases

An operator $A \in \mathcal{L}(X, Y)$ is called finite-dimensional if $\operatorname {dim} R(A) < \infty$. Show that $A$ is finite-dimensional if and only if there exist $f_{1}, \ldots , f_{n} \in X^{*}$ and $y_{1}, \ldots , y_{n} \in Y$ such that

Establish the general form of a finite-dimensional operator

Let $\left\{ y_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{ z_{k} \mid 1 \leq k \leq n\right\}$ be two orthonormal systems in a Hilbert space $H$. Find the norm of the operator $A$, where $A x = \sum_{k = 1}^{n} c_{k}\left(x, y_{k}\right) z_{k}$, $c_{k} \in \mathbb {C}$.

Prove that the following integral operators are linear and continuous. Find their norms.

Prove that the following operators $A: l_{2} \rightarrow l_{2}$ are linear and continuous. Find their norms.

Prove that the following operators are continuous and not linear:

Let $X, Y$ be Banach spaces, $A \in \mathcal{L}(X, Y)$ be an operator, for which there exists $c > 0$ such that $\left\| A x\right\| \geq c\left\| x\right\| , x \in X$. Prove that

Let $X, Y$ be normed linear spaces, and $A: X \rightarrow Y$ be a linear operator, for which $\operatorname {dim} R(A) < +\infty$, and $\operatorname {Ker} A$ is a closed set in $X$. Prove that $A \in \mathcal{L}(X, Y)$.

Let $H$ be a Hilbert space, $G \subset H$ be a subspace, $P x = \operatorname {pr}_{G} x$. Prove that the projection operator $P$ is linear and continuous. Find the norm of $P$.

Let $A$ be a linear operator in an NLS $X$. Prove that $A$ is continuous if and only if the set $\left\{ x \in X \mid \left\| A x\right\| < 1\right\}$ has interior points.

Let $A$ be a linear operator in an NLS $X$, which satisfies one of the following conditions:

Let $X$ be an NLS, $A: X \rightarrow X$. Prove that $A \in \mathcal{L}(X)$ if and only if for every $f \in X^{*}$ the functional $f_{0}(x) = f(A x), x \in X$, is linear and continuous.

Let $X_{1}, X_{2}$ be normed linear spaces, $\left\{ x, x_{n}: n \geq 1\right\} \subset X_{1}$ and $A \in \mathcal{L}\left(X_{1}, X_{2}\right)$. Prove that if $x_{n} \xrightarrow {w} x, n \rightarrow \infty$, then $A x_{n} \xrightarrow {w} A x, n \rightarrow \infty$.

Let $X$ be a nonseparable NLS. Prove that the space $\mathcal{L}(X)$ is nonseparable.

Let $X$ be a Banach space, $A \in \mathcal{L}(X)$. Are the following functions norms on $X$ :

Let $X$ be an NLS, $Y$ be a Banach space, the linear set $M \subset X$ is dense in $X$, and $A \in \mathcal{L}(M, Y)$. Prove that the operator $A$ admits a unique extension $\bar{A} \in \mathcal{L}(X, Y)$, and $\left\| \bar{A}\right\| = \left\| A\right\|$.

Let $H$ be a Hilbert space, $A: H \rightarrow H$ be a linear operator. Prove that $A \in \mathcal{L}(H)$ if and only if $\exists C > 0 \forall x, y \in H:\left|(A x, y)\right| \leq C\left\| x\right\| \cdot \left\| y\right\|$.

Moreover, $\left\| A\right\| = \sup_{\left\| x\right\| = 1,\left\| y\right\| = 1}\left|(A x, y)\right|$.

A system of sets $\mathcal{A} = \left\{ A_{\alpha }: \alpha \in T\right\}$ is said to be linked, if every pair of sets from $\mathcal{A}$ has a nonempty intersection. A Banach space $X$ is a space of type $\mathcal{M}$, if every linked system of closed balls in $X$ has a nonempty intersection.

Prove that

Let $A, B$ be linear operators in the Hilbert space $H$ such that $(A x, y) = (x, B y)$ for all $x, y \in H$. Prove that $A, B \in \mathcal{L}(H)$.