Linear operators

Let $X$ and $Y$ be normed spaces, $T \in \mathcal{L}(X, Y)$. Prove that the following conditions are equivalent:

1. $T$ is continuous at every point of the space $X$;

2. $T$ is continuous at zero;

3. $T$ is bounded;

4. $T$ is uniformly continuous on $X$ (see Definition 1.22).

Definition 1.22: A mapping $f: X \rightarrow Y$ of a metric space $(X, \rho )$ into a metric space $(Y, d)$ is called uniformly continuous if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for every pair of points $x_{1}, x_{2} \in X$ with $\rho \left(x_{1}, x_{2}\right) < \delta$ one has $d\left(f\left(x_{1}\right), f\left(x_{2}\right)\right) < \varepsilon$.

Let $A: C[0,1] \rightarrow C[0,1]$ be a positive linear operator, i.e., $(A x)(t) \geqslant 0$ for every nonnegative function $x \in C[0,1]$ and every $t \in [0,1]$. Prove that $A$ is continuous and $\left\| A\right\| = \left\| A\left(x_{0}(t) \equiv 1\right)\right\|$. (Cf. problem 5.54.)

Prove that in a Hilbert space $H$ the norm of an arbitrary bounded operator $T \in \mathcal{B}(H)$ can be found by the formula $\left\| T\right\| = \sup_{\left\| x\right\| = \left\| y\right\| = 1}\left|(T x, y)\right|$.

Let $X$ and $Y$ be linear spaces, $X_{1} = \left(X,\left\| \cdot \right\|_{X_{1}}\right)$, $X_{2} = \left(X,\left\| \cdot \right\|_{X_{2}}\right), Y_{1} = \left(Y,\left\| \cdot \right\|_{Y_{1}}\right)$, and $Y_{2} = \left(Y,\left\| \cdot \right\|_{Y_{2}}\right)$, where $\left\| \cdot \right\|_{X_{1}} \sim \left\| \cdot \right\|_{X_{2}}$, and $\left\| \cdot \right\|_{Y_{1}} \sim \left\| \cdot \right\|_{Y_{2}}$. Prove that

What can be asserted if $\left\| \cdot \right\|_{X_{1}} \prec \left\| \cdot \right\|_{X_{2}}$, and $\left\| \cdot \right\|_{Y_{1}} \prec \left\| \cdot \right\|_{Y_{2}}$?

Let $X$ and $Y$ be Banach spaces, $X_{0}$ an everywhere dense subspace of $X$, and $A \in \mathcal{B}\left(X_{0}, Y\right)$. Prove that there exists a unique bounded linear operator $\tilde{A} \in B(X, Y)$ that coincides with the operator $A$ on $X_{0}$. Prove that $\left\| \tilde{A}\right\|_{\mathcal{B}(X, Y)} = \left\| A\right\|_{\mathcal{B}\left(X_{0}, Y\right)}$.

Prove the boundedness and find the norm of the following embedding operators:

Let $X$ and $Y$ be normed spaces, $A \in \mathcal{B}(X, Y)$. Prove that $\operatorname {Ker} A$ is a closed subspace of the space $X$. Give an example showing that $\operatorname {Im} A$ need not be a closed subspace of $Y$.

Prove that a linear operator $P$ in a normed space $X$ is a projection operator if and only if $P^{2} = P$.

Let $X$ be a normed space, $X_{0}$ and $X_{1}$ linear subspaces of it, such that $X$ is the linear direct sum of $X_{0}$ and $X_{1}$. Let $P$ be the projection operator onto $X_{0}$ along $X_{1}$. Prove that the operator $P$ is bounded if and only if the direct sum of $X_{0}$ and $X_{1}$ is a topological direct sum (see Definition 2.21). Prove that if at least one of the subspaces $X_{0}$ or $X_{1}$ is finite-dimensional, then $P$ is bounded.

Definition 2.21: Let $X$ be a normed space and $X_{1}$, $X_{2}$ two linear subspaces of $X$ with $X_{1} \cap X_{2} = \left\{ 0\right\}$. The linear direct sum of these subspaces is called a topological direct sum if there exist constants $C_{1}$ and $C_{2}$ such that for every $x = x_{1}+x_{2}$, $x_{1} \in X_{1}$, $x_{2} \in X_{2}$, one has $\left\| x_{1}\right\| \leqslant C_{1}\left\| x\right\|$, $\left\| x_{2}\right\| \leqslant C_{2}\left\| x\right\|$.

In the space $C[-1,1]$ consider the operator

Prove that this operator is a bounded projector. Find its image, kernel, and norm.

Let a linear subspace $X_{0}$ of a normed space $X$ be complemented. Prove that then the subspace $X_{0}$ itself and the complementary subspace $X_{1}$ are closed.

Does there exist in the space $C[0,1]$ a bounded projection operator

Let $X_{0}$ be a finite-dimensional subspace of a normed space $X$. Prove that $X_{0}$ is complemented.

Let $X_{0}$ be a closed subspace of finite codimension in a normed space $X$. Prove that $X_{0}$ is complemented.

Let $X$ and $Y$ be normed spaces. Prove that an operator $A \in \mathcal{B}(X, Y)$ is a finite-rank operator if and only if it admits the following representation: $A x = \sum_{k = 1}^{n} f_{k}(x) y_{k}$, where $f_{k} \in X^{*}$, and $y_{k} \in Y$. Prove that if $X$ and $Y$ are Hilbert spaces, then an operator $A \in \mathcal{B}(X, Y)$ is a finite-rank operator if and only if it admits the representation $A x = \sum_{k = 1}^{n} s_{k}\left(x, \psi_{k}\right) \varphi_{k}$, where $\left\{ \psi_{k}\right\}_{1}^{n}$ is an orthonormal system in $X,\left\{ \varphi_{k}\right\}_{1}^{n}$ is an orthonormal system in $Y$, and $s_{k} \in \mathbb {C}$ (compare with problem 12.79).

Find the norm, image, and kernel of the operator $A \in \mathcal{B}(X, Y), A x = f(x) y$, where $f \in X^{*}$ is a fixed nonzero functional, and $y \in Y$ is a fixed nonzero vector.

Prove that the following operators are finite-rank operators:

Find the exact value of the norm of the operator $A$ from Example 6.2.

Example 6.2: the operator $A$ on $L_{2}[-1,1]$ is given by $(A x)(t) = \int_{-1}^{1}(1+t s) x(s) \, d s$; it is a finite-rank operator with $\operatorname {Im} A = \operatorname {Lin}(1, t)$ and $\left\| A\right\| \leqslant \frac{\sqrt{40}}{3}$.

Prove that the operator $A\left(x_{1}, x_{2}, \ldots \right) = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots \right)$ is bounded in the space $V$ if and only if $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$. Find its norm and kernel in each of the spaces, depending on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty }$.

Find the norm of the operator

Prove that the operator from problem 6.19 is surjective in each of the spaces а)-в) if and only if $\inf_{n \geqslant 1}\left|\lambda_{n}\right| > 0$. In cases

Prove that the operator from problem 6.19 is a projection operator in each of the spaces а)-в) if and only if $\lambda_{n} \in \left\{ 0,1\right\}$ for every $n \in \mathbb {N}$.

Prove that the operators of right and left shift

are bounded in the spaces $l_{p}, p \in [1, \infty ]$, c, and $c_{0}$. Find the norm, image, and kernel of these operators in each of the spaces.

Prove that the operator of multiplication by a continuous function $a(t)$ is bounded in the space

Prove that the operator of multiplication by a measurable function $a(t)$ is bounded in the spaces $L_{p}[0,1], p \in [1, \infty ]$, if and only if $a \in L_{\infty }[0,1]$. Find its norm and kernel in each of the spaces.

Let $A$ be the operator of multiplication by a continuous function $a(t)$, acting in the space $C[0,1]$. Prove that if the function $a$ does not vanish on the interval $[0,1]$, then the operator $A$ is surjective, while if $a$ has zeros, then the operator $A$ is not surjective and, moreover, the closure of its image does not coincide with the space $C[0,1]$.

Let $A$ be the operator of multiplication by a function $a \in L_{\infty }[0,1]$, acting in the space $L_{p}[0,1]$, where $p \in [1, \infty )$. Prove that if $\underset {t \in [0,1]}{\operatorname {ess} \inf }\left|a(t)\right| > 0$, then the operator $A$ is surjective. ¹ Prove that if $\underset {t \in [0,1]}{\operatorname {ess} \inf }\left|a(t)\right| = 0$, but $\mu (\left\{ t \in [0,1]: a(t) = 0\right\} ) = 0$, then the operator $A$ is not surjective, but its image is everywhere dense in $L_{p}[0,1]$. Prove that if $\mu (\left\{ t \in [0,1]: a(t) = 0\right\} ) > 0$, then $\overline{\operatorname {Im} A} \neq L_{p}[0,1]$.

Let the operator of multiplication by the independent variable $(A x)(t) = t x(t)$ act in the space

Prove that the integration operator $(A x)(t) = \int_{0}^{t} x(s) d s$ is bounded

Prove that the differentiation operator $A x = x^{\prime }$ is bounded as an operator

Prove that there is no normed space containing entire functions¹, on which the differentiation operator is continuous.

Prove that the operator $\left(A_{K} x\right)(t) = \int_{a}^{b} K(t, s) x(s) d s$, $K(t, s) \in C\left([a, b]^{2}\right)$, is bounded

Consider the Hardy integral operator

For which real $r$ is the Hardy operator (6.4) bounded in the space $L_{2}[0,1]$?

For which real $r$ is the Riemann-Liouville operator (6.3) bounded in the space $L_{2}[0,1]$?

Let the function $K(t, s)$, defined for $t, s \in [0,+\infty )$, be nonnegative and homogeneous of degree of homogeneity -1 (i.e., $K(\lambda t, \lambda s) = \lambda^{-1} K(t, s)$ for every $\lambda > 0$). Let $p > 1,1 / p+1 / q = 1$ and

Prove that the integral operator $A: x(t) \mapsto \int_{0}^{\infty } K(t, s) x(s) d s$ is bounded in the space $L_{p}[0, \infty )$ and $\left\| A\right\| \leqslant k$.

Prove that the operator $(A x)(t) = \int_{0}^{\infty } x(s) e^{-t s} d s$ is bounded in the space $L_{2}[0, \infty )$, and estimate its norm.

Let $a \in C[-1,1]$. Prove that the convolution operator $A: x \mapsto \int_{0}^{1} x(s) a(t-s) d s$ is bounded in the space $C[0,1]$, and find its norm.

Prove that the shift operator $(A x)(t) = x(t+a)$ is bounded as an operator in the space

Find the norm, image, and kernel of the trace operator:

Find the norm and kernel of the operator $A: C[0,1] \rightarrow C[0,1],(A x)(t) = (x \circ \varphi )(t) = x(\varphi (t))$, where $\varphi$ is a fixed continuous function mapping the interval $[0,1]$ into itself. Prove that the operator $A$ is surjective if and only if the mapping $\varphi$ is injective.

Let $X$ be a normed space, and $X_{0}$ its closed subspace. Prove the equivalence of the following conditions:

1. the subspace $X_{0}$ is complemented;

2. the identity operator on $X_{0}$ admits an extension to a bounded operator from $X$ to $X_{0}$;

3. every bounded operator from $X_{0}$ to $Y$, where $Y$ is an arbitrary normed space, admits an extension to some bounded operator from $X$ to $Y$.

Prove that a Banach space $X$ has the following property: for all $Z \subset Y$ and for every bounded linear operator $A: Z \rightarrow X$ an extension $\tilde{A}: Y \rightarrow X$ with $\left\| \tilde{A}\right\| = \left\| A\right\|$ exists if and only if $X$ is 1-complemented in every Banach space $X_{1}$ containing it (i.e., there exists a linear projector from $X_{1}$ onto $X$ of unit norm).

Let $A \in \mathcal{B}\left(X_{0}, Y\right)$ be a finite-dimensional operator, where $X_{0}$ and $Y$ are normed spaces, $X_{0}$ a linear subspace of a normed space $X$. Prove that $A$ admits an extension to a bounded operator $\tilde{A} \in \mathcal{B}(X, Y)$, and moreover $\tilde{A}$ can be chosen so that $\operatorname {Im}(\tilde{A}) = \operatorname {Im}(A)$.

Let the operator $A$ be defined on the subspace $c_{0}$ of the space $c$ as the identity operator. Prove that this operator cannot be extended to a bounded operator $\tilde{A}$ from $c$ to $c_{0}$ without increasing the norm. Exhibit an extension of $A$ to a bounded operator $\tilde{A}$ from c to $c_{0}$ with an increase in norm.

(R. Phillips, 1940). Prove that the closed subspace $c_{0}$ of the Banach space $l_{\infty }$ is not complemented.

Let $A \in \mathcal{B}(X, Y)$, where $X$ is a reflexive Banach space, $Y$ an arbitrary Banach space, and rank $A < \infty$. Prove that $A$ attains its norm on the unit ball, i.e., there is a vector $x \in X$ such that $\left\| x\right\|_{X} = 1$ and $\left\| A x\right\|_{Y} = \left\| A\right\|$.

Let $X$ be a finite-dimensional normed space, and $Y$ an arbitrary normed space. Prove that $\mathcal{L}(X, Y) = \mathcal{B}(X, Y)$ and every operator attains its norm on the unit ball.

Give an example of a bounded operator in a reflexive Banach space that does not attain its norm on the unit ball.

Give an example of an unbounded operator $A \in \mathcal{L}(X)$ in a Banach space $X$ that is defined everywhere on $X$ and has closed kernel.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{L}(X, Y), \operatorname {Ker} A$ closed in $X$, and $\operatorname {rank} A < \infty$. Prove that the operator $A$ is bounded.

Give an example of a nonzero operator in an infinite-dimensional Banach space whose square is equal to zero.

Give an example of an operator $A \in \mathcal{B}(X)$ in an infinite-dimensional Banach space $X$ such that $\operatorname {Ker} A \subsetneq \operatorname {Im} A$.

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

For an arbitrary $n \in \mathbb {N}$ give an example of operators $A$ and $B$ in an infinite-dimensional Banach space $X$ with the properties

Let $X$ be a Banach space and $A \in \mathcal{B}(X)$ an operator. Prove that if $A$ has finite ascent $n$ and finite descent $m$, then $n = m$ and $X = \operatorname {Ker}\left(A^{m}\right) \oplus \overline{\operatorname {Im}\left(A^{m}\right)}$.

Prove that if for $A, B \in \mathcal{L}(X)$ the commutation relation $A B-B A = I$ holds, then $A B^{n}-B^{n} A = n B^{n-1}$. Prove that either $A$ or $B$ is unbounded. Deduce from this that the matrix equation $A B-B A = I$, where $A$ and $B$ are unknown $n \times n$ matrices, has no solutions.

Let $X$ be a Banach space, $A \in \mathcal{B}(X)$. Under what conditions on the operator $A$ do the expressions

Let $X$ be a Banach space, $A$ an injective bounded operator in $X$. For the norms from the previous problem prove that $\left\| \cdot \right\|_{1} \prec \left\| \cdot \right\|_{X}$, and $\left\| \cdot \right\|_{2} \sim \left\| \cdot \right\|_{X}$. Prove that $\left\| \cdot \right\|_{1} \sim \left\| \cdot \right\|_{X}$ if and only if $\inf_{\left\| x\right\| = 1}\left\| A x\right\| \geqslant r > 0$.

(S. Mazur, S. Ulam, 1932). Let $X$ and $Y$ be normed spaces, and let the map $J$ be an isometry, i.e., for all $x, y \in X$ we have $\left\| J(x)-J(y)\right\|_{Y} = \left\| x-y\right\|_{X}$ and $J(0) = 0$. Prove that $J$ is a linear isometric isomorphism of the normed spaces $X$ and $Y$.

Let $J$ be an isometry of the real space $c_{0}$ or $l_{p}, 1 \leqslant p < \infty , p \neq 2$, with $J(0) = 0$. Prove that there exist a set of signs $\varepsilon_{n} \in \left\{ \pm 1\right\}$ and a permutation $\sigma$ of the natural numbers such that $J\left(x_{1}, \ldots , x_{n}, \ldots \right) = \left(\varepsilon_{1} x_{\sigma (1)}, \ldots , \varepsilon_{n} x_{\sigma (n)}, \ldots \right)$.

Let $P \in \mathcal{B}(H)$ be a projector of a Hilbert space $H$ onto some subspace of it. Prove that $P$ is an orthogonal projector if and only if for any $x, y \in H$ the following equality holds: $(P x, y) = (x, P y).{ }$¹

Definition 8.3: An operator $A \in \mathcal{B}(H)$ is called self-adjoint if $A = A^{*}$.

Let $P$ be an orthogonal projector in a Hilbert space $H$. Prove that it is bounded, and find $\left\| P\right\|$.

Let $P_{1}$ and $P_{2}$ be orthogonal projectors in a Hilbert space $H$. Denote the subspaces $H_{1} = \operatorname {Im} P_{1}$ and $H_{2} = \operatorname {Im} P_{2}$. Prove that if $H_{1} \subset H_{2}$ (or, conversely, $H_{2} \subset H_{1}$) and $\left\| P_{1}-P_{2}\right\| < 1$, then $P_{1} = P_{2}$. Give an example of two such orthogonal projectors $P_{1} \neq P_{2}$ that $\left\| P_{1}-P_{2}\right\| < 1$.

In the notation of the previous problem, prove that if $\left\| P_{2}-P_{1}\right\| < 1$, then $\operatorname {dim} H_{1} = \operatorname {dim} H_{2}$. Give an example of two such orthogonal projectors $P_{1}$ and $P_{2}$ that $\left\| P_{2}-P_{1}\right\| = 1$ and $\operatorname {dim} H_{1} \neq \operatorname {dim} H_{2}$.

Let $H$ be a Hilbert space, $P$ the orthogonal projector onto a proper subspace $H_{0} \subset H$. What can be asserted about $\operatorname {Im} A$ and Ker $A$ of an operator $A \in \mathcal{B}(H)$, if

Prove that if, under the conditions of the previous problem, the equality $A P = P A$ holds, then $A\left(H_{0}\right) \subset H_{0}$. Prove that the converse is false.

Let $H$ be a Hilbert space, and $B: H \times H \rightarrow \mathbb {C}$ a form that is linear in the first argument, conjugate-linear in the second argument, and continuous in each argument. Prove that there is an operator $A \in \mathcal{B}(H)$ such that $B(x, y) = (A x, y)$ for all $x, y \in H$.

For any $M > 0$ give an example of an operator $A$ in a Hilbert space $H$ such that for some orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ all the norms $\left\| A e_{n}\right\|$ are bounded by one, but $\left\| A\right\| > M$.