Different properties of operators

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 $J\left(x_{1}, \ldots , x_{n}, \ldots \right) = \left(\varepsilon_{1} x_{\sigma (1)}, \ldots , \varepsilon_{n} x_{\sigma (n)}, \ldots \right)$.