Operator space

Prove Theorem 6.1.

Theorem 6.1: The set $\mathcal{B}(X, Y)$ of all continuous linear operators from a normed space $X$ into a Banach space $Y$, with the norm $\left\| A\right\| = \sup_{x \neq 0} \frac{\left\| A x\right\|_{Y}}{\left\| x\right\|_{X}}$, is a Banach space.

Let $X, Y$ and $Z$ be normed spaces, $A \in \mathcal{B}(Y, Z)$, $B \in \mathcal{B}(X, Y)$. Prove the inequality $\left\| A B\right\| \leqslant \left\| A\right\| \cdot \left\| B\right\|$. Prove that if one of the operators $A$ or $B$ is an isometric isomorphism of the corresponding spaces, then $\left\| A B\right\| = \left\| A\right\| \cdot \left\| B\right\|$. Give an example of a Banach space $X$ and operators $A$ and $B$ from $\mathcal{B}(X)$ such that $\left\| A B\right\| < \left\| A\right\| \cdot \left\| B\right\|$.

Prove that the space $C[0,1]$ with the usual operation of function multiplication is a unital commutative Banach algebra.

Prove that for an arbitrary Banach space $X$ the set of bounded operators $\mathcal{B}(X)$ is a unital noncommutative (if $\operatorname {dim} X > 1$) Banach algebra with respect to the composition operation.

Let $X$ and $Y$ be normed spaces, an operator $A \in \mathcal{B}(X, Y)$. Which of the following statements hold for every operator $A \in \mathcal{B}(X, Y)$?

Let $X$ and $Y$ be normed spaces, $X_{0}$ a closed subspace of $X$. Set

Is $M$ a closed subspace of $\mathcal{B}(X, Y)$?

Let $X$ and $Y$ be normed spaces, $X_{0}$ a closed subspace of $X$. Set

Is $M$ a closed subspace of $\mathcal{B}(X, Y)$?

Let $X$ be a normed space, $A$ a fixed operator from $\mathcal{B}(X)$. Set $M = \left\{ B \in \mathcal{B}(X): A B = 0\right\} , N = \left\{ B \in \mathcal{B}(X): A B = B A\right\}$. Prove that $M$ and $N$ are closed subspaces of $\mathcal{B}(X)$.

Let $H$ be an infinite-dimensional Hilbert space. Prove that the space $\mathcal{B}(H)$ is not separable.

Prove that the space $\mathcal{B}\left(l_{p}\right), p \in [1, \infty ]$, is not separable.

Let $X$ and $Y$ be Banach spaces, with $Y = Y_{1} \oplus Y_{2}$ the direct sum of two closed subspaces. Prove that $\mathcal{B}(X, Y) = \mathcal{B}\left(X, Y_{1}\right) \oplus \mathcal{B}\left(X, Y_{2}\right)$.

Let $X$ be a Banach space, $\left\{ P_{n}\right\}_{1}^{\infty } \in \mathcal{B}(X)$, where all the operators $P_{n}$ are projection operators and $P_{n} \rightrightarrows P$. Must $P$ necessarily be a projection operator?

Let $X$ be a Banach space, $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}(X)$, where all the operators $A_{n}$ are operators of the same finite rank: $\operatorname {rank} A_{n} = r < \infty$ and $A_{n} \rightrightarrows A$. Is it true that rank $A \leqslant r$?

In the space $X$, where $X$ is $c_{p}$ or $l_{p}, p \in [1, \infty ]$, consider the sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ of operators of multiplication by sequences $\lambda_{n} \in l_{\infty }$. Prove that the sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ has a limit in the space $\mathcal{B}(X)$ if and only if the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty }$ has a limit in the space $l_{\infty }$. Prove that in the case when the limit $A = \lim_{n \rightarrow \infty } A_{n}$ exists, it is also equal to the operator of multiplication by the sequence $\lambda = \lim_{n \rightarrow \infty } \lambda_{n}$.

In the space $X$, where $X$ is $c_{0}$, c, or $l_{p}, p \in [1, \infty ]$, consider two sequences of operators $\left\{ T_{r}^{n}\right\}_{1}^{\infty }$ and $\left\{ T_{l}^{n}\right\}_{1}^{\infty }$:

• the operators of right and left shift. Prove that both of these sequences have no limit in the space $\mathcal{B}(X)$.