Compact operators in specific spaces

For which $\lambda \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \ldots , \lambda_{n} x_{n}, \ldots \right)$ a compact operator in $l_{p}(1 \leqslant p < \infty )$? Under what conditions on the sequence $\lambda$ is this operator nuclear?

For which $a \in L_{\infty }[a, b]$ is the operator $(A x)(t) = a(t) x(t)$ a compact operator in $L_{p}[0,1], p \in [1, \infty ]$? For which $a \in C[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ compact in $C[0,1]$?

Prove that any bounded operator acting from a reflexive space into the space $l_{1}$ is compact. In particular, if $A \in \mathcal{B}\left(l_{p}, l_{1}\right), 1 < p < \infty$, then $A \in \mathcal{K}\left(l_{p}, l_{1}\right)$.

Prove that any bounded operator acting from the space $c_{0}$ into a reflexive space is compact.

Let $A \in \mathcal{B}\left(L_{2}[0,1]\right)$ and $\operatorname {Im} A \subset C[0,1]$. Prove that $A \in \mathcal{K}\left(L_{2}[0,1]\right)$ (cf. problem 9.18).

Prove that the operators of right and left shift in any of the spaces $l_{p}, p \in [1, \infty ], c$ or $c_{0}$ are not compact.

Prove that if $K \in C\left([a, b]^{2}\right)$, then the integral operator $A: L_{p}[a, b] \rightarrow L_{r}[a, b],(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$, is compact for any $p, r \in [1, \infty )$.

Prove that if $K \in L_{q}\left([a, b]^{2}\right)$, then the integral operator $A: L_{p}[a, b] \rightarrow L_{q}[a, b], \frac{1}{p}+\frac{1}{q} = 1, q < \infty ,(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$, is compact.

Prove that if the function $K(t, s)$ is continuous on the square $[a, b]^{2}$ except for a finite number of continuous curves of the form $s = s_{k}(t), k = 1, \ldots , n, t \in [a, b]$, then the integral operator $A: C[a, b] \rightarrow C[a, b],(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$, is compact.

Prove that the differentiation operator $A x = x^{\prime }$ is not compact when acting:

Prove that the shift operator $(A x)(t) = x(t-a)$ is not compact in the spaces:

Using problem 11.21, prove that the Hardy operator

in the space $L_{p}[0,1], p \in (1, \infty )$, is not compact.

Give an example of a non-compact bounded operator $A$ in the space $L_{p}[0,1], p \in (1, \infty )$, and a sequence $\left\{ x_{n}\right\}_{1}^{\infty } \subset L_{p}[0,1]$ with the properties

1. $x_{n} \rightharpoonup 0$;

2. $\left\| x_{n}\right\|_{L_{p}} = 1$ for all $n \in \mathbb {N}$;

3. $\left\| A x_{n}\right\|_{L_{p}} \rightarrow 0$ (cf. problem 11.22).

Let $X$ be one of the spaces: $l_{p}, p \in (1, \infty )$, $c$ or $c_{0}$. Let $A \in \mathcal{K}(X)$, and $\left\{ e_{n} = (0,0, \ldots , 0,\underset {n}{1},0,0, \ldots )\right\}_{n = 1}^{\infty }$. Prove that $A e_{n} \rightarrow 0$. Give an example of a compact operator $A$ in the space $l_{1}$ such that $A e_{n} \nrightarrow 0$.

Prove that the operator $A$ of the form

in the spaces $l_{p}, p \in (1, \infty )$, $c$, and $c_0$, is bounded, not compact, but $\left\| A e_{n}\right\| \rightarrow 0$.

Is the operator $(A x)(t) = \frac{1}{2}(x(t)+x(-t))$ compact in the space $C[-1,1]$?

Let $\varphi (t)$ be a continuous map of the segment $[0,1]$ into itself. Prove that the composition operator $A \in \mathcal{B}(C[0,1]),(A x)(t) = x(\varphi (t))$, is compact in $C[0,1]$ if and only if $\varphi (t) \equiv$ const.

(Theorem of I. K. Daugavet, 1963). Prove that for every compact operator $A$ in $C[0,1]$ the equality $\left\| A+\lambda I\right\| = \left\| A\right\| +\left|\lambda \right|(\lambda \in \mathbb {C})$ holds.