Compact operators in Hilbert spaces

Let, in a Hilbert space $H$, the operator $A^{*} A$ be compact. Prove that the operator $A$ is compact.

Prove that in a Hilbert space the operator $A$ is compact if and only if $A^{*}$ is compact.

Let $A$ be a self-adjoint nonnegative operator in a Hilbert space $H$. Prove that $A$ is compact if and only if $\sqrt{A}$ is compact.

Let $A, B$ and $C$ be self-adjoint operators in a Hilbert space, $B \leqslant A \leqslant C$, where $B$ and $C$ are compact. Prove that $A$ is also compact.

For which values of the parameter $r$ is the Hardy operator

(see problem 6.33), compact?

Prove that if $A \in \mathcal{K}(H)$, and $\left\{ e_{n}\right\}_{n = 1}^{\infty }$ is an orthonormal basis of the space $H$, then $\left\| A e_{n}\right\| \rightarrow 0$.

Prove the compactness of the operator $A: l_{2} \rightarrow l_{2}$,

Is the operator $A$ compact in $l_{2}$ under the condition $\lim_{i \rightarrow \infty } \sum_{j = 1}^{+\infty }\left|a_{i j}\right|^{2} = 0$?

Is the Hilbert operator (see problem 6.72), defined by the matrix $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$, where $a_{i j} = \frac{1}{i+j}$, compact in the space $l_{2}$?

Give an example of a compact operator $A$ in the space $l_{2}$ whose matrix entries in the standard basis satisfy the relation $\sum_{k = 1}^{\infty } \sum_{n = 1}^{\infty }\left|a_{k n}\right|^{2} = \infty$.

Let $T_{r}:\left(x_{1}, x_{2}, \ldots \right) \mapsto \left(0, x_{1}, x_{2}, \ldots \right)$ be the right shift operator in the space $l_{2}$. Prove that if a compact operator $K$ commutes with $T_{r}$, then $K = 0$.

Prove that the operator $(A x)(t) = \int_{-\infty }^{+\infty } e^{-\left|t-s\right|} x(s) d s$ is not compact in the space $L_{2}(\mathbb {R})$.

Prove the compactness in the space $L_{2}[a, b]$ of the integral operator $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$ with kernel $K \in L_{2}\left([a, b]^{2}\right)$.

For which $\alpha \in (0,1)$ is the operator

compact?

Prove the compactness of the operator $A: L_{2}[0,1] \rightarrow L_{2}[0,1]$, $(A x)(t) = \int_{0}^{1} \frac{1}{t-s} \ln \frac{1+t}{1+s} x(s) d s$.

Let $H$ be a Hilbert space, $A \in \mathcal{K}(H)$. Prove that there exists a vector $x \in H,\left\| x\right\| = 1$, such that $\left\| A x\right\| = \left\| A\right\|$ (cf. problem 11.25).

Let $H$ be a Hilbert space, $A \in \mathcal{K}(H)$, a vector $x \in H,\left\| x\right\| = 1$, and $\left\| A x\right\| = \left\| A\right\|$. Prove that the operator $A$ carries $\left\{ x\right\}^{\perp }$ into $\left\{ A x\right\}^{\perp }$.