Membership in Hilbert-Schmidt and trace-class via singular numbers

Let $H$ be a separable Hilbert space, $A \in S_{\infty }(H),\left\{ s_{n}(A): n \geq 1\right\}$ be a sequence of singular values of the operator $A$, numbered with respect to their multiplicities and, possibly, supplemented by zeros. Prove that $A$ is a Hilbert-Schmidt operator if and only if $\sum_{n = 1}^{\infty } s_{n}^{2}(A) < +\infty$. Moreover $\left\| A\right\|_{2} = \left(\sum_{n = 1}^{\infty } s_{n}^{2}(A)\right)^{1 / 2}$.

Let $H$ be an infinite-dimensional separable Hilbert space, $A \in \mathcal{K}(H),\left\{ s_{k}\right\}_{1}^{\infty }$ the $s$-numbers of the operator $A,\left\{ e_{n}\right\}_{1}^{\infty }$ an arbitrary orthonormal basis in $H$. Prove that $\sum_{k = 1}^{\infty } s_{k}^{2} = \sum_{n = 1}^{\infty }\left\| A e_{n}\right\|^{2}$ (i.e., either both series diverge, or both series converge and their sums are equal).

Let $H$ be an infinite-dimensional separable Hilbert space, $A \in \mathcal{K}(H)$. Prove the equivalence of the following statements:

1. $A \in \mathcal{S}_{2}(H)$, i.e., the series $\sum_{k = 1}^{\infty } s_{k}^{2}(A)$ converges;

2. for some orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ in the space $H$ the series $\sum_{n = 1}^{\infty }\left\| A e_{n}\right\|^{2}$ converges;

3. for every orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ in the space $H$ the series $\sum_{n = 1}^{\infty }\left\| A e_{n}\right\|^{2}$ converges;

4. for the matrix entries $a_{j k} = \left(A e_{j}, e_{k}\right)$ of the operator $A$ in some orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ of the space $H$ the inequality $\sum_{j, k = 1}^{\infty }\left|a_{j k}\right|^{2} < \infty$ holds;

5. for the matrix entries $a_{j k} = \left(A e_{j}, e_{k}\right)$ of the operator $A$ in an arbitrary orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ of the space $H$ the inequality $\sum_{j, k = 1}^{\infty }\left|a_{j k}\right|^{2} < \infty$ holds. Prove that if the operator $A$ satisfies these conditions, then the sums of all the indicated series coincide.

Prove that for an operator $A \in \mathcal{S}_{1}(H) = \operatorname {Sp}(H)$ the equalities

hold, where $\left\{ e_{k}\right\}_{k = 1}^{\infty }$ is an arbitrary orthonormal basis (for the definition of $\left\| A\right\|_{\mathrm{Sp}(H)}$ see problem 11.27).