Hilbert–Schmidt theorem

Prove Theorem 12.3.

Theorem 12.3 (D. Hilbert, 1906; E. Schmidt, 1907): Let $H$ be an infinite-dimensional separable Hilbert space and $A = A^{*} \in \mathcal{K}(H)$. Then there exists an orthonormal basis $\left\{ e_{n}\right\}_{n = 1}^{\infty }$ of $H$ consisting of eigenvectors of $A$: $A e_{n} = \lambda_{n} e_{n}$. The nonzero eigenvalues are real and can be numbered in nonincreasing order of absolute value: $\left|\lambda_{1}\right| \geqslant \left|\lambda_{2}\right| \geqslant \ldots \geqslant \left|\lambda_{n}\right| \geqslant \ldots$, and moreover $\left|\lambda_{1}\right| = \left\| A\right\|$ and $\lim_{n \rightarrow \infty } \lambda_{n} = 0$.

(D. Hilbert, 1906; E. Schmidt, 1907). Prove that in a separable Hilbert space a compact operator can be represented in the form

where $N \leqslant \infty ,\left\{ \psi_{i}\right\}$ is an orthonormal basis, $\left\{ \varphi_{i}\right\}$ is an orthonormal system.

Let $H$ be a separable Hilbert space, $A \in \mathcal{K}(H)$. Prove that the numbers $s_{k}(A)$ are the distances from the operator $A$ to the set of all $(k-1)$-dimensional operators, i.e.,

In particular, $s_{1} = \left\| A\right\|$.

(Courant-Fischer variational principle). Prove that for every compact self-adjoint nonnegative operator $A$ in a Hilbert space $H$ the equalities

hold, where $\lambda_{n}$ are the eigenvalues of the operator $A$, numbered in non-increasing order counting multiplicity, and the minimum is taken over all subspaces of dimension $n-1$.

Let $A$ and $B$ be compact self-adjoint operators in a Hilbert space and $0 \leqslant A \leqslant B$. Prove that $\lambda_{n}(A) \leqslant \lambda_{n}(B)$, where $\lambda_{n}$ are the eigenvalues, numbered in non-increasing order counting multiplicity.

Let $A$ and $B$ be compact operators in a Hilbert space $H$, with $\operatorname {Im} A \subset \operatorname {Im} B$. Prove that there exists a number $C > 0$ such that $s_{n}(A) \leqslant C s_{n}(B)$ for all $n \in \mathbb {N}$.

Let $A \in \mathcal{B}\left(L_{2}[0,1]\right)$ and $\operatorname {Im} A \subset \dot{W}_{2}^{1}[0,1]$. Prove that $A \in \mathcal{K}\left(L_{2}[0,1]\right)$ and $s_{n}(A) \leqslant C n^{-1}$ for some $C > 0$ and every $n \in \mathbb {N}$.

Given a sequence $s_{k} \geqslant 0, s_{k} \searrow 0$. Prove that there exists an operator $A \in \mathcal{K}\left(l_{2}\right)$ such that $s_{k}(A) = s_{k}$ for every $k \geqslant 1$.

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 $H_{1}$ and $H_{2}$ be separable Hilbert spaces, $A \in \mathcal{K}\left(H_{1}, H_{2}\right),\left\{ e_{n}\right\}_{1}^{\infty }$ an orthonormal basis in $H_{1}$. Prove that $\left\| A\right\|^{2} \leqslant \sum_{n = 1}^{\infty }\left\| A e_{n}\right\|^{2}$.

Let $H_{1}$ and $H_{2}$ be separable Hilbert spaces, $A \in \mathcal{B}\left(H_{1}, H_{2}\right),\left\{ e_{n}\right\}_{1}^{\infty }$ an orthonormal basis in $H_{1}$. Prove that if the series $\sum_{n = 1}^{\infty }\left\| A e_{n}\right\|^{2}$ converges, then $A$ is compact.

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.

Let $H$ be a separable infinite-dimensional Hilbert space. Prove that the set $\mathcal{S}_{2}(H)$ is a non-closed (in the norm of $\mathcal{B}(H)$) linear subspace of the space $\mathcal{K}(H)$.

Let $H$ be a separable infinite-dimensional Hilbert space. Prove that the norm in $\mathcal{S}_{2}(H)$ is generated by the inner product

where $a_{j k} = \left(A e_{j}, e_{k}\right)$ and $b_{j k} = \left(B e_{j}, e_{k}\right)$ are the matrix entries of the operators $A$ and $B$ in an arbitrary orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ of the space $H$. Prove that the same inner product can be given by the rule

Let $H$ be a separable Hilbert space. Prove that $\mathcal{S}_{2}(H)$ is a Hilbert space with the inner product defined in the previous problem.

Prove that an operator $A$ in the space $L_{2}[0,1]$ is a Hilbert-Schmidt operator if and only if

where $K \in L_{2}\left([0,1]^{2}\right)$. Prove that the map $A \mapsto K$ is a unitary isomorphism of the spaces $\mathcal{S}_{2}\left(L_{2}[0,1]\right)$ and $L_{2}\left([0,1]^{2}\right)$ (in particular, $\left\| A\right\|_{\mathcal{S}_{2}\left(L_{2}[0,1]\right)}^{2} = \int_{0}^{1} \int_{0}^{1}|K(x, t)|^{2} d x d t$).

Prove that every bounded linear operator in the space $L_{2}[0,1]$ has the form

for some function $K \in L_{2}\left([0,1]^{2}\right)$.

Prove that the class $\operatorname {Sp}(X)$ of nuclear operators in a Banach space $X$ (see Definition 11.2) coincides with the class $\mathcal{S}_{1}(H)$, if the space $X = H$ is Hilbert.

Definition 11.2: Let $X$ and $Y$ be Banach spaces. An operator $A \in \mathcal{B}(X, Y)$ is called nuclear, or a trace-class operator ($A \in \operatorname {Sp}(X, Y)$), if there exist one-dimensional operators $A_{k} \in \mathcal{B}(X, Y)$ such that $A = \sum_{k = 1}^{\infty } A_{k}$, where the series converges absolutely in $\mathcal{B}(X, Y)$.

Here (Definition 12.7), for $A \in \mathcal{K}(H)$ the $s$-numbers of $A$ are $s_{n}(A) := \sqrt{\lambda_{n}\left(A^{*} A\right)}$, $n = 1,2, \ldots$ (numbered in nonincreasing order, counting multiplicity), and $A \in \mathcal{S}_{1}(H)$ if $\left\{ s_{n}\right\}_{n = 1}^{\infty } \in l_{1}$.

Let $H$ be a separable Hilbert space and $A \in \mathcal{S}_{1}(H)$. Prove that for an arbitrary orthonormal basis $\left\{ e_{k}\right\}_{k = 1}^{\infty }$ the series $\sum_{k = 1}^{\infty }\left(A e_{k}, e_{k}\right)$ converges, and its sum does not depend on the choice of the basis $\left\{ e_{k}\right\}_{k = 1}^{\infty }$.

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).

Let $H$ be a separable Hilbert space. Prove that $\mathcal{S}_{p}(H)$ are linear subspaces of $\mathcal{K}(H)$ and of $\mathcal{B}(H)$. Prove that $\mathcal{S}_{p}(H)$ are non-closed two-sided ideals in $\mathcal{B}(H)$.

Let $A, B \in \mathcal{S}_{2}(H)$. Prove that $A B \in \mathcal{S}_{1}(H)$.

Let $T$ be an integral operator in the space $L_{2}[0,1]$, $(T x)(t) = \int_{0}^{1} K(t, s) x(s) d s$, where $K(\cdot , \cdot ) \in L_{2}\left([0,1]^{2}\right)$ and for almost every $s \in [0,1]$ the function $t \mapsto K(t, s)$ satisfies the Lipschitz condition: $\left|K\left(t_{2}, s\right)-K\left(t_{1}, s\right)\right| \leqslant C(s)\left|t_{2}-t_{1}\right|$ for all $t_{1}, t_{2} \in [0,1]$, where the function $C(\cdot ) \in L_{2}[0,1]$. Prove that $T$ is a nuclear operator. Prove that if the function $K$ is moreover continuous on the square $[0,1]^{2}$, then $\operatorname {tr}(T) = \int_{0}^{1} K(t, t) d t$.

(T. Carleman's example). Give an example of a non-nuclear integral operator in the space $L_{2}[0,1]$ with continuous kernel.

Do there exist:

Let $H$ be a separable Hilbert space and $A \in \mathcal{K}(H), A \geqslant 0$. Let $\lambda_{k}(A)$ be the eigenvalues of the operator $A$, numbered in non-increasing order. Prove that $\lambda_{k}(A) = s_{k}(A)$.

Let $A$ and $B$ be compact operators in a Hilbert space $H$, with $\operatorname {Im} A \subset \operatorname {Im} B$. Prove that if $B \in \mathcal{S}_{p}(H)$, then $A \in \mathcal{S}_{p}(H)$ as well.

Find the norm of the integration operator

in $L_{2}[0,1]$:

Find the polar decomposition $A = W S$ for the operator from problem 12.105.

Find the norms of the operators in $L_{2}[-b, b]$:

Find the spectrum of the integral operators in $L_{2}[0, \pi ]$ with kernel

Find the spectrum of the integral operators in $L_{2}[0,1]$ with kernel