Operators in Hilbert spaces

Let $P \in \mathcal{B}(H)$ be a projector of a Hilbert space $H$ onto some subspace of it. Prove that $P$ is an orthogonal projector if and only if for any $x, y \in H$ the following equality holds: $(P x, y) = (x, P y).{ }$¹

Definition 8.3: An operator $A \in \mathcal{B}(H)$ is called self-adjoint if $A = A^{*}$.

Let $P$ be an orthogonal projector in a Hilbert space $H$. Prove that it is bounded, and find $\left\| P\right\|$.

Let $P_{1}$ and $P_{2}$ be orthogonal projectors in a Hilbert space $H$. Denote the subspaces $H_{1} = \operatorname {Im} P_{1}$ and $H_{2} = \operatorname {Im} P_{2}$. Prove that if $H_{1} \subset H_{2}$ (or, conversely, $H_{2} \subset H_{1}$) and $\left\| P_{1}-P_{2}\right\| < 1$, then $P_{1} = P_{2}$. Give an example of two such orthogonal projectors $P_{1} \neq P_{2}$ that $\left\| P_{1}-P_{2}\right\| < 1$.

In the notation of the previous problem, prove that if $\left\| P_{2}-P_{1}\right\| < 1$, then $\operatorname {dim} H_{1} = \operatorname {dim} H_{2}$. Give an example of two such orthogonal projectors $P_{1}$ and $P_{2}$ that $\left\| P_{2}-P_{1}\right\| = 1$ and $\operatorname {dim} H_{1} \neq \operatorname {dim} H_{2}$.

Let $H$ be a Hilbert space, $P$ the orthogonal projector onto a proper subspace $H_{0} \subset H$. What can be asserted about $\operatorname {Im} A$ and Ker $A$ of an operator $A \in \mathcal{B}(H)$, if

Prove that if, under the conditions of the previous problem, the equality $A P = P A$ holds, then $A\left(H_{0}\right) \subset H_{0}$. Prove that the converse is false.

Let $H$ be a Hilbert space, and $B: H \times H \rightarrow \mathbb {C}$ a form that is linear in the first argument, conjugate-linear in the second argument, and continuous in each argument. Prove that there is an operator $A \in \mathcal{B}(H)$ such that $B(x, y) = (A x, y)$ for all $x, y \in H$.

For any $M > 0$ give an example of an operator $A$ in a Hilbert space $H$ such that for some orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$ all the norms $\left\| A e_{n}\right\|$ are bounded by one, but $\left\| A\right\| > M$.

Let $\sum_{i = 1}^{\infty } \sum_{j = 1}^{\infty }\left|a_{i j}\right|^{2} < \infty$, and let $\left\{ e_{i}\right\}_{1}^{\infty }$ be an arbitrary orthonormal basis in a Hilbert space $H$. Prove that there is an operator $A \in \mathcal{B}(H)$ such that $\left(A e_{i}, e_{j}\right) = a_{i j}$. Give an example of a bounded linear operator in a Hilbert space whose matrix (in any orthonormal basis) does not satisfy the indicated inequality.

Let an operator $A$ act in the space $l_{2}$, and in some orthonormal basis let its matrix have the form $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$, $a_{i j} \geqslant 0$ for all $i, j$. Prove that if for this matrix there exist numbers $p_{j} > 0, j = 1,2, \ldots$, and $\alpha , \beta > 0$, such that

(Schur's inequalities), then the operator is bounded, and $\left\| A\right\|^{2} \leqslant \alpha \beta$.

Prove that the operator $A$ with matrix $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$, where $a_{i j} = \frac{1}{i+j}$ (the Hilbert operator), is bounded in $l_{2}$, and $\left\| A\right\| \leqslant \pi$.

Let the matrix $\left\{ a_{i j} = K(i, j)\right\}_{i, j = 1}^{\infty }$ be given by a function $K(x, y)$ satisfying the following conditions:

1. $K$ is nonnegative, with $K(x, x) = 0$, symmetric, and homogeneous of degree of homogeneity -1, i.e., $K(a x, a y) = a^{-1} K(x, y)$;

2. $\int_{0}^{\infty } K(1, y) \frac{d y}{\sqrt{y}} = k$;

3. $\frac{K(1, y)}{\sqrt{y}}$ is decreasing for $y \in (0, \xi ) \cup (1,+\infty )$ and increasing for $y \in (\xi , 1)$, where $\xi \in [0,1]$. Prove that the matrix $\left\{ a_{i j}\right\}$ defines a bounded operator $A$ in the space $l_{2}$, and $\left\| A\right\| \leqslant k.$¹

Prove that the operator $A$ with matrix $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$, where $a_{i j} = \frac{1}{(i+j)^{1-\alpha }|i-j|^{\alpha }}\left(a_{i j}: = 0\right.$ for $\left.i = j\right)$ with parameter $\alpha \in (0,1)$, is bounded in $l_{2}$.

Prove that for any operator $A$ in a separable Hilbert space $H$ one can choose an orthonormal basis $\left\{ e_{j}\right\}_{1}^{\infty }$ such that for every $j \in \mathbb {N}$ only finitely many entries of the $j$-th column of the matrix $\left\{ a_{i j} = \left(A e_{i}, e_{j}\right)\right\}_{i, j = 1}^{\infty }$ are nonzero.

Prove that a matrix $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$ is the matrix of some operator from $\mathcal{B}\left(l_{2}, c_{0}\right)$ if and only if

1. there is $M$ such that $\sum_{j = 1}^{\infty }\left|a_{i j}\right|^{2} < M$ for every $i \in \mathbb {N}$;

2. $\lim_{i \rightarrow \infty } a_{i j} = 0$ for every $j \in \mathbb {N}$.