Adjoint operators

Prove Theorem 8.1.

Theorem 8.1 (F. Riesz, 1910): the norm of the operator $A^{\prime }$ equals the norm of the operator $A$, i.e., the conjugation operation preserves the norm.

Let $X, Y, Z$ be normed spaces, $A, B \in \mathcal{B}(X, Y)$, $C \in \mathcal{B}(Y, Z), \alpha , \beta \in \mathbb {C}$. Prove that

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$, and let a sequence $g_{n} \stackrel{*}{\rightharpoonup } g$ in $Y^{*}$. Prove that $A^{\prime } g_{n} \stackrel{*}{\rightharpoonup } A^{\prime } g$ in $X^{*}$.

Find the adjoint operators in the following cases.

Find the adjoint operator $A^{\prime }$ for the operator $A$ in the spaces $l_{p}, p \in [1, \infty )$, and $c_{0}$:

Find the adjoint operator $A^{\prime }$ for the following operators $A$:

Find the adjoint operator $A^{\prime }$ for the operator $A$ in the space $C[a, b]:(A x)(t) = \sum_{i = 1}^{n} \alpha_{i}(t) f_{i}(x)$, where $\alpha_{i} \in C[a, b]$, $f_{i} \in (C[a, b])^{*}$, $f_{i}(x) = \int_{a}^{b} x(t) d w_{i}(t), w_{i} \in B V_{0}[0,1]$.

Let the operator $S$ in $c_{0}^{*} = l_{1}$ be given by the formula $S:\left(x_{1}, x_{2}, x_{3}, \ldots \right) \mapsto \left(\sum_{k = 1}^{\infty } x_{k}, x_{2}, x_{3}, \ldots \right)$. Prove that $S$ is a linear isomorphism in $l_{1}$, but is not the adjoint of any operator $T \in \mathcal{B}\left(c_{0}\right)$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$. Denote the canonical embeddings $\pi_{X}: X \hookrightarrow X^{* *}, \pi_{Y}: Y \hookrightarrow Y^{* *}$. Prove that $A^{\prime \prime } \circ \pi_{X} = \pi_{Y} \circ A$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$. Prove that

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$. Prove that $(\operatorname {Im} A)^{\perp } = \operatorname {Ker} A^{\prime }$, and $(\operatorname {Ker} A)^{\perp } \supseteq \overline{\operatorname {Im} A^{\prime }}$ (recall that for an arbitrary set $M \subset X$ the set $M^{\perp } = \left\{ f \in X^{*}: f(x) = 0 \text{ for every } x \in M\right\}$). Give an example where the last inclusion is strict.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$. Prove that if $A$ is a surjection, then $(\operatorname {Ker} A)^{\perp } = \operatorname {Im} A^{\prime }$.

Let $X$ and $Y$ be Banach spaces, $A \in \mathcal{B}(X, Y)$. Prove that $\left(\operatorname {Im} A^{\prime }\right)^{\perp } = \operatorname {Ker} A,\left(\operatorname {Ker} A^{\prime }\right)^{\perp } = \overline{\operatorname {Im} A}$ (here for a set $M \subset X^{*}$ we denote $M^{\perp } = \left\{ x \in X: f(x) = 0\text{ for every }f \in M\right\}$).

Let $X$ and $Y$ be Banach spaces, $A_{n} \xrightarrow {s} A$ in $\mathcal{B}(X, Y)$. Does the sequence $\left\{ A_{n}^{\prime }\right\}_{n = 1}^{\infty }$ converge strongly to $A^{\prime }$ in $\mathcal{B}\left(Y^{*}, X^{*}\right)$? Does it converge weakly? Answer the same questions under the assumption that $X$ or $Y$ is reflexive.

Prove the following properties of the adjoint operation (here $H_{1}$ and $H_{2}$ are Hilbert spaces, $A \in \mathcal{B}\left(H_{1}, H_{2}\right)$):

Using the Riesz isomorphism, establish for an arbitrary operator $A \in \mathcal{B}\left(H_{1}, H_{2}\right)$ the relation $A^{*} = J_{H_{1}}^{-1} A^{\prime } J_{H_{2}}$, i.e., prove the commutativity of the following diagram:

Find the adjoint operators $A^{*}$ for the following operators in the space $L_{2}[0,1]$:

Let the operator $A \in \mathcal{B}\left(l_{2}\right)$ be given in some orthonormal basis by the matrix $\left\{ a_{i j}\right\}_{i, j = 1}^{\infty }$. Prove that $A^{\prime }$ in the same basis is given by the matrix $\left\{ a_{j i}\right\}_{i, j = 1}^{\infty }$, and $A^{*}$ is given by the matrix $\left\{ \bar{a}_{j i}\right\}_{i, j = 1}^{\infty }$.

Find the adjoint operators $A^{*}$ for the following operators in the space $l_{2}$:

Find the Hermitian adjoint operators for the operators of left and right shift in the space $l_{2}(\mathbb {Z})$.

Find the adjoint operators $A^{\prime }$ and $A^{*}$ to the integral operator $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s$ in the space $L_{2}[a, b]$, where $K \in L_{2}\left([a, b]^{2}\right)$. Find the conditions on the function $K$ under which the operator $A$ is self-adjoint.

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ self-adjoint in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(B x)(t) = a(t) x(t)$ self-adjoint in the space $L_{2}[0,1]$?

Find the adjoint operator $A^{*}$ to the change-of-variable operator

Find the adjoint operators $A^{*}$ for the following operators:

Let $H_{0}$ be a subspace of a Hilbert space $H$, invariant with respect to an operator $A \in \mathcal{B}(H)$. Prove that $H_{0}^{\perp }$ is invariant with respect to $A^{*}$.

Let a Hilbert space $H$ be decomposed into the direct sum of closed subspaces $H = H_{0} \oplus H_{1}$, and let a projector $P$ onto the subspace $H_{0}$ along $H_{1}$ be given. Prove that $H_{0} \perp H_{1} \Leftrightarrow P = P^{*}$ (compare with problem 6.62).

Let $H$ be a Hilbert space, and let $P_{1}$ and $P_{2}$ be orthoprojectors onto the subspaces $H_{1}$ and $H_{2}$. Prove the equivalence of the following statements:

1. $P_{1} \leqslant P_{2}$ (i.e., $\left(\left(P_{2}-P_{1}\right) x, x\right) \geqslant 0$ for every $x \in H$);

2. $H_{1} \subseteq H_{2}$;

3. $P_{1} P_{2} = P_{2} P_{1} = P_{1}$;

4. $P_{2}-P_{1}$ is an orthoprojector.

Let $P_{1}$ and $P_{2}$ be orthoprojectors in a Hilbert space $H$ onto the subspaces $H_{1}$ and $H_{2}$. Under what conditions on $P_{1}$ and $P_{2}$ (under what conditions on $H_{1}$ and $H_{2}$) are the following operators also orthoprojectors:

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$ and $\operatorname {dim} \operatorname {Im} A = n < \infty$. Prove that $\operatorname {dim} \operatorname {Im}\left(A^{*}\right) = n$.

Let $H_{1}$ and $H_{2}$ be Hilbert spaces, with the space $H_{2}$ separable. Prove that if there exists an injective operator $A \in \mathcal{B}\left(H_{1}, H_{2}\right)$, then the space $H_{1}$ is also separable. Is this statement true for Banach spaces?

Let $H_{1}$ and $H_{2}$ be Hilbert spaces, $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}\left(H_{1}, H_{2}\right)$ and $A_{n} \rightrightarrows A \in \mathcal{B}\left(H_{1}, H_{2}\right)$. Prove that $A_{n}^{*} \rightrightarrows A^{*}$. Prove that if $A_{n} \rightharpoonup A \in \mathcal{B}\left(H_{1}, H_{2}\right)$, then $A_{n}^{*} \rightharpoonup A^{*}$.

Give an example of a sequence of operators $\left\{ A_{n}\right\}_{1}^{\infty } \subset \mathcal{B}\left(H_{1}, H_{2}\right)$ in Hilbert spaces $H_{1}$ and $H_{2}$ such that $A_{n} \xrightarrow {s} A \in \mathcal{B}\left(H_{1}, H_{2}\right)$, but the sequence $\left\{ A_{n}^{*}\right\}_{1}^{\infty }$ has no strong limit.

Let $H_{1}$ and $H_{2}$ be Hilbert spaces. Prove that an operator $U \in \mathcal{B}\left(H_{1}, H_{2}\right)$ is unitary (see Definition 8.4) if and only if both equalities $U U^{*} = I_{H_{2}}$ and $U^{*} U = I_{H_{1}}$ hold. Give an example of a non-unitary operator for which one of these equalities holds.

Definition 8.4: An operator $U \in \mathcal{B}\left(H_{1}, H_{2}\right)$, where $H_{1}$ and $H_{2}$ are Hilbert spaces, is called unitary if it is isometric and bijective.

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ unitary in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ unitary in the space $L_{2}[0,1]$?

Prove that the shift operator $(A x)(t) = x(t-a)$ is unitary in $L_{2}(\mathbb {R})$.

Prove that the operators of left and right shift are unitary in the space $l_{2}(\mathbb {Z})$.

Let $H$ be a Hilbert space, and $U \in \mathcal{B}(H)$. Prove that if $\left\{ e_{n}\right\}_{1}^{\infty }$ is some orthonormal basis in $H$ and the system $\left\{ U e_{n}\right\}_{1}^{\infty }$ is also an orthonormal basis, then $U$ is unitary. Conversely, prove that if the operator $U$ is unitary, then it takes any orthonormal basis into an orthonormal basis.

Let $H_{1}$ and $H_{2}$ be Hilbert spaces, $W \in \mathcal{B}\left(H_{1}, H_{2}\right)$. Prove that $W$ is a partially isometric operator if and only if $W W^{*} W = W$. How does the operator $W^{*}$ act?

The Hilbert transform of a function $x \in L_{2}[0,2 \pi ]$ is the function $y(t) = \frac{1}{2 \pi } \int_{0}^{2 \pi } x(s) \operatorname {ctg} \frac{t-s}{2} d s.{ }$¹ Prove that the operator $A: x \mapsto y$ is a partial isometry.

Prove that the operators of left and right shift are partial isometries of the space $l_{2}$.

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ a partial isometry in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ a partial isometry in the space $L_{2}[0,1]$?

Let $H$ be a Hilbert space. Prove that the set

Prove that both the weak and the strong limit of self-adjoint operators (if it exists) is a self-adjoint operator. Prove that both the weak and the strong limit of unitary operators is a unitary operator, and the strong limit of orthogonal projectors is an orthogonal projector. Give an example showing that the weak limit of orthogonal projectors need not be a projector.

Let $H$ be a Hilbert space

Let $A$ and $B$ be two self-adjoint operators in a complex Hilbert space $H$. For every $x \in H, x \neq 0$, define

the uncertainty constant corresponding to the pair of operators $A, B$ for the element $x$ (if the denominator of the fraction equals zero, then $U C(x): = \infty$). Prove that the inequality $U C(x) \geqslant \frac{1}{2}$ always holds.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator. Can the image $\operatorname {Im} A$ be non-closed?

Let $H$ be a Hilbert space, $A$ a self-adjoint operator. Prove that $\overline{\operatorname {Im} A} = H$ if and only if $\operatorname {Ker} A = \left\{ 0\right\}$.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator, $x \notin \operatorname {Ker} A$. Prove that the sequence $\alpha_{n} = \left\| A^{n+1} x\right\| /\left\| A^{n} x\right\|$ converges.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that $\operatorname {Ker}\left(A A^{*}\right) = \operatorname {Ker} A^{*}, \operatorname {Ker}\left(A^{*} A\right) = \operatorname {Ker} A$. Is it true that

Let $A = A^{*}$. Prove that $\left\| A\right\| = \sup_{\left\| x\right\| = 1}\left|(A x, x)\right|$. Give an example of a non-self-adjoint operator for which this formula is false.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that $\left\| A A^{*}\right\| = \left\| A^{*} A\right\| = \left\| A\right\|^{2}$.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that there exists a unique pair of self-adjoint operators $A_{\text{re }}$ and $A_{\text{im }}$ such that $A = A_{\text{re }}+i A_{\text{im }}$.

Let $H$ be a Hilbert space. Prove that the following statements hold for self-adjoint operators in it:

Prove that for a positive operator $A$ the generalized Cauchy-Bunyakovsky inequality holds:

Let a sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ of positive operators converge weakly to zero. Prove that $A_{n}$ converge to zero strongly.

Let $A$ and $B$ be bounded operators in a Hilbert space $H$. Prove that the inequality $A^{*} A \leqslant B^{*} B$ is equivalent to the system of inequalities $\left\| A x\right\| \leqslant \left\| B x\right\|$ for every $x \in H$.

Let $H$ be a Hilbert space, $A \in \mathcal{B}(H)$. Prove that $A A^{*} \geqslant 0$.

Let $H$ be a complex Hilbert space, $A \in \mathcal{B}(H)$. Prove that if for every $x \in H$ the quadratic form $(A x, x)$ is real (in particular, if the operator is positive), then $A^{*} = A$.

Let $H$ be a Hilbert space, $A$ and $B$ self-adjoint operators and $A \geqslant 0$. Prove that $B A B \geqslant 0$.

Let $H$ be a Hilbert space, $A$ and $B$ self-adjoint operators, $0 \leqslant A \leqslant B$ and $A B = B A$. Prove that $A B \geqslant 0$ and $A^{2} \leqslant B^{2}$. Give examples showing that the condition $A B = B A$ in this statement is essential.

Let $H$ be a Hilbert space, and let a sequence $\left\{ A_{n}\right\}_{1}^{\infty }$ of self-adjoint operators from $\mathcal{B}(H)$ be monotonically non-decreasing, i.e., $A_{1} \leqslant A_{2} \leqslant A_{3} \leqslant \ldots$ Prove that such a sequence has a limit in the sense of strong operator convergence, but does not necessarily have a limit in the sense of uniform operator convergence.

Let $H$ be a Hilbert space, $A$ a self-adjoint operator and $A \geqslant 0$. Prove that there exists a unique self-adjoint nonnegative operator $B$ for which $B^{2} = A$.

For the following operators $A$ find $\sqrt{A}$:

Let $H$ be a Hilbert space, $A$ a self-adjoint operator and $A \geqslant 0$. Prove that $\left\| \sqrt{A}\right\| = \sqrt{\left\| A\right\| }$.