The Adjoint of an Operator

Prove Proposition 2.5: If $U \in \mathcal{B}(\mathcal{H}, \mathcal{K})$, then $U$ is an isomorphism if and only if $U$ is invertible and $U^{-1} = U^{*}$.

Prove Proposition 2.6: If $A, B \in \mathcal{B}(\mathcal{H})$ and $\alpha \in \mathbb {F}$, then:

1. $(\alpha A + B)^{*} = \overline{\alpha } A^{*} + B^{*}$.

2. $(AB)^{*} = B^{*} A^{*}$.

3. $A^{**} \equiv (A^{*})^{*} = A$.

4. If $A$ is invertible in $\mathcal{B}(\mathcal{H})$ and $A^{-1}$ is its inverse, then $A^{*}$ is invertible and $(A^{*})^{-1} = (A^{-1})^{*}$.

Verify the statement in Example 2.8: Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $M_{\phi }$ be the multiplication operator with symbol $\phi$ (1.5). Then $M_{\phi }^{*}$ is $M_{\overline{\phi }}$, the multiplication operator with symbol $\overline{\phi }$.

Verify the statement in Example 2.9: If $K$ is the integral operator with kernel $k$ as in (1.6), then $K^{*}$ is the integral operator with kernel $k^{*}(x,y) \equiv \overline{k(y,x)}$.

Find the adjoint of a diagonal operator (Exercise 1.8).

Exercise 1.8 (context): Let $\left\{ e_{n}\right\}$ be the usual basis for $l^{2}$ and let $\left\{ \alpha_{n}\right\}$ be a sequence of scalars. Show that there is a bounded operator $A$ on $l^{2}$ such that $Ae_{n} = \alpha_{n} e_{n}$ for all $n$ if and only if $\left\{ \alpha_{n}\right\}$ is uniformly bounded, in which case $\left\| A\right\| = \sup \left\{ \left|\alpha_{n}\right|: n \geq 1\right\}$. This type of operator is called a diagonal operator or is said to be diagonalizable.

Let $S$ be the unilateral shift and compute $SS^{*}$ and $S^{*}S$. Also compute $S^{n} S^{*n}$ and $S^{*n} S^{n}$.

Compute the adjoint of the Volterra operator $V$ (1.7) and $V+V^{*}$. What is $\operatorname {ran}(V+V^{*})$?

Example 1.7 (context, defining $V$): Let $k: [0,1] \times [0,1] \rightarrow \mathbb {R}$ be the characteristic function of $\left\{ (x,y): y<x\right\}$. The corresponding operator $V: L^{2}(0,1) \rightarrow L^{2}(0,1)$ defined by $Vf(x) = \int_{0}^{1} k(x,y) f(y) \, dy$ is called the Volterra operator. Note that

Where was the hypothesis that $\mathcal{H}$ is a Hilbert space over $\mathbb {C}$ used in the proof of Proposition 2.12?

Proposition 2.12: If $\mathcal{H}$ is a $\mathbb {C}$-Hilbert space and $A \in \mathcal{B}(\mathcal{H})$, then $A$ is hermitian if and only if $\left\langle Ah, h \right\rangle \in \mathbb {R}$ for all $h$ in $\mathcal{H}$.

Suppose $A = B+iC$, where $B$ and $C$ are hermitian, and prove that $B=(A+A^{*})/2$, $C=(A-A^{*})/2i$.

Prove Proposition 2.15: If $\mathcal{H}$ is a $\mathbb {C}$-Hilbert space and $A \in \mathcal{B}(\mathcal{H})$ such that $\left\langle Ah, h \right\rangle = 0$ for all $h$ in $\mathcal{H}$, then $A=0$.

If $A$ and $B$ are self-adjoint, show that $AB$ is self-adjoint if and only if $AB=BA$.

Let $\sum_{n=0}^{\infty } \alpha_{n} z^{n}$ be a power series with radius of convergence $R$, $0<R \leq \infty$. If $A \in \mathcal{B}(\mathcal{H})$ and $\left\| A\right\| < R$, show that there is an operator $T$ in $\mathcal{B}(\mathcal{H})$ such that for any $h,g$ in $\mathcal{H}$, $\left\langle Th, g \right\rangle = \sum_{n=0}^{\infty } \alpha_{n} \left\langle A^{n} h, g \right\rangle$.

Let $A$ and $T$ be as in Exercise 12 and show that $\left\| T-\sum_{k=0}^{n} \alpha_{k} A^{k}\right\| \rightarrow 0$ as $n \rightarrow \infty$. If $BA=AB$, show that $BT=TB$.

If $f(z) = \exp z = \sum_{n=0}^{\infty } z^{n}/n!$ and $A$ is hermitian, show that $f(iA)$ is unitary.

If $A$ is a normal operator on $\mathcal{H}$, show that $A$ is injective if and only if $A$ has dense range. Give an example of an operator $B$ such that $\ker B = (0)$ but $\operatorname {ran} B$ is not dense. Give an example of an operator $C$ such that $C$ is surjective but $\ker C \neq (0)$.

Let $M_{\phi }$ be a multiplication operator (1.5) and show that $\ker M_{\phi } = 0$ if and only if $\mu (\left\{ x: \phi (x)=0\right\} ) = 0$. Give necessary and sufficient conditions on $\phi$ that $\operatorname {ran} M_{\phi }$ be closed.