Elementary Properties and Examples

Prove Proposition 1.1: Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces and $A: \mathcal{H} \rightarrow \mathcal{K}$ a linear transformation. The following statements are equivalent.

1. $A$ is continuous.

2. $A$ is continuous at $0$.

3. $A$ is continuous at some point.

4. There is a constant $c > 0$ such that $\left\| Ah\right\| \leq c \left\| h\right\|$ for all $h$ in $\mathcal{H}$.

Prove Proposition 1.2: (a) If $A$ and $B \in \mathcal{B}(\mathcal{H}, \mathcal{K})$, then $A+B \in \mathcal{B}(\mathcal{H}, \mathcal{K})$, and $\left\| A+B\right\| \leq \left\| A\right\| +\left\| B\right\|$.

(b) If $\alpha \in \mathbb {F}$ and $A \in \mathcal{B}(\mathcal{H}, \mathcal{K})$, then $\alpha A \in \mathcal{B}(\mathcal{H}, \mathcal{K})$ and $\left\| \alpha A\right\| = \left|\alpha \right| \left\| A\right\|$.

(c) If $A \in \mathcal{B}(\mathcal{H}, \mathcal{K})$ and $B \in \mathcal{B}(\mathcal{K}, \mathcal{L})$, then $BA \in \mathcal{B}(\mathcal{H}, \mathcal{L})$ and $\left\| BA\right\| \leq \left\| B\right\| \left\| A\right\|$.

Suppose $\left\{ e_{1}, e_{2}, \ldots \right\}$ is an orthonormal basis for $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum \left\| Ae_{n}\right\| < \infty$. Show that $A$ has a unique extension to a bounded operator on $\mathcal{H}$.

Proposition 1.2 says that $d(A,B) = \left\| A-B\right\|$ is a metric on $\mathcal{B}(\mathcal{H}, \mathcal{K})$. Show that $\mathcal{B}(\mathcal{H}, \mathcal{K})$ is complete relative to this metric.

Show that a multiplication operator $M_{\phi }$ satisfies $M_{\phi }^{2} = M_{\phi }$ if and only if $\phi$ is a characteristic function.

Theorem 1.5 (context for $M_\phi$): Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and put $\mathcal{H} = L^{2}(X, \Omega , \mu ) \equiv L^{2}(\mu )$. If $\phi \in L^{\infty }(\mu )$, define $M_{\phi }: L^{2}(\mu ) \rightarrow L^{2}(\mu )$ by $M_{\phi } f = \phi f$. Then $M_{\phi } \in \mathcal{B}(L^{2}(\mu ))$ and $\left\| M_{\phi }\right\| = \left\| \phi \right\|_{\infty }$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $k_{1}, k_{2}$ be two kernels satisfying the hypothesis of (1.6):

Theorem 1.6 (context): Let $(X, \Omega , \mu )$ be a measure space and suppose $k: X \times X \rightarrow \mathbb {F}$ is an $\Omega \times \Omega$-measurable function for which there are constants $c_{1}$ and $c_{2}$ such that

If $K: L^{2}(\mu ) \rightarrow L^{2}(\mu )$ is defined by

then $K$ is a bounded linear operator and $\left\| K\right\| \leq (c_{1} c_{2})^{1/2}$.

Define

If $(X, \Omega , \mu )$ is a measure space and $k \in L^{2}(\mu \times \mu )$, show that $k$ defines a bounded integral operator.

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 $\left\{ \alpha_{ij}\right\}_{i,j=1}^{\infty }$ be an infinite matrix such that $\alpha_{ij} \geq 0$ for all $i,j$ and such that there are scalars $p_{i} > 0$ and $\beta , \gamma > 0$ with

for all $i, j \geq 1$. Show that there is an operator $A$ on $l^{2}(\mathbb {N})$ with $\left\langle Ae_{j}, e_{i} \right\rangle = \alpha_{ij}$ and $\left\| A\right\|^{2} \leq \beta \gamma$.

Show that $\left\langle Ae_{j}, e_{i} \right\rangle = (i+j+1)^{-1}$ for $0 \leq i,j < \infty$ defines a bounded operator on $l^{2}(\mathbb {N} \cup \left\{ 0\right\} )$ with $\left\| A\right\| \leq \pi$. (See also Choi [1983] and Redheffer and Volkmann [1983].)

If $A = \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]$, put $\alpha = [\left|a\right|^{2}+\left|b\right|^{2}+\left|c\right|^{2}+\left|d\right|^{2}]^{1/2}$ and show that $\left\| A\right\| = \frac{1}{2}(\alpha^{2}+\sqrt{\alpha^{4}-4\delta^{2}})$, where $\delta^{2} = \operatorname {det}\left(A^{*}A\right)$.

Let $\left\{ \mathcal{H}_{i}\right\}$ be a collection of Hilbert spaces and let $\mathcal{H} = \oplus_{i} \mathcal{H}_{i}$. Suppose $A_{i} \in \mathcal{B}(\mathcal{H}_{i})$ for all $i$. Show that there is a bounded operator $A$ on $\mathcal{H}$ such that $A|\mathcal{H}_{i} = A_{i}$ for all $i$ if and only if $\sup_{i} \left\| A_{i}\right\| < \infty$. In this case, $\left\| A\right\| = \sup_{i} \left\| A_{i}\right\|$. The operator $A$ is called the direct sum of the operators $\left\{ A_{i}\right\}$ and is denoted by $A = \oplus_{i} A_{i}$.

Call a sequence of vectors $\left\{ f_{n}\right\}$ in a Hilbert space $\mathcal{H}$ a Bessel sequence if $\sum \left| \left\langle f, f_{n} \right\rangle \right|^{2} < \infty$ for every $f$ in $\mathcal{H}$. Show that a sequence $\left\{ f_{n}\right\}$ of vectors in $\mathcal{H}$ is a Bessel sequence if and only if the infinite matrix $( \left\langle f_{m}, f_{n} \right\rangle )$ defines a bounded operator on $l^{2}$. (Also see Shapiro and Shields [1961], p.524.)