Exercises 3

Exercise 3.1:

In this exercise, we prove the first generalisation of Schur's test. Prove that under the same conditions as Theorem 3.2, the operator $K$ is in $\mathcal{B}(L^p(\Omega,\mu))$ for every $p \in [1,\infty]$, with Schur bound:

Here we use the convention that $\frac{1}{\infty} = 0$. You may want to treat the cases $p \in \{1,\infty\}$ separately.

Exercise 3.2:

In this exercise, we prove a second generalisation of Schur's test. For a measure space $(\Omega,\mu)$, $p,q \in [1,\infty]$ and $k: \Omega \times \Omega \rightarrow \mathbb{C}$ define:

and:

where for $p=\infty$ we interpret $(\int |f|^p d\mu)^{\frac{1}{p}} = \text{ess }\sup_x |f(x)|$. In this notation, we can also read $m_1 = m_{1,\infty,1} < \infty$ and $m_2 = m_{2,\infty,1} < \infty$.

Let $K$ be the integral operator with integral kernel $k$.

• Prove that if $p > q$, $m_{1,\infty,1} < \infty$ and $m_{2,\frac{p}{p-q},1} < \infty$, the $K \in \mathcal{B}(L^p(\Omega),L^q(\Omega))$ and:
  $$\|K\| \leq m_{1,\infty,1}^{1-\frac{1}{q}}m_{2,\frac{p}{p-q},1}^{\frac{1}{q}}.$$

Hint: Write $kf = k^{1-\frac{1}{q}}(k^{\frac{1}{q}}f)$ and use Hölder's inequality with exponents $(\frac{q}{q-1},q)$.

• Prove that if $q > q$, $m_{1,\infty,\frac{p(q-1)}{q(p-1)}} < \infty$ and $m_{2,\infty,1} < \infty$, then $K \in \mathcal{B}(L^p(\Omega), L^q(\Omega))$ and:
  $$\|K\| \leq m_{1,\infty,\frac{p(q-1)}{q(p-1)}^{1-\frac{1}{q}}} m_{2,\infty,1}^{\frac{1}{q}}.$$

Hint: There is the following generalisation of Hölder's inequality: if $u \in L^r$, $v \in L^s$ and $w \in L^t$ for $r^{-1}+s^{-1}+t^{-1}=1$, then $uvw \in L^1$ and:

Write $kf = (k^{\frac{1}{q}}f^{\frac{p}{q}})f^{\frac{q-p}{q}}k^{1-\frac{1}{q}}$ and use the generalised Hölder inequality with exponents $(q,\frac{pq}{q-p},\frac{p}{p-1})$.

Exercise 3.3:

Consider the operator $V$ acting on $L^2(0,1)$ given by:

$V$ is called the Volterra operator. Prove that $V$ is bounded. What is the integral kernel of $V$?

Exercise 3.4:

Under what condition on the function $r$ is the convolution operator self-adjoint? Normal?

Exercise 3.5:

Give an alternative proof of Schur's bound by estimating $|(Tf,g)|$ for any $f,g \in L^2$.

Exercise 3.6:

Construct an example of an operator whose Hilbert-Schmidt norm is strictly greater than the operator norm. Hint: consider operators on $\mathbb{C}^2$.

Exercise 3.7:

Construct an example of an operator for which the inequality in the Schur bound is strict. Hint: consider operators on $\mathbb{C}^2$.

Exercise 3.8:

Let $A$ be a self-adjoint operator in $\mathcal{B}(\mathcal{H})$. Prove that:

and find an example which shows that it does not necessarily hold if $A$ is not self-adjoint.

Exercise 3.9:

Show that $T \in \mathcal{B}(\mathcal{H})$ is normal if and only if $\|Tx\| = \|T^* x\|$ for all $x \in \mathcal{H}. Using this, show that for a normal$T$,$|T^2| = |T|^2$.

Exercise 3.10:

Let $\mathcal{H}$ be a finite dimensional Hilbert space, $\mathcal{H} = \mathbb{C}^N$. For $A \in \mathcal{B}(\mathcal{H})$, consider the expression $\|A\|_{HS}^2 = \text{Tr}(A^*A)$.

1. Prove that $\|\cdot\|_{HS}$ is a norm on $\mathcal{B}(\mathcal{H})$.
2. Prove that $\|\cdot\|_{HS}$ coincides with the Hilbert-Schmidt norm defined in Example 3.9.
3. Prove that the Hilbert-Schmidt norm is equivalent to the usual operator norm, i.e. there exist constants $C_1,C_2$ such that for all $A \in \mathcal{B}(\mathcal{H})$:
   $$C_1 \|A\| \leq \|A\|_{HS} \leq C_2 \|A\|$$
   (the first inequality always holds but the second inequality is only true for the finite-dimensional case!)
4. Find the optimal constants $C_1,C_2$.