Definition 10.3.1.
The Hardy space \(H^2\) is the space of functions
\begin{equation*}
f(z) = \sum_{n=0}^\infty a_n z^n \text{ where } (a_n) \in \ell^2.
\end{equation*}
Section 10.3 \(H^2\) and \(H^\infty(\D)\)
We’ll now set up a classically important and fun space of functions derived from \(\ell^2\)
We should note that there is a more standard setup for \(H^2\) involving extending continuous \(L^2\) functions off of the unit circle. The inner product on \(H^2\) is given by
\begin{equation*}
\ip{f}{g}_{H^2} = \ip{\sum a_n z^n}{\sum b_n z^n} = \sum a_i \cc{b_i},
\end{equation*}
under which \(H^2\) is a Hilbert space. In fact, the Hardy space is more.
Definition 10.3.2.
The Szegő kernel is the function
\begin{equation*}
s_\la(z) = \frac{1}{1 - \cc{\la}z}.
\end{equation*}
The Szegő kernel \(s_\la\) has the interesting property that it represents evaluation of a function in the \(H^2\) inner product. That is, for \(f \in H^2\text{,}\)
\begin{equation*}
f(\la) = \ip{f}{s_\la}.
\end{equation*}
We will not prove this here, but it will be used extensively. Encoding function evaluation at a point as an inner product is a very powerful idea.
Along with the Hardy space, we also consider the space of bounded analytic functions on the disk \(\D\text{,}\) which is denoted by \(H^\infty(\D)\text{.}\) This space is a Banach algebra under the supremum norm on the disk. The unit ball in \(H^\infty\) is the Schur class, which is going to provide the connection between Question 10.1.1 and operators by way of the observation that \(H^\infty(\D)\) is a space of multipliers on \(H^2\text{.}\)
Lemma 10.3.3.
Let \(\ph \in H^\infty(\D)\text{.}\) then \(\ph f \in H^2\) when \(f \in H^2\text{.}\) That is, for \(\ph in H^\infty(\D)\text{,}\) the operator \(M_\ph :H^2 \to h^2\) defined by \(M_\ph f = \ph f\) is in \(\mathcal{L}(H^2)\text{.}\) Furthermore, \(\norm{M_\ph} \leq \norm{\ph}\text{.}\)
The connection between the operator \(M_\ph\) and the function \(\ph\) is provided by the Szegő kernel. In particular, the Szegő kernel \(s_\la\) is an eigenvector for the adjoint of \(M_\ph\text{.}\)
Lemma 10.3.4.
If \(\ph \in H^\infty(\D)\) and \(\la \in \D\text{,}\) then
\begin{equation*}
M_\ph\ad = \cc{\ph(\la)} s_\la.
\end{equation*}
Proof.
Pick \(\ph \in H^\infty(\D)\) and \(\la \in \D\text{.}\) If \(f\in H^2\text{,}\)
\begin{align*}
\ip{f}{M_\ph\ad s_\la} \amp= \ip{M_\ph f}{s_\la}\\
\amp = \ip{\ph f}{s_\la}\\
\amp = \ph(\la) f(\la)\\
\amp = \ph(\la) \ip{f}{s_\la}\\
\amp = \ip{f}{\cc{\ph(\la)}s_\la}.
\end{align*}
Since this holds for all \(f \in H^2\text{,}\) we conclude that \(M_\ph s_\la = \cc{\ph(\la)}s_\la\text{.}\)
In fact we can say more.
Lemma 10.3.5.
If \(\la \in D\text{,}\) then
\begin{equation*}
\ker(\cc\la - M_z\ad) = \C s_\la.
\end{equation*}
