The spaces now called de Brange-Rovnyak spaces were introduced by de
Branges and Rovnyak in 1966. De Branges-Rovnyak spaces are
subspaces of \(H^2\) the standard Hardy space of the open unit disk
\(\mathbb{D}\). To give their definition we denote by \(T_{\chi}, \
\chi\in L^{\infty}(\mathbb{T}),\) where
\(\mathbb{T}=\partial\mathbb{D}\), the bounded Toeplitz operator on
\(H^2\), that is, \(T_{\chi}f=P(\chi f)\), where \(P\) is the orthogonal
projection of \(L^{2}(\mathbb{T})\) onto \(H^2\). Given a function \(b\)
in the unit ball of \(H^{\infty}\), the de Branges-Rovnyak
space \(\mathcal{H}(b)\) is the image of \(H^2\) under the operator
\((I-T_bT_{\overline{b}})^{1/2}\). The space \(\mathcal{H}(b)\) is
given the Hilbert space structure that makes the operator
\((I-T_bT_{\overline{b}})^{1/2}\) a coisometry of \(H^2\) onto
\(\mathcal{H}(b)\), namely
\[\langle(I-T_bT_{\overline{b}})^{1/2}f,(I-T_bT_{\overline{b}})^{1/2}g\rangle_b=\langle f ,g\rangle_2\quad (f,g\in(\mathrm{ker}(I-T_bT_{\overline{b}})^{1/2})^{\perp}).\]
It turns out that if \(b\) is an inner function, then \(
\mathcal{H}(b)=(bH^2)^{\bot}\). Here we deal with the case when \(b\)
is not an extreme point of the unit ball of \(H^{\infty}\). We
describe the structure of some spaces \(\mathcal{H}(b)\) and their
connections with the generalized Dirichlet spaces defined below.
For \(\lambda\in\mathbb{T}\) we define the local Dirichlet integral of
\(f\) at \(\lambda\) by
\[D_{\lambda}(f)=\frac1{2\pi}\int_0^{2\pi}\left|\frac{f(\lambda)-f(e^{it})}{\lambda-e^{it}}\right|^2dt.\]
where \(f(\lambda)\) is the nontangential limit of \(f\) at \(\lambda\).
If \(f(\lambda)\) does not exist, then we set \(D_{\lambda}(f)=\infty\).
Let \(\mu\) be a positive Borel measure on \(\mathbb T\). The
generalized Dirichlet space \(\mathcal{D}(\mu)\) consists of those
functions \(f\in H^2\) for which
\[D_{\mu}(f)=\int_{\mathbb{T}}D_{\lambda}(f)d\mu(\lambda)<\infty.\]
In 1997 D. Sarason showed that \(\mathcal{D}(\delta_{\lambda})\),
where \(\delta_{\lambda}\) is the unit mass at \(\lambda\), can be
identified with \(\mathcal{H}(b_{\lambda})\), where
\(b_{\lambda}(z)=(1-w_0)\overline{\lambda}z/(1-w_0\overline
{\lambda}z), \) and \( w_0=(3-\sqrt 5)/2\). Further results showing
connection between the spaces \(\mathcal{H}(b)\) and \(D({\mu})\) have
been recently obtained by T. Ransford, D. Guillot, N. Chevrot and C. Costara.
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