A note on multipliers between model spaces (Q1743748)

The authors give, under suitable hypothesis, necessary and sufficient conditions for the set of multipliers from one model space to another to be non-trivial. Let \(\mathcal{H}^{2}=\mathcal{H}^{2} (\mathbb{C}_{+})\) be the Hardy space of the upper half-plane. As usual one identifies functions in \(\mathcal{H}^{2}\) with their boundary values on \(\mathbb{R}\). An inner function \(U\) on \(\mathbb{C}_{+}\) is a bounded analytic function on \(\mathbb{C}_{+}\) with boundary values of modulus one almost everywhere on \(\mathbb{R}\). When \(U\) can be extended to a meromorphic function on \(\mathbb{C}\), one says that \(U\) is a meromorphic inner function (MIF). Associated to an inner function \(U\) on \(\mathbb{C}_{+}\) the model space \(K_{U}\) is defined by \[ K_{U}=\mathcal{H}^{2}\cap (U\mathcal{H}^{2})^{\perp}=\mathcal{H}^{2}\cap (U\overline{\mathcal{H}^{2}}), \] where \(\overline{\mathcal{H}^{2}}\) can be regarded as the Hardy space of the lower half-plane. Recall that for every \(\varphi\in L^{\infty}(\mathbb{R})\) the corresponding Toeplitz operator \(T_{\varphi}: \mathcal{H}^{2}\to\mathcal{H}^{2}\) is given by \[ T_{\varphi}(f)=P_{+}(\varphi f),\quad f\in \mathcal{H}^{2}, \] where \(P_{+}\) is the orthogonal projection of \(L^{2}(\mathbb{R})\) onto \(\mathcal{H}^{2}\). Then the model space \(K_{U}\) associated to \(U\) is given by \[ K_{U}=\text{ker} T_{\overline{U}}. \] For a pair of inner functions \(U\) and \(V\) on \(\mathbb{C}_{+}\), the multiplier set \(\mathcal{M}(U,V)\) is the set of analytic functions \(\Phi\) on \(\mathbb{C}_{+}\) such that \[ \Phi K_{U}\subset K_{V}. \] A basic question is whether or not \(\mathcal{M}(U,V)\neq \{0\}\). The noninjectivity of a certain Toeplitz operator is necessary for the set of multipliers to be nontrivial. The problem of injectivity of Toeplitz operators is a classical problem in analysis, being related to completeness of exponential systems on \(L^{2}(0,2\pi)\). In this paper the authors give a class of MIFs \(U\) and \(V\) for which the triviality of \(\mathcal{M}(U,V)\) can be reduced to the injectivity of the Toeplitz operator \(T_{U\overline{V}}\). We denote by \(\Pi\) the Poisson measure on \(\mathbb{R}\) \[ d\Pi(t)=\frac{dt}{1+t^{2}} \] and set \(L^{1}_{\Pi}=L^{1}(\mathbb{R},\Pi)\). The Hilbert transform of a function \(h\in L^{1}_{\Pi}\) is defined as the singular integral \[ \widetilde{h}(x)=\lim_{\varepsilon\to 0}\frac{1}{\pi} \int_{|x-t|>\varepsilon}\left[ \frac{1}{x-t}+\frac{t}{1+t^{2}}\right] h(t)\,dt. \] The elementary Blaschke factor on \(\mathbb{C}_{+}\) with zero at \(i\) is \[ b_{i}(z)=\frac{z-i}{z+i}. \] The following characterization of nontriviality of \(\mathcal{M}(U,V)\) is proved. \smallskip \(\bullet\) Let \(U\) and \(V\) be MIFs with \(|U'| \asymp 1\) on \(\mathbb{R}\) and let \[ m:=\text{arg} (U)-\text{arg}(Vb_{i}) \] on \(\mathbb{R}\). Suppose that either \(m\notin \widetilde{L}^{1}_{\Pi}\) or if \(m=\widetilde{h}\) for some \(h\in L^{1}_{\Pi}\), then \(e^{-h}\notin L^{1}(\mathbb{R})\). Then the following three conditions are equivalent. (1) \(\dim\text{ker} T_{U\overline{V b_{i}}}\geq 2\), (2) \(\text{ker} T_{U\overline{V}}\neq \{0\}\), (3) \(\mathcal{M}(U,V)\neq \{0\}\). Finally the authors give some examples of MIFs \(U\) and \(V\) satisfying the hypothesis of the above result, where \(U\), \(V\) are of the form \(U=S^{a}B_{\Lambda_{1}}\), \(V=S^{b}B_{\Lambda_{2}}\), where \(S(z)\) is the singular inner function \(e^{iz}\) and \(B_{\Lambda_{1}}\), \(B_{\Lambda_{2}}\) are Blaschke products with zero sets \(\Lambda_{1}\) \(\Lambda_{2}\) with appropriate Beurling-Malliavin densities and the constants \(a\), \(b\) are conveniently chosen.