Outer preserving linear operators
The authors investigate bounded linear operators acting on the Hardy space \(H^2\) of analytic functions on the unit disc \(\mathbb{D}\) which preserve the set of \textit{shifted outer functions}. Recall that \(H^2\cong \ell^2(\mathbb{Z}_+)\), the isomorphism is given by \(\vec{a} = (a_k) \leftrightarrow f: f(z) = \sum_k a_k z^k\). Outer functions are functions \(f\in H^2\), representable as \[ z\mapsto \lambda\cdot \exp\left((1/2\pi)\int_{-\pi}^\pi (e^{i\theta} + z)/(e^{i\theta} - z)g(\theta)\,d\theta \right) \] with unimodular constant \(\lambda\) and real integrable \(g\). Equivalently, \(f\) is an outer function if \(\{U^n f\}\) is dense in \(H^2\), where the shift operator \(U\) on \(H^2\) is defined by \(Uf(z)=z\cdot f(z)\). Furthermore, \(\mathcal{S}:= \left\{U^n f: n\geq 0, \text{ for outer functions } f \right\}\) is the set of shifted outer functions. Analytic functions \(\phi\) act as composition operators on \(H^2\), \(C_\phi(f):=f\circ \phi\), preserving \(\mathcal{S}\). In Theorem~2 resp.\ 4, the authors describe the shape of linear bounded operators \(A\) on \(H^2\) which preserve \(\mathcal{S}\): For \(\phi, \psi\in H^2\), \(p,q\geq 0\) (under suitable boundedness conditions), the operator \(A: f\mapsto (U^q\psi)\cdot C_{U^p \phi}(f)\) preserves \(\mathcal{S}\), and conversely, if \(A\) preserves \(\mathcal{S}\), then \(A\) is representable as \(f\mapsto (U^q\psi_0)\cdot C_{U^p \phi}(f)\), where \((\psi_n)\) is a sequence of outer functions defined by \(A(z^n)=z^{k(n)}\cdot \psi_n(z)\), \(k(n)\geq 0\), and the outer function \(\phi:=\psi_1/\psi_0\) is bounded by \(1\). In Section~5, the authors discuss applications to signal transforms under certain geo-physical processes, as, e.g., seismic wave propagations.
scientific article; zbMATH DE number 5978465
scientific article