Analytic Hilbert spaces over the complex plane
Let \(X\) denote a Hilbert space of entire functions on the complex plane \(\mathbb{C}\). We say that \(X\) is an analytic Hilbert space in case (i) the polynomials are dense in \(X\) and (ii) the evaluation mapping \(E_\lambda (f)= f(\lambda)\) for all \(f\in X\) is continuous on \(X\) for each \(\lambda\in \mathbb{C}\). The point of departure for the present paper is the result that if \(M\neq \{0\}\) is a closed subspace of \(X\) and \(fM\subset M\) for some entire function \(f\), then \(f\) is a constant. Following the work of \textit{K. Guo} and \textit{D. Zheng} [J. Funct. Anal. 187, 308-342 (2001; Zbl 1045.47022)] for the Fock space \(L^2_a (\mathbb{C})\), the authors put \(X_0=\{f\in X:z f\in X\}\) and call any \(M\) such that \(X_0\cap M\) is dense in \(M\) a quasi-invariant subspace (QIS) in case \(zM\cap X\subset M\). For \(f\in X\) let \([f]\) be the intersection of all QIS's containing \(f\). The authors show that if the QIS \(M_1\) has finite codimension, then \(M_1=[p]\) for some polynomial \(p\). They define a QIS \(M_2\) to be similar to \(M_1\) in case some bounded invertible operator \(A\) maps \(M_1\) onto \(M_2\) in such a way that \(A(zf)= zA(f)\) and \(A^{-1} (zg)=zA^{-1} (g)\) for all \(f\in M_1\) and \(g\in M_2\). They show that if \(M_2\) is similar to \(M_1= [p]\), then \(M_2= [q]\) for some polynomial \(q\) with the same degree as \(p\). These results are extended to the case where \(A\) and \(A^{-1}\) are unitary. A natural order between entire functions \(f\) and \(g\) is defined, and \(X\) is shown to be ordered if and only if for any polynomial \(p\), \(pf\in X\) implies \(f\in X\). They show for an ordered analytic Hilbert space \(X\) that the QIS's \([p]\) and \([q]\) are similar if and only if \(p\) and \(q\) haye the same degree.
scientific article; zbMATH DE number 1763482