Krein spaces of analytic functions
An inner product space C is called a Krein space in case it is the orthogonal sum of a Hilbert space \(C_+\) and the anti-space \(C_-\) of another Hilbert space (in which the new inner product is the negative of the original one). The Pontryagin index of C is defined to be the dimension of \(C_-\). Let \(C_+(z)\) denote the Hilbert space of square summable power series \(f(z)=\sum a_ nz^ n\) with coefficients \(a_ n\) in \(C_+\) and z a complex variable. Let \(C_-(z)\) be defined in like manner and C(z) denote the Krein space spanned by the isometric and orthogonal subspaces \(C_+(z)\), \(C_-(z)\). The author studies invariant subspaces of C(z) in the course of generalizing his results [J. Math. Anal. Appl. 29, 163-200 (1970; Zbl 0167.133)] and those of himself and \textit{J.Rovnyak} [Perturbation theory and its applications in quantum mechanics, pp. 295-391 (1966)]. Among them we may quote the following: Let C have finite dimension and M be a Krein space contained contractively in C(z) with the property that the transformation A defined by \(Af=zf(z)\) maps M isometrically into itself; let N be the orthogonal complement in M of AM; then dimension N \(\leq\) dimension C. A similar result is obtained for the Pontryagin indices of N and C. Subspaces invariant under the difference quotient transformation D defined by \(Df=(f(z)-f(0))/z\) play a prominent role in these results. Generalizations of a theorem of Beurling and \textit{P. D. Lax} [Acta Math. 101, 163-178 (1959; Zbl 0085.091)] are also included.
scientific article; zbMATH DE number 4090163
scientific article