Krein-type theorems and ordered structure for Cauchy–de Branges spaces

Abstract

We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.

Spectral theory of rank one perturbations of normal compact operators

“…In particular, in the cases Z, Π and A γ an ordering theorem for nearly invariant subspaces of the backward shift (see the definition in Section 7) in H(T, A, µ) was obtained similar to the classical de Branges' Ordering Theorem [14,Theorem 35]. On the other hand, it is shown in [1] that in general there is no ordered structure for nearly invariant subspaces in Cauchy-de Branges spaces.…”

“…In [1] the theory of Cauchy-de Branges spaces H(T, A, µ) (see the definition in Subsection 2.5) was developed which generalizes the theory of classical de Branges spaces. In particular, in the cases Z, Π and A γ an ordering theorem for nearly invariant subspaces of the backward shift (see the definition in Section 7) in H(T, A, µ) was obtained similar to the classical de Branges' Ordering Theorem [14,Theorem 35].…”

“…This follows, e.g., from the axiomatic description of de Branges spaces [14,Theorem 23]. In a recent preprint [1] certain properties of de Branges spaces (e.g., ordered structure of subspaces) are extended to a class of Cauchy-de Branges spaces.…”

Fractional Paley–Wiener and Bernstein spaces

“…Let g ∈ L 2 a (| | 2s ) and define f as in (4). Then, since 0 < s < 1 2 , for z = x + iy, Therefore, f is well-defined, is clearly entire and belongs to E a . We wish to show that f 0 ∈ E s,2 .…”

“…Indeed, this is the case for the reconstruction formula 17, but we cannot really improve (18). For simplicity, we now restrict ourselves to the case 0 < s < 1 2 .…”