The Fock Space as a De Branges–Rovnyak Space

Abstract

We show that de Branges-Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy-Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.

Fock and Hardy spaces: Clifford Appell case

“…\end{equation*}Remark We note that, by specifying the sequence c , $$\mathcal {HS}_c$$ include different spaces of slice hyperholomorphic functions such as Fock, Hardy, Dirichlet and generalized Fock spaces. Such spaces are the quaternionic counterpart of the complex version introduced in [3]. …”

Commutators on Fock spaces

“…When the sequence ϕ n is increasing and ϕ 0 = 1 the space H(ϕ) is a de Branges Rovnyak space. See [1].…”

“…We note that inequality (3.8) is the structure inequality which characterizes de Branges Rovnyak spaces; see [2, Theorem 3.1.2, p. 83] and [1] for further on this point.…”

“…that is to say, D (1) z n = d n−1 z n (under the convention that d −1 = 0) and D (−1) z n = d n+1 z n . We note that where P := diag (0, 1, 1, 1, .…”

Preliminaries

“…It is interesting to note that the Fock space restricted to the unit disk is a special case of a P(S) space; see [8].…”