Two classes of vector valued de Branges spaces

Abstract

There are two classes of reproducing kernel Hilbert spaces of m×1 vector valued functions that are commonly referred to as de Branges spaces. The reproducing kernel of the first is expressed in terms of an m×m signature matrix J and a meromorphic m×m matrix valued function Θ(λ) that is J-inner with respect to the open upper-half plane C+ asKω(λ)=J−Θ(λ)JΘ(ω)⁎−2πi(λ−ω‾)(λ≠ω‾) The reproducing kernel of the second is expressed in terms of a pseudo-meromorphic m×2m matrix valued function E(λ)=[E−(λ)E+(λ)] with m×m blocks E±(λ) asKω(λ)=E+(λ)E+(ω)⁎−E−(λ)E−(ω)⁎−2πi(λ−ω‾)(λ≠ω‾), wherein E+−1E− is an m×m inner function with respect to C+. We shall refer to these two classes of spaces as H(Θ) spaces and B(E) spaces, respectively. It was recently observed that every H(Θ) space is automatically a B(E) space. Although the proof is easy, the result is somewhat surprising because at first glance the known characterizations of these spaces look quite different. In this report we shall present another characterization that makes this connection more transparent. We shall also develop analogous results for spaces H(Θ) and B(E) that are defined with respect to the open unit disc D rather than C+ and shall present additional formulas for some finite dimensional examples.