On a new class of structured reproducing kernel spaces

A class of reproducing kernel spaces with reproducing kernels of the form K_ω(λ) = (J - Θ(λ)JΘ(ω)*)/ρ_ω (λ) with pω(λ) = a(λ)a(ω)* is characterized in terms of invariance under a pair of generalized shift operators and a structural identity. This incorporates a characterization of de Branges for the “line” case and a later analogue due to Ball for the “circle” case, as well as many other possibilities, by specializing the choice of ρ. These results also permit the extension of some earlier characterizations by the authors of finite dimensional spaces with reproducing kernels of the form given above to the infinite dimensional case. The non-Hermitian case is also considered.