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We use reproducing kernel Hilbert space methods to solve a Carathéodory–Fejér type interpolation problem with mixed derivatives (corresponding to a so-called lower inclusive set) in the class of Schur multipliers.

We solve a boundary interpolation problem in the reproducing kernel Hilbert space of functions analytic in the unit ball of CN with reproducing kernel 1/(1−∑1Nzkwk*). We introduce the notion of Brune factor (or Blaschke–Potapov factor of the third kind) in this setting.

We use the theory of reproducing kernel Hilbert spaces to solve a Caratheodory–Fejer interpolation problem in the class of Schur multipliers of the reproducing kernel Hilbert space of functions analytic in the unit ball of C N with reproducing kernel 1 1−∑ 1 N z k w k * .

Recently Bolotnikov and Rodman characterized finite dimensional backward shift invariant subspaces of the Arveson space. In this note we study finite dimensional backward shift invariant Hilbert spaces contractively included in the Arveson space. To that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions.

Natural connections between positive semidefinite solutions X of homogeneous algebraic Riccati equations and finite dimensional reproducing kernel de Branges spaces based on a J-inner proper rational square matrix valued functions are known. In this paper analogous connections between the positive semidefinite solutions X of nonhomogeneous algebraic Riccati equations and finite dimensional reproducing kernel Hilbert spaces based on rectangular (J,J∼)-coinner proper rational matrix valued functions Θ(λ) are developed and are then applied to obtain factorization formulas for Θ(λ) in terms of elementary factors. Enroute, formulas for the factors in a version of a theorem of Leech are also obtained.