Rational Cayley inner Herglotz–Agler functions: Positive-kernel decompositions and transfer-function realizations

Abstract

The Bessmertnyĭ class consists of rational matrix-valued functions of d complex variables representable as the Schur complement of a block of a linear pencil A(z)=z1A1+⋯+zdAd whose coefficients Ak are positive semidefinite matrices. We show that it coincides with the subclass of rational functions in the Herglotz–Agler class over the right poly-halfplane which are homogeneous of degree one and which are Cayley inner. The latter means that such a function is holomorphic on the right poly-halfplane and takes skew-Hermitian matrix values on (iR)d, or equivalently, is the double Cayley transform (over the variables and over the matrix values) of an inner function on the unit polydisk. Using Agler–Knese's characterization of rational inner Schur–Agler functions on the polydisk, extended now to the matrix-valued case, and applying appropriate Cayley transformations, we obtain characterizations of matrix-valued rational Cayley inner Herglotz–Agler functions both in the setting of the polydisk and of the right poly-halfplane, in terms of transfer-function realizations and in terms of positive-kernel decompositions. In particular, we extend Bessmertnyĭ's representation to rational Cayley inner Herglotz–Agler functions on the right poly-halfplane, where a linear pencil A(z) is now in the form A(z)=A0+z1A1+⋯+zdAd with A0 skew-Hermitian and the other coefficients Ak positive semidefinite matrices.