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

The Bessmertny\u{\i} class consists of rational matrix-valued functions of $d$ complex variables representable as the Schur complement of a block of a linear pencil $A(z)=z_1A_1+\cdots+z_dA_d$ whose coefficients $A_k$ 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 $(i\mathbb{R})^d$, 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 Bessmertny\u{\i}'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 $A(z)=A_0+z_1A_1+\cdots +z_dA_d$ with $A_0$ skew-Hermitian and the other coefficients $A_k$ positive semidefinite matrices.