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

摘要

The Bessmertny\u{\i} class consists of rational matrix-valued functions of$d$ complex variables representable as the Schur complement of a block of alinear pencil $A(z)=z_1A_1+\cdots+z_dA_d$ whose coefficients $A_k$ are positivesemidefinite matrices. We show that it coincides with the subclass of rationalfunctions in the Herglotz-Agler class over the right poly-halfplane which arehomogeneous of degree one and which are Cayley inner. The latter means thatsuch a function is holomorphic on the right poly-halfplane and takesskew-Hermitian matrix values on $(i\mathbb{R})^d$, or equivalently, is thedouble Cayley transform (over the variables and over the matrix values) of aninner function on the unit polydisk. Using Agler-Knese's characterization of rational inner Schur-Agler functionson the polydisk, extended now to the matrix-valued case, and applyingappropriate Cayley transformations, we obtain characterizations ofmatrix-valued rational Cayley inner Herglotz-Agler functions both in thesetting of the polydisk and of the right poly-halfplane, in terms oftransfer-function realizations and in terms of positive-kernel decompositions.In particular, we extend Bessmertny\u{\i}'s representation to rational Cayleyinner Herglotz-Agler functions on the right poly-halfplane, where a linearpencil $A(z)$ 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.