Rational Cayley inner Herglotz-Agler functions: positive-kernel decompositions and transfer-function realizations
Rational Cayley inner Herglotz-Agler functions: positive-kernel decompositions and transfer-function realizations
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Rational Cayley 内 Herglotz-Agler 函数：正核分解和传递函数实现
DOI：
10.1016/j.laa.2013.10.022
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Dmitry S. Kaliuzhnyi
中科院分区：
文献类型：
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作者：
J. Ball;Dmitry S. Kaliuzhnyi
Bessmertnyĭ 类由 d 个复变量的有理矩阵值函数组成，可表示为线性铅笔 A (z)= z 1 A 1+⋯+ z d A d 的 Schur 补，其系数 A k 是正半定矩阵。我们证明它与右多半平面上 Herglotz-Agler 类中的有理函数子类一致，该子类是一阶齐次且是凯莱内函数。后者意味着这样的函数在右多半平面上是全纯的，并且在 (i R) d 上采用斜埃尔米特矩阵值，或者等效地，是单位多圆盘上内部函数的双凯莱变换（在变量上和在矩阵值上）。使用 Agler-Knese 对多圆盘上有理内 Schur-Agler 函数的表征，现在扩展到矩阵值情况，并应用适当的凯莱变换，我们在多圆盘和右多半平面的设置下，在传递函数实现和正核分解方面获得了矩阵定值有理 Cayley 内 Herglotz-Agler 函数的表征。特别是，我们将 Bessmertnyĭ 的表示扩展到右聚半平面上的有理 Cayley 内 Herglotz-Agler 函数，其中线性铅笔 A (z) 现在的形式为 A (z)= A 0+ z 1 A 1+⋯+ z d A d，其中 A 0 偏厄米特矩阵和其他系数 A k 正半定矩阵。
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)= z 1 A 1+⋯+ z d A 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 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 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)= A 0+ z 1 A 1+⋯+ z d A d with A 0 skew-Hermitian and the other coefficients A k positive semidefinite matrices.