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Homogeneous Herglotz class versus homogeneous Herglotz-Agler class Ball, Joseph


The Herglotz class is defined as the class of holomorphic functions mapping the $d$-fold Cartesian product of the right half-plane with itself into the right half-plane. The Herglotz-Agler class is defined as the space of such holomorphic functions such that (under any reasonable functional calculus) maps a $d$-tuple of commuting accretive operators to an accretive operator (accretive operator here meaning having positive-definite real part). Integral representation formulas and transfer-function realization formulas for the Herglotz class are rather involved as soon as $d > 2$ while such formulas are more like the single-variable case for the Herglotz-Agler class. A routine observation is that the Herglotz-Agler class is a sub-class of the Herglotz class with equality for $d=1,2$ and strict inclusion for $d>2$. Also of interest are the homogeneous Herglotz class and the homogeneous Herglotz-Agler class where here homogeneous means "homogeneous of degree 1": $f(\lambda z_1, \dots, \lambda z_d) = \lambda f(z_1, \dots, z_d)$ for all complex scalars $\lambda$. This class has motivation from electrical circuit theory and has been studied at length by V. Bessmertnyi and D. Kaliuzhnyi-Verbovetskyi. A parallel picture exists for the homogeneous Herglotz and homogeneous Herglotz-Agler classes: there is a better structural understanding of the homogeneous Herglotz-Agler class than for the general homogeneous Herglotz class, and it is an easy observation that the homogeneous Herglotz-Agler class is a subclass of the homogeneous Herglotz class. However there remains an open question: for what values of $d$ (the number of variables) is it the case that the homogeneous Herglotz class and the homogeneous Herglotz-Agler class are actually the same We make some observations on this question and discuss a possible approach to resolving this question.

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