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A new approach to linear stochastic systems
Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We present stability theorems for these systems. We also discuss state space theory. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior. The talk is based on joint works with David Levanony and Ariel Pinhas.
Michael A. Dritschel
Realization techniques for obtaining a bounded functional calculus for certain sectorial operators
We sketch a way of using the realization theorem to get a bounded functional for Hilbert space operators whose imaginary powers are bounded when the powers lie in an interval of the real line, so giving a result along the lines of McIntosh's theorem for operators with bounded imaginary powers. Included are a Herglotz-type representation and transfer function representation. We also indicate how these results might be extended to the multivariable case.
Factorizations of nonnegative operators and their extremal nonnegative extensions
The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle
Fixed points of holomorphic mappings
In this joint work with Simeon Reich and David Shoikhet, we study the structure of the fixed point set of a holomorphic mapping defined on a (not necessarily bounded) domain in a complex Banach space, by means of ergodic theory and a nonlinear numerical range.