In this paper a definition is given for an unbounded Toeplitz-like operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.2010 Mathematics Subject Classification. Primary 47B35, 47A53; Secondary 47A68.

The operator-valued Schur-class is defined to be the set of holomorphic functions S mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by S defines a contraction operator between two Hardy Hilbert spaces, S satisfies a von Neumann inequality, a certain operator-valued kernel associated with S is positive-definite, and S can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently, one of the most encompassing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence E σ associated with a W *correspondence E over a W * -algebra A together with a * -representation σ of A. The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences associated with a completely positive analogue of the classical Szegö kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems.1991 Mathematics Subject Classification. 47A57.

No abstract

For the strictly positive case (the suboptimal case) the maximum entropy solution X to the Leech problem G(z)X(z) = K(z) and X ∞ = sup |z|≤1 X(z) ≤ 1, with G and K stable rational matrix functions, is proved to be a stable rational matrix function. An explicit state space realization for X is given, and X ∞ turns out to be strictly less than one. The matrices involved in this realization are computed from the matrices appearing in a state space realization of the data functions G and K. A formula for the entropy of X is also given.1991 Mathematics Subject Classification. Primary 47A57; Secondary 47A68, 93B15, 47A56.

No abstract