zbMATH — the first resource for mathematics

Admissibility and exact observability of observation operators for semigroups. (English) Zbl 1159.47302
Authors’ abstract: A characterization of the admissibility of an observation operator for a linear semigroup system in terms of certain rational functions of the infinitesimal generator is given, extending work of Grabowski-Callier and Gao-Hu. The same functions are then used to give new necessary and sufficient conditions for admissibility and exact observability, in both infinite and finite time. In the special case of the right shift semigroup on \(L^2((0,\infty),{\mathcal K})\), where \(\mathcal K\) is a Hilbert space, this translates into necessary and sufficient conditions for boundedness of vectorial Hankel operators, including a formulation in terms of test functions. This leads finally to a characterization of operator-valued functions in the dual of trace-class valued \(H^1\) spaces in terms of BMO-type conditions.

47D06 One-parameter semigroups and linear evolution equations
93B28 Operator-theoretic methods
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))