Admissible observation operators for the right-shift semigroup. (English) Zbl 0966.93033

Partington, Jonathan R.; Weiss, George

Math. Control Signals Syst. 13, No. 3, 179-192 (2000).

The main result in this paper is that if \(A\) is a generator of the right-shift semigroup on \(L^2[0,\infty]\) and \(C\) is an observation operator, mapping \(D(A)\) into the complex numbers, then \(C\) is infinite-time admissible if and only if \(\|C(sI- A)^{-1}\|\leq m/\sqrt{\text{Re }s}\) for all \(s\) in the open right half-plane. The authors proved this theorem by using Fefferman’s theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For a normal semigroup the conjecture is known to be true and then it is equivalent to the Carleson measure theorem.

Reviewer: J.Y.Park (Pusan)

MSC:

93B25	Algebraic methods
93B07	Observability
93C25	Control/observation systems in abstract spaces

Keywords:

admissibility; right-shift semigroup; observation operator; bounded mean oscillation; Hankel operators