The weighted Weiss conjecture for admissible observation operators B. Jacob 1 , E. Rydhe 2 , and A. Wynn 3 Abstract— The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterised by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space H 2 (D) (discrete time) or the right-shift semigroup on L 2 (R+) (continuous time). To contrast and complement these counterexamples, in this talk positive results are presented characterising weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis. I. INTRODUCTION Consider an infinite dimensional control system ˙ x(t)= Ax(t), y(t)= Cx(t), t ≥ 0, x(0) = x 0 ∈ X where A is the generator of a C 0 -semigroup (T (t)) t≥0 on a Hilbert space X and the observation operator satisfies C ∈ L(D(A), C). For the system to be well-posed, in the sense of [9], a necessary condition is that C is admissible for A, that is, there exists k> 0 such that ‖CT (·)x 0 ‖ L 2 (R+) ≤ k‖x 0 ‖ X , x 0 ∈ D(A). An important consequence of admissibility is that the output y can be well defined even in the case that C is un- bounded. In particular, admissibility implies that the map x 0 → CT (·)x 0 ∈ L 2 (R + ), defined initially on D(A), has a continuous extension to the whole space X, meaning that the output is well defined for any initial condition x 0 ∈ X. A generalization of admissibility, first considered in [1], is to require that the output is an element of a weighted L 2 - space. For β> −1, C is said to be β-admissible for A if there exists a constant k> 0 such that  ∞ 0 t β |CT (t)x 0 | 2 dt ≤ k 2 ‖x 0 ‖ 2 , x 0 ∈ D(A). (1) To test whether a given system is β-admissible, a frequency- domain characterization is convenient and, to this end, it is not difficult to show that β-admissibility implies the resolvent growth condition sup λ∈C+ (Reλ) 1+β 2 ‖CR(λ, A) -(1+β) ‖ X * < ∞, (2) 1 B. Jacob is with the Fachbereich C - Mathematik und Naturwis- senschaften, Bergische Universit¨ at Wuppertal, Gaußstraße 20, 42119 Wup- pertal, Germany bjacob@uni-wuppertal.de 2 E. Rydhe is with the Matematikcentrum Lunds Universitet, 22100 Lund, Sweden eskil.rydhe@math.lu.se 2 A. Wynn is with the Department of Aeronautics, Impe- rial College London, London, SW7 2AZ, United Kingdom a.wynn@imperial.ac.uk where R(λ, A) := (λI − A) -1 denotes the resolvent of the semigroup generator A, and C + := {λ ∈ C : Reλ> 0} is the right-half plane. The question of whether the converse statement (2) ⇒ (1) holds, commonly referred to as a (weighted) Weiss conjecture, is much more subtle. Existing results concerning the conjecture are discussed below, but we first describe a discrete time version of the Weiss conjecture, introduced in [2], which will also be studied in this talk. A discrete-time linear control system on a Hilbert space X has the form x n+1 = Tx n , y n = Cx n , x 0 ∈ X, n ∈ N, where T ∈L(X) and C ∈ X * . In this case, for β> −1, the observation functional C is said to be (discrete) β-admissible for T if there exists k> 0 such that ∞  n=0 (1 + n) β |CT n x| 2 ≤ k 2 ‖x‖ 2 X , x ∈ X. (3) Analogous to continuous time systems, the resolvent condi- tion sup ω∈D (1 −|ω| 2 ) 1+β 2 ‖C (I − ¯ ωT ) -(1+β) ‖ X * < ∞ (4) is necessary for (3) and the discrete time form of the weighted Weiss conjecture is to ask when the converse implication is true. The Weiss conjecture is superficially easier to study in discrete time due to the boundedness of the operators involved. However, it should be noted that it is sometimes possible to translate positive and negative results concerning the conjecture via the Cayley transform [2], [10]. The continuous time conjecture (2) ⇒ (1) was originally posed [7] in the unweighted case β =0. In this situation, the conjecture is true if A generates a C 0 -semigroup of contractions [3], which extends the results that the conjecture holds if A is normal [8] and if A is the generator of the right- shift semigroup on L 2 (R + ) [5]. The discrete time version (4) ⇒ (3) for β =0 and T a contraction was shown in [2]. For β = 0, the behaviour of the conjecture is more complicated. In the case that A is normal, the continuous time conjecture (2) ⇒ (1) is true [12] for positive weight exponents β ∈ (0, 1), but false [11] in the case that β ∈ (−1, 0). Analogous results also hold for the discrete time conjecture conjecture when T is normal [11], [12]. Fur- thermore, both continuous and discrete time conjectures are not true for general contraction operators when β ∈ (0, 1): in continuous time, the right-shift semigroup on L 2 (R + ) provides the counterexample [10]; while in discrete time (4) ⇒ (3) fails if T is the unilateral shift on the Hardy space H 2 (D) [11]. 21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. Groningen, The Netherlands ISBN: 978-90-367-6321-9 1170