Disk algebra

Abstract: Robust approximation and identification of stable shiftinvariant systems is studied in the H∞ sense using a stable perturbation set-up. Issues of model set selection tion are addressed using the n-width concept: a concrete result establishes a priori knowledge for which a certain rational model set is optimal in the n-width sense. A general construction of interest to identification theory using ϵ-nets provides near-optimal identification methods tuned to the a priori knowledge about the system. A notion of robust convergence is defined so that any untuned identification method satisfying it has a generic well-posedness property for systems in the disk algebra. The existence of robustly convergent identification methods based on any complete model set in the disk algebra is established. It is also shown that the classical Fejer and de la Vallee-Poussin polynomial approximation operators provide robustly convergent identification methods. Furthermore, a result is given for optimal Hankel norm model reduction from experimentally obtained models.

Abstract: Let X and Y be two Banach spaces. We show that a bounded bilinear form m: X x Y -C is Arens regular if it is weakly compact. This result permits us to find very short proofs of some known results as well as some new results. Some of them are: Any C*-algebra, the disk algebra and the Hardy class H?? are Arens regular under every possible product. We also characterize the Arens regularity of certain bilinear mappings. Introduction. The theory of Arens multiplications is usually taken to be about Banach algebras [4, 6, 9, 21]. In fact Arens himself presented his theory principally in terms of bilinear mappings [2], though logically the two are equivalent (see ?2 below). Bilinear mappings can be naturally identified with certain linear mappings, and in this paper we show that, by shifting the emphasis to these linear maps, the proofs of most of the main results about the regularity of Arens extensions can be made almost trivial-or perhaps we should say that the real work can be delegated to standard theorems of analysis-and many new results can be obtained. In particular, the theory of weakly compact operators provides new insights, for example the regularity of a Banach algebra is sometimes a property of the geometry of the underlying Banach space rather than of the particular multiplication. This work is organized as follows. In ?1 we have collected basic definitions, results and notations we will need. In ?2 we prove that a bounded bilinear form defined on the product of two Banach spaces X and Y is Arens regular iff it is weakly compact iff it factors through a reflexive Banach space. This theorem combined with the well-known theorems of analysis, produces, as corollaries, most of the known results about Arens regularity as well as the following results which seem to be new: (i) under every possible product, any C*-algebra, the disk algebra A(D) and the Hardy class H? are Arens regular. These results extend a result, and answer a question, of [12]. (ii) A bounded bilinear form m: X x Y -. C is Arens regular iff its higher Arens extensions m3n are Arens regular. In ?3 we characterize the Arens regularity of certain bilinear mappings one encounters very often. In particular, we show that, A being a Banach algebra with a unit element, the bilinear mapping m: (x, f) e A x A' -x f e A' is Arens regular if A is reflexive. The author would like to express his thanks to Professor J. S. Pym for his encouragement and is grateful to him for valuable conversations. Received by the editors December 3, 1985 and, in revised form, September 12, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46H05; Secondary 46H99.