# Introduction to the Theory of Banach Representations of by Yurii I. Lyubich

By Yurii I. Lyubich

The thought of team representations performs an immense roie in sleek arithmetic and its applica~ions to normal sciences. within the obligatory college curriculum it's incorporated as a department of algebra, facing representations of finite teams (see, for instance, the textbook of A. I. Kostrikin [25]). The illustration conception for compact, in the neighborhood compact Abelian, and Lie teams is co vered in graduate classes, focused round sensible research. the writer of the current boo~ has lectured for a few years on practical research at Khar'kov collage. He therefore con tinued those lectures within the kind of a graduate direction at the idea of crew representations, during which detailed realization was once dedicated to a retrospective exposition of operator thought and harmo nic research of capabilities from the viewpoint of illustration idea. during this procedure it was once average to think about not just uni tary, but additionally Banach representations, and never simply representations of teams, but in addition of semigroups.

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**Example text**

Consider the Banach space A of all scalar functions analytic in the disk IAI < 1 and continuous in the closed disk I A I ~ 1, endowed with the norm II ¢II = max I ¢ (A) I . IAI=l Let A denote the operator of multiplication by A acting in A. The spectrum of A coincides with the closed disk ID = {AI IAI~ l}. INVARIANT SUBSPACES Sec. 4 Q c ID Let be a nontrivial spectral compact set and let the corresponding spectral subspace. , L 0, € L = O.

Raikov, 1940 ; M. G. Krein, 1949). All Banach algebras considered in this section are tacitly assumed to be commutative. Concerning the norm we assume that IIxyllo;;; IlxllllYIl and lie II = 1, where e is the unit of the algebra this can be always achieved by replacing the given norm with an equivalent norm. Let A be a Banach algebra. The 6pec~4um spec x of the element x E A is, by definition, the spectrum of the operator given by Rx Y = xy. ~y (or the 4e¢otven~ ¢e~) reg x of x is defined in similar manner.

Dunford ([13] theory of such operators). is devoted to the 32 ELEMENTS OF SPECTRAL THEORY The operator T € L(B) is called an Chap. 1 ope~ato~ w~th 4epa~able if the family of its spectral compact sets is a basis for ~pect~um the topology of spec T, meaning that every set open in is a union of interiors of spectral compact sets. spec T For this to happen it suffices that every compact set which is the closure of its interior be a spectral compact set. original terminology of the author. [Here we followed the The closely related notion used in the western literature is that of a see, for example, [8] and decompo~able ope~ato~ ; transl.