Pelczyński, A.

Institute of Mathematics AS CR, v. v. i.

WSAA 7A. PełczyńskiCommutative harmonic analysis and Banach spacesIn: Zdeněk Frolík (ed.): Abstracta. 7th Winter School on Abstract Analysis.Czechoslovak Academy of Sciences, Praha, 1979. pp. 48--52.Persistent URL: http://dml.cz/dmlcz/701147Terms of use:© Institute of Mathematics of the Academy of Sciences of the Czech Republic,1979Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copy of anypart of this document must contain these Terms of use.This paper has been digitized, optimized for electronic deliveryand stamped with digital signature within the project DML-CZ:The Czech Digital Mathematics Library http://project.dml.cz48 SEVENTH WINTER SCHOOL (1979) COMMUTATIVE HARMONIC ANALYSIS AND BANACH SPACES A. Pelczynski Preliminaries* Let G be a compact abelian group, P its dual, ra-the normalized Haar measure on G • The symbols C(G), LP(G) (0<p<oo) denote as usual the spaces of the continuous scalar valued functions o.. G , respectively of m-equivalence classes of measurable p-absolutely integrable functions on G • For a £ G , r denotes the operator of translation by a acting on functions on G by the ru*o VQf(x) a f(x-a) . A linear space X of (equivalence classes of) functions is called translation invariant if ra(X)C X for all a E G . A linear operator acting between translation invariant spaces is translation invariant if it commutes v/ith all 'T a A translation invariant Banach space * is regular if (a) X consists of equivalence classes of absolutely integra -i le functions on G ; the inclusion XC->L (G) is a one o one continuous operator; (b) t : X—»X is an isometry for all o(EG • (c) given f €F X the map a—*Taf iQ a co tinuous fu etion from G into X . The elements of P are called characters, A trigonometric polynomial i3 a finite linear combination of the characters, "Measure" means here a complex valued Borel treasure on G whose total variation is bounded. For fEL (G) , resp* for a measure A A IUV the Fourier transforms arc the functions f , resp, M/ on I A , _ A , __ / r defined by f(i^) n I f jr dm , resp. /u,( if) = f^f^ ^or For a A C T , C A denotes the closed linear subspace of C(G) generated by A . 49 Main results presented in the lectures Lecture I Theorem 1.1« Let A C P # Then , 1° A is a Cohen set (i.e. there is a measure whose Fou-rier transform is the characteristic function of A ) iff C^ is an cC space in the sense of Lindenstrauss and Pelczynski [LP]. 2° A is a Sidon set (i.e. there exists a k> 0 such that for every trigonometric polynomial f «-»y ' , c^ w , r € A " I IMIOO" X L I cjr-| k ) i f f 9A i s a n * * i s P a c e i n t n e ©anse of [LP] • Part 2° is due to Varopoulos [vj . The proof presented in the Lecture bases on the following (cf. |_KPj). Proposition 1.2. Let C^ be such that every finite dimensional operator from the dual space of Cŷ  into C ^ factors through a Hilbert space. Then A is a Sidon set. Corollary 1.3. (cf. [ K P ] and [pi]). A C P is a Sidon ot iff C ^ i3 a Banach space of cotype 2 (cf.e.g. [_Mj for the defini tion of the cotype). Lecture II Theorem 2.1. Every regular translation invariant Banach pace X has the invariant uniform approximation property; precisely for every £ > 0 thero is a function m—*q (m) such that gi-ven a finite dimensional translation invariant subspace E o X there exists a translation invariant operator uE such that (1) uE(e) « e for e £ E , (2) || u£ || < 1 + £ , (3) dim uE(X)<q (dim E) . Theorem 2.1 follows immediately (In fact i3 equivalent to) from the next one 50 Theorem 2.2. For every E > 0 there is a function ra —-> q (m) such that given a finite set M C P there is a trigonometric polynomial g£ such that (i) g(^) - * for |-GM . (ii) ||gE|| X<1 +<E . (iii) |s(g)| < qG(|M|) . Here S(g) « { j ^ E P : g ( r ) / O J and |A| denotes the number of elements of a finite set A . Theorem 2.1 and 2.2 are taken from the paper by M. Bozej-ko and A. Pelczynski CBP.] • Lecture III Definition 3.1. A set 7V_ C V is a Marcinkiewicz set if the 2 2 orthogonal projection P^ : L (G)—->L (G) , defined by P^ f a 1 7 A a 2. , f(jh)jT • regarded as the operator on trigonometric re A d 6 </ polynomials is (l.p) bounded for some (equivalently for all) p with 0 < p < l , i.e. there is a k> 0 such that ( J | P^ (f )| pdm) p -£ k J | f |dm (f-trigonometric polynomial) G G Recall that an operator u : X—>Y (XfY-Banach spaces) is said to be p-absolutely summing (0<p<oo) if there exists a constant C>0 such that for every f in i te set FCX Z L | | u x | | p < C s u p ^ L , | x * ( x ) | p x E F 1 1 " x G F 1 where the supremum is taken over all x* in the unit ball of the dual of X . Theorem 3.2. [ K P ] . If V\_ is a Marcinkiewicz set then every p translation invariant operator u : L (G) —* C^ has the one--absolutely summing adjoint. Theorem 3.2 can be regarded as a generalization for trans-lation invariant operators of Grothendieck's -Fundamental Theo-51 rem in Metric Theory of Tensor Products" (cf. [G] , [ L P ] ) . The proof presented in the lecture bases upon the following fact essentially proved in [_KPJ • Theorem 3.3. Let A be a Marcinkiewicz set, 0 < p < l , X-a regular translation invariant Banach space. Then every p-absolu-tely summing translation invariant operator u : C^—-> X is integral; precisely there exists a bounded linear operator v : L (G)—>X such that the diagram C(G) i-*L1(G) 4 V c — y — * x is commutative, where j is the natural isometric embedding and i is the natural injection which assigns to each f in C(G) its m-equivalence class in L (G) . References [BPJ M. Bozejko and A. Pelczynski: An analogue in commutative Harmonic Analysis of the uniform bounded approxima-tion property of Banach spaces. Séminaire d'Analyse Fonctionelle 1978-9, Exposé No. IX, Ecole Polytech-nique, Palaiseau 1979 [GJ A. Grothendieck: Résumé de la théorie métrique des pro-ducts tensoricls topologiquos, Bol.Soc.Matem., 8 (1956), 1-79 [KPJ S. Kwapien and A. Pelczynski: Absolutely summing operators and translation invariant spaces of functions on compact abelian groups, Math.Nachr., to appear; Inst, of Math.PolishtAcad.Sci. Preprint, Warsaw 1978 [_LPj 0. Lindenstrauss and A. Pelczynski: Absolutely summing operators in c/ spaces and their applications, Studia Math., p 29 (1968), 275-326 [MJ B. Maurey: Théorèmes de factorisations pour les opéra-teurs linéaires à valeurs dans les espaces Lp , Astérisque, 11 (1974) [PiJ G. Pisier: Séminaire d'Analyse Fonctionelle 1977-8, Ecole Polytechnique, Palaiseau 1978 52 [_VJ N.T. Varopoulos: Sous espaces de C(G) invariant par translation et de type c\± , Séminaire Maurey--Schwartz 1975/6, Expose 12, Ecole Polytechni-que, Palaiseau 1976 Institute of Mathematics Polish Academy of Sciences, Sniadeckich 8, I p. , 00950 Warszavva, Poland 

English

Commutative harmonic analysis and Banach spaces

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/701147/WSAA_07-1979-1_15.pdf

Commutative harmonic analysis and Banach spaces

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library