Geometry of Müntz Spaces

Transcription

1 WDS'12 Proceedngs of Contrbuted Papers, Part I, 31 35, 212. ISBN MATFYZPRESS Geometry of Müntz Spaces P. Petráček Charles Unversty, Faculty of Mathematcs and Physcs, Prague, Czech Republc. Abstract. Ths artcle s dvded nto three sectons. In the frst one we present the defnton of Müntz spaces as well as some key Müntz-type theorems. The second secton contans three orgnal results. In the frst we prove that no Müntz space s reflexve. In the second we prove that a specal type of Müntz spaces does not have the Radon-Nkodym property. The thrd result characterzes the Choquet boundary of Müntz spaces. In the last secton we menton some of the open problems. 1. Introducton The classcal Weerstrass theorem states that the set of all fnte lnear combnatons of the functons x n ; n N (whch we denote span{1, x, x 2,... }) s dense n the space C([, 1]) endowed wth the supremum norm. Weerstrass also presented the queston whether t s possble to characterze real sequences Λ = {λ } for whch the set,... } s dense n C([, 1]). Ths problem was frst solved for the case of strctly ncreasng sequences of exponents by Herman Müntz (1914) and Otto Szász (1916) (see [Pnkus, [25]]). It s now referred to as the Müntz theorem: Theorem 1.1. Müntz theorem Let Λ = {λ } =1 be a strctly ncreasng sequence of postve real numbers. Then the set,... } s dense n the space C([, 1]) endowed wth the supremum norm f and only f the seres 1 dverges. λ =1 We wll need the followng more general theorem (see [Almra, [27]]): Theorem 1.2. Full Müntz theorem Let Λ = {λ } =1 be a sequence of dstnct postve real numbers. Then the set,... } s dense n the space C([, 1]) f and only f the seres dverges. =1 λ λ Ths Theorem allows us to defne Müntz spaces ntutvely as follows: Defnton 1.3. Müntz space For an nfnte set Λ = {λ 1, λ 2,... } of dstnct postve real numbers for whch the seres =1 λ λ

2 converges, the proper closed subspace of C([, 1]) defned as the closure of the set s called the Müntz space M Λ.,... } Pror to dscussng the geometrc propertes of Müntz spaces we should note that there are many theorems analogous to the prevous one. A verson of the Full Müntz theorem on a general nterval [a, b] exsts (see [Pnkus, [25]]), but t s nterestng to note that the case / [a, b] s substantally more dffcult to prove. The concept can be translated ntutvely to the settng of L p spaces (see [Borwen, Erdely, [1995]]) and also to the L p µ spaces, where µ stands for a non-negatve Borel measure on [, 1] (see [Chalendar et al., [21]]). 2. Geometrc propertes of Müntz spaces Müntz spaces as we have defned them represent a specal case of separable Banach spaces and treatng them as such we can study ther varous geometrc propertes. At ths pont, we feel oblgated to menton the publcaton Geometry of Müntz spaces and related questons [Gurary, Lusky, [25]] where the authors study the topc extensvely. However, they do not address any of the problems we present n ths artcle. Reflexvty of Müntz spaces One of the basc geometrc propertes of a Banach space s reflexvty. The followng theorem answers the queston of reflexvty of Müntz spaces. Theorem 2.4. Reflexvty of Müntz spaces Müntz spaces are non-reflexve. Proof. Consder a Müntz space M Λ where Λ = {λ 1, λ 2,... }. Accordng to the Full Müntz theorem(1.2) the seres λ λ converges. It follows that at least one of the followng equaltes must hold Assume frst that nf Consder the followng operator =1 nf λ =, sup λ =. λ =. Then there exsts a subsequence {λ k } of Λ such that T : f lm λ k =. k f(t)dt f(), f M Λ. It s readly verfed that T s a contnuous lnear operator on M Λ whose norm s less or equal to 2. Observng that lm T 2 k (2xλ k 1) = lm k λ k + 1 = 2 32

3 we obtan that the norm of T s n fact equal to 2. However, T cannot attan ts norm. Suppose that there exsts a functon φ M Λ, φ = 1 such that T (φ) = 2. Then φ(t)dt = φ() = 1, whch s a contradcton. The case sup λ = can be treated smlarly. We pck a subsequence {λ l } of Λ such that and consder a contnuous lnear operator lm λ l = l S : f f(1) f(t)dt, f M Λ. Its norm s equal to two, whch can be seen from the followng: ( lm l S(2xλ l 1) = lm ) = 2. l λ l + 1 Agan, S cannot attan ts norm. Suppose there exsts a functon ψ M Λ, ψ = 1 such that Then ψ must satsfy the followng: S(ψ) = 2. ψ(t)dt = ψ(1) = 1, whch s a contradcton. We have shown that for every Müntz space there exsts at least one contnuous lnear operator whch does not attan ts norm on the unt sphere. Accordng to the James theorem (see [James, [1972]]) ths mples that there s no reflexve Müntz space. Radon-Nkodym property of Müntz spaces Another queston s whether any Müntz spaces have the Radon-Nkodym property. Ths queston was one of the author s motvatons for the study of reflexvty of Müntz spaces. The followng theorem holds (see [Phllps, [194]]): Theorem 2.5. Let X be a reflexve Banach space. Then X has the Radon-Nkodym property. Accordng to the Theorem(2.4) Müntz spaces are non-reflexve. That s why we have to use other characterstcs f we wsh to study ths property of Müntz spaces. Snce t s too dffcult to usefully characterze Müntz spaces n general, we wll (for the moment) focus our attenton on a specal case of Müntz spaces, the quas-lacunary Müntz spaces. Defnton 2.6. We call a strctly ncreasng sequence {λ } of postve real numbers quas-lacunary f there exsts a strctly ncreasng subsequence of ndces { k } and q > 1 such that nf k λ k+1 λ k and sup( k+1 k ) <. k If M Λ s a Müntz space and Λ s a quas-lacunary sequence, we call M Λ a quas-lacunary Müntz space. q 33

4 The reason for studyng ths specfc type of Müntz spaces s the followng theorem (see [Gurary, Lusky, [25]]): Theorem 2.7. Each quas-lacunary Müntz space s somorphc to c. It s well known that c does not have the Radon-Nkodym property (see for example [Destel, Uhl, [1977]]). Snce ths property s preserved under somorphsms, we can conclude the followng: Theorem 2.8. Quas-lacunary Müntz spaces do not have the Radon-Nkodym property. The condton of quas-lacunarty s qute restrctve. However, t seems that n order for the Müntz spaces to have some useful propertes, at least some restrcton on the behavour of the sequence Λ s necessary. Take for example the Müntz spaces M Λ where Λ satsfes the gap condton, that s nf (λ +1 λ ) >. Every such space s somorphc to a subspace of c (see [Gurary, Lusky, [25]]). The author would be nterested n fndng out f there exsts such a space wth the Radon-Nkodym property. Müntz spaces and Choquet theory One of the man concepts of Choquet theory are the Choquet boundary and smplcalty. We wll now brefly comment on these propertes of Müntz spaces. When studyng the Choquet boundary of Müntz space the followng theorem wll be useful (see [Lukeš et al., [21]]): Theorem 2.9. Let H C([, 1]) be a functon space and let x [, 1] be H-exposed. Then x les n the Choquet boundary of H. The next theorem descrbes the Choquet boundary of a general Müntz space. Theorem 2.1. Let M Λ be a Müntz space. Then every pont of [, 1] s M Λ -exposed. Hence the Choquet boundary of M Λ s equal to [, 1]. Proof. It s enough to show the frst part of our clam that s that every pont of [, 1] s M Λ -exposed. The second part then follows from the Theorem (2.9). Pck λ 1, λ 2 two dfferent elements of Λ. We can assume wthout loss of generalty that λ 2 > λ 1. The functon x λ 1 clearly exposes the pont whle the functon 1 x λ 1 exposes the pont 1. Let t n (, 1) be arbtrary. We want to fnd a functon n M Λ that exposes t. Let φ(x) = λ 1 λ 2 t (λ 1 λ 2 ) x λ 2 + (1 x λ 1 ) (t λ 1 ( λ 1 λ 2 1) + 1). Then φ s an element of M Λ, φ(t) = and t s the pont of global mnmum of φ on [, 1]. In other words, φ exposes t. 3. Open problems Apparently there are stll open questons related to the varous propertes of Müntz spaces. We present some of them here for the nterested reader. One of the corollares of the Theorem (2.7) s that quas-lacunary Müntz spaces have a Schauder bass. However, n [Gurary, Lusky, [25]] the authors pose a queston whether there exsts a non-quas-lacunary Müntz space that does not have a Schauder bass. They menton a conjecture due to Schektman that ths mght be the case for the space M Λ for Λ = { 2 } =1. If 34

5 ths conjecture proved to be true, we would obtan an nterestng example of a separable Banach space wthout a Schauder bass. In 1982 Newman proved the exstence of a Müntz space that has no topologcal complement n C([, 1]) (see [Newman, [1984]]). He also mentoned that there s a Müntz space that has a topologcal complement n C([, 1]). A relatvely new result (see [Al Alam, [28]]) proves the exstence of a Müntz space that has no topologcal complement n L 1 ([, 1]). It s stll an open problem whether the Müntz spaces that have a topologcal complement ether n C([, 1]) or n L 1 ([, 1]) can be characterzed. References Al Alam, A., A Müntz space havng no complement n L 1, Proc. Amer. Math. Soc. 136, , 28. Almra, J. M., Müntz type theorems I, Surveys n Approxmaton Theory 3, , 27. Borwen, P., Erdély, T., Polynomals and polynomal nequaltes, Sprnger-Verlag, New York, Inc., 172, 24-25, Chalendar, I., Frcan, F., Tmotn, D., Embeddng theorems for Müntz spaces, Destel, J., Uhl, J. J., Vector measures, Amercan Mathematcal Soc., 6, Erdély, T., The full Clarkson-Erdös-Schwartz theorem on the closure of non-dense Müntz spaces, Studa Math. 155, , 23. Erdély, T., Johnson, W., The Full Müntz theorem n L p ([, 1]) for < p <, Journal d Analyse Math., , 21. Gurary, V., Lusky, W., Geometry of Müntz spaces and related questons, Sprnger-Verlag, Berln Hedelberg, 25. James, R. C., Reflexvty and the sup of lnear functonals, Israel J. Math. 13 (3 4), 289 3, Lukeš, J., Malý, J., Netuka, I., Spurný, J., Integral representaton theory, Applcatons to convexty, Banach spaces and potental theory, Walter de Gruyter GmbH, 21. Newman, D. J., A Müntz space havng no complement, Journal of Approxmaton theory 4, , Pnkus, A., Densty n Approxmaton Theory, Surveys n Approxmaton Theory 1, 25. Phllps, R. S., On lnear transformatons, Trans. Amer. Math. Soc. 48, ,