Contractive Projections in Variable Lebesgue Spaces

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

Home Editorial board Publication ethics Peer review guidelines Current Archive Rules for authors Contacts Address: Markusa st. 22, Vladikavkaz, 362027, RNO-A, Russia Phone: (8672)50-18-06 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2017.1.5827 Contractive Projections in Variable Lebesgue Spaces Tasoev B. B. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 1. Abstract: In this article we describe the structure of positive contractive projections in variable Lebesgue spaces \(L_{p(\cdot)}\) with \(\sigma\)-finite measure and essentially bounded exponent function \(p(\cdot)\). It is shown that every positive contractive projection \(P:L_{p(\cdot)}\rightarrow L_{p(\cdot)}\) admits a matrix representation, and the restriction of \(P\) on the band, generated by a weak order unite of its image, is weighted conditional expectation operator. Simultaneously we get a description of the image \(\mathcal{R}(P)\) of the positive contractive projection \(P\). Note that if measure is finite and exponent function \(p(\cdot)\) is constant, then the existence of a weak order unit in \(\mathcal{R}(P)\) is obvious. In our case, the existence of the weak order unit in \(\mathcal{R}(P)\) is not evident and we build it in a constructive manner. The weak order unit in the image of positive contractive projection plays a key role in its representation. Keywords: conditional expectation operator, contractive projection, variable Lebesgue space, Nakano space, \(\sigma\)-finite measure. Language: Russian Download the full text For citation: Tasoev B. B. Contractive Projections in Variable Lebesgue Spaces. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.72-78. DOI 10.23671/VNC.2017.1.5827 + References 1. Kantorovich L. V., Akilov G. P. Funkcional'nyj analiz [Functional Analysis]. St. Petersburg: Nevsky Dialect; BHV-Petersburg, 2004. 812 p. [in Russian]. 2. Abramovich Y. A., Aliprantis C. D., O. Burkinshaw O. An elementary proof of Douglas theorem on contractive projections on \(L_1\) spaces. J. Math. Anal. Appl., 1993, vol. 177, pp. 641-644. 3. Abramovich Y. A., Aliprantis C. D. An Invitation to Operator Theory. Providence, R.I.: Amer. Math. Soc., 2002. (Graduate Stud. in Math., 50). 4. Ando T. Contractive projections in \(L_p\) spaces. Pacific J. Math., 1966, vol. 17, pp. 391-405. 5. Bernau S. J., Lacey E. H. The range of contractive projection on an \(L_p\) space. Pacific J. Math., 1974, vol. 53, no. 1, pp. 21-41. 6. Cruz-Uribe D. V., Fiorenza A. Variable Lebesgue Spaces. Springer, 2013. 7. Dodds P. G., Huijsmans C. B., de Pagter B. Characterizations of conditional expectation-type operators. Pacific J. Math., 1990, vol. 141, no. 1, pp. 55--77. 8. Douglas R. G. Contractive projections on an \(L_1\) space. Pacific J. Math., 1965, vol. 15, no. 2, pp. 443-462. 9. Wulbert D. E. A note on the characterization of conditional expectation operators. Pacific J. Math., 1970, vol. 34, no. 1, pp. 285-288. ← Contents of issue


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