Contractive Projections in Variable Lebesgue Spaces

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 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. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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