Essential norm of Cesàro operators on \(L^p\) and Cesàro spaces. (English) Zbl 1445.47026
Summary: In this paper, we consider the Cesàro-mean operator \(\Gamma\) acting on some Banach spaces of measurable functions on \((0, 1)\), as well as its discrete version on some sequences spaces. We compute the essential norm of this operator on \(L^p([0, 1])\) for \(p \in(1, + \infty]\) and show that its value is the same as its norm: \(p /(p - 1)\). The result also holds in the discrete case. On Cesàro spaces, the essential norm of \(\Gamma\) turns out to be 1. At last, we introduce the Müntz-Cesàro spaces and study some of their geometrical properties. In this framework, we also compute the value of the essential norm of the Cesàro operator and the multiplication operator restricted to those Müntz-Cesàro spaces.
MSC:
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
Cesàro spaces; Cesàro operator; Müntz spaces; compact operator; essential norm; multiplication operatorReferences:
| [1] | Al Alam, I., A Müntz space having no complement in \(L_1\), Proc. Amer. Math. Soc., 136, 1, 193-201, (2008) · Zbl 1171.41003 |
| [2] | Albanese, A.; Bonet, J.; Ricker, W., The Cesàro operator in the Fréchet spaces \(l^{p +}\) and \(L^{p -}\), Glasg. Math. J., 59, 273-287, (2017) · Zbl 06713810 |
| [3] | Al Alam, I.; Habib, G.; Lefèvre, P.; Maalouf, F., Essential norms of Volterra and Cesàro operators on Müntz spaces, Colloq. Math., 151, 157-169, (2018) · Zbl 1502.47049 |
| [4] | Al Alam, I.; Lefèvre, P., Essential norms of weighted composition operators on \(L^1\)-Müntz spaces, Serdica Math. J., 151, 157-169, (2018) · Zbl 1502.47049 |
| [5] | Aghajani, A.; Raj, K.; Mursaleen, M., Multiplication operators on Cesàro function spaces, Fac. Sci. Math., Univ. Niš, Serbia, 1175-1184, (2016) · Zbl 1465.47024 |
| [6] | Astashkin, S.; Maligranda, L., Structure of Cesàro function spaces, Indag. Math. (N.S.), 20, 3, 329-379, (2009) · Zbl 1200.46027 |
| [7] | Borwein, P.; Erdélyi, T., Polynomials and polynomial inequalities, (1995), Springer Berlin, Heidelberg, New York · Zbl 0840.26002 |
| [8] | Clarkson, A.; Erdös, P., Approximation by polynomials, Duke Math. J., 10, 5-11, (1943) · Zbl 0063.00919 |
| [9] | Chalendar, I.; Fricain, E.; Timotin, D., Embedding theorems for Müntz spaces, Ann. Inst. Fourier, 61, 6, 2291-2311, (2011) · Zbl 1255.46013 |
| [10] | Curbera, G.; Ricker, W., Abstract Cesàro spaces: integral representations, J. Math. Anal. Appl., 441, 25-44, (2016) · Zbl 1355.46031 |
| [11] | Gaillard, L.; Lefèvre, P., Lacunary Müntz spaces: isomorphisms and Carleson embeddings, Ann. Inst. Fourier (Grenoble), (2018), in press · Zbl 1412.30005 |
| [12] | L. Gaillard, P. Lefèvre, Asymptotic isometries for lacunary Müntz spaces and applications (2018), submitted for publication.; L. Gaillard, P. Lefèvre, Asymptotic isometries for lacunary Müntz spaces and applications (2018), submitted for publication. |
| [13] | Gurariy, V. I.; Lusky, W., Geometry of Müntz spaces and related questions, Lecture Notes in Mathematics, (2005), Springer Berlin, Heidelberg, New York · Zbl 1094.46003 |
| [14] | Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, (1934), Cambridge University Press London · JFM 60.0169.01 |
| [15] | Lefèvre, P., The Volterra operator is finitely strictly singular from \(L^1\) to \(L^\infty\), J. Approx. Theory, 214, 1-8, (2017) · Zbl 1369.47042 |
| [16] | Müntz, Ch. H., Über den approximationssatz von Weierstrass, 303-312, (1914), H.A. Schwarz’s Festschrift Berlin · JFM 45.0633.02 |
| [17] | Schwartz, L., Etude des sommes d’exponentielles, (1959), Hermann Paris · Zbl 0092.06302 |
| [18] | Wojtaszczyk, P., Banach spaces for analysts, (1991), Cambridge University Press · Zbl 0724.46012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
