Embeddings of Müntz Spaces in $L^{\infty }(\unicode[STIX]{x1D707})$. Issue 1 (9th January 2019)

• Record Type: Journal Article
• Authors: Al Alam, Ihab; Lefèvre, Pascal
• Is Part Of: Canadian mathematical bulletin =. Volume 62:Issue 1(2019)
• Page Start: 1
• Page End: 9
• Publication Date: 2019-01-09
• Subjects: 32C22, -- 47B33, -- 30B10; Müntz space, -- embedding, -- essential norm, -- compact operator; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2018-031-8
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: In this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0, 1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1}, \unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$ .