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Embeddings of Müntz Spaces in $L^{\infty }(\unicode[STIX]{x1D707})$

Part of: Series expansions Special classes of linear operators Analytic spaces

Published online by Cambridge University Press:  09 January 2019

Ihab Al Alam  and
Pascal Lefèvre
Ihab Al Alam
Affiliation:
Lebanese University, Faculty of Sciences II, Department of Mathematics, Fanar-Matn 90656, Lebanon Email: [email protected]
Pascal Lefèvre
Affiliation:
Laboratoire de Mathématiques de Lens, Université Artois, 62307 Lens, France Email: [email protected]
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Abstract

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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 $\unicode[STIX]{x1D707}_{1}$.

Keywords

Müntz spaceembeddingessential normcompact operator

MSC classification

Primary: 32C22: Embedding of analytic spaces
Secondary: 47B33: Composition operators 30B10: Power series (including lacunary series)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 1 - 9
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work is part of the project CEDRE ESFO. The authors would like to thank the program PHC CEDRE and the Lebanese University for their support.

References

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