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A Gleason–Kahane–Żelazko theorem for modules and applications to holomorphic function spaces
Author(s) -
Mashreghi Javad,
Ransford Thomas
Publication year - 2015
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdv080
Subject(s) - mathematics , hardy space , holomorphic function , pure mathematics , endomorphism , bergman space , open mapping theorem (functional analysis) , function space , zero (linguistics) , space (punctuation) , function (biology) , interpolation space , functional analysis , mathematical analysis , banach space , eberlein–šmulian theorem , lp space , linguistics , philosophy , biochemistry , chemistry , evolutionary biology , gene , bounded function , biology
We generalize the Gleason–Kahane–Żelazko theorem to modules. As an application, we show that every linear functional on a Hardy space that is non‐zero on the outer functions is a multiple of a point evaluation. A further consequence is that every linear endomorphism of a Hardy space that maps outer functions to nowhere‐zero functions is a weighted composition operator. In neither case is continuity assumed. We also consider some extensions to other function spaces, including the Bergman, Dirichlet and Besov spaces, the little Bloch space and VMOA .