On Uncountable Frames and Riesz Bases in Nonseparable Banach Spaces
Journal: Sahand Communications in Mathematical Analysis (Vol.19, No. 2)Publication Date: 2022-06-15
Authors : Migdad Ismailov;
Page : 149-170
Keywords : Nonseparable Banach space; Uncountable unconditional basis; $K$-Bessel and $K$-Hilbert systems; $K$-frames; uncountable unconditional $K$-Riesz bases;
Abstract
Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space $K$ of systems of scalars are considered in this work. The concepts of uncountable $K$-Bessel, $K$-Hilbert systems, $K$-frames and $K^{*} $-Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable $K$-Besselianness, $K$-Hilbertianness for systems, $K$-frames and unconditional $K^{*} $-Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of $K$-Besselian systems which are not minimal are given. It is proved that every $K$-Hilbertian systems is minimal. The case where $K$ is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.
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