Schauder Bases and Operator Theory
Yang Cao, Geng Tian, Bingzhe Hou
TL;DR
This paper introduces a matrix approach to bases in separable Hilbert spaces, corrects a mistake in Olevskii's work, and demonstrates that diagonal compact operators can transform orthonormal bases into conditional bases.
Contribution
It provides a new matrix-based perspective on Hilbert space bases, corrects existing literature errors, and reveals novel properties of diagonal compact operators.
Findings
Matrix approach to Hilbert space bases
Correction of Olevskii's paper mistake
Diagonal compact operators can produce conditional bases
Abstract
In this paper, we firstly give a matrix approach to the bases of a separable Hilbert space and then correct a mistake appearing in both review and the English translation of the Olevskii's paper. After this, we show that even a diagonal compact operator may map an orthonormal basis into a conditional basis.
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Taxonomy
TopicsHolomorphic and Operator Theory · Advanced Operator Algebra Research · Matrix Theory and Algorithms