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- Title
Perturbation of linear operators on Banach spaces: some applications to Schauder bases and frames.
- Authors
Navascués, M. A.; Viswanathan, P.
- Abstract
In earlier papers we introduced and studied the notion of fractal operator that maps a continuous scalar valued function on a compact interval in R to its fractal (self-referential) analogue. Further, it has been observed that by perturbing known Schauder bases via the bijective bicontinuous fractal operator, Schauder bases consisting of self-referential functions can be constructed for standard function spaces. Motivated by the theory and applications of the fractal operator, in this note we propose a new kind of perturbation of linear operators. We consider a pair of linear operators L and T on a Banach space X such that ‖ T x - x ‖ ≤ k ‖ L x - x ‖ for all x ∈ X and some k > 0 . We establish that the property of being bounded below and that of having a closed range are stable under the perturbation considered herein. As an immediate application, it is deduced that a given Schauder basis in X can be transformed to Schauder sequences. Perturbation of linear operators via an equation of the form ‖ T x - x ‖ = k ‖ T x - L x ‖ is also alluded and its application in constructing new Schauder bases from a given one is considered.
- Subjects
SCHAUDER bases; BANACH spaces; FUNCTION spaces; OPERATOR equations; SET-valued maps; LINEAR operators
- Publication
Aequationes Mathematicae, 2020, Vol 94, Issue 1, p13
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-019-00681-6