Perturbation of linear operators on Banach spaces: some applications to Schauder bases and frames

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 ${\mathbb {R}}$ 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

$$\begin{aligned} \Vert Tx-x\Vert \le k \Vert Lx-x\Vert \end{aligned}$$

for all $x \in X$ 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

$$\begin{aligned} \Vert Tx-x\Vert = k \Vert Tx-Lx\Vert \end{aligned}$$

is also alluded and its application in constructing new Schauder bases from a given one is considered.