Bases of Banach and Hilbert Spaces Transformed by Linear Operators

Abstract

The construction of bases comprising of elements with special structures in various Banach and Hilbert spaces is a major enterprise in many applications. The current article is targeted to study what we call perturbed bases and frames for Banach and Hilbert spaces. To this end, we consider a pair of bounded linear operators related via an inequality of a special form. Our results are in spirit close to a family of bounded linear operators known in fractal approximation theory. Hence, the proposed theory exhibits, in particular, bases and frames consisting of fractal (self-referential) functions for some standard spaces of functions. In the last part of the article we study the structure of the space of (strictly) relatively bounded operators.