On Approximate Operator Representations of Sequences in Banach Spaces

摘要

Generalizing results by Halperin et al., Grivaux recently showed that any linearly independent sequence {f(k)}(k=1)(infinity) in a separable Banach space X can be represented as a suborbit {T-alpha(k)phi}(k=1)(infinity) of some bounded operator T : X -> X. In general, the operator T and the powers alpha(k) are not known explicitly. In this paper we consider approximate representations {f(k)}(k=1)(infinity) approximate to {T-alpha(k)phi}(k=1)(infinity) of certain types of sequences {f(k)}(k=1)(infinity) in contrast to the results in the literature we are able to be very explicit about the operator T and suitable powers alpha(k), and we do not need to assume that the sequences are linearly independent. The exact meaning of approximation is defined in a way such that {T-alpha(k)phi}(k=1)(infinity) keeps essential features of {f(k)}(k=1)(infinity), e.g., in the setting of atomic decompositions and Banach frames. We will present two different approaches. The first approach is universal, in the sense that it applies in general Banach spaces; the technical conditions are typically easy to verify in sequence spaces, but are more complicated in function spaces. For this reason we present a second approach, directly tailored to the setting of Banach function spaces. A number of examples prove that the results apply in arbitrary weighted l(p)-spaces and L-p-spaces.