On approximate operator representations of sequences in Banach spaces
Abstract
Generalizing results by Halperin et al., Grivaux recently showed that any linearly independent sequence \fk\k=1∞ in a separable Banach space X can be represented as a suborbit \Tα(k)\k=1∞ of some bounded operator T: X X. In general, the operator T and the powers α(k) are not known explicitly. In this paper we consider approximate representations \fk\k=1∞ ≈ \Tα(k)\k=1∞ of certain types of sequences \fk\k=1∞. In contrast to the results in the literature we are able to be very explicit about the operator T and suitable powers α(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 \Tα(k)\k=1∞ keeps essential features of \fk\k=1∞, 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 p-spaces and Lp-spaces.