On Approximate Operator Representations of Sequences in Banach Spaces

Generalizing results by Halperin et al., Grivaux recently showed that any linearly independent sequence {f_k}∞_k=1 in a separable Banach spaceX can be represented as a suborbit {T^α(k)ϕ}∞_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 {f_k}∞_k=1 ≈ {T^α(k)ϕ}∞_k=1 of certain types of sequences {f_k}∞_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 notneed to assume that the sequences are linearly independent. The exact meaning of approximation is defined in a way such that {T^α(k)ϕ}∞_k=1 keeps essential features of {f_k}∞_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 insequence 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.