On Approximate Operator Representations of Sequences in Banach Spaces

Abstract

Generalizing results by Halperin et al., Grivaux recently showed that any linearly independent sequence $\{f_k\}_{k=1}^\infty$ in a separable Banach space X can be represented as a suborbit $\{T^{\alpha (k)}\varphi \}_{k=1}^\infty$ of some bounded operator $T: X\rightarrow X.$ In general, the operator T and the powers $\alpha (k)$ are not known explicitly. In this paper we consider approximate representations $\{f_k\}_{k=1}^\infty \approx \{T^{\alpha (k)}\varphi \}_{k=1}^\infty$ of certain types of sequences $\{f_k\}_{k=1}^\infty ;$ 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 $\{T^{\alpha (k)}\varphi \}_{k=1}^\infty$ keeps essential features of $\{f_k\}_{k=1}^\infty ,$ 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 $\ell ^p$-spaces and $L^p$-spaces.