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.