BASISITY PROBLEM AND WEIGHTED SHIFT OPERATORS


GÜRDAL M. , GARAYEV M. T. , SALTAN S.

ACTA MATHEMATICA SCIENTIA, vol.34, no.5, pp.1655-1660, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 34 Issue: 5
  • Publication Date: 2014
  • Doi Number: 10.1016/s0252-9602(14)60111-9
  • Title of Journal : ACTA MATHEMATICA SCIENTIA
  • Page Numbers: pp.1655-1660

Abstract

We investigate a basisity problem in the space l(A)(p)(D) and in its invariant sub-spaces. Namely, let W denote a unilateral weighted shift operator acting in the space l(A)(p)(D), 1 <= p < infinity, nu Wz(n) = lambda(n)z(n+1), n >= 0, with respect to the standard basis {z(n)}(n >= 0). Applying the so-called "discrete Duhamel product" techique, it is proven that for any integer k >= 1 the sequence {(w(i+nk))(-1) (W vertical bar E-i)(kn) f}(n >= 0) is a basic sequence in E-i := span (z(i+n) : n >= 0} equivalent to the basis {z(i+n)}(n >= 0) if and only if <(f)over cap>(i) not equal 0. We also investigate a Banach algebra structure for the subspaces E-i, i >= 0.