STRONGLY SPLITTING WEIGHTED SHIFT OPERATORS ON BANACH SPACES AND UNICELLULARITY

KARAEV M. T., GÜRDAL M.

OPERATORS AND MATRICES, vol.5, no.1, pp.157-171, 2011 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 5 Issue: 1
  • Publication Date: 2011
  • Doi Number: 10.7153/oam-05-11
  • Journal Name: OPERATORS AND MATRICES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.157-171
  • Keywords: Banach space, weighted shift operator, strong splitting operator, invariant subspace, Duhamel product, EXTENDED EIGENVALUES, EIGENVECTORS
  • Süleyman Demirel University Affiliated: Yes

Abstract

We introduce the notion of strong splitting operator on a separable Banach space, and prove a structure theorem for this operator. We consider the weighted shift operator T, Te-n = lambda(n)e(n+1), n >= 0, acting in the Banach space X with basis {e(n)}(n >= 0). We give some sufficient conditions for X and for the weight sequence {lambda(n)}(n >= 0) under which the operator is unicellular, that is, every nontrivial invariant subspace E of T has the form E = X-i := Span {e(k) : k >= i} for some i >= 1; and prove that the restricted operators T vertical bar X-i (i >= 1) are strong splitting. Moreover, we describe in terms of so-called discrete Duhamel operator and diagonal operator all extended eigenvectors of the operators T vertical bar X-i (i >= 1).