QUASITRIANGULARITY OF THE WEIGHTED SHIFT OPERATORS WITH SPECIAL OPERATOR WEIGHTS

Abstract

In this paper are given necessary and sufficient conditions for quasitriangularity of certain weighted shift operators with operator weights, in terms of the weight sequence, approximate point spectrum, essential spectrum and Weyl’s spectrum. The main result is Theorem 2, where it is shown that if \(T\) is a unilateral weighted shift with weights \(A_i\), \(i \in N\) such that \(A_i/\|A_i\|\) are isometry, then

\[
T \text{ is quasitriangular if and only if } \liminf_{i \in N}\|A_i\| = 0
\]