ON BACKWARD ALUTHGE ITERATES OF COMPLEX SYMMETRIC OPERATORS

Abstract

For a nonnegative integer k, an operator T is an element of L(H) is called a backward Aluthge iterate of a complex symmetric operator of order k if the kth Aluthge iterate (T) over tilde ((k)) of T is a complex symmetric operator, denoted by T is an element of BAIC(k). In this paper, we study several properties of the backward Aluthge iterate of a complex symmetric operator. We show that every nilpotent operator of order k + 2 belongs to BAIC(k). Moreover. we prove that if T belongs to BAIC(k), then T has the property (beta) if and only if T is decomposable. Finally, we show that. under some conditions. operators in BAIC(k) have nontrivial hyperinvariant subspaces and we consider Weyl type theorems for such operators.