Janko Bračič

SIMPLE MULTIPLIERS ON BANACH MODULES

In this paper we introduce simple multipliers, a special subclass of multipliers on a Banach module. We show that, from a local spectral point of view, these multipliers behave like multipliers on a commutative Banach algebra. Our definition of simple multipliers relies on the notion of point multipliers. These multipliers were studied earlier. However our approach gives new insight into this topic and therefore could be of some interest by itself. 2000 Mathematics Subject Classification. Primary 46H25; Secondary 47B40.

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ⊆ B(H1, H2) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ1, ψ2) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ1(P,Q) and Q ≤ ψ2(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H1, H2) lies in M if and only if ψ2(P,Q)Tψ1(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

ON THE INVERTIBILITY OF LENGTH TWO ELEMENTARY OPERATORS

CIy C ‡ Abstract. Let X be a complex Banach space and L(X ) be the algebra of all bounded linear operators on X. For a given elementary operatorof length 2 on L(X ), necessary and sufficient conditions for the existence of a solution of the equation X� = 0 in the algebra of all elementary operators on L(X ) are determined. The proposed approach allows the characterization of some invertible elementary operators of length 2 whose inverses are elementary operators.

Algebraic reflexivity for semigroups of operators

Algebraic reflexivity of sets and semigroups of linear transformations are studied in this paper. Some new examples of algebraically reflexive sets and semigroups of linear transformations are given. Using known results on algebraically orbit reflexive linear transformations, those linear transformations on a complex Banach space that are determined by their invariant subsets are characterized.

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.