More elementary operators that are spectrally bounded

We discuss some necessary and some sufficient conditions for an elementary operator x↦∑ⁿ_i=1a_ixb_i on a Banach algebra A to be spectrally bounded. In the case of length three, we obtain a complete characterisation when A acts irreducibly on a Banach space of dimension greater than three.

Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Elementary operators-still not elementary?

Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.

Norms of basic elementary operators on algebras of regular operators

We show that if E is an atomic Banach lattice with an ordercontinuous norm, A, B ∈ L^r(E) and M_A,_B is the operator on L^r(E) defined by M_A,_B(T) = AT B then ||M_A,_B||r = ||A||_r||B||_r but that there is no real α > 0 such that ||MA,B || ≥ α ||A||r||B ||r.