m-symmetric Operators with Elementary Operator Entries.

Given Banach space operators A, B, let δ A , B denote the generalised derivation δ (X) = (L A - R B) (X) = A X - X B and ▵ A , B the length two elementary operator ▵ A , B (X) = (I - L A R B) (X) = X - A X B . This note considers the structure of m-symmetric operators δ ▵ A 1 , B 1 , ▵ A 2 , B 2 m (I) = (L ▵ A 1 , B 1 - R ▵ A 2 , B 2 ) m (I) = 0 . It is seen that there exist scalars λ i ∈ σ a (B 1) , 1 ≤ i ≤ 2 , such that δ λ 1 A 1 , λ 2 A 2 m (I) = 0 . Translated to Hilbert space operators A and B this implies that if δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 , then there exists λ ¯ ∈ σ a (B ∗) such that δ (λ A) ∗ , λ A m (I) = 0 = δ λ ¯ B , λ B ∗ m (I) . We prove that the operator δ ▵ A ∗ , B ∗ , ▵ A , B m is compact if and only if (i) there exists a real number α and finite sequnces (i) { a j } j = 1 n ⊆ σ (A) , { b j } j = 1 n ⊆ σ (B) such that a j b j = 1 - α , 1 ≤ j ≤ n ; (ii) decompositions ⊕ j = 1 n H j and ⊕ j = 1 n H J of H such that ⊕ j = 1 n (A - a j I) | H j and ⊕ j = 1 n (B - b j I) | H j are nilpotent. If δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 implies δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 to imply δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 is that λ A and λ ¯ B satisfy the commutativity property for scalars lambda ¯ ∈ σ a (B ∗) . An analogous result is seen to hold for the operators ▵ δ A ∗ , B ∗ , δ A , B m and ▵ δ A ∗ , B ∗ , δ A , B m (I) . Perturbation by commuting nilpotents is considered. [ABSTRACT FROM AUTHOR]