m-symmetric Operators with Elementary Operator Entries

Given Banach space operators A, B, let delta(A,B) denote the generalised derivation delta(X)=(L-A-R-B)(X)=AX-XB and triangle(A,B) the length two elementary operator triangle(A,B)(X)=(I-LARB)(X)=X-AXB. This note considers the structure of m-symmetric operators delta(m)triangle(A1,B1),triangle(A2,B2 )(I)=(L-triangle A1,L-B1-R-triangle A2,R-B2)(m)(I)=0. It is seen that there exist scalars lambda(i)is an element of sigma a(B-1), 1 <= i <= 2, such that delta(m)lambda(1)A(1),lambda(2)A(2)(I)=0. Translated to Hilbert space operators A and B this implies that if delta(m)triangle(A & lowast;,B)& lowast;,triangle(A,B)(I)=0, then there exists lambda<overline>is an element of sigma a(B & lowast;) such that delta(m)(lambda(A))& lowast;,lambda(A)(I)=0=delta(m)(lambda<overline>B,lambda B)& lowast;(I). We prove that the operator delta(m)triangle(A & lowast;,B)& lowast;,triangle(A,B) is compact if and only if (i) there exists a real number alpha and finite sequnces (i) {aj}j=1n subset of sigma(A), {bj}(j=1)(n)subset of sigma(B) such that ajb(j=1)(-alpha), 1 <= j <= n; (ii) decompositions circle plus(n)(j=1)Hj and circle plus(n)(j=1)HJ of H such that circle plus(j=1)n(A-ajI)| Hj and circle plus(j=1)(n)(B-bjI)|Hj are nilpotent. If delta(m)triangle(A & lowast;,B & lowast;,triangle A,B)(I)=0 implies delta(m)triangle(A & lowast;,B & lowast;,triangle A,B)(I)=0, then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for delta(m)triangle(A & lowast;,B & lowast;,triangle A,B)(I)=0 to imply delta(m)triangle(A & lowast;,B & lowast;,triangle A,B)(I)=0 is that lambda A and lambda<overline>B satisfy the commutativity property for scalars lambda<overline>is an element of sigma a(B & lowast;). An analogous result is seen to hold for the operators triangle delta(m)(A & lowast;,B & lowast;,delta A,Bm) and triangle delta(m)(A & lowast;,B & lowast;,delta A,Bm)(I). Perturbation by commuting nilpotents is considered.