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 \(\triangle _{A,B}(X)=(I-L_AR_B)(X)=X-AXB\). This note considers the structure of m-symmetric operators \(\delta ^m_{\triangle _{A_1,B_1},\triangle _{A_2,B_2}}(I)=(L_{\triangle _{A_1,B_1}} - R_{\triangle _{A_2,B_2}})^m(I)=0\). It is seen that there exist scalars \(\lambda _i\in \sigma _a(B_1)\), \(1\le i\le 2\), 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 \(\delta ^m_{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\), then there exists \({\overline{\lambda }}\in \sigma _a(B^*)\) such that \(\delta ^m_{(\lambda A)^*,\lambda A}(I)=0=\delta ^m_{{\overline{\lambda }}B,\lambda B^*}(I)\). We prove that the operator \(\delta ^m_{\triangle _{A^*,B^*},\triangle _{A,B}}\) is compact if and only if (i) there exists a real number \(\alpha \) and finite sequnces (i) \(\{a_j\}_{j=1}^n\subseteq \sigma (A)\), \(\{b_j\}_{j=1}^n\subseteq \sigma (B)\) such that \(a_jb_j=1-\alpha \), \(1\le j\le n\); (ii) decompositions \(\oplus _{j=1}^n {\mathcal {H}}_j\) and \(\oplus _{j=1}^n{\texttt {H}_J}\) of \({\mathcal {H}}\) such that \(\oplus _{j=1}^n{(A-a_j I)|_{\ H_j}}\) and \(\oplus _{j=1}^n{(B-b_j I)|_{\texttt {H}_j}}\) are nilpotent. If \(\delta ^{m}_{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) implies \(\delta _{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\), then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for \(\delta ^{m}_{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) to imply \(\delta _{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) is that \({\lambda }A\) and \({\overline{\lambda }}B\) satisfy the commutativity property for scalars \(\overline{lambda} \in \sigma _a(B^*)\). An analogous result is seen to hold for the operators \(\triangle ^m_{\delta _{A^*,B^*},\delta _{A,B}}\) and \(\triangle ^m_{\delta _{A^*,B^*},\delta _{A,B}}(I)\). Perturbation by commuting nilpotents is considered.