m-symmetric Operators with Elementary Operator Entries

Given Banach space operators A, B, let \(\delta _{A,B}\) ${\delta }_{A,B}$ denote the generalised derivation \(\delta (X)=(L_{A}-R_{B})(X)=AX-XB\) $\delta (X)=(L_A-R_B)(X)=AX-XB$ and \(\triangle _{A,B}\) ${▵}_{A,B}$ the length two elementary operator \(\triangle _{A,B}(X)=(I-L_AR_B)(X)=X-AXB\) ${▵}_{A,B}(X)=(I-L_A{R}_B)(X)=X-AXB$ . This note considers the structure of m-symmetric operators \(\delta ^m_{\triangle _{A_1,B_1},\triangle _{A_2,B_2}}(I)=(L_{\triangle _{A_1,B_1}} - R_{\triangle _{A_2,B_2}})^m(I)=0\) ${\delta }_{{▵}_{A_1,B_1},{▵}_{A_2,B_2}}^m(I)={(L_{{▵}_{A_1,B_1}}-R_{{▵}_{A_2,B_2}})}^m(I)=0$ . It is seen that there exist scalars \(\lambda _i\in \sigma _a(B_1)\) ${\lambda }_i\in {\sigma }_a(B_1)$ , \(1\le i\le 2\) $1\le i\le 2$ , such that \(\delta ^m_{\lambda _1 A_1,\lambda _2 A_2}(I)=0\) ${\delta }_{{\lambda }_1{A}_1,{\lambda }_2{A}_2}^m(I)=0$ . Translated to Hilbert space operators A and B this implies that if \(\delta ^m_{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) ${\delta }_{{▵}_{A^{\ast },B^{\ast }},{▵}_{A,B}}^m(I)=0$ , then there exists \({\overline{\lambda }}\in \sigma _a(B^*)\) $\overline{\lambda }\in {\sigma }_a(B^{\ast })$ such that \(\delta ^m_{(\lambda A)^*,\lambda A}(I)=0=\delta ^m_{{\overline{\lambda }}B,\lambda B^*}(I)\) ${\delta }_{{(\lambda A)}^{\ast },\lambda A}^m(I)=0={\delta }_{\overline{\lambda }B,\lambda B^{\ast }}^m(I)$ . We prove that the operator \(\delta ^m_{\triangle _{A^*,B^*},\triangle _{A,B}}\) ${\delta }_{{▵}_{A^{\ast },B^{\ast }},{▵}_{A,B}}^m$ is compact if and only if (i) there exists a real number \(\alpha \) $\alpha$ and finite sequnces (i) \(\{a_j\}_{j=1}^n\subseteq \sigma (A)\) ${\{a_j\}}_{j=1}^n\subseteq \sigma (A)$ , \(\{b_j\}_{j=1}^n\subseteq \sigma (B)\) ${\{b_j\}}_{j=1}^n\subseteq \sigma (B)$ such that \(a_jb_j=1-\alpha \) $a_j{b}_j=1-\alpha$ , \(1\le j\le n\) $1\le j\le n$ ; (ii) decompositions \(\oplus _{j=1}^n {\mathcal {H}}_j\) ${\oplus }_{j=1}^n{\mathcal{H}}_j$ and \(\oplus _{j=1}^n{\texttt {H}_J}\) ${\oplus }_{j=1}^n{\mathtt{H}}_J$ of \({\mathcal {H}}\) $\mathcal{H}$ such that \(\oplus _{j=1}^n{(A-a_j I)|_{\ H_j}}\) ${\oplus }_{j=1}^n(A-a_{j}I){|}_{H_j}$ and \(\oplus _{j=1}^n{(B-b_j I)|_{\texttt {H}_j}}\) ${\oplus }_{j=1}^n(B-b_{j}I){|}_{{\mathtt{H}}_j}$ are nilpotent. If \(\delta ^{m}_{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) ${\delta }_{{▵}_{A^{\ast },B^{\ast }},{▵}_{A,B}}^m(I)=0$ implies \(\delta _{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) ${\delta }_{{▵}_{A^{\ast },B^{\ast }},{▵}_{A,B}}(I)=0$ , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for \(\delta ^{m}_{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) ${\delta }_{{▵}_{A^{\ast },B^{\ast }},{▵}_{A,B}}^m(I)=0$ to imply \(\delta _{\triangle _{A^*,B^*},\triangle _{A,B}}(I)=0\) ${\delta }_{{▵}_{A^{\ast },B^{\ast }},{▵}_{A,B}}(I)=0$ is that \({\lambda }A\) $\lambda A$ and \({\overline{\lambda }}B\) $\overline{\lambda }B$ satisfy the commutativity property for scalars \(\overline{lambda} \in \sigma _a(B^*)\) $\overline{\mathit{lambda}}\in {\sigma }_a(B^{\ast })$ . An analogous result is seen to hold for the operators \(\triangle ^m_{\delta _{A^*,B^*},\delta _{A,B}}\) ${▵}_{{\delta }_{A^{\ast },B^{\ast }},{\delta }_{A,B}}^m$ and \(\triangle ^m_{\delta _{A^*,B^*},\delta _{A,B}}(I)\) ${▵}_{{\delta }_{A^{\ast },B^{\ast }},{\delta }_{A,B}}^m(I)$ . Perturbation by commuting nilpotents is considered.