On the range inclusion of normal derivations: variations on a theme by Johnson, Williams and Fong. (Q2766414)

This paper is concerned with bounded derivations of the type NEWLINE\[NEWLINE \delta_S(X)=[S,X]=SX-XS NEWLINE\]NEWLINE on Schatten-von Neumann ideals \(C_p\), \(1\leq p < \infty\), of compact operators on a Hilbert space \(H\), where \(S \in B(H)\). The starting point for this investigation is the problem of the inclusion of ranges of two such derivations \(\delta_T(C_p) \subseteq\delta_S(C_p)\), which in the case \(p=\infty\) was previously studied by \textit{B. E. Johnson} and \textit{J. P. Williams} [Pac. J. Math. 58, 105--122 (1975; Zbl 0275.47010)]. It is shown, in particular, that for normal \(S \in B(H)\) the range inclusion \(\delta_T(C_p) \subseteq\delta_S(C_p)\) is equivalent to the existence of \(D > 0\) such that \(\| \delta_T(X)\| _p\leq D\| \delta_S(X)\| _p\) for all \(X \in C_p\). Furthermore, it is established that if, in addition, the set of isolated eigenvalues of the operator \(S\) is empty, then it follows from the inclusion above that \(T=g(S)\), where \(g\) is \(C_p\)-Lipschitzian function, i.e., such that NEWLINE\[NEWLINE \| g(A)-g(B)\| _p \leq C\| A-B\| _p NEWLINE\]NEWLINE for all normal \(A,B \in B(H)\) with the spectra \(\sigma(A),\;\sigma(B)\) contained in the domain of the (complex-valued) function \(g\) and some constant \(C > 0\).