ON THE RANGE INCLUSION OF NORMAL DERIVATIONS: VARIATIONS ON A THEME BY JOHNSON, WILLIAMS AND FONG

Abstract

This paper considers various extensions of results of Johnson and Williams and of Fong on the range inclusion of normal derivations. Let $C_p$, with $p \in [1,\infty)$, be the Schatten ideals of compact operators on a Hilbert space $H$ with norms $|\ |_p$, $C_\infty$ be the ideal of all compact operators, and $C_b$ the algebra $B(H)$ of all bounded operators. Any $S \in B(H)$ defines a bounded derivation $\delta_S$ on all $C_p$: $\delta_S(X) = SX - XS$, for $X \in C_p$. Johnson and Williams proved that, for normal $S$, the inclusion of ranges $\delta_T(C_b) \subseteq \delta_S (C_b)$ implies $T = g(S)$, where $g$ is a Lipschitz, differentiable function on $\sigma(S)$. Fong showed that the condition $T = g(S)$ is equivalent to the range inclusion $\delta_T(C_1) \subseteq \delta_S (C_b)$. This paper studies the range inclusion \begin{equation} \delta_T(C_p) \subseteq \delta_S (C_p) \end{equation} for normal $S$ and $p \in [1,\infty] \cup b$, and the classes of functions for which $T = g(S)$. Set $$ p_- = \min\bigg(p, \frac{p}{p - 1}\bigg),\quad p_+ = \max\bigg(p, \frac{p}{p - 1}\bigg), \quad \text{for } p \in (1, \infty), $$ and $$ p_- = 1,\quad p_+ = b, \quad \text{for } p \in \{1,\infty,b\}. $$ This paper shows that condition (1) implies: \begin{enumerate} \item[(i)] the range inclusions $\delta_T(C_r) \subseteq \delta_S (C_r)$, for $r \in [p_-,p_+]$; \item[(ii)] that there exists $D > 0$ such that $|\delta_T(X)|_p \leq D|\delta_S (X)|_p$, for $X \in B(H)$ ($|X|_p = \infty$ if $X \notin C_p$); \item[(iii)]the range inclusions $\delta_{g(A)}(C_p) \subseteq \delta_A (C_p)$ for any normal operator $A$ with $\sigma(A) \subseteq \sigma(S)$. \end{enumerate} It establishes that (1) implies that the function $g$ (in $T = g(S)$) is $C_p$-Lipschitzian on $\sigma(S)$, that is, there is $D > 0$ such that $|g(A) - g(B)|_p \leq D|A - B|_p$ for all normal $A$ and $B$ with spectra in $\sigma(S)$. Conversely, it is proved that, for any selfadjoint $S$ and $C_p$-Lipschitz function $g$ on $\sigma(S)$, $\delta_{g(S)}(C_p) \subseteq \delta_S(C_p)$. The paper also extends the above results of Johnson and Williams to bounded derivations of C$^*$-algebras. 2000 Mathematics Subject Classification: 46L57, 47B47, 58C07.