Derivations, derivatives and chain rules (Q1970441)

Let \({\mathcal B}( {\mathcal H}) \) be the space of bounded linear operators on a Hilbert space \({\mathcal H} \) and \(f\) be a function mapping \({\mathcal B}( {\mathcal H}) \) into itself. The authors establish formulae that relate the \(n-\)th derivative \(D^{n}f(A)\) of \(f\) at the point \(A\) and the \(n\)-th order commutator \(\delta^{[n]}(A)(X)=[\delta^{[n-1]}(A)(X),X]\), \(\delta(A)(X)=[A,X]=AX-XA,\) which are interesting for the calculus of operator functions. These formulae are useful in obtaing bounds for norms of generalized commutators \(f(A)X-Xf(B)\) and their higher order analogues.