Analytic functions, ideals, and derivation ranges

Abstract:

When $A$ is in the Banach algebra $\mathcal {B}(\mathcal {H})$ of all bounded linear operators on a Hilbert space $\mathcal {H}$, the derivation generated by $A$ is the bounded operator ${\Delta _A}$ on $\mathcal {B}(\mathcal {H})$ defined by ${\Delta _A}(X) = AX - XA$. It is shown that (i) if $B$ is an analytic function of $A$, then the range of ${\Delta _B}$ is contained in the range of ${\Delta _A}$; (ii) if $U$ is a nonunitary isometry, then the range of ${\Delta _U}$, contains nonzero left ideals; (iii) if $U$ and $V$ are isometries with orthogonally complemented ranges, then the span of the ranges of the corresponding derivations is all of $\mathcal {B}(\mathcal {H})$.