On the ascent and the angle between the null space and the range of elementary operators

Abstract

We study the angle between the null space and the range of elementary operators of length one or two acting on $\mathcal{B}(\mathscr{X})$, the Banach algebra of all bounded linear operators on a complex Banach space $\mathscr{X}$. For the multiplication operator $μ_{A,B}(X) = AXB$, we characterize positivity of this angle in terms of the corresponding angles for $A$ and $B^*$. For elementary operators of length two $Δ_{\boldsymbol{A},\boldsymbol{B}} = μ_{A_1,B_1} - μ_{A_2,B_2}$, we establish conditions under which the angle is positive, and the ascent of $Δ_{\boldsymbol{A},\boldsymbol{B}}$ equals one. Finally, for a generalized derivation $δ_{A,B}$ and an injective holomorphic function $f$ on a neighborhood of $σ(A)\cupσ(B)$, we show that the angle between the null space and the range of $δ_{f(A),f(B)}$ is positive whenever the angle between the null space and the range of $δ_{A,B}$ is positive.