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arXiv · 2606.18930

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

Janko Bračič·Bojan Kuzma·Hranislav Stanković
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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.

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Janko Bračič, Bojan Kuzma, Hranislav Stanković. 2026-06-17. On the ascent and the angle between the null space and the range of elementary operators. https://arxiv.org/abs/2606.18930

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