# Quasi-nilpotent operators and non-Euclidean metrics on resolvent set

#### Dr Yuxia Liang (Tianjin Normal University, Visiting Research Fellow at Leeds). Part of the Analysis and Applications Seminar Series.

Given a linear operator $A$ that is densely defined on a Hilbert space $H$, the operator-valued 1-form $w_A(z)=-(A-z)^{-1}dz$ is analytic on the resolvent set $\rho(A).$ In this report, we introduce the $A$-dependent non-Euclidean metric on $\rho(A)$ through the coupling of the operator-valued (1,1)-form with the vector state $\phi_f$, where $f\in H$ with $\|f\|=1$. Formulae for the length of circles and the energy functionals over circles are presented under this non-Euclidean Hermitian metric and their extremal equations are obtained. For some classical examples, we show that the extremal equations have a relationship with these operators' invariant subspaces. A particular example is when $A$ is quasi-nilpotent,in which case the metric lives on the punctured complex plane $\mathbb{C}\setminus\{0\}.$ Interestingly, the metric's blow up rate at 0 is linked with $A$'s hyper-invariant subspace.

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