Energy functional of the Volterra operator

Abstract

We define the energy functional $E_{f,A}$ for a bounded linear operator $A$ acting on a Hilbert space $\mathcal{H}$ through a newly defined non-Euclidean metric $g_f\left(z\right)|dz{|}^2$ on its resolvent set $\rho (A)$, where the vector $f\in \mathcal{H}$. We investigate the extremal values of $E_{f,A}$ with respect to the change of $f$. We conduct an in-depth study of the case when $A$ is the classical Volterra operator $V$ on $L^2([0,1])$. Our main theorem suggests a likely connection between the energy functional and the invariant subspace problem.