Quasinilpotent operators and non-Euclidean metrics

The power set Λ(V) of quasinilpotent operator V on a Hilbert space H is defined in [4] to study the singularity of the non-Euclidean metrics ‖(V−z)⁻¹x‖²dz⊗dz¯ at σ(V)={0}, and it is shown that if Λ(V) contains more than one point then V has a nontrivial hyperinvariant subspace. This paper first proves that the Volterra integral operator on the classical Hardy–Hilbert space has singleton power set, thus answering a question raised in [4]. Then, it studies the length of circles under the metrics and its connection with power set. In particular, it determines the maximal length of the unit circle with respect to the change of x in the metrics. Moreover, it shows that an extremal value integral equation is able to detect representing functions for all the invariant subspaces of the Volterra operator.