Hermitian Metrics on the Resolvent Set and Extremal Arc Length

Abstract

For a bounded linear operator A on a complex Hilbert space \({\mathcal H}\), the functions gx(z) = ∥(A − z)−1x∥2, where \(x\in {\mathcal H}\) with ∥x∥ = 1, defines a family of Hermitian metrics on the resolvent set ρ(A). Thus the arc length of a fixed circle C ⊂ ρ(A) with respect to the metric gx is dependent on the choice of x. This paper derives an integral equation for the extremal values of the arc length. Solution x of the equation, if exists, has particular properties as related to A. In the case A is the unilateral shift operator on the Hardy space \({ H}^2({\mathbb D})\), the paper proves that the arc length of C is maximal if and only if x is an inner function.