Hermitian Metric and the Infinite Dihedral Group

Abstract

For a tuple A = (A1,A2,…, An) of elements in a unital Banach algebra B, the associated multiparameter pencil is A(z) = z1A1 + z2A2 + … + znAn. The projective spectrum P(A) is the collection of z ∈ ℂn such that A(z) is not invertible. Using the fundamental form ΩA = −ω * ∧ ωA, where ωA(z) = A−1(z) dA(z) is the Maurer–Cartan form, R. Douglas and the second author defined and studied a natural Hermitian metric on the resolvent set Pc(A) = ℂn \ P(A). This paper examines that metric in the case of the infinite dihedral group, D∞ = , with respect to the left regular representation λ. For the non-homogeneous pencil R(z) = I + z1λ(a) + z2λ(t), we explicitly compute the metric on Pc(R) and show that the completion of Pc(R) under the metric is ℂ2 \ {(±1, 0), (0, ±1)}, which rediscovers the classical spectra σ(λ(a)) = σ(λ(t)) = {± 1}. This paper is a follow-up of the papers by R. G. Douglas and R. Yang (2018) and R. Grigorchuk and R. Yang (2017).