Hermitian Geometry on Resolvent Set

For a tuple A = (A1, A2, ..., An) of elements in a unital Banach algebra B, its projective joint spectrum P (A) is the collection of z ∈ C such that A(z) = z1A1 +z2A2 + · · ·+znAn is not invertible. It is known that the B-valued 1-form ωA(z) = A−1(z)dA(z) contains much topological information about the joint resolvent set P (A). This paper defines Hermitian metric on P (A) through the B-valued fundamental form ΩA = −ω∗ A ∧ ωA and its coupling with faithful states φ on B. The connection between the tuple A and the metrics is the main subject of this paper. A notable feature of this metric is that it has singularities at the joint spectrum P (A). So completeness of the metric is an important issue. When this construction is applied to a single operator V , the metric on the resolvent set ρ(V ) adds new ingredients to functional calculus. An interesting example is when V is quasi-nilpotent, in which case the metric live on the punctured complex plane. It turns out that the blow up rate of the metric at the origin 0 is directly linked with V ’s lattice of hyper-invariant subspaces. 0. INTRODUCTION In [6], the first author and Cowen introduced geometric concepts such as holomorphic bundle and curvature into Operator Theory. This gave rise to complete and computable invariants for the Cowen-Douglas operators. This idea was followed up in a series of papers in the study of Hilbert modules in analytic function spaces, where a curvature invariant is defined for some natural tuples of commuting operators. We refer readers to [12, 13, 14] and the references therein for this line of 2010 Mathematics Subject Classification: 47A13.