Quantization of Yang--Mills metrics on holomorphic vector bundles

Abstract: We investigate quantization properties of Hermitian metrics on holomorphic vector bundles over homogeneous compact K\"ahler manifolds. This allows us to study operators on Hilbert function spaces using vector bundles in a new way. We show that Yang--Mills metrics can be quantized in a strong sense and for equivariant vector bundles we deduce a strong stability property which supersedes Gieseker-stability. We obtain interesting examples of generalized notions of contractive, isometric, and subnormal operator tuples which have geometric interpretations related to holomorphic vector bundles over coadjoint orbits.