2016/04/26 by Md. Ramiz Reza, Reza, Md. Ramiz
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1604.07758
Fix a bounded planar domain \Ω. If an operator T, in the\nCowen-Douglas class B1(\Ω), admits the compact set \\Ω as a\nspectral set, then the curvature inequality mathcal KT(w) \≤ - 4 \π2\nS_\Ω(w,w)2, where S_\Ω is the S "zego kernel of the domain\n\Ω, is evident. Except when \Ω is simply connected, the existence\nof an operator for which mathcal KT(w) = 4 \π2 S_\Ω(w,w)2 for all\nw in \Ω is not known. However, one knows that if w is a fixed but\narbitrary point in \Ω, then there exists a bundle shift of rank 1, say\nS, depending on this w, such that mathcal KS^*(w) = 4 \π2\nS_\Ω(w,w)2. We prove that these em extremal operators are uniquely\ndetermined: If T1 and T2 are two operators in B1(\Ω) each of which\nis the adjoint of a rank 1 bundle shift and \KT1(w) = -4\π\n2 S(w,w)2 = \KT2(w) for a fixed w in \Ω, then T1 and\nT2 are unitarily equivalent. A surprising consequence is that the adjoint of\nonly some of the bundle shifts of rank 1 occur as extremal operators in\ndomains of connectivity greater than 1. These are described explicitly.\n