Harnack parts for some truncated shifts

Abstract

The purpose of this paper is to analysis the Harnack part of some truncated shifts whose ρ-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the ρ-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given. * Corresponding author. 2010 Mathematics Subject Classification. Primary 47A12, 47A20, 47A65; Secondary 15A60.

A note on a spectral constant associated with an annulus

“…C ρ was introduced by Sz.-Nagy and Foiaş [31] (see also [30, Chapter 1]) and has subsequently been investigated by many authors (see e.g. [9] and the references therein). It is known that C 1 is precisely the set of contractions on H [29], while C 2 is the set of operators whose numerical range W (T ) = { T x, x | ||x|| = 1} is contained in D = {|z| ≤ 1} [4], [6].…”