We show that the q-numerical range and the real q-numerical range of the unilateral shift and bilateral shift and their adjoints are the closed unit disc when jqj < 1. We also consider some weighted shifts.

A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form [ aI cT dT ∗ bI ] . It reduces to a quadratic operator if d = 0. In this paper, spectra, norms, and various kinds of numerical ranges of generalized quadratic operators are determined. Some operator inequalities are also obtained. In particular, it is shown that for a...