Spectra, norms and numerical ranges of generalized quadratic operators

Abstract

A bounded linear operator acting on a Hilbert space is a generalized quadratic operator if it has an operator matrix of the form

It reduces to a quadratic operator if d = 0. In this article, 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 given generalized quadratic operator, the rank-k numerical range, the essential numerical range and the q-numerical range are elliptical discs; the c-numerical range is a sum of elliptical discs. The Davis–Wielandt shell is the convex hull of a family of ellipsoids unless the underlying Hilbert space has dimension 2.

Keywords:

• spectra
• norms
• elliptical ranges
• generalized numerical ranges
• Davis–Wielandt shell

AMS Subject Classifications:

• Primary 47A12
• 15A60

Acknowledgements

Research of the first two authors was supported by USA NSF. The first author was also supported by the William and Mary Plumeri Award. He is an Honorary Professor of the University of Hong Kong and an Honorary Professor of the Taiyuan University of Technology.