On Generalized Numerical Ranges of Quadratic Operators

Abstract

Abstract. It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the essential numerical range. The latter is described quantitatively, and based on that sufficient conditions are established under which the c-numerical range also is an ellipse. Several examples are considered, including singular integral operators with the Cauchy kernel and composition operators.

Self-commutators of automorphic composition operators on the Dirichlet space

“…Recall that the numerical range of C ϕ is defined as W (C ϕ ) := { C ϕ f, f : f ∈ D, f = 1}. Repeating the arguments of Proposition 5 of [2] we obtain the following corollary, which has the same result as before (for example see [13]). …”

Spectra, norms and numerical ranges of generalized quadratic operators

“…We then give the description of various generalized numerical ranges of A in Section 4. The results cover those in [3,22,27] and the reference therein. Additional remarks and further research are discussed in Section 5.…”

“…In fact, it is known that an operator A satisfies (1.2) if and only if it has an operator matrix of the form (1.1) with d = 0, by a suitable choice of orthonormal basis. Motivated by theory and applied problems, there has been considerable interest in studying the norms and generalized numerical ranges (see the definition in later sections) of operators of the form (1.1) under the additional assumptions that (i) a, b, c, d are nonnegative, or (ii) d = 0; see [3,13,22,27] and the references therein. In this paper, a complete description is given to the spectrum, the norm, and various types of generalized numerical ranges of an operator of the form (1.1).…”

On the self-inverse operators

“…Some of the earliest occurrences are [1,6,4] and [12] more recent treatments include [3,5,8,9,11,15,16] and [17].…”