Positivity (2011) 15:639–659
Spectrum of weighted composition operators: part 1.
Weighted composition operators on C( K ) and uniform
A. K. Kitover
Received: 16 September 2010 / Accepted: 23 November 2010 / Published online: 8 December 2010
© Springer Basel AG 2010
Abstract The paper contains a complete description of spectrum of disjointness pre-
serving operators on C(K ). As an application we fully describe spectrum of weighted
automorphisms of unital uniform algebras.
Keywords Disjointness preserving operators · Spectrum
Mathematics Subject Classiﬁcation (2000) Primary 47B33; Secondary
47B48 · 46B60
This paper is dedicated to the memory of the remarkable mathematician and not less
remarkable human being Charalambos D. (“Roko”) Aliprantis who is badly missed
by all his friends.
The current paper is the ﬁrst one in a series of papers on spectral properties of
weighted composition operators. It deals with disjointness preserving operators (DPO)
on C(K ) and weighted automorphisms of uniform algebras. The choice of C(K )-
spaces is not arbitrary. We have a pretty complete description of spectrum of DPO on
Dedekind complete Banach lattices under the mild condition that the Lorentz semi-
norm is a norm (see ), but for Banach lattices which are not Dedekind complete such
a description is missing though some important results were obtained by Arendt 
(seealso and two of the author’s old papers [12,13]). Meanwhile, the knowledge
Dedicated to the memory of Charalambos D. Aliprantis.
A. K. Kitover (
Community College of Philadelphia, 1700 Spring Garden St., Philadelphia, PA, USA
e-mail: firstname.lastname@example.org; email@example.com