Positivity (2013) 17:655–676
Spectrum of weighted composition operators. Part II:
weighted composition operators on subspaces of Banach
A. K. Kitover
Received: 9 May 2012 / Accepted: 13 August 2012 / Published online: 5 November 2012
© Springer Basel AG 2012
Abstract We describe the spectrum of weighted d-isomorphisms of Banach lattices
restricted on closed subspaces that are “rich” enough to preserve some “memory” of
the order structure of the original lattice. The examples include (but are not limited
to) weighted isometries of Hardy spaces on the polydisk and unit ball in C
Keywords Disjointness preserving operators · Spectrum
Mathematics Subject Classiﬁcation (2000) Primary 47B33;
Secondary 47B48 · 46B60
This paper is a continuation of the study of spectrum of weighted composition operators
attempted in [2, Part 3] and . The results of the current paper heavily depend on
thosein and , and the reader is referred to  for the notations not explained
here. The main goal is to establish a connection between the spectrum of a weighted
d-isomorphism T on a Banach lattice X and the spectrum of its restriction on a
closed T -invariant subspace Y of X . Surely, for such a connection to exist and be
meaningful we must assume some “richness” of the subspace Y . The reader is referred
to Deﬁnitions 2 and 13 for details, but Hardy spaces H
considered as subspaces of the
spaces on the unit circle, or a unital uniform algebra as a subspace of
the space of all continuous functions on its Shilov boundary represent typical examples
of “rich” subspaces.
The following notations will be used throughout the paper.
X: a Banach lattice over the ﬁeld of complex numbers C.
Z(X): the center of the Banach lattice X.
A. K. Kitover (
Community College of Philadelphia, 1700 Spring Garden St., Philadelphia, PA, USA