Type-1 Shifts on C(X)
by
Jeffrey Norden
Tennessee Tech University
Coauthors: Ramesh Garimella, Andrzej Gutek

Let X be a compact Hausdorff space, and let C(X) denote the Banach Space of (Real or Complex valued) continuous functions on X. A type-1 Shift on C(X) is a certain type of linear isometry T:C(X) --> C(X), as defined in ``Shift operators on Banach spaces, '' by Gutek, Hart, Jamison, and Rajagopalan, J. Funct. Anal., 1991. The prototypical example is to let X be the one-point compactification of the integers, identify C(X) with the convergent sequences, and let T( < a₁, a₂, a₃, ... > )= < 0, a₁, a₂, ... > . An open problem is whether there exists a non-separable X for which C(X) admits a type-1 shift.

We provide a ``structure-type theorem'' for analyzing shifts by characterizing the functions in the range of Tⁿ. One immediate consequence is that the space X must be ccc. Underlying any type-1 shift is homeomorphism \psi from X-{p} onto X, where p is an isolated point of X. It is therefore natural to look at the set D consisting of p, \psi^-1(p), \psi^-2(p), etc. It is known that D need not be dense in X, but we are able to provide additional examples of this type. In the most interesting one, the space X consists of the discrete sum of the Stone-Cech and one-point compactifications of the integers. For this space, any \psi will have the property that there are infinitely many isolated points of X which are not in D.

Date received: July 13, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caap-03.