Compact weighted endomorphisms of $C(X)$

Abstract

A weighted endomorphism of an algebra is defined to be a linear operator which is equal to an endomorphism followed by a multiplier. Thus, if B is an algebra with unit, then T is a weighted endomorphism of B if there is an element u in B and an endomorphism S of B such that T: f u Sf. The problem considered in this note is to characterize compact weighted endomorphisms of C(X), the Banach algebra of continuous functions on a compact Hausdorff space X. If X is a compact Hausdorff space, then it is easy to see that every nonzero endomorphism of C(X) has the form Sf = f o qp for some continuous function 9p from X into X. Consequently, weighted endomorphisms of C(X) have the form f(x) -* u(x)f(qp(x)) for some u E C(X) and continuous function 9p from X into X. This map will be denoted by uCcp and its spectrum by a(uC,). In related papers, the spectra of weighted automorphisms of various algebras were considered by Kitover ([6] and [7]) and in a lattice theoretic setting by Arendt ([1] and [2]) and Schaeffer, Wolff and Arendt [8]. Results on weighted automorphisms and endomorphisms of the disc algebra may be found in [4] and [5]. We begin with some notation and terminology. If B, and B2 are Banach spaces, then a linear map T: B1 -B2 is compact if it maps bounded sets into sequentially compact sets. If X is a set and qp: X -* X, then we let qPn denote the nth iterate of qT, i.e. T0(x) = x and %, (x) = p(P(, 1(x)) for n > 0, x E X. Also, if qp: X -* X, then a point c in X is called a fixed point of 9p of order n if n is a positive integer, Tpn(c) = c and q)k(c) =-c, k = 1, . . . , n1. The principal result of this note will be the following theorem.