Compact Weighted Composition Operators and Fixed Points in Convex Domains

• Main Author: Clahane Dana D
• Format: Article
• Language: English
• Published: SpringerOpen 2007-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2007/028750

<p/> <p>Let <inline-formula><graphic file="1687-1812-2007-028750-i1.gif"/></inline-formula> be a bounded, convex domain in <inline-formula><graphic file="1687-1812-2007-028750-i2.gif"/></inline-formula>, and suppose that <inline-formula><graphic file="1687-1812-2007-028750-i3.gif"/></inline-formula> is holomorphic. Assume that <inline-formula><graphic file="1687-1812-2007-028750-i4.gif"/></inline-formula> is analytic, bounded away from zero toward the boundary of <inline-formula><graphic file="1687-1812-2007-028750-i5.gif"/></inline-formula>, and not identically zero on the fixed point set of <inline-formula><graphic file="1687-1812-2007-028750-i6.gif"/></inline-formula>. Suppose also that the weighted composition operator <inline-formula><graphic file="1687-1812-2007-028750-i7.gif"/></inline-formula> given by <inline-formula><graphic file="1687-1812-2007-028750-i8.gif"/></inline-formula> is compact on a holomorphic, functional Hilbert space (containing the polynomial functions densely) on <inline-formula><graphic file="1687-1812-2007-028750-i9.gif"/></inline-formula> with reproducing kernel <inline-formula><graphic file="1687-1812-2007-028750-i10.gif"/></inline-formula> satisfying <inline-formula><graphic file="1687-1812-2007-028750-i11.gif"/></inline-formula> as <inline-formula><graphic file="1687-1812-2007-028750-i12.gif"/></inline-formula>. We extend the results of J. Caughran/H. Schwartz for unweighted composition operators on the Hardy space of the unit disk and B. MacCluer on the ball by showing that <inline-formula><graphic file="1687-1812-2007-028750-i13.gif"/></inline-formula> has a unique fixed point in <inline-formula><graphic file="1687-1812-2007-028750-i14.gif"/></inline-formula>. We apply this result by making a reasonable conjecture about the spectrum of <inline-formula><graphic file="1687-1812-2007-028750-i15.gif"/></inline-formula> based on previous results.</p>