Weighted composition operators between Lp-spaces
(Uncorrected OCR) Abstract of thesis entitled WEIGHTED COMPOSITION OPERATORS BETWEEN L p-SPACES submitted by LO Ching-on for the degree of Master of Philosophy at The University of Hong Kong in October 2002 Let (X, S,? and (Y,G,?) be two s-finite and complete measure spaces. For a G-measurable function u : Y ? C and a non-singular measurable transformation ? : Y ? X, a linear map uC? from Lp(? (1 < p < oo) into the linear space consisting of (equivalence classes of) G-measurable functions on Y was defined by uC?(f)(y) := u(y)f(?(y)) for any f G Lp(? and y EY. This map is known as a weighted composition operator. We were interested in properties of such class of operators between Lp-spaces. Weighted composition operators appear in various contexts in the literature. However, there are relatively few results about these operators from Lp(? into Lq(?), when p and q are distinct. The boundedness, closedness of ranges and compactness of these operators for 1 < p, q < oo were completely characterized. The invertible weighted composition operators from Lp(? onto Lp(?) were also studied. When the Lp-spaces are weighted sequence spaces, the invertibility of uC? and ? are related and it was proved that the inverse of uC? is a weighted composition map. The case that either p = oo or q = oo was considered. In particular, it was shown that when (X, S,? is non-atomic, the only compact weighted composition map between Lp-spaces is the zero operator. The spectrum of a compact weighted composition operator on L?(? was also determined.
School:The University of Hong Kong
School Location:China - Hong Kong SAR
Source Type:Master's Thesis
Keywords:composition operators lp spaces
Date of Publication:01/01/2002