The study of weighted composition operators on various function spaces has received considerable attention in past decades. Characterizations, which usually involve interplay of symbol functions, for certain types of weighted composition operators have been obtained. In this thesis, we study Fredholmness and compactness of these operators on Lebesgue spaces L^p and on Hardy spaces H^p of the unit disk. ^{p p}...[ Read more ]

The study of weighted composition operators on various function spaces has received considerable attention in past decades. Characterizations, which usually involve interplay of symbol functions, for certain types of weighted composition operators have been obtained. In this thesis, we study Fredholmness and compactness of these operators on Lebesgue spaces L^p and on Hardy spaces H^p of the unit disk.

For 1 ≤ p < ∞ and non-atomic measure spaces, we show that Fredholm weighted composition operators on L^p are precisely the invertible ones. Our result does not require boundedness of the corresponding composition and multiplication operators. This was assumed in Takagi's work. By investigating invertible weighted composition operators on H^p, we also characterize the Fredholm ones explicitly and obtain their Fredholm indices.

Characterizations of compact weighted composition operators on H^p, 1 ≤ p ≤ ∞, in the literature are less tractable. We give some necessary and/or sufficient conditions for compactness with connection to function theory of analytic functions. These results are applicable in constructing examples of (non-)compact weighted composition operators on H^p.

Relations among compact, completely continuous, weakly compact and M-weakly compact weighted composition operators between L^p-spaces (1 ≤ p ≤ ∞) are completely described. Some (or even all) of these four classes of operators coincide under certain cases; in other occasions, some properties are satisfied by bounded weighted composition operators.