A study of weighted translation operators on l2 µ

Abstract

The composition operators on 2 L ( ) and#955; when the underlying space ( ,, ) X s and#955; is a general measure space have appeared in Koopman s (1931) work on classical mechanics. Earlier works were concentrated mainly on the operators induced by invertible measure preserving transformation. Choksi (1969) gave a necessary and sufficient condition for a unitary operator on a Hilbert space H to be associated with an operator induced by an invertible measure preserving nonsingular transformation in some representation of H as [0,1] 2 L . Later the composition operators on 2 L ( ) and#955; induced by general nonsingular measurable transformation from X in to it self were taken up. Ridge (1965) and Singh (1972) have invented the condition under which the composition transformation CT induced by a measurable non-singular transformation T, a bounded operator on 2 L ( ) and#955; . It has also have proved that for a composition operator CT on ( ) 2 L and#956; , and#8734; = 0 2 C f T , where 0f denotes the RadonNikodym derivative of the measure and#8722;1 and#955;T with respect to the measure and#955; and and#8734; induced the essential sup. norm. Ridge (1973) has started working on composition operators on 2 L ( ) and#955; of a sigma finite measure space. He characterized composition operators on 2 L ( ) and#955; and proved that invertibility of CT implies the invertibility of Sign (1976) has proved that CT on 2 L ( ) and#955; is invertible if and only if T is invertible and and#8722;1 T C is a composition operator on 2 L ( ) and#955; induced by the inverse and#8722;1 T of T. Sign (1975) characterized Hermitian composition operators and proved that every Hermitian operator is an isometry. Later he proved that no composition operator on 2 L ( ) and#955; of a non-atomic measure space is compact. He also has characterized quasi-normal composition operators on 2 L ( ) and#955; and proved the very important result that 0 CTCT = M f and#8727; , where f is the Radon-Nikodym derivative of the measure and#8722;1 and#955;T with respect to the measure and#955; . Singh (1978) gave the necessary and sufficient conditions for a rational function to induce a composition operator on (and#956;) P L , when and#956; is a Lebeague measure on the Borel subset of the real line. Singh and Kumar (1977) have characterized composition operators with closed ranges. Further they characterized normal, quasi-normal, invertible and partial-isometric composition operators on 2 L ( ) and#955; when underlying measure space both atomic and non atomic Parts. newline