Traces of operators with integrable kernels

Abstract

Trace class operators are compact operators whose singular values are absolutely summable. C. Brislawn proved in [1] and [2] that a trace formula for trace class operator on L2(X) is determined by it's corresponding kernel where X Rn {u100000} in [1] or X is countably generated measure space {u100000} in [2] . We will bene t from [1] and [2] in this thesis. Yet the main aim of this thesis is to establish tools in order to nd a trace formula for trace class operators on L2(X) with a kernel and to be given the proofs in detail by utilizing these tools. Once X Rn, proofs make use of Hardy-Littlewood Maximal Theorem; otherwise, when X is countably generated measure space, Hardy- Littlewood Maximal Theorem should be replaced by Doob's Maximal Theorem for martingales which is used to generalize the former results.