My Research Journey on Cesaro Averaging and New Invariant Banach Limits on L∞ and L∞(0,∞)

Abstract:

I will present joint work with my advisor Chris Lennard, and his former student Jeromy Sivek. While discussing the results we obtained, I will also talk about the different stages we went through during our research for this project. I will start on the sequence space l^∞, where we constructed Banach limits that are invariant under the Cesàro Averaging operator. Next, I will discuss our results on the function space L^∞(0,∞). Here we defined a new operator J^α , for α > 0. This new operator extends the definition of Jⁿ , with n ∈ N, which is the operator obtained by composing the Cesàro Averaging operator with itself n times. We showed that the family of operators {J^α}_α>0 has the semigroup property, and we also constructed Banach limits that are invariant under these operators.