A Banach Algebra Structure of Some Function Spaces and Related Topics

We will talk mainly about Duhamel Banach algebras and their operators. We focus on some kind of integration operators, including double integration operator, $q$-integration operator and "weighted" Hardy operator, and discuss some of their properties. In particular, extended eigenvalues and extended eigenfunctions are discussed. Also, we talk about cyclic vectors, multiplicity of spectrum, invariant subspaces, and hyperinvariant subspaces of operators associated with integration operators. Some other questions will also be discussed.