S-Mixing Tuple of Operators on Banach Spaces

We consider the question: what is the appropriate formulation of Godefroy-Shapiro criterion for tuples of operators? We also introduce a new notion about tuples of operators, S-mixing, which lies between mixing and weakly mixing. We also obtain a sufficient condition to ensure a tuple of operators to be S-mixing. Moreover, we study some new properties of S-mixing operators on several concrete Banach spaces