SJ -class operators and supercyclicity

In this paper, we suggest a novel approach to linear dynamics based on the idea that supercyclicity can be localized. A nonzero vector x in a Banach space X is called an SJ-class vector for an operator T∈L(X) provided for every open neighborhood Ux of x and every nonempty open subset V of X and also for every N∈N, there exists a complex number λ and an integer n>N such that λTn(Ux)∩V≠∅. When an operator T has at least one SJ-class vector, it is called an operator of SJ-class. As a consequence, if T is a supercyclic operator, then it falls into the SJ-class. Aside from investigating the SJ-class operators, we provide some examples demonstrating that the SJ-class is a new class in L(X). We would like to emphasize that despite the fact that the non-separable Banach space ℓ∞(N) does not admit supercyclic operators, it does admit operators of the SJ-class. Also, we provide a characterization of the SJ-class operators on the Banach space X⊕C in terms of the J-class operators on the Banach space X.