Jℱ-Class Weighted Backward Shifts
Abstract
In this article Jℱ-class operators are introduced and some basic properties of Jℱ-vectors are given. The Jℱ-class operators include the J-class operators and Jmix-class operators introduced by Costakis and Manoussos in 2008. This class also includes the Jwmix-class and Jerg-class operators defined by Zhang [2012]. Furthermore, for the unilateral weighted backward shifts on a Fréchet sequence space, we establish a criterion under which the shift operators belong to the Jℱ-class. From the criterion it is easy to obtain the existing criteria of hypercyclic backward shifts and of the topological mixing backward shifts. The obtained criterion also reveals the characteristic of Jℱ-class shift operators by the recurrence property. Meanwhile, we obtain infinite topological entropy when the shifts have stronger recurrence property, which generalizes the related results by Brian et al. in 2017.
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