TWO CRITERIA FOR A PATH OF OPERATORS TO HAVE COMMON HYPERCYCLIC VECTORS

We offer two conditions for a path of bounded linear operators on a Banach space to have a dense G(delta) set of common hypercyclic vectors. One of them is an equivalent condition and the other one is a generalization of the hypercyclicity criterion. Using the conditions, we show that between any two hypercyclic unilateral weighted backward shifts, there exists a path of such operators having a dense G(delta) set of common hypercyclic vectors. Furthermore, we prove that such a set of vectors exists for a path of scalar multiples of the unweighted shift, reproducing a result of Abakurnov and Gordon, and of Costakis and Sambarino. Motivated by our results, we provide ail example of a path of unilateral weighted backward shifts that fails to have any common hypercyclic vector. Lastly, we adopt the main results to bilateral weighted shifts.