Common supercyclic vectors for a path of operators (Q2382737)

Let \(X\) be an infinite-dimensional separable Banach space and let \(I\) be an interval of real numbers. A family \((F_t : t \in I) \subset B(X)\) of continuous linear operators on \(X\) is a path of operators if the map \(F:I \rightarrow B(X)\) is continuous when \(B(X)\) is endowed with the operator norm. The present paper investigates the existence of common supercyclic vectors for paths of operators. Abakumov, Gordon, Bayart, Matheron, Costakis, Sambarino and the present authors have investigated the existence of common hypercyclic vectors for families of operators. Here, a sufficient condition for a path of operators to have a \(G_{\delta}\) set of common supercyclic vectors is presented. This condition implies that every operator in the path must satisfy the supercyclicity criterion of \textit{H.\,N.\thinspace Salas} [Stud.\ Math.\ 135, No.\,1, 55--74 (1999; Zbl 0940.47005)]. This result is applied to paths of unilateral weighted backward shifts on \(\ell_2\), in particular when the path consists of all the convex combinations of two shifts. The behaviour of hypercyclicity is different: there are two hypercyclic unilateral weighted backward shifts \(T_0\) and \(T_1\) on \(\ell_2\) such that \((1/2)T_0 + (1/2)T_1\) is not hypercyclic. An example is given of a path of unilateral weighted backward shifts on \(\ell_2\) that fails to have a common supercyclic vector. The authors also construct a path with a dense \(G_{\delta}\) set of common supercyclic vectors between a hypercyclic unilateral weighted backward shift and an operator which fails Salas's supercyclicity criterion.