Notes about the structure of common supercyclic vectors

In this paper, we give a sufficient condition for the existence of a common universal subspace for various countable families of universal sequences of linear operators. Besides, we show that if σp(T⁎)=∅σp(T⁎)=∅, the unitary orbit of the supercyclic operator T contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and yet every nonzero vector in the linear span of the orbit of a given supercyclic vector is a common supercyclic vector for the entire path.