Hypercyclic Conjugate Operators

Abstract

We prove that for any weighted backward shift B = Bw on an infinite dimensional separable Hilbert space H whose weight sequence w = (wn) satisfies \( \sup_{n} {\left| {w_{1} w_{2} \ldots w_{n} } \right|} = \infty \), the conjugate operator \( C_{B} :S \mapsto BSB^{*} \) is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an \( S \in S(H) \) such that \( \{ C^{n}_{B} (S) = B^{n} SB^{*n}\} _{{n \geq 0}} \) is dense in S(H). We generalize the result to more general conjugate maps \( S \mapsto TST^{*} \), and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.