We introduce q-frequently hypercyclic operators and derive a sufficient criterion for a continuous operator to be q-frequently hypercyclic on a locally convex space. Applications are given to obtain q-frequently hypercyclic operators with respect to the norm-, F -norm-and weak*-topologies. Finally, the frequent hypercyclicity of the non-convolution operator T µ defined by T µ (f )(z) = f ′ (µz), |µ| ≥ 1 on the space H(C) of entire functions equipped with the compact-open topology is shown. * Corresponding author. 2010 Mathematics Subject Classification. Primary 47A16; Secondary 46A45. Key words and phrases. q-frequently hypercyclic operator, q-frequent hypercyclicity criterion, unconditional convergence, symmetric Schauder basis, backward shift operator.

We give a complete characterization of invariant subspaces for (M z1 , . . . , M zn ) on the Hardy space H 2 (D n ) over the unit polydisc D n in C n , n > 1. In particular, this yields a complete set of unitary invariants for invariant subspaces for (M z1 , . . . , M zn ) on H 2 (D n ), n > 1. As a consequence, we classify a large class of n-tuples, n > 1, of commuting isometries. All of our results hold for vector-valued Hardy spaces over D n , n > 1. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.

A sufficient criterion for the map CA,B(S) = ASB to be supercyclic on certain algebras of operators on Banach spaces is given. If T is an operator satisfying the Supercyclicity Criterion on a Hilbert space H, then the linear map CT (V ) = T V T * is shown to be norm-supercyclic on the algebra K(H) of all compact operators, COT-supercyclic on the real subspace S(H) of all self-adjoint operators and weak * -supercyclic on L(H) of all bounded operators on H. Examples including operators of the form CB w ,Fμ are provided, where Bw and Fµ are respectively backward and forward shifts on Banach sequence spaces. Mathematics Subject Classification. Primary 47A16; Secondary 47L05.

Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for M * z , the adjoint of M z , on vector-valued analytic reproducing kernel Hilbert spaces H in terms of the derivatives of kernel functions on the open unit disc D in C. Here M z denotes the multiplication operator by the coordinate function z, that is, for all f ∈ H and w ∈ D. We analyze the special case of quasi-scalar reproducing kernel Hilbert spaces. We also present a complete characterization of hypercyclicity of M * z on tridiagonal reproducing kernel Hilbert spaces and some special classes of vector-valued analytic reproducing kernel Hilbert spaces.∂ 2n k ∂z n ∂w n (0, 0) = 0.

Abstract. We provide conditions for a linear map of the form CR,T (S) = RST to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-Frequent Hypercyclicity Criterion, then the map CR(S)=RSR * is shown to be q-frequently hypercyclic on the space K(H) of all compact operators and the real topological vector space S(H) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for CR,T to be q-frequently hypercyclic on the Schatten von Neumann classes Sp(H). We also characterize frequent hypercyclicity of CM * ϕ ,M ψ on the trace-class of the Hardy space, where the symbol Mϕ denotes the multiplication operator associated to ϕ.