Hypercyclicity of weighted composition operators on $L^p$-spaces

چکیده مقاله

Let (X,Σ, μ) be a σ-finite measure space and W = uCϕ be
a weighted composition operator on Lp(Σ) (1 ≤ p < ∞), defined by
W : f → u.(f ◦ ϕ), where ϕ : X → X is a measurable transformation
and u is a weight function on X. In this paper, we study the hypercyclicity
of W in terms of u, using the Radon–Nikodym derivatives and
the conditional expectations. First, it is shown that if ϕ is a periodic
nonsingular transformation, then W cannot be hypercyclic. The necessary
conditions for the hypercyclicity of W are then given in terms
of the Radon–Nikodym derivatives provided that ϕ is non-singular and
finitely non-mixing. For the sufficient conditions, we also require that
ϕ is normal. The weakly mixing and topologically mixing concepts are
also studied for W. Moreover, under some specific conditions, we establish
the subspace hypercyclicity of the adjoint operator W
∗
with respect
to the Hilbert subspace L2(A). Finally, to illustrate the results, some
examples are given.