Hypercyclicity of adjoint of convex weighted shift and multiplication operators on Hilbert spaces

A bounded linear operator $T$ on a Hilbert space $\mathfrak{H}$ is convex, if $$\|\mathfrak{T}^{2}v\|^2-2\|\mathfrak{T}v\|^2+\|v\|^2 \geq 0.$$ In this paper, sufficient conditions to hypercyclicity of adjoint of unilateral (bilateral) forward (backward) weighted shift operator is given. Also, we present some examples of convex operators such that it s adjoint is hypercyclic. Finally, the spectrum of convex multiplication operators is obtained and an example of convex, multiplication operators is given such that it s adjoint is hypercyclic.