Common non supercyclic vectors for commuting contractions

Let T be an operator on a separable Hilbert space H , then it is called supercyclic if there exists an x ∊ H , (called supercyclic vector for T ) such that the set { λTnx : λ ∊ ℂ} is dense in H . Let T = ( T1 , ..., TN ) be a system of N commuting contractions defined on a separable Hilbert space, in this article we will show that if there exists at least a point of the Harte spectrum on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}^N$$ \end{document} (where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document} is the unit circle), then there exists a vector such that is not supercyclic for any of the N -contractions. This result complements recent results of M. Kosiek and A. Octavio (see [4]) and extend results in [7].