On joint spectral radius of commuting operators in Hilbert spaces

Our aim in this paper is to give a new formula of the joint spectral radius of commuting d-tuples of operators on a complex Hilbert space. Also we show that r(T)≤ω(T) for a commuting d-tuple of operators T, where r(T) and ω(T) denote respectively the joint spectral radius and the joint numerical radius of T. This generalizes the well known relation between the spectral and the numerical radii of Hilbert space operators which is proved in [10].