Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces

Recently, Audenaert (2010)12], Horn and Zhang (2010) [15], Huang (2011) [16] and Schep (2011) [22,23] proved inequalities between the spectral radius rho of Hadamard product (denoted by o) of finite and infinite non-negative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product. We extend these results to the generalized and the joint spectral radius of bounded sets of such operators. Moreover, we prove new inequalities even in the case of the usual spectral radius of non-negative matrices. In particular, we prove that rho(A o B) <= rho(1/2)((A o A)(B o B) <= rho(AB o AB)(1/4) <= rho(AB) and rho(A o B) <= rho(1/2)(AB o BA) <= rho(AB o AB)(1/4) rho(BA o BA)(1/4) <= rho (AB). We also obtain related results in max algebra. (C) 2012 Elsevier Inc. All rights reserved.