The spectral radius of matrix continuous refinement operators

A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space L2 (R),m ≥ 1 and s ≥ 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and the matrix function c defining the corresponding refinement operator. A similar representation is valid for the continuous refinement operators considered on spaces Lp for p ∈ [1,∞), p ̸= 2. However, additional restrictions on the kernel c are imposed in this case.