Norm estimates of ω-circulant operator matrices and isomorphic operators for ω-circulant algebra

An n x n omega-circulant matrix which has a specific structure is a type of important matrix. Several norm equalities and inequalities are proved for omega-circulant operator matrices with = e(i theta) (0 <= theta < 2 pi) in this paper. We give the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norms. Pinching type inequality is also proposed for weakly unitarily invariant norms. Meanwhile, we present that the set of.-circulant matrices with complex entries has an idempotent basis. Based on this basis, we introduce an automorphism on the omega-circulant algebra and then show different operators on linear vector space that are isomorphic to the omega-circulant algebra. The function properties, other idempotent bases and a linear involution are discussed for omega-circulant algebra. These results are closely related to the special structure of omega-circulant matrices.