A Characterization of a Special Class of Operator Matrices

Abstract

The classi cation of operator matrices has attracted a great deal of research in the recent past. To the class of such matrices belongs; the Normal, Binormal, Hypernormal, Hamadard, Toeplitz, Pythagorean matrix operators among others. These operator matrices demonstrate many classical properties on dealing with them in connection to algebraic structural properties. In the case of pythagorean matrices in which the column entries are the entries of the triplets(right triangle) of consecutive integers, the shift operator matrix preserves the order and nature of the original matrix. These classes of operators have been studied before to a fair extent, however, from the documented literature, normal operator matrices that result from matrix products in direct sums of Hilbert spaces have not been characterized before. In particular, there is no mention in literature of a classi cation of normal matrices resulting from a combination of Hadamard and Khatri-Rao decompositions on Hilbert spaces. On the other hand, the matrix products have found applications in Information Science( signal sensing), Coding Theory(quantum error computation) among other areas. In this paper, we characterize a special class of normal operator matrices of pythagorean type, which are newly constructed as the Khatri-Rao(which generalizes Hadamard products) products whose entries are the block matrices of pythagorean triplets of class C1 and extend the ndings to an arbitrary Cn completed normal matrix of the same category. We provide detailed survey on the normality and subnormality conditions, positivity and boundedness, and prove new forms of numerical and spectral radii properties as well as the inherent structural relationships of the constructed matrix operator.