The Images of Completely Homogeneous Polynomials on 2 × 2 Upper Triangular Matrix Algebras

The purpose of this paper is to initiate the study of the images of non-multilinear polynomials on upper triangular matrix algebras. We shall give a complete description of the images of completely homogenous polynomials on 2 × 2 upper triangular matrix algebras. As a consequence, we show that the image of some special completely homogenous polynomials on 2 × 2 upper triangular matrix algebras are not vector spaces.