On two-isometries in finite-dimensional spaces

A linear bounded operator $A$ in a complex Hilbert space $H$ is called a 2-isometry if $A^{*2}A^2-2A^*A+I=0$. In particular, the class of 2-isometries contains conventional isometries. It is shown that in the finite-dimensional case, the concept of a 2-isometry has no new content, that is, 2-isometries of a finite-dimensional unitary space are conventional unitary operators.