Polynomially partial isometric operators

In order to extend the notion of semi-generalized partial isometries 
and partial isometries, we introduce a new class of operators called polynomially partial isometries. 
Since this new class of operators contains semi-generalized partial isometries, partial isometries, 
isometries and co-isometries, we proposed a wider class of operators. 
Several basic properties of polynomially partial isometries and some invariant subspaces of corresponding operators are presented. 
We study decomposition theorems and spectral theorems for polynomially partial isometries, generalizing some well-known results for partial isometries and 
semi-generalized partial isometries to polynomially partial isometries. Applying polynomially partial isometries, we solve some equations.