  We introduce a class of linear bounded invertible operators on Banach spaces,
called shift operators, which comprises weighted backward shifts and models
finite products of weighted backward shifts and dissipative composition
operators. We classify vast families of these shift operators, including the
ones generated by orthogonal, diagonalizable, rotation or hyperbolic matrices.
and this classification yields verifiable conditions which we use to construct
concrete examples of shift operators with a variety of dynamical properties. As
a consequence, we show that, for large classes of shift operators, generalized
hyperbolicity is equivalent to the shadowing property.
