The characterization of A-m-isometric weighted shift operators in semi-Hilbertian spaces

In this paper, we will study the characterization of [Formula: see text]-[Formula: see text]-isometric unilateral and bilateral weighted shift operators. We shall prove that any such operator is a [Formula: see text]-Hadamard product of [Formula: see text]-[Formula: see text]-isometries and [Formula: see text]-[Formula: see text]-isometries. We also study weighted shift operators whose powers are [Formula: see text]-[Formula: see text]-isometric.