We focus this note on operator inequalities related to Beurling type theorem and Wold decompositions. We first introduce the notion of the Cauchy dual for operators with closed range. We show that some operator inequalities, used by several authors to prove Beurling type theorem for left invertible operators, are in fact, equivalent in the more general setting of operators with closed range.

wandering spaces, Beurling type theorem, Moore-Penrose inverse, operator inequalities.