Unbounded representations of discrete abelian semigroups

This survey article contains the lecture delivered on the conference; it is based on the papers [5]-[9], where complete proofs can be found. In the first section we present results concerning single operators, and the second section is devoted to the study of representations of general discrete abelian semigroups. The main idea in both sections is to apply an appropriate normalizing gauge function in order to get connection with an isometry, or isometric representation.