The subject of this work is intended to give an overview of some important classes of operators semigroups, frequently used in many applications. In Part 1 we consider the so called C₀ semigroups managed by the Hille-Yosida Theorem and the Lumer Phillips Theorem (see [6])\n Part 2 is devoted to a special class of semigroups of operators S_{t},t≥0, called Markov semigroups and acting on the Banach space C(X) of all real continuous functions on compact metric space X. Such semigroups are closely related to the notion of transition function also called markovian kernel [4].\n Part 3 contains a somewhat detailed description of the important class of contraction C₀ semigroups and their generators properties. For such semigroups operating on Hilbert spaces, we point out an interesting property of their extension (dilation) by a unitary group of operators.

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