In this chapter we present an introductory treatment of the theory of semigroups of linear operators over a Hilbert space, emphasizing those aspects which are of importance in applications. As a rule we shall not strive for generality and instead shall dwell on special classes of semigroups such as compact semigroups and Hilbert-Schmidt semigroups. Semigroup theory is generally accepted as an integral part of functional analysis and is included in most standard treatises on functional analysis which should be consulted for details if necessary. We have taken some pains to illustrate the application to partial differential equations; the abstract parts of the theory are in many ways easier than the specialization to partial differential equations. Nevertheless the abstract formulation has the advantage that it provides a direct generalization of finite dimensional models and makes the transition more transparent, especially in the application to control problems.
KeywordsLinear Operator Point Spectrum Continuous Semigroup Infinitesimal Generator Analytic Semigroup
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