Classification of Solution Operators Semigroups for Abstract Cauchy Problems

The paper is devoted to studying solution operators semigroups and its generators for abstract Cauchy problem in Banach space. It is considered two types of families — "classical" that defined on whole Banach space and possesses the semigroup property, and "regularized" that can be defined on some subspace, it doesn't possess the semigroup property but some their transformation possesses. Among the classical semigroups are considered semigroups of class $C_0$, Cesaro-summable and Abel-summable semigroups, semigroups of classes $C_k$ and $\mathfrak{C}_k$, semigroups of growth $\alpha$. Among the regularized semigroups are considered integrated semigroups, $R$-semigroups, convoluted semigroups. For each kind of regularized semigroups it's described the regularization method that allows to consider the amended semigroup property defined on whole Banach space. Also for each kind of regularized semigroups are considered the definition of its generator and in addition the exponentially bounded and local versions of semigroups.
The paper deduces the diagram of solution operators semigroups inclusions. Implications that involve regularized semigroups are by embedding of generators. Implication with pair of classical semigroups are by embedding of semigroups themeselves and as a consequence by embedding of generators too. Particular attention is paid for giving an examples that prove strictness for some embeddings. For the simplicity of the main diagram the relationship between Abel-summable semigroups (i.e. semigroups of classes $Ab$, $(0,Ab)$, $(1,Ab)$) and their relationship with semigroups of class $C_k$ are taken out into separate diagram.