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A short course on operator semigroups. (English) Zbl 1106.47001
Universitext. New York, NY: Springer (ISBN 0-387-31341-9/hbk). x, 247 p. (2006).
Semigroup theory, initiated by Hille, Yosida, Phillips, Miyadera and Feller some sixty years ago, deals with the initial value problem \[ u'(t) = Au(t), \quad u(0) = u, \tag{1} \] in a linear (usually Banach) space \(E\), where \(A\) is a linear operator (usually densely defined and unbounded) in \(E\). The abstract equation (1) models partial differential equations (where \(A\) is a differential operator in the space variables), but also other equations where \(A\) is a nonlocal operator. If (1) is well-posed, which means that the solution \(u(t)\) depends continuously on the initial condition \(u\), then the solution operator or propagator defined by \(S(t)u = u(t)\) is strongly continuous and satisfies the semigroup equations \(S(s + t) = S(s)S(t)\), \(S(0) = I\). Conversely, a strongly continuous semigroup is always the propagator of a well-posed initial value problem (1).
Semigroup theory was an active research field during the 1950s and 1960s, when it became clear that the semigroup language is, if not “the language of”, at least one of the main dialects in many different disciplines such as probability, partial differential equations and infinite-dimensional control theory (to name only three). In the 1960s, semigroup theory branched in various different directions. In one, the operator \(A\) in (1) depends on time (evolution equations). In another direction, \(A\) is a nonlinear perturbation of a linear operator (semilinear and quasilinear equations). These equations are partly included in the fully nonlinear semigroup theory developed in the 1970s, which, however, only generalizes the contraction case \(\| S(t)\| \leq 1\).
Applications were many and important; for instance, the Navier–Stokes equations can be modeled as as semilinear equation in a vector \(L^p\) space, and many fundamental results were obtained through the theory of the resulting Navier–Stokes semigroup. In yet another different direction, the semigroup equation was generalized to the realm of vector-valued distributions (distribution semigroups) to cover weak notions of well-posedness of (1), and various subclasses such as integrated semigroups, were staked out.
Investigations in semigroup theory proper continue; one recent field of study is that of maximal regularity, where smoothness properties of the solution \(u(t)\) of the nonhomogeneous equation \(u'(t) = Au(t) + f(t)\) are derived from corresponding properties of \(f(t)\), a theory whose tools are vector-valued singular integrals and smoothness of Banach spaces. Nowadays, semigroup theory is a subject that no graduate student in analysis, probability or many branches of applied mathematics can safely ignore.
This book furnishes an introduction to the classical theory of linear semigroups. It includes the basic generation theorems (characterizing operators \(A\) for which is (1) well-posed), perturbation theory, the study of special classes of semigroups that correspond to parabolic or hyperbolic partial differential equations modeled by (1), and stability (behavior as \(t \to \infty)\) of semigroups. There are numerous examples and additional topics.

MSC:
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
47D03 Groups and semigroups of linear operators
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
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