A Short Course on Operator Semigroups
The book gives a streamlined and systematic introduction to strongly continuous semigroups of bounded linear operators on Banach spaces. It treats the fundamental Hille-Yosida generation theorem as well as perturbation and approximation theorems for generators and semigroups. The special feature is its treatment of spectral theory leading to a detailed qualitative theory for these semigroups. This theory provides a very efficient tool for the study of linear evolution equations arising as partial differential equations, functional differential equations, stochastic differential equations, and others. Therefore, the book is intended for those wanting to learn and apply functional analytic methods to linear time dependent problems arising in theoretical and numerical analysis, stochastics, physics, biology, and other sciences. It should be of interest to graduate students and researchers in these fields.
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A G C A-bounded abstract Cauchy problem adjoint analytic semigroup approximate assertions are equivalent assume Banach lattice Banach space bounded operator characterization closed operator continuous function continuous semigroup T(t))t>o contraction semigroup convergence Corollary Deﬁnition densely deﬁned differentiable dissipative dissipative operator domain D(A eigenvalue equation estimate Example Exercise exists ﬁrst following assertions formula function q G D(A growth bound hence Hilbert space Hint holds implies integral inverse Lemma linear operator matrix Moreover multiplication operator multiplication semigroup norm-continuous obtain operator Mq operators T(t Paragraph perturbation Proof Proposition prove quasi-compact rescaling satisﬁes satisfying Section semi sequence Sobolev spaces spectral bound spectral mapping theorem spectral radius spectral theory strictly positive strong operator topology strongly continuous semigroup subspace Tn(t Tq(t translation semigroup unbounded uniform boundedness principle uniformly continuous uniformly exponentially stable WSMT yields