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Vector-Valued Laplace Transforms and Cauchy Problems
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In this article, linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms, and the central tool is an integrated version of Widder's theorem (characterizing Laplace transform of bounded functions).Abstract:
Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central tool is an “integrated version” of Widder’s theorem (characterizing Laplace transforms of bounded functions). It holds in any Banach space (whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the Radon-Nikodým property). The Hille-Yosida theorem and other generation theorems are immediate consequences. The method presented here can be applied to operators whose domains are not dense.read more
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Observation and Control for Operator Semigroups
Marius Tucsnak,George Weiss +1 more
TL;DR: The main topics of interest about observation and control operators are admissibility, observability, controllability, stabilizability and detectability as discussed by the authors, which is a mature area of functional analysis, which is still very active.
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The Functional Calculus for Sectorial Operators
TL;DR: In this article, the basic theory of sectorial operators is developed along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved and a panorama of functional calculi is developed.
Introduction to stochastic partial differential equations
Mihály Kovács,Stig Larsson +1 more
TL;DR: In this paper, the Hilbert space-valued Wiener process and the corresponding stochastic integral of Ito type were introduced and used together with semigroup theory to obtain existence and uniqueness of weak solutions of linear and semilinear stochiastic evolution problems in Hilbert space.
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Limit theorems for continuous-time random walks with infinite mean waiting times
TL;DR: In this article, the scaling limit of a continuous-time random walk is shown to be an operator Levy motion subordinated to the hitting time process of a classical stable subordinator.
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Optimal polynomial decay of functions and operator semigroups
Alexander Borichev,Yuri Tomilov +1 more
TL;DR: In this article, the polynomial decay of orbits of Hilbert space C 0-semigroups in resolvent terms was studied and the results of the same type for general Banach space semigroups and functions were shown to be sharp.
References
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Book
Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Book
Classical Banach spaces
Joram Lindenstrauss,L. Tzafriri +1 more
TL;DR: Springer-Verlag is reissuing a selected few of these highly successful books in a new, inexpensive sofcover edition to make them easily accessible to younger generations of students and researchers.
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Banach Lattices and Positive Operators
TL;DR: In this paper, the authors propose the use of linear operators on positive matrices and apply it to non-positive matrices, including the case of positive projections. But they do not consider the case where positive projections are defined by a linear operator.
Book
One-parameter Semigroups of Positive Operators
Wolfgang Arendt,Annette Grabosch,Günther Greiner,Ulrich Moustakas,Rainer Nagel,Ulf Schlotterbeck,Ulrich Groh,Heinrich P. Lotz,Frank Neubrander +8 more
TL;DR: In this article, Spectral theory of positive semigroups on Co(X) and asymptotics of positive operators on Co (X) were presented. But the results on the spectral properties of the positive operators were not considered.