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Journal of Operator Theory

Volume 88, Issue 1, Summer 2022  pp. 205-244.

Locally eventually positive operator semigroups

Authors:  Sahiba Arora
Author institution: Technische Universitaet Dresden, Institut fuer Analysis, Fakultaet fuer Mathematik, 01062 Dresden, Germany

Summary:  We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of the domain, after a sufficiently large time. A drawback of the present theory of eventually positive $C_0$-semigroups is that it is applicable only when the leading eigenvalue of the semigroup generator has a strongly positive eigenvector. We weaken this requirement and give sufficient criteria for individual and uniform local eventual positivity of the semigroup. This allows us to treat a larger class of examples by giving us more freedom on the domain when dealing with function spaces $-$ for instance, the square of the Laplace operator with Dirichlet boundary conditions on $L^2$ and the Dirichlet bi-Laplacian on $L^p$-spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.

Keywords: one parameter semigroups of linear operators, semigroups on Banach lattices, eventually positive semigroup, locally eventually positive semigroup, positive spectral projection, eventually positive resolvent, locally eventually positive resolvent, antimaximum principle

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