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Locally Eventually Positive Operator Semigroups

Research paper by Sahiba Arora

Indexed on: 28 Jan '21Published on: 27 Jan '21Published in: arXiv - Mathematics - Functional Analysis



Abstract

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.