Previous issue ·  Next issue ·  Recently posted articles ·  Most recent issue · All issues   
Home Overview Authors Editorial Contact Subscribe

Journal of Operator Theory

Volume 88, Issue 1, Summer 2022  pp. 203-242.

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.

DOI: http://dx.doi.org/10.7900/jot.2021jan26.2316
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

Contents   Full-Text PDF