Searcharxiv⌕ Search

arXiv subjects

Ali BenAmor

Publications and source records attributed to Ali BenAmor.

15 recordsLinked to original sources
arXiv2025

Strongly continuous fields of operators over varying Hilbert spaces

After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g.~the tangent bundle of Wasserstein space), without referring to identification operators at all.

math.FA↗
arXiv2024

Feller property and convergence for semigroups of time-changed processes

We give a substitute to Feller property for semigroups of time-changed processes; under some conditions this leads to establish sufficient (new) conditions for the semigroups to be Feller. Moreover, given a standard process and a sequence of measures converging vaguely to a final measure, under some assumptions, we establish convergence of the sequence of the semigroups and the resolvents of the corresponding time changed-processes. Some applications are given: convergence of solutions of evolution equations and convergence of finite time distributions, as well as weak convergence of the related processes.

math.PR↗
arXiv2024

Continuity aspects for traces of Dirichlet forms with respect to monotone weak convergence of G-Kato measures

We investigate some analytic properties of traces of Dirichlet forms with respect to measures satisfying Hardy-type inequality. Among other results we prove convergence of spectra, ordered eigenvalues, eigenfunctions as well as convergence of resolvents on appropriate spaces, for traces of Dirichlet forms when the speed measure is the monotone weak limit of G-Kato measures. Some quantitative estimates are also given. As applications we show continuity of stationary solutions of some elliptic operators with measure-valued coefficients with respect to the coefficients and give an approximation procedure for the eigenvalues of one-dimensional graph Laplacian as well as for the Laplacian on annular thin sets with mixed Neumann- Wentzell boundary conditions.

math.FA↗
arXiv2022

Essential spectrum and Feller type properties

We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call \emph{weak Feller property}. Our characterization involves potential theoretic as well as probabilistic aspects and seems to be new even in the symmetric case. As a consequence, in the symmetric case, we obtain a new variant of a decomposition principle of the essential spectrum for (the self-adjoint operators induced by) regular symmetric Dirichlet forms and a Persson type theorem, which applies e.g. to Cheeger forms on $\mathsf{RCD^*}$ spaces.

math.FA↗
arXiv2020

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. For singular Dirichlet-to-Neumann operators we will establish Laurent expansion near singularities as well as Mittag--Leffler expansion for the related quadratic form. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the $L^2$ setting. The obtained results are supported by some examples on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boundary may affect positivity and derive Mittag-Leffler expansion for the eigenvalues of singular Dirichlet-to-Neumann operators.

math.AP↗
arXiv2019

Decomposition formulae for Dirichlet forms and their corollaries

We provide decompositions of Dirichlet forms into recurrent and transient parts as well as into conservative and dissipative parts, in the framework of Hausdorff state spaces. Combining both formulae we write every Dirichlet form as the sum of a recurrent, dissipative and transient conservative Dirichlet forms. Besides, we prove that Mosco convergence preserves invariant sets and that a Dirichlet form shares the same invariants sets with its approximating Dirichlet forms E(t) and E(?). Finally we show the equivalence between conservativeness (resp. dissipativity) of a Dirichlet form and the conservativeness (reps. dissipativity) of E(t) and E(?). The elaborated results are enlightened by some examples.

math.FA↗
arXiv2019

On the construction and convergence of traces of forms

We elaborate a new method for constructing traces of quadratic forms in the framework of Hilbert and Dirichlet spaces. Our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part. We give various situations where the trace can be described more explicitly and compute it for some illustrating examples. We then show that Mosco convergence of Dirichlet forms implies Mosco convergence of a subsequence of their approximating traces and that asymptotic compactness of Dirichlet forms yields asymptotic compactness of their traces.

math.FA↗
arXiv2019

Computations and global properties for traces of Bessel's Dirichlet form

We compute explicitly traces of the Dirichlet form related to the Bessel process with respect to discrete measures as well as measures of mixed type. Then some global properties of the obtained Dirichlet forms, such as conservativeness, irreducibility and compact embedding for their domains are discussed.

math.AP↗
arXiv2015

The Robin Laplacian in the large coupling limit: Convergence and spectral asymptotic

We study convergence modes as well as their respective rates for the resolvent difference of Robin and Dirichlet Laplacian on bounded smooth domains in the large coupling limit. Asymptotic expansions for the resolvent, the eigenprojections and the eigenvalues of the Robin Laplacian are performed. Finally we apply our results to the case of the unit disc.

math.AP↗
arXiv2012

Pointwise estimates for the ground states of singular Dirichlet fractional Laplacian

We establish sharp pointwise estimates for the ground states of some singular fractional Schrödinger operators on relatively compact Euclidean subsets. The considered operators are of the type $(-Δ)^{α/2}|_\Om-c|x|^{-α}$, where $(-Δ)^{α/2}|_\Om$ is the fraction-Laplacien on an open subset $\Om$ in $\R$ with zero exterior condition and $0<c\leq(\frac{d-α}{2})^2$. The intrinsic ultracontractivity property for such operators is discussed as well and a sharp large time asymptotic for their heat kernels is derived.

math.SP↗
arXiv2010

Pointwise estimates for the ground states of some classes of positivity preserving operators

We establish pointwise estimates for the ground states of some classes of posi- tivity preserving operators. The considered operators are negatively perturbed (by measures) strongly local Dirichlet operators. These estimates will be written in terms of the Green's kernel of the considered operators, whose existence will be proved. In many circumstances our estimates are even sharp so that they recover known results about the subject. The results will deserve to obtain large time heat kernel estimates for the related operators.

math.FA↗
arXiv2009

Hardy's inequality in the scope of Dirichlet forms

We revisit Hardy's inequality in the scope of regular Dirichlet forms following an analytical method. We shall give an alternative necessary and sufficient condition for the occurrence of Hardy's inequality. A special emphasis will be given for the case where the Dirichlet form under consideration is strongly local, extending therefore some known results in the Euclidean case.

math.FA↗