Sahiba Arora
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View article: Eventually positive semigroups: spectral and asymptotic analysis
Eventually positive semigroups: spectral and asymptotic analysis Open
The spectral theory of semigroup generators is a crucial tool for analysing the asymptotic properties of operator semigroups. Typically, Tauberian theorems, such as the ABLV theorem, demand extensive information about the spectrum to deriv…
View article: Smoothing of operator semigroups under relatively bounded perturbations
Smoothing of operator semigroups under relatively bounded perturbations Open
We investigate a smoothing property for strongly-continuous operator semigroups, akin to ultracontractivity in parabolic evolution equations. Specifically, we establish the stability of this property under certain relatively bounded pertur…
View article: Input-to-state stability in integral norms for linear infinite-dimensional systems
Input-to-state stability in integral norms for linear infinite-dimensional systems Open
We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in $L^p$-spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded in…
View article: A universal example for quantitative semi-uniform stability
A universal example for quantitative semi-uniform stability Open
We characterise quantitative semi-uniform stability for $C_0$-semigroups arising from port-Hamiltonian systems, complementing recent works on exponential and strong stability. With the result, we present a simple universal example class of…
View article: Admissible operators for sun-dual semigroups
Admissible operators for sun-dual semigroups Open
We extend classical duality results by Weiss on admissible operators to settings where the dual semigroup lacks strong continuity. This is possible using the sun-dual framework, which is not immediate from the duality of the input and outp…
View article: Eventually positive semigroups: spectral and asymptotic analysis
Eventually positive semigroups: spectral and asymptotic analysis Open
The spectral theory of semigroup generators is a crucial tool for analysing the asymptotic properties of operator semigroups. Typically, Tauberian theorems, such as the ABLV theorem, demand extensive information about the spectrum to deriv…
View article: The lattice structure of negative Sobolev and extrapolation spaces
The lattice structure of negative Sobolev and extrapolation spaces Open
It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the sp…
View article: Irreducibility of eventually positive semigroups
Irreducibility of eventually positive semigroups Open
In the context of positive infinite-dimensional linear systems, we systematically study $L^p$-admissible control and observation operators with respect to the limit-cases $p=\infty$ and $p=1$, respectively. This requires an in-depth unders…
View article: Domination of semigroups generated by regular forms
Domination of semigroups generated by regular forms Open
We give a representation for regular forms associated with dominated -semigroups which, in turn, characterizes the domination of -semigroups associated with regular forms. In addition, we prove a relationship between the positivity of (dom…
View article: A characterization of the individual maximum and anti-maximum principle
A characterization of the individual maximum and anti-maximum principle Open
Abstract approaches to maximum and anti-maximum principles for differential operators typically rely on the condition that all vectors in the domain of the operator are dominated by the leading eigenfunction of the operator. We study the n…
View article: Domination of semigroups on standard forms of von Neumann algebras
Domination of semigroups on standard forms of von Neumann algebras Open
Consider $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ as real $C_0$-semigroups generated by closed and symmetric sesquilinear forms on a standard form of a von Neumann algebra. We provide a characterisation for the domination of the semigroup $(T…
View article: Irreducibility of eventually positive semigroups
Irreducibility of eventually positive semigroups Open
Positive $C_0$-semigroups that occur in concrete applications are, more often than not, irreducible. Therefore a deep and extensive theory of irreducibility has been developed that includes characterizations, perturbation analysis, and spe…
View article: Stability of (eventually) positive semigroups on spaces of continuous functions
Stability of (eventually) positive semigroups on spaces of continuous functions Open
We present a new and very short proof of the fact that, for positive -semigroups on spaces of continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of Vogt, makes the role of the underlying s…
View article: Locally eventually positive operator semigroups
Locally eventually positive operator semigroups Open
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 …
View article: Criteria for eventual domination of operator semigroups and resolvents
Criteria for eventual domination of operator semigroups and resolvents Open
We consider two $C_0$-semigroups $(e^{tA})_{t \ge 0}$ and $(e^{tB})_{t \ge 0}$ on function spaces (or, more generally, on Banach lattices) and analyse eventual domination between them in the sense that $|e^{tA}f| \le e^{tB}|f|$ for all suf…
View article: Domination of semigroups generated by regular forms
Domination of semigroups generated by regular forms Open
We give a representation for regular forms associated with dominated $C_0$-semigroups which, in turn, characterises domination of $C_0$-semigroups associated with regular forms. In addition, we prove a relationship between the positivity o…
View article: Spectrum and convergence of eventually positive operator semigroups
Spectrum and convergence of eventually positive operator semigroups Open
An intriguing feature of positive $$C_0$$ -semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence…
View article: Uniform (anti-)maximum principles and eventual positivity
Uniform (anti-)maximum principles and eventual positivity Open
Maximum principles and uniform anti-maximum principles are a ubiquitous topic in PDE theory that is closely tied to the Krein--Rutman theorem and kernel estimates for resolvents.
We take up a classical idea of Takac - to prove (anti-)max…
View article: Locally Eventually Positive Operator Semigroups
Locally Eventually Positive Operator Semigroups Open
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 …