Bi-continuous Semigroups Research Articles

Sun Dual Theory For Bi-Continuous Semigroups

Abstract The sun dual space corresponding to a strongly continuous semigroup is a known concept when dealing with dual semigroups, which are in general only weak $$^*$$ ∗ -continuous. In this paper we develop a corresponding theory for bi-continuous semigroups under mild assumptions on the involved locally convex topologies. We also discuss sun reflexivity and Favard spaces in this context, extending classical results by van Neerven.

On bi-continuous cosine families

In this article, we introduce and investigate bi-continuous cosine families. Those are inspired by the theory of bi-continuous semigroups which had an impact in the area of evolution equations in the last decades. The idea here is again that we equip the underlying Banach space with an additional locally convex topology. We will analyze both the general behavior as well as generators and resolvents of bi-continuous cosine families.

Mixed topologies on Saks spaces of vector-valued functions

We study Saks spaces of functions with values in a normed space and the associated mixed topologies. We are interested in properties of such Saks spaces and mixed topologies which are relevant for applications in the theory of bi-continuous semigroups. In particular, we are interested if such Saks spaces are complete, semi-Montel, C-sequential or a (strong) Mackey space with respect to the mixed topology. Further, we consider the question whether the mixed and the submixed topology coincide on such Saks spaces and seek for explicit systems of seminorms that generate the mixed topology.

How to Approach Stability of Bi-Continuous Semigroups?

This paper serves as a kick-off to address the question of how to define and investigate the stability of bi-continuous semigroups. We will see that the mixed topology is the key concept in this framework.

Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time T>0 by taking into account the orbit of the initial value under the semigroup for tin [0,T], measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauß–Weierstraß semigroup and the Ornstein–Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the case of Banach spaces.

A note on the Lumer–Phillips theorem for bi-continuous semigroups

Given a Banach space X and an additional coarser Hausdorff locally convex topology \tau on X , we characterise the generators of \tau -bi-continuous semigroups in the spirit of the Lumer–Phillips theorem, i.e. by means of dissipativity with respect to a directed system of seminorms and a range condition.

A Lumer–Phillips type generation theorem for bi-continuous semigroups

The famous 1960s Lumer–Phillips theorem states that a closed and densely defined operator A\colon \operatorname{D}(A)\subseteq X \to X on a Banach space X generates a strongly continuous contraction semigroup if and only if (A,\operatorname{D}(A)) is dissipative and the range of \lambda-A is surjective in X for some \lambda>0 . In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks.

On equicontinuity and tightness of bi-continuous semigroups

In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefully clarify on mutual relations between previously studied variants of these notions. The abstract results—exploiting techniques from topological vector spaces—are thoroughly discussed by means of several example classes, such as semigroups on spaces of bounded continuous functions.

Positive Desch-Schappacher perturbations of bi-continuous semigroups on AM-spaces

Positive Desch-Schappacher perturbations of bi-continuous semigroups on AM-spaces

Positive Miyadera–Voigt perturbations of bi-continuous semigroups

We discuss positive Miyadera–Voigt type perturbations for bi-continuous semigroups on \(\mathrm {AL}\)-spaces with an additional locally convex topology generated by additive seminorms. The main example of such spaces is the space of bounded Borel measures (on a Polish space).

Bi-Continuous semigroups for flows on infinite networks

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Strongly continuous and locally equi-continuous semigroups on locally convex spaces

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces \((X,\tau )\) that are also equipped with a ‘suitable’ auxiliary norm. We introduce the set \(\mathcal {N}\) of \(\tau \)-continuous semi-norms that are bounded by the norm. If \((X,\tau )\) has the property that \(\mathcal {N}\) is closed under countable convex combinations, then a number of Banach space results can be generalised in a straightforward way. Importantly, we extend the Hille–Yosida theorem. We relate our results to those on bi-continuous semigroups and show that they can be applied to semigroups on \((C_b(E),\beta )\) and \((\mathcal {B}(\mathfrak {H}),\beta )\) for a Polish space \(E\) and a Hilbert space \(\mathfrak {H}\) and where \(\beta \) is their respective strict topology.

Pointwise Saturation Theorem of Bi-Continuous Semigroups

By means of Riemann-Stieltjes stochastic process, moment-generating functions and operator-Valued mathematical expectation,the problem of probabilistic approximation for bi-continuous semigroups was studied and the saturation theorem of probabilistic representations of semigroups are obtained .

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space Cb(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on Cb(Ω). We also prove that the class of bi-continuous semigroups on Cb(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C b (ℝN) and to evolution operators associated with nonautonomous second-order differential operators in C b (ℝN) with time-periodic coefficients.

Rational approximation schemes for bi-continuous semigroups

This paper extends the Hille–Phillips functional calculus and rational approximations results due to R. Hersh, T. Kato, P. Brenner, and V. Thomée to generators of bi-continuous semigroups. The method yields error estimates for rational time-discretization schemes for such semigroups, in particular for dual semigroups, Feller semigroups such as the Ornstein–Uhlenbeck semigroup, the heat semigroup, semigroups induced by nonlinear flows, implemented semigroups, and evolution semigroups. Furthermore, the results provide error estimates for a new class of inversion formulas for the Laplace transform.

Cores of Second Order Differential Linear Operators with Unbounded Coefficients on ℝN

We consider the problem of the existence of bi-cores for some classes of second order elliptic differential operators with unbounded coefficients generating bi-continuous semigroups on the space of bounded continuous functions on ℝN.

Positivity of Perturbed Ornstein-Uhlenbeck Semigroups on Cb(H)

In this paper we further develop the abstract theory of bi-continuous semigroups by combining perturbation and approximation techniques. The results are then applied to show the positivity for a class of perturbed Ornstein-Uhlenbeck semigroups on the Banach lattice Cb(H) of bounded continuous functions on a separable Hilbert space H.

Perturbations of bi-continuous semigroups

initial value problems (Cauchy problems) are usually studied via operator semigroups. In many cases, the well-developed theory of C0-semigroups, i.e., one-parameter operator semigroups which are strongly continuous for the norm on a Banach space X , suffices and provides a powerful machinery to study such problems. The applications range from partial differential equations, Volterra integro-differential equations and dynamic boundary problems to delay equations. It seems that a linear (autonomous) evolution equation like { u′(t) = Au(t) u(0) = x, (EE) that is an equation that describes a system evolving from an initial state ”in time”, can be handled via the theory of C0-semigroups. However, as the most trivial example shows this is not the case. Consider the left shift semigroup S on the space of bounded, continuous functions Cb(R) S(t)f(s) := f(t+ s). It is clear that the orbit t 7→ S(t)f is continuous for the supremum norm, if and only if f is uniformly continuous. This shows that the shift semigroup does not fit into the framework of C0-semigroups, nevertheless S clearly describes an evolution system on R. The situation is not so bad either. If we replace the norm topology by the topology τc of uniform convergence on compact sets, then the orbits become τc-continuous. This leads us to weakening the notion of C0-semigroups, thus allowing to consider such ”pathological” cases, these therefore turn out to be less unpleasant than thought previously. There are numerous generalisations of the theory of C0-semigroups. Among these we find the approach of introducing new continuity notions, this method is we want to follow. Investigations on semigroups on locally convex spaces were started fairly long ago, see e.g., [14], [51], [52] and [64]. A nice exposition on the historical aspects can be found in [56]. To exploit the Banach space structure and the same time to introduce coarser topologies, the notion of bi-continuous semigroups was introduced recently by Kuhnemund in [56]. The theory developed therein covers a large part of previously known results, and puts these concrete examples in an abstract framework. The general theory is then applicable in concrete cases.

Trotter–Kato theorems for bi-continuous semigroups and applications to Feller semigroups

We give results on the convergence of bi-continuous semigroups introduced and studied by Kühnemund. As a consequence, we obtain a Lie–Trotter product formula and apply it to Feller semigroups generated by second order elliptic differential operators with unbounded coefficients in C b( R N) .