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1.
We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup
extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous
for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L
1-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity
for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and
consequently obtain a direct sum decomposition. 相似文献
2.
Summary In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the case is considered of a finite-dimensional Hilbert space (n-level quantum system) and of instruments defined on a finite-dimensional Lie group. Then, the generator of a continuous semigroup of (quantum) probability operators is characterized. In this way a quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. 相似文献
3.
Luigi Manca 《Journal of Evolution Equations》2008,8(2):231-262
We consider a semigroup of operators in the Banach space C
b
(H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply
the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.
相似文献
4.
James C. S. Wong 《Semigroup Forum》1986,35(1):227-236
LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly
continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10].
Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups
of probability measures on a locally compact semigroup or group.
Research Supported by NSERC of Canada Grant No. A8227. 相似文献
5.
It is proved that a general non-differentiable skew convolution semigroup associated with a strongly continuous semigroup of linear operators on a real separable Hilbert space can be extended to a differentiable one on the entrance space of the linear semigroup. A càdlàg strong Markov process on an enlargement of the entrance space is constructed from which we obtain a realization of the corresponding Ornstein–Uhlenbeck process. Some explicit characterizations of the entrance spaces for special linear semigroups are given. 相似文献
6.
Existence and nonexistence of hypercyclic semigroups 总被引:1,自引:0,他引:1
L. Bernal-Gonzá lez K.-G. Grosse-Erdmann 《Proceedings of the American Mathematical Society》2007,135(3):755-766
In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from--and considerably shorter than--the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.
7.
Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein–Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on a real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein–Uhlenbeck processes in different topologies is also discussed. 相似文献
8.
The present paper is mainly devoted to the study of initial boundary problems associated with a wide class of degenerate second-order differential operators on real intervals, in the framework of weighted continuous function spaces. Such operators are of particular interest, since they often occur, for instance, while building up theoretical models in Mathematical Finance. In order to develop our approach, essentially based on semigroup theory, we provide here some general tools which, perhaps, cover an interest on their own, being concerned with the generation of positive strongly continuous semigroups and their deep connection with Markov processes, in the setting of weighted spaces of continuous functions on a locally compact space. Due to its length, the paper is split up into two parts; the second part will appear in this same journal. 相似文献
9.
We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint. 相似文献
10.
J.W. Neuberger 《Applicable analysis》2013,92(3-4):223-231
For a jointly continuous semigroup of transformations on a complete separable metric space X an induced semigroup of linear transformations on an appropriate space of measures is defined. A complete characterization of generators of such semigroups is given and it is shown how to construct a jointly continuous semigroup on X from a generator taken from this characterized collection. 相似文献
11.
R. H. W. Hoppe 《Applied Mathematics and Optimization》1982,9(1):263-290
In this paper we are concerned with the approximate solution of time-optimal control problems in a nonreflexive Banach SpaceE by sequences of similar problems in Banach spacesE
n which are assumed to approximateE in a fairly general sense. The problems under consideration are such that the solution operator of the associated evolution equation is a strongly continuous holomorphic contraction semigroup and the class of controls is taken from the dual of the Phillips adjoint space with respect to the infinitesimal generator of that semigroup. The main object is to establish convergence of optimal controls, transition times and corresponding trajectories of the approximating control problems which can be done by means of some results from the theory of approximation of semigroups of operators. Finally, these abstract convergence results will be applied to time-optimal control problems arising from heat transfer and diffusion processes.Research supported in part by the Deutsche Forschungsgemeinschaft (DFG) 相似文献
12.
Interpretation, derivation and application of a variation of constants formula for measure-valued functions motivate our investigation
of properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals.
Spaces of measures densely embed into these preduals. The metric space embeds continuously in these preduals, even isometrically
in a specific case. Under mild conditions, a semigroup of Lipschitz transformations on the metric space then embeds into a
strongly continuous semigroups of positive linear operators on these Banach spaces generated by measures.
相似文献
13.
研究有界线性算子强连续半群在非线性Lipschitz扰动下的正则性质保持问题.具体地,我们证明:如果强连续半群是直接范数连续的,则非线性扰动半群是直接Lipschitz范数连续的.结论推广了线性算子半群的范数连续性质保持,丰富和完善了非线性算子半群的理论. 相似文献
14.
Feng-Yu Wang 《Mathematische Zeitschrift》2004,246(1-2):359-371
We study the essential spectrum and the semigroup property for self-adjoint operators on abstract Hilbert spaces by using functional inequalities. Some known results obtained on the L
2
-space w.r.t. a measure space are generalized. The functional inequality is also used to study non-symmetric semigroups.
Mathematics Subject Classification (2000): 49R20, 58F19.Research supported in part by NNSFC (10121101, 10025105), TRAPOYT and the 973-Project. 相似文献
15.
Let
be a dense sub-semigroup of ℝ+, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of
operators on X over
can be extended to a weakly continuous semigroup over ℝ+. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized
semigroups which are extendable to semigroups over ℝ+.
O.M. Shalit was partially supported by the Gutwirth Fellowship. 相似文献
16.
Yu. V. Plyushchev 《Mathematical Notes》1975,17(1):34-38
The concepts of the homogeneously continuable semigroup of operators, and of infinitesimal and reproducing families of a semigroup, are introduced. The class of strongly continuous homogeneously continuable semigroups of commuting linear operators is discussed. This class contains in particular the class (C0) of homogeneous semigroups. An analog of the Hill-Yosida theorem is proved for it. 相似文献
17.
Ramón Bruzual 《Integral Equations and Operator Theory》1993,17(3):301-321
A notion of two-parameter local semigroups of isometric operators in Hilbert space is discussed. It is shown that under certain conditions such a semigroup can be extended to a strongly continuous two-parameter group of unitary operators in a larger Hilbert space. As an application a simple proof of the Eskin bidimensional version of the Krein extension theorem is given. 相似文献
18.
This paper considers semigroups of operators generated by pseudodifferential operators in weighted L
p
-spaces of vector functions on
\mathbbRn {\mathbb{R}^n} (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic
are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal
generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established. 相似文献
19.
Grega Cigler Roman Drnovšek Damjana Kokol-Bukovšek Matja? Omladi? Thomas J. Laffey Heydar Radjavi Peter Rosenthal 《Journal of Functional Analysis》1998,160(2):245
T. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A, B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable. 相似文献
20.
Sen-Yen Shaw 《Journal of Functional Analysis》1982,49(2):152-169
The aim of this paper is to study ergodic properties (i.e., properties about the limit of Cesàro averages) of a semigroup of bounded linear operators on a Banach space X, which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of X. Then the results are applied to particular examples, such as locally strongly integrable semigroups, their dual semigroups, and the tensor product semigroup of two (C0)-semigroups. 相似文献