Vector-valued Littlewood-Paley-Stein theory for semigroups

We develop a generalized Littlewood-Paley theory for semigroups acting on L^p-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein g-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on Rⁿ, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calderón-Zygmund singular integral operators.     

Strongly continuous semigroups of operators generated by systems of pseudodifferential operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

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.     

Rotenberg模型中迁移半群的本质谱

在Lp（1（≤）p＜＋∞）空间中,本文运用半群理论研究了Rotenberg模型中具光滑边界条件的迁移半群的本质谱.采用半群方法和比较算子等方法,证明了对任意的t＞0,s＞0,算子[UH（t）-U0（t）]U0（s）[UH（t）-U0（t）]在Lp（1＜p＜＋∞）在空间中紧和在L1空间弱紧,得到迁移半群VH（t）与V0（t）有相同的本质谱型.     

On approximation properties of Balázs-Szabados operators their Kantorovich extension

In this paper we deal with a sequence of positive linear operatorsR _n^[β] approximating functions on the unbounded interval [0, t8) which were firstly used by K. Balázs and J. Szabados. We give pointwise estimates in the framework of polynomial weighted function spaces. Also we establish a Voronovskaja type theorem in the same weighted spaces for K_n^[β] operators, representing the integral generalization in Kantorovich sense of the R_n^[β].     

Cryptic Semigroup Varieties

We describe all minimal noncryptic periodic semigroup [monoid] varieties. We prove that there are exactly three distinct maximal cryptic semigroup [monoid] varieties contained in the variety determined by xⁿ ≈ x ^n+m, n ≥ 2, m ≥ 2. Analogous results are obtained for pseudovarieties: there are exactly three maximal cryptic pseudovarieties of semigroups [monoids]. It is shown that if a cryptic variety or pseudovariety of monoids contains a nonabelian group, then it consists of bands of groups only. Several characterizations are given of the cryptic overcommutative semigroup [monoid] varieties.     

C_0半群在非线性Lipschitz扰动下的范数连续性保持

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.     

Necessary and sufficient conditions for conservativeness of dynamical semigroups

Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let be a Hilbert space andA a von Neumann algebra. A dynamical semigroup P_t is a -weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup P_t possessing the property of preserving the identityIA is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA ₊. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra of (R ⁿ ) yields a series of new regularity conditions for both diffusion and jump processes.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 149–184, 1990.     

Functional Calculus for Infinitesimal Generators of Holomorphic Semigroups

We give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on Banach spaces, which involves the Bochner–Riesz kernels of such generators. The rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup. The strength of the formula is tested on Poisson and Gauss semigroups inL¹(Rⁿ) andL¹(G), for a stratified Lie groupG. We give also a self-contained theory of smooth absolutely continuous functions on the half line [0, ∞).     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Quantum stochastic calculus and quantum Gaussian processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy [9]. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space Γ(? ⁿ ) over ? ⁿ . These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn [19]. They were recently investigated in the context of quantum information theory by Heinosaari et al. [7]. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

Nonlinear semigroup approach to age structured proliferating cell population with inherited cycle length

This paper deals with a nonlinear semigroup approach to semilinear initial-boundary value problems which model nonlinear age structured proliferating cell population dynamics. The model involves age-dependence and cell cycle length, and boundary conditions may contain compositions of nonlinear functions and trace of solutions. Hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. A family of equivalent norms is introduced to discuss local quasidissipativity of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators. The resultant solution operators are obtained as nonlinear semigroups which are not quasicontractive but locally equi-Lipschitz continuous.     

Inhomogeneous semigroups of commuting operators

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.     

Hypercontractivity, Nash inequalities and subordination for classes of nonlinear semigroups

A suitable notion of hypercontractivity for a nonlinear semigroup {T _t } is shown to imply Nash-type inequalities for its generator H, provided a subhomogeneity property holds for the energy functional (u,Hu). We use this fact to prove that, for semigroups generated by operators of p-Laplacian-type, hypercontractivity implies ultracontractivity. Then we introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is f(x)=x ^α , and write an explicit formula for the associated generator. It is shown that hypercontractivity still holds for the subordinated semigroup and, hence, that Nash-type inequalities hold as well for the subordinated generator.     

An extension of a Phillips's theorem to Banach algebras and application to the uniform continuity of strongly continuous semigroups

In this work we present an extension to arbitrary unital Banach algebras of a result due to Phillips [R.S. Phillips, Spectral theory of semigroups of linear operators, Trans. Amer. Math. Soc. 71 (1951) 393-415] (Theorem 1.1) which provides sufficient conditions assuring the uniform continuity of strongly continuous semigroups of linear operators. It implies that, when dealing with the algebra of bounded operators on a Banach space, the conditions of Phillips's theorem are also necessary. Moreover, it enables us to derive necessary and sufficient conditions in terms of essential spectra which guarantee the uniform continuity of strongly continuous semigroups. We close the paper by discussing the uniform continuity of strongly continuous groups (T(t))_t∈R acting on Banach spaces with separable duals such that, for each t∈R, the essential spectrum of T(t) is a finite set.     

Decomposition of operator semigroups on W*-algebras

We consider semigroups of operators on a W^∗-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a “stable” and “reversible” part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W^∗-algebras.     

On the theory of semigroups of operators on locally convex spaces

The behavior of strongly continuous one-parameter semigroups of operators on locally convex spaces is considered. The emphasis is placed on semigroups that grow too rapidly to be treated by classical Laplace transform methods.A space

of continuous E-valued functions is defined for a locally convex space E, and the generalized resolvent $\text{R}$ of an operator A on E is defined as an operator on

. It is noted that $\text{R}$ may exist when the classical resolvent (λ ? A)^?1 fails to exist. Conditions on $\text{R}$ are given that are necessary and sufficient to guarantee that A is the generator of a semigroup T(t). The action of $\text{R}$ is characterized by convolution against the semigroup, and the semigroup is computed as the limit of $\text{R}$ acting on an approximate identity.Conditions on an operator B are introduced that are sufficient to guarantee that A + B is the generator of a semigroup whenever A is. A formula is given for the perturbed semigroup.Two characterizations of semigroups that can be extended holomorphically into some sector of the complex plane are given. One is in terms of the growth of the derivative $\text{(}\text{d}\text{dt}\text{) T(t)}$ as t approaches 0, the other is in terms of the behavior of $\text{R}$ⁿ, the powers of the generalized resolvent.Throughout, the generalized resolvent plays a role analogous to the role of the classical resolvent in the work of Hille, Phillips, Yosida, Miyadera, and others.     

A uniformly differentiable approximation scheme for delay systems using splines

A new spline-based scheme is developed for linear retarded functional differential equations within the framework of semigroups on the Hilbert spaceR ⁿ ×L ². The approximating semigroups inherit in a uniform way the characterization for differentiable semigroups from the solution semigroup of the delay system (e.g., among other things the logarithmic sectorial property for the spectrum). We prove convergence of the scheme in the state spacesR ⁿ ×L ² andH ¹. The uniform differentiability of the approximating semigroups enables us to establish error estimates including quadratic convergence for certain classes of initial data. We also apply the scheme for computing the feedback solutions to linear quadratic optimal control problems.Work done by K. Ito was supported by AFOSR under Contract No. F-49620-86-C-0111, by NASA under Grant No. NAG-1-517, and by NSF under Grant No. UINT-8521208. Work done by F. Kappel was supported by AFOSR under Grant No. 84-0398 and by FWF(Austria) under Grants S3206 and P6005.