Strong stability of quasi-affine transforms of contraction semigroups and the steady-state riccati equation

This paper studies strong stability of the class of strongly continuous Hilbert space semigroups which are quasi-affine transforms of contraction semigroups. Sufficient conditions for such a semigroup to be approximately strong stable are given. Applications to the stabilizability problem of Hilbert space semigroups, using a feedback involving a solution of the steady-state Riccati equation, are made. Our key tool is the generalization of the LaSalle invariance principle by Hale.This research was supported by the United States Air Force Office of Scientific Research, Directorate of Mathematical and Information Sciences, under Grant No. AFOSR-79-0053D.     

Stabilities of Hilbert space contraction semigroups revisited

A necessary and sufficient condition for exponential stability of Hilbert space contraction semigroups is obtained in terms of an inequality involving the dissipative norm associated with the generator of the semigroup and with the Hilbert space norm.     

关于C_0半群族的等度指数稳定性

本文给出了Hilbert空间中C0半群的一个表示公式，利用它讨论C0半群算子族的等度指数稳定性．     

Unitary extensions of two-parameter local semigroups of isometric operators and the Krein extension theorem

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.     

Contraction Integrated Semigroups and Their Application to Continuous-time Markov Chains

We introduce the notion of the contraction integrated semigroups and give the Lumer-Phillips characterization of the generator, and also the charaterazied generators of isometric integrated semigroups. For their application, a necessary and sufficient condition for q-matrices Q generating a contraction integrated semigroup is given, and a necessary and sufficient condition for a transition function to be a Feller-Reuter-Riley transition function is also given in terms of its q-matrix.     

On the reduction of a contraction semigroup to a completely non unitary semigroup

According to a Theorem of B. Sz.-Nagy and C. Foia?, every strongly continuous semigroup of contraction operators on a Hilbert space, can be decomposed into a completely non unitary part and a unitary part. In this note we wish to show that by appropriately perturbing its generator, a contraction semigroup can be reduced to a completely non unitary one. In control theory, such a perturbation is related to the so called state feedback, and the reduction presented here has application in the problem of stabilizing linear control systems on a Hilbert space. This will be briefly discussed.     

The structure of ?*-inverse semigroups

The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.     

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.This research is supported by the Indian National Science Academy under Young Scientist Project.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

Semigroups of partial isometries

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.     

Closed sectorial forms and one-parameter contraction semigroups

Suppose thats[u, v] is a closed sesquilinear sectorial form with vertex at zero, half-angle α ∈ [0, π/2), and dense domainD(s) in a Hilbert spaceH, S is them-sectorial operator associated withs, S _R is the real part ofS, andT(t)=exp(−tS) is the contraction semigroup with generator −S, holomorphic in the sector |argt|<π/2−α. We characterizes in terms ofT(t). In particular, we prove that the following conditions a`2) the function ‖T(t)u‖ is differentiable at zero; 3) the function (T(t)u, u) is differentiable at zero. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 643–654, May, 1997. Translated by V. E. Nazaikinskii     

正规密码H~#-富足半群的结构

证明了H~#-富足半群S是正规密码H~#-富足半群当且仅当它是完全J~#-单半群的强半格.该结果也是正规密码超富足半群和正规密码群并半群分别在超富足半群和完全正则半群上的相应结构定理的推广.     

Robustness of strongly and polynomially stable semigroups

In this paper we study the robustness properties of strong and polynomial stability of semigroups of operators. We show that polynomial stability of a semigroup is robust with respect to a large and easily identifiable class of perturbations to its infinitesimal generator. The presented results apply to general polynomially stable semigroups and bounded perturbations. The conditions on the perturbations generalize well-known criteria for the preservation of exponential stability of semigroups. We also show that the general results can be improved if the perturbation is of finite rank or if the semigroup is generated by a Riesz-spectral operator. The theory is applied to deriving concrete conditions for the preservation of stability of a strongly stabilized one-dimensional wave equation.     

基半群为0-J-严格单半群的零直并的代数

设Ａ是代数闭域ｋ上的一个具乘基Ｂ的有限维含幺结合代数，称半群Ｂ∪｛０｝为Ａ的基半群．本文给出了０ Ｊ 严格单半群的定义．对于基半群为０ Ｊ 严格单半群的零直并的代数，完全研究了它的代数表示型     

Non-Differentiable Skew Convolution Semigroups and Related Ornstein–Uhlenbeck Processes

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.     

正规密码o-超富足半群的结构

证明了ο-超富足半群S是正规密码ο-超富足半群当且仅当它是完全Jο-单半群的强半格.该结果也是正规密码超富足半群和正规密码群并半群分别在超富足半群和完全正则半群上的相应结构定理的推广。     

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

Markov structures on Clifford algebras

The minimal unitary dilations of contraction semigroups on Hilbert spaces naturally yield systems of orthogonal projections with pre-Markovian properties. Antisymmetric second quantization is a functorial construction on Hilbert space contractions which takes semigroups into doubly Markovian contraction semigroups on a scale of Banach spaces associated with certain Clifford algebras. Multiplicative functionals are introduced which are related to perturbations of these semigroups by a formula of the Feynman-Kac-Nelson type.     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

Singular unitary dilations

We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Some of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) everyC ₀ contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and aC ₀(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.