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.     

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     

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.     

Approximations to fixed points of contraction semigroups in hilbert spaces

Sequences (or curves) are constructed to approximate common fixed points of a pair of nonex-pansive mappings (or contraction semigroups) in Hilbert spaces. The obtained results extend the previously known results from a single mapping to a family of mappings.     

Inverse subsemigroups and classes of finite aperiodic semigroups

We prove a number of results related to finite semigroups and their inverse subsemigroups, including the following. (1) A finite semigroup is aperiodic if and only if it is a homomorphic image of a finite semigroup whose inverse subsemigroups are semilattices. (2) A finite inverse semigroup can be represented by order-preserving mappings on a chain if and only if it is a semilattice. Finally, we introduce the concept of pseudo-small quasivariety of finite semigroups, generalizing the concept of small variety.     

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．     

Eigenvalues and stability properties of multiplication operators and multiplication semigroups

We investigate uniform, strong, weak and almost weak stability of multiplication semigroups on Banach space‐valued ‐spaces. We show that, under certain conditions, these properties can be characterised by analogous ones of the pointwise semigroups. We prove a spectral mapping theorem for the point spectra of the pointwise and global semigroups and apply this as a major tool for determining almost weak stability.     

On the convergence of rational approximations of semigroups on intermediate spaces

We generalize a result by Brenner and Thomée on the rate of convergence of rational approximation schemes for semigroups. Using abstract interpolation techniques we obtain convergence on a continuum of intermediate spaces between the Banach space and the domain of a certain power of the generator of the semigroup. The sharpness of the results is also discussed.

     

Strong convergence of a general viscosity explicit rule for the sum of two monotone operators in Hilbert spaces

In this paper, we study a general viscosity explicit rule for approximating the solutions of the variational inclusion problem for the sum of two monotone operators. We then prove its strong convergence under some new conditions on the parameters in the framework of Hilbert spaces. As applications, we apply our main result to the split feasibility problem and the LASSO problem. We also give some numerical examples to support our main result. The results presented in this paper extend and improve the corresponding results in the literature.     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).     

无限族渐近k-严格伪压缩映象的强收敛定理

在H ilbert空间框架下,给出了迭代逼近无限族渐近k-严格伪压缩映象的公共不动点的杂交投影算法,并证明了一个强收敛定理.此结果改进和推广了原有的相关结果.     

Analytic contractions, nontangential limits, and the index of invariant subspaces

Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of .

We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0.

It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent:

(1) for some ,

(2) for all , ,

(3) has nonzero Lebesgue measure,

(4) every nonzero invariant subspace of has index 1, i.e., satisfies dim .

     

Banach空间中的积分算子不动点定理及其应用

利用不动点指数理论得到了若干个积分算子的不动点定理,这些结论可用于研究许多边值问题.利用这些结论研究了Banach空间中二阶两点边值问题正解的存在性,特别的,在有些情况下去掉了Banach空间中的锥必须是正规的这一条件限制,而这一条件限制在许多文献中是必须的.     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

Stability of solutions of an equation arising in hydromechanics

We study the strong stability of the equation describing small oscillations of an elastic pipe conveying an ideal fluid. We prove the existence and uniqueness theorem for the generalized solution and find sufficient conditions for strong stability in the case of a pulsing fluid. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 15–24, January, 2000.     

The space of -additive mappings on semigroups

In this paper, we introduce the concept of -pseudoadditive mappings from a semigroup into a Banach space, and we provide a generalized solution of Ulam's problem for approximately additive mappings.

     

拟非扩张映像族的公共不动点的迭代方法

引入了修正的杂交投影迭代算法,用来构造Hilbert空间中拟非扩张映像族的公共不动点.使用新的算法证明了几个强收敛定理.新算法的优点是不要求映像具有次闭性质.     

On the H-property of functionals in Sobolev spaces

In this paper, we consider special classes of strongly convex functionals in Sobolev spaces. It is proved that functionals from such classes have the so-called H-property: weak convergence of sequences of arguments and convergence of such sequences with respect to a given functional imply strong convergence.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 378–394.Original Russian Text Copyright © 2005 by A. S. Leonov.This revised version was published online in April 2005 with a corrected issue number.