Nonzero fixed points of power-bounded linear operators

This paper provides a variety of sufficient conditions for the existence of a nonzero fixed point of a power-bounded linear operator defined on a real Banach space. In the case of power-bounded positive operators on a Banach lattice, among the conditions we provide are not being strongly stable along with commuting with a compact operator or being quasicompact. These results apply directly to Markov operators. In the case of an arbitrary power-bounded operator on a Hilbert space, being uniformly asymptotically regular and not strongly stable guarantees the existence of a nonzero fixed point.

     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.     

Universal and chaotic multipliers on spaces of operators

We use tensor product techniques to study universality, hypercyclicity and chaos of multipliers defined on operator ideals and of multiplication operators on the space of all continuous and linear operators, thus continuing the work of Kit Chan. We also obtain the first examples of outer multipliers on a Banach algebra which are chaotic in the sense of Devaney, and prove sufficient conditions for the existence of closed subspaces of universal vectors for operators between Fréchet spaces.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

Hypercyclic subspaces of a Banach space

Recently a lot of research has been done on hypercyclicity of a bounded linear operator on a Banach space, based on the hypercyclicity criterion obtained by Kitai in 1982, and independently by Gethner and Shapiro in 1987. By combining this criterion with one extra condition, Montes-Rodríguez obtained in 1996 a sufficient condition for the operator to have a closed infinite dimensional hypercyclic subspace, with a very technical proof. Since then, this result has been used extensively to generate new results on hypercyclic subspaces. In the present paper, we give a simple proof of the result of Montes-Rodríguez, by first establishing a few elementary results about the algebra of operators on a Banach space.     

广义正交分解定理与Tseng度量广义逆

本文将Banach空间中广义正交分解定理从线性子空间拓广至非线性集—太阳集,分别给出了一算子为度量投影算子和一度量投影算子为有界线性算子的充要条件;得到了判别Banach空间中子空间广义正交可补的充要条件;建立了王玉文和季大琴(2000年)新近引入的Banach空间中的线性算子的Tseng度量广义逆存在的特征刻划条件;这些工作本质地把王玉文等人的新近结果从自反空间拓广至非自反空间的情形.     

Nonlinear impulsive systems on infinite dimensional spaces

In this paper we consider two different classes of nonlinear impulsive systems one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures. The later class is easily extended to systems driven by general vector measures. The principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of Banach spaces called Gelfand triple. The other nonlinear operators are more regular, non-monotone continuous operators with respect to suitable Banach spaces. We present here a new result on compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.     

Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity

A new continuity theorem of minimum selection is presented for a continuous set-valued operator from a topological space into a Banach space with some uniform convexity. As applications, some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Banach空间的Lipschitz对偶及其应用

本文引进Banach空间E的一个全新对偶空间概念—Lipschitz对偶空间,并证明：任何Banach空间的Lipschitz对偶空间是某个包含E的Banach空间的线性对偶空间,以所引进的新对偶空间为框架,本文定义了非线性Lipschitz算子的Lipshitz对偶算子,证明：任何非线性Lipschitz算子的Lipschitz对偶算子是有界线性算子．所获结果为推广线性算子理论到非线性情形（特别,运用线性算子理论研究非线性算子的特性）开辟了一条新的途径．作为例证,我们应用所建立的理论证明了若干新的非线性一致Lipschitz映象遍历收敛性定理.     

Random commutation

We investigate the commutation between a continuous linear random operator and a continuous linear deterministic operator on a Banach space. From this we obtain probabilistic versions of theorems by Fuglede and Putnam, both of them dealing with the commutation between continuous linear operators with continuous normal operators on a Hilbert space.     

Disintegration of Gaussian measures and average-case optimal algorithms

It is shown that a Gaussian measure in a given infinite-dimensional Banach space always admits an essentially unique Gaussian disintegration with respect to a given continuous linear operator. This covers a similar statement made earlier in [Lee and Wasilkowski, Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2(1) (1986) 12–43.] for the case of finite-rank operators.     

On the concept of generalized solution of operator equations in banach spaces

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9. pp. 1286–1290, September, 1996.     

A new concept of a generalized solution of operator equations in a Banach space

We define a new notion of a generalized solution of an operator equation with a closed linear operator in a Banach space as an element of the completion of this space with respect to some locally convex topology. We prove a theorem on the existence and uniqueness of the generalized solution. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 93–99.     

Frames for vector spaces and affine spaces

A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields.     

A model for operators with cyclic adjoint

It is proved that every bounded linear operator on a complex Banach space whose adjoint has aw ^*-cyclic vector, is similar to the differentiation operator on a Banach space of entire functions of finite exponential type. The relation of this model to the existence of non-trivial invariant subspaces is discussed.     

Dixmier's theorem for sequentially order continuous Baire measures on compact spaces

We prove that a Baire measure (or a regular Borel measure) on a compact Hausdorff space is sequentially order continuous as a linear functional on the Banach space of all continuous functions if and only if it vanishes on meager Baire subsets, a result parallel to a much earlier theorem of Dixmier. We also give some results on the relation between sequentially order continuous measures on compact spaces and countably additive measures on Boolean algebras.

     

Banach空间中有限秩拟线性投影算子的逼近问题

证得:在Banach空间中,相对紧集上的恒等算子可由一列有限秩连续拟线性投影算子一致逼近.由此得到:线性算子为紧线性算子必须且仅须它可由一列有限秩连续齐性算子一致逼近.     

Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.     

Dilations for systems of imprimitivity acting on Banach spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-known result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space. This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.