Vitali's and Harnack's type results for vector-valued functions

In this brief note, we extend Vitali's theorem for holomorphic functions obtained by Arendt and Nikolski to nets of functions of sheaves of smooth vector-valued functions. As a consequence we also extend a Harnack's theorem for compact operator-valued harmonic functions recently obtained by Enflo and Smithies to bounded operator-valued harmonic functions, avoiding the assumption that the Hilbert space H where the operators are defined is separable.     

A quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups

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.     

Operator approach to weighted estimates of singular integrals

In this paper we give a geometric approach (using only the theory of operators in Hilbert space) to L²-weighted estimates of singular integral operators. In this way we are able to get an abstract operator theorem, a special case of which is the familiar theorem of Koosis, and also a generalization of Koosis' theorem to the case of operator-valued weights.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 150–174, 1984.     

Linear equations over cones with interior: a solvability theorem with applications to matrix theory

The solvability of linear equations with solutions in the interior of a closed convex cone is characterized. Corollaries include Lyapunov's theorem characterizing stable matrices and a generalization of Stiemke's theorem of the alternative for complex linear inequalities.     

An analogue of Hunt's representation theorem in quantum probability

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, which now are defined on spaces of functions with values in a von Neumann algebra. We consider a semigroup of probability operators with a continuity property weaker than uniform continuity, and we succeed in characterizing its infinitesimal generator under the additional hypothesis that twice differentiable functions belong to the domain of the generator. Such hypothesis can be proved in some particular cases. In this way a partial quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. Our result provides also a closed characterization of generators of a new class of not norm continuous quantum dynamical semigroups.     

Yosida-Hewitt type decompositions for weakly compact operators and operator-valued measures

We derive Yosida-Hewitt type decompositions for weakly compact operators from Köthe-Bochner function spaces to Banach spaces. As an application, we obtain a Yosida-Hewitt type decomposition for strongly bounded operator-valued measures.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Szegö's limit theorem: The higher-dimensional matrix case

The continuous version of Szegö's theorem gives the first two terms of the asymptotics as α → ∞ of the determinants of certain convolution operators on L₂(0, α) with scalar-valued kernels. Generalizations are known if the kernel is matrix valued or if the interval (0, α) is replaced by αΩ with Ω a bounded set in Rⁿ with smooth boundary. This paper treats the higher-dimensional matrix case. The coefficient in the interesting (second) term is an integral over the contangent bundle of ?Ω of the correponding coefficients of one-dimensional problems.     

Estimates for Elements of Inverse Matrices for a Class of Operators with Matrices of Special Structure

In this paper, we consider questions related to the structure of inverse matrices of linear bounded operators acting in infinite-dimensional complex Banach spaces. We obtain specific estimates of elements of inverse matrices for bounded operators whose matrices have a special structure. Matrices are introduced as special operator-valued functions on an index set. The matrix structure is described by the behavior of the given function on elements of a special partition of the index set. The method used for deriving the estimates is based on an analysis of Fourier series of strongly continuous periodic functions.     

Generalized controllability and inertia theory

Recently the idea of controllability has been used to generalize Lyapunov's theorem and the main inertia theorem. Corresponding results are established in this paper for a large class of linear transformations on the space of n×n Hermitian matrices.     

Generation of semigroups,and two-dimensional quantum lattice systems

We consider two types of conditions on an operator on a Banach space which ensure that it is the generator of a semigroup of contractions. First, S. Sakai's concept of commutative normal ¹-derivations of UHF algebras is generalized to “approximately commutative operators” on Banach spaces. Next we consider the situation in which the domain of the operator contains an increasing sequence of “approximately invariant subspaces,” and generalize results of A. Kishimoto and P. E. T. Jørgensen. A corollary is the existence of time development for two-dimensional quantum lattice models when the average surface energy per unit surface is uniformly bounded in the volume.     

Spectra of singular measures as multipliers on Lp

Stein's theorem on the interpolation of a family of operators between two analytic spaces is generalized, both to a multiply connected domain and to an interpolation between more than two spaces. The theorem is then applied to get setwise upper bounds for spectra of convolution operators on L^p of the circle. In particular the spectra of operators given by convolution by Cantor-Lebesgue-type measures on L^p are determined. The same is done for certain Riesz products. These results are used to derive a result on translation-invariant subspaces of L^p of the circle.     

白噪声分析中广义算子值函数的Bochner-Wick积分

白噪声广义算子在白噪声分析理论及其应用中起着十分重要的作用. 本文主要讨论了白噪声广义算子值函数的积分及相关问题. 主要工作有: 引入了广义算子值测度的概念, 分别讨论了这种测度在象征和算子p-范数意义下的变差及相互关系; 借助于广义算子的Wick积运算, 引入了广义算子值函数关于广义算子值测度的一种积分---Bochner-Wick积分, 讨论了这种积分的性质, 建立了相应的收敛定理并且展示了其在量子白噪声理论中的应用; 探讨了Bochner-Wick积分的Fubini定理及相关问题.     

A remark on commutativity of the image of A-Valued analytic function

Let BL(?) be a complex Banach algebra of all bounded linear operators acting on the Hilbert space ?. J. Globevnik and I. Vidav proved that when the range of an operator-valued analytic function with domain of definition D ? ? consists of the normal operators, then f(D) is a commutative set in the algebra BL(?). The paper strengthens this result for the topological algebras.     

On an upper bound for the number of characteristic values of an operator function

We prove a theorem on an upper bound for the number of characteristic values of an operator-valued function that is holomorphic and bounded in a domain. This estimate is similar to the well-known inequality for zeros of a number function that is holomorphic and bounded in a domain. We derive several corollaries of the theorem proved, in particular, a statement on an estimate of the number of characteristic values of polynomial bundles of operators that lie in a given disk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 211–224, February, 1998. This work was financially supported by the Foundation for Fundamental Research of the Ukrainian Ministry of Science.     

Operators associated with the Jacobi semigroup

The aim of this paper is to introduce some operators induced by the Jacobi differential operator and associated with the Jacobi semigroup, where the Jacobi measure is considered in the multidimensional case.In this context, we introduce potential operators, fractional integrals, fractional derivates, Bessel potentials and give a version of Carleson measures.We establish a version of Meyer’s multiplier theorem and by means of this theorem, we study fractional integrals and fractional derivates.Potential spaces related to Jacobi expansions are introduced and using fractional derivates, we give a characterization of these spaces. A version of Calderon’s Reproduction Formula and a version of Fefferman’s theorem are given.Finally, we present a definition of Triebel–Lizorkin spaces and Besov spaces in the Jacobi setting.     

Operator-valued measures and linear operators

We study operator-valued measures , where L(X,Y) stands for the space of all continuous linear operators between real Banach spaces X and Y and Σ is a σ-algebra of sets. We extend the Bartle-Dunford-Schwartz theorem and the Orlicz-Pettis theorem for vector measures to the case of operator-valued measures. We generalize the classical Vitali-Hahn-Saks theorem to sets of operator-valued measures which are compact in the strong operator topology.