On totally *-paranormal operators

In this paper we study some properties of a totally *-paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally *-paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally *-paranormal operators through the local spectral theory. Finally, we show that every totally *-paranormal operator satisfies an analogue of the single valued extension property for W ²(D, H) and some of totally *-paranormal operators have scalar extensions.     

Strongly irreducible operators on Banach spaces

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.     

Products of positive operators

A new, very simple proof is given of a result of P. Y. Wu which asserts that every unitary operator on an infinite-dimensional Hilbert space is a product of positive operators.

     

非游荡算子的性质*

本文研究一类具有混纯性质的线性算子:非游荡算子,该类算子仅在无穷维线性空间中.我们给出非游荡算子紧集上的超循环分解.     

The spectrum is continuous on the set of quasi-n-hyponormal operators

In this paper it is shown that the spectrum σ, a set-valued function, is continuous when the function is restricted to the set of all ‘quasi-n-hyponormal’ operators acting on an infinite-dimensional separable Hilbert space, where a quasi-n-hyponormal operator is defined to be unitarily equivalent to an n×n upper triangular operator matrix whose diagonal entries are hyponormal operators.     

Sums of hypercyclic operators

Every bounded operator on a complex infinite-dimensional separable Hilbert space can be written as the sum of two hypercyclic operators, and also as the sum of two chaotic operators.     

Super-Ergodic operators

The aim of this work is to study operators naturally connected to Ergodic operators in infinite-dimensional Banach spaces, such as Uniform-Ergodic, Cesaro-bounded and Power-bounded operators, as well as stable and superstable operators. In particular, super-Ergodic operators are introduced and shown to be strictly between Ergodic and Uniform-Ergodic operators, and that any power bounded operator is super-Ergodic in a superreflexive space. New relationships between these operators are shown, others are proven to be optimal or can be ameliorated according to structural properties of the Banach space, such as the superreflexivity or with unconditional basis.

     

Spectral representations and density operators for infinite-dimensional homogeneous random fields

A theory of spectral representations and spectral density operators of infinite-dimensional homogeneous random fields is established. Some results concerning the form of the spectral representation are given in the general infinite-dimensional case, while the results pertaining to the density operator are confined to Hilbert space valued fields. The concept of a purely non-deterministic (p.n.d.) field is defined, and necessary and sufficient conditions for the property of p.n.d. are obtained in terms of the spectral density operator. The theory is developed using some isomorphisms induced by families of self-adjoint operators in the linear second order space associated with the field. The method seems to lead to more direct results also in the random process case, and it sheds new light on concepts such as multiplicity of the field and rank of the spectral density operator.     

To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

Small Transitive Families of Dense Operator Ranges

In complex, separable, infinite-dimensional Hilbert space there exist 5 proper dense operator ranges with the property that every operator leaving each of them invariant is a scalar multiple of the identity. The algebra of operators leaving a pair of proper dense operator ranges invariant can have an infinite nest of invariant subspaces. A slight extension of Foiaş' Theorem shows that it can also have a non-trivial reducing subspace. Submitted: July 13, 2001? Revised: December 6, 2001.     

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.     

Approximation by chaotic operators and by conjugate classes

We show that every bounded linear operator on a separable, infinite-dimensional Hilbert space H is the sum of two operators in the norm-closure of the set of operators on H that are chaotic in the sense of Devaney. We also observe that the density of several classes of cyclic operators, with respect to the strong operator topology, may be derived from a result by Hadwin et al. (Proc Amer. Math. Soc. 76 (1979) 250-252).     

Suboptimality Estimates for the Semi-Discretized LQR Problem for Parabolic PDEs

The linear quadratic regulator problem (LQR) for parabolic partial differential equations (PDEs) has been understood to be an infinite-dimensional Hilbert space equivalent of the finite-dimensional LQR problem known from mathematical systems theory. The matrix equations from the finite-dimensional case become operator equations in the infinite-dimensional Hilbert space setting. A rigorous convergence theory for the approximation of the infinite-dimensional problem by Galerkin schemes in the space variable has been developed over the past decades. Numerical methods based on this approximation have been proven capable of solving the case of linear parabolic PDEs. Embedding these solvers in a model predictive control (MPC) scheme, also nonlinear systems can be handled. Convergence rates for the approximation in the linear case are well understood in terms of the PDE's solution trajectories, as well as the solution operators of the underlying matrix/operator equations. However, in practice engineers are often interested in suboptimality results in terms of the optimal cost, i.e., evaluation of the quadratic cost functional. In this contribution, we are closing this gap in the theory. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A short proof of existence of disjoint hypercyclic operators

We give a short proof of existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Fréchet space. Similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.     

A Thomson’s Principle and a Rayleigh’s Monotonicity Law for Nonlinear Networks

We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.     

关于无穷维算子的有限维最佳逼近

A method of approaching to the infinite-dimensional linear operators by the finite-dimensional operators is discussed. It is shown that,for every infinite-dimensional operator A and every natural number n,there exists an n-dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n-dimensional optimal approximations to be unique is obtained.     

The solutions to some operator equations with corresponding operators not necessarily having closed ranges

In this paper, we study the solvability of the operator equations in the general setting of infinite-dimensional Hilbert space with corresponding operators no necessarily having closed range. We get the necessary and sufficient conditions for the existences of the solutions and obtain formulae in each case for the general selfadjoint, positive and real positive solutions to these operator systems.     

Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from--and considerably shorter than--the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

     

Noncommutative Grassmannian <Emphasis Type="Italic">U</Emphasis>(1) sigma model and a Bargmann-Fock space

We consider a Grassmannian version of the noncommutative U(1) sigma model specified by the energy functional E(P) = ‖[a, P]‖ _HS ² , where P is an orthogonal projection operator in a Hilbert space H and a: H → H is the standard annihilation operator. With H realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator [a, P] is densely defined in H for a certain class of projection operators P with infinite-dimensional ranges and kernels. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 347–357, December, 2007.     

二次算子族及非负无穷维Hamilton算子的谱分布

本文讨论了一类在弦和梁的微小振动中出现的二次算子族L(λ)=λ~2MλK-A的谱分布问题,进而将所得结论与无穷维Hamilton算子联系起来,利用无穷维Hamilton算子的特殊结构,得到了一类非负无穷维Hamilton算子的谱分布,这为无穷维Hamilton算子的半群方法提供了理论保证.