Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S‐spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Navier–Stokes equations with free convection in three‐dimensional unbounded triangular channels

The quaternionic calculus is a powerful tool for treating the Navier–Stokes equations very elegantly and in a compact form, through the evaluation of two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. In this paper, we use special variants of quaternionic‐holomorphic multiperiodic functions in order to obtain explicit formulas for unbounded three‐dimensional parallel plate channels, rectangular block domains and regular triangular channels. Copyright © 2007 John Wiley & Sons, Ltd.     

Relative oscillation theory for Dirac operators

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.     

Geometric properties of root vectors for equation of nonhomogeneous damped string: transformation operators method

The spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space is obtained in the paper. These operators are the dynamics generators for the systems governed by 1–dim hyperbolic equations with spatially nonhomogeneous coefficients containing first order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and also damped at the boundary points. The main result leading to the spectral decomposition is the fact that the generalized eigenvectors (root vectors) of the above operators form Riesz bases in the corresponding energy spaces. The proofs are based on the transformation operators method. The classical concept of transformation operators is extended to the equation of damped string. Originally, this concept was developed by I. M. Gelfand, B. M. Levitan and V. A. Marchenko for 1–dim Schrödinger equation in connection with the inverse scattering problem. In the classical case, the transformation operator maps the exponential function (stationary wave function of the free particle) into the Jost solution of the perturbed Schrödinger equation. For the equation of a nonhomogeneous damped string, it is natural to introduce two transformation operators (outgoing and incoming transformation operators). The terminology is motivated by an analog with the Lax—Phillips scattering theory. The transformation operators method is used to reduce the Riesz bases property problem for the generalized eigenvectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators. The latter problem is solved based on the spectral asymptotics and known facts about exponential families. The main result presented in the paper means that the generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of non—selfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.     

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.     

Abstract Volterra operators

We propose a new general definition of Volterra operators. Several types of evolutionary operators, including Volterra ones in the sense of A.N. Tikhonov, satisfy this definition. For equations with generalized Volterra operators we introduce the notions of local, global, and maximally extended solutions. For solutions to nonlinear equations we formulate the existence, uniqueness, and extendability conditions. The theorems proved in this paper imply both known and new results on the solvability of concrete equations. We adduce an example of the application of obtained results to the study of the Cauchy problem for functional differential equations.     

Absolute continuity of spectral shift

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.     

Two classes of operators with irreducibility and the small and compact perturbations of them

This paper gives the concepts of finite dimensional irreducible operators((FDI) operators)and infinite dimensional irreducible operators((IDI) operators). Discusses the relationships of(FDI)operators,(IDI) operators and strongly irreducible operators((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an(FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in(Σ FDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in(Σ FDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where(ΣFDI)(X) := {T∈B(X) : T=Σki=1Ti, Ti ∈(FDI), k ∈ N}.     

Error estimates for successive approximations and spectral properties of linear operators

The aim of this paper is to describe some relations between the convergence speed of successive approximations to solutions of linear operator equations, on the one hand, and various spectral properties of the corresponding operators, on the other. We shall show, in particular, that the estimates for the convergence speed of successive approximations is basically determined by certain properties of the pheripheral spectrum of the operator involved (recall that the peripheral spectrum is that part of the spectrum which lies on the boundary, i.e. consists of numbers with absolute values equal to the spectral radius). Equivalently, the convergence speed is characterized by the growth of the (Fredholm) resolvent when approaching the peripheral spectrum. Interestingly, these properties are essentially different for Volterra and non-Volterra operators, where by Volterra operator we mean, as usual, an operator whose spectrum consists only of zero.     

On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Similarity of Perturbed Multiplication Operators

Let S be the multiplication operator by an independent variable x in L ₂(0,1), and let V be an integral operator of Volterra type. In this note, we find sufficient conditions for the similarity of the operators T := S + V and S and discuss some generalizations to an abstract setting of the results obtained.     

Degenerate integro-differential operators in Banach spaces and their applications

In this paper we consider linear integro-differential equations in Banach spaces with Fredholm operators at the highest-order derivatives and convolution-type Volterra integral parts. We obtain sufficient conditions for the unique solvability (in the classical sense) of the Cauchy problem for the mentioned equations and illustrate the abstract results with pithy examples. The studies are carried out in classes of distributions in Banach spaces with the help of the theory of fundamental operator functions of degenerate integro-differential operators. We propose a universal technique for proving theorems on the form of fundamental operator functions.     

On the double commutation method

We provide a complete spectral characterization of the double commutation method for general Sturm-Liouville operators which inserts any finite number of prescribed eigenvalues into spectral gaps of a given background operator. Moreover, we explicitly determine the transformation operator which links the background operator to its doubly commuted version (resulting in extensions and considerably simplified proofs of spectral results even for the special case of Schrödinger-type operators).

     

p-ω-亚正规算子的一些性质

In this paper,let T be a bounded linear operator on a complex Hilbert H.We give and prove that every p-w-hyponormal operator has Bishop's property(β)and spectral properties;Quasi-similar p-w-hyponormal operators have equal spectra and equal essential spectra.Finally,for p-w-hyponormal operators,we give a kind of proof of its normality by use of properties of partial isometry.     

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

     

On the Similarity between Volterra Operators and Transformation Operators for Integro-Differential Operators of Fractional Order

We establish the similarity between certain Volterra integral operators and the Riemann--Liouville fractional integration operator as well as the existence of a triangular transformation operator for integro-differential equations of fractional order. The results obtained are consistent with similar results for the case of integer order.     

An unusual self-adjoint linear partial differential operator

In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

     

Cauchy Integral Formulae in Quaternionic Hermitean Clifford Analysis

The theory of complex Hermitean Clifford analysis was developed recently as a refinement of Euclidean Clifford analysis; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean Dirac operators constituting a splitting of the traditional Dirac operator. In this function theory, the fundamental integral representation formulae, such as the Borel?CPompeiu and the Clifford?CCauchy formula have been obtained by using a (2 ×?2) circulant matrix formulation. In the meantime, the basic setting has been established for so-called quaternionic Hermitean Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitean monogenic functions, of four Hermitean Dirac operators in a quaternionic Clifford algebra setting. In this paper we address the problem of establishing a quaternionic Hermitean Clifford?CCauchy integral formula, by following a (4?× 4) circulant matrix approach.     

Elements of the Theory of Linear Volterra Operators in Banach Spaces

The new definition of Volterra operator introduced in [5] allows specification of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations.     

两不同部件并联可修系统解的稳定性

用强连续算子半群理论证明了两不同部件并联可修系统解的存在唯一性和非负性，并通过研究相应算子的谱特征得到了该系统的稳定性。