Reduction of operator families to integral form

We establish a condition for reducibility of a family of operators to integral form by an isometric operator the same for the whole family.     

Norm of a multilinear integral operator from product of weighted-type spaces to weighted-type space

We investigate the following multilinear integral operator $$T_K^m(\overrightarrow{f})(x)={\int }_0^{\infty }\dots {\int }_0^{\infty }K(x,t_1,\dots ,t_m){\displaystyle \prod _{j=1}^m}{f}_j(t_j)d{t}_1\dots d{t}_m,$$ where $m\in ?$ and $K:{?}_{+}^{m+1}\to {?}_{+}$ is a continuous kernel function satisfying the condition $$K(x,g_1(x)s_1,\dots ,g_m(x)s_m)=h(x)K(1,s_1,\dots ,s_m),$$ for some functions $g_j,j=\stackrel{￣}{1,m}$, which are continuous, increasing, $g_j({?}_{+})={?}_{+},j=\stackrel{￣}{1,m}$, and a function $h:{?}_{+}\to {?}_{+}$, from a product of weighted-type spaces to weighted-type spaces of real functions. We calculate the norm of the operator, extending and complementing some results in the literature. We also give an explanation for a relation between integrals of an L^p integrable function and its radialization on ${?}^n$.     

Characterizations of positive selfadjoint extensions

The set of all positive selfadjoint extensions of a positive operator (which is not assumed to be densely defined) is described with the help of the partial order which is relevant to the theory of quadratic forms. This enables us to improve and extend a result of M. G. Krein to the case of not necessarily densely defined operators .

     

The Noor integral operator and strongly close-to-convex functions

The purpose of the present paper is to introduce some new subclasses of strongly close-to-convex functions defined by using the Noor integral operator and study their inclusion relationships with the integral preserving properties. Our results include several previous known results as special cases.     

On the norm of an integral operator and applications

In this paper, the norm of an integral operator (r>1) is obtained. As applications, a new bilinear integral operator inequality with the norm and the equivalent forms are given, and some new Hilbert's type inequalities with the best constant factors as the particular cases are established.     

A characterization of semibounded selfadjoint operators

For a class of closed symmetric operators with defect numbers it is possible to define a generalization of the Friedrichs extension, which coincides with the usual Friedrichs extension when is semibounded. In this paper we provide an operator-theoretic interpretation of this class of symmetric operators. Moreover, we prove that a selfadjoint operator is semibounded if and only if each one-dimensional restriction of has a generalized Friedrichs extension.

     

Some subclasses of analytic functions involving the generalized Noor integral operator

In this paper, we define the generalized Noor integral operator by using convolution. By applying this operator, we introduce some subclasses , and of analytic functions and study their subordinate relations, inclusion relations, the integral operator, the sufficient conditions for a function to be in the class and the radius problems.     

On a p-adic Hilbert-type integral operator and its applications

In this note, we deal with a $p$-adic Hilbert-type integral operator induced by a symmetric homogeneous kernel of degree $-1$ and obtain the expression of the norm of this operator. As applications, we establish some new $p$-adic Hilbert-type inequalities with best constant factors.     

On the nonnegativity of operator products

Summary A bounded, not necessarily everywhere defined, nonnegative operator A in a Hilbert space <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathfrak{H}$ is assumed to intertwine in a certain sense two bounded everywhere defined operators B and C. If the range of A is provided with a natural inner product then the operators B and C induce two new operators on the completion space. This construction is used to show the existence of selfadjoint and nonnegative extensions of B*A and C*A.     

一类奇异积分算子的加权估计

假设A是一个Young函数,MD#为广义sharp极大函数.本文首先引进了LA-Hrmander条件,对于满足LA-Hrmander条件的算子T,得到了与T相关的广义sharp极大函数的估计.然后,再利用该估计得到了算子T的加权Lp范数被Hardy-Littlewood极大函数和与Aˉ相关的极大函数的加权Lp范数所控制(0相似文献   

On integral means of star-like functions

We study univalent holomorphic functions in the unit diskU = {z: |z| < 1} of the formf(z)=z+∑ _n=2 ^∞ a _n z ⁿ that satisfy the condition Re zf’(z)/f(z) > α with α∈ [0, 1) inU. Some integral means of such funcions are estimated.     

Some classes of analytic functions involving Noor integral operator

The object of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions defined by using the Noor integral operator. The integral preserving properties in connection with the operator are also considered. Relevant connections of the results presented here with those obtained in earlier works are pointed out.     

On Bernardi's integral operator and the Briot-Bouquet differential subordination

Let A,B,D,E∈[−1,1]. Conditions on A,B,D and E are determined so that

     

On the Integral Operators of the Third Kind

We establish some necessary and sufficient conditions for an operator in the space of square summable functions to be representable as a sum of multiplication by a bounded function and an integral operator.     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

Notes on Jung-Kim-Srivastava integral operator

The object of the present paper is to investigate some properties of certain integral operator Q_β^α introduced and studied recently by Jung, Kim, and Srivastava [J. Math. Anal. Appl. 176 (1993) 138-147].     

On convergence of functions of normal operators in strong operator topology

The theorem in [E. B. Davies, J. London Math. Soc. (2), 4, 435–436 (1972)] is extended. As a corollary, Kadison’s theorem on strong continuity of operator functions is refined.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

About one system of integral equations in kinetic theory

One system of integral convolution equations is considered on a half-line with an noninvertible matrix integral operator whose symbol has a fourth-order zero. The application of a special factorization method makes it possible to distinguish noninvertible factors in the original noninvertible operator and reduce the system to a new system with a nonsingular integral operator. The structural theorem of the existence of a solution to the original system is proved.