Decomposition of a positive operator

Dedicated to Yuri Grigor'evich Reshetnyak on his sixtieth birthday.     

Classification of positive solutions to an integral system with the poly-harmonic extension operator

In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.     

On the extension of a generalized differentiation operator

A generalized differentiation operator introduced by A. F. Leont'ev is considered in this note. The concept of an L_D-operator is introduced, and necessary and sufficient conditions are determined for the extension of a generalized differentiation operator to some L_D-operator.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 425–436, October, 1969.The present work was completed under the direction of A. F. Leont'ev to whom the author expresses much gratitude.     

Certain classes of extension of a lacunary Hermitian operator

Certain classes of extensions of a lacunary Hermitian operator are described in terms of abstract boundary conditions. The connection between the asymptotic behavior of eigenvalues of an extension near the boundary of lacuna and the asymptotic of the negative spectrum of the corresponding boundary operator is found.Candidate of Phisicomathematical Science.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 215–233, February, 1992.     

The generalized Roper-Suffridge extension operator

Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z₁,z₂,…,z_n), z₁∈D, , p_i?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z₁),f′(z₁)^1/p₂z₂,…,f′(z₁)^1/p_nz_n) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z₁))^1/p_i|_z1=0=1, i=2,…,n. Some other related results are proved.     

On the roper-suffridge extension operator

In 1995, Roper and Suffridge defined an extension operator which maps a locally biholomorphic function on the unit diskD in ℂ to a locally biholomorphic mapping on the unit ballB ⁿ in ℂⁿ. This extension operator preserves many important properties, e.g., convexity and starlikeness, etc. In this note, we introduce the family ofε starlike mappings, and prove that the Roper-Suffridge extension operator preserves theε starlikeness on some Reinhardt domains. This result includes many known results and solves an open problem of Graham and Kohr. Project supported by the National Science Foundation of China.     

A positive compact operator

I give an example of a positive compact operator on L²[0,1] which is not representable by any Lebesgue measurable function on [0,1]². This example can be adapted to answer a question of H.H.Schaefer (§4 below).     

A BIVARIATE EXTENSION OF BLEIMANN－BUTZER－HAHN OPERATOR

Let C( R ^2) be a class of continuous functions f on R ^2.A bivariate extension Ln(f,x,y)of Bleiman-Butzer-Hahn operator is defined and its standard convergence properties are given.Moreover,a local analogue of Voronovskaja theorem is also give for a subclass of C(R ^2).     

A bivariate extension of Bleimann-Butzer-Hahn operator

Let C(R2+ ) be a class of continuous functions f on R2+. A bivariate extension L.(f ,x,y) of BleimannButzer-Hahn operator is defined and its standard convergence properties are given. Moreover, a local analogue of Voronovskaja theorem is also given for a subclass of C(R2+).     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

On the square root of a positive selfadjoint operator

We provide a short, elementary proof of the existence and uniqueness of the square root in the context of unbounded positive selfadjoint operators on real or complex Hilbert spaces.     

Approximate computation of the positive eigenvalue of a positive operator with a nonlinear occurrence of a parameter

A method is indicated for the approximate determination of the positive eigenvalue of the problem x–Qx=0, >0, xK, x0, whereK is a cone in Banach space and Q is an operator-valued function positive relative toK.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 37–39, 1976.