A representation theorem of harmonic functions and its application to BMO on manifolds

Abstract. In this paper a representation theorem of harmonic functions on manifolds is set up.As an application, a characterization of BMO in terms of Carleson measures is obtained.     

An extension of Tarski's fixed point theorem and its application to isotone complementarity problems

Tarski's fixed point theorem is extended to the case of set-valued mappings, and is applied to a class of complementarity problems defined by isotone set-valued operators in a complete vector lattice.     

An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces

The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations with continuous jointly in and of class . We also prove a bifurcation theorem for critical points of the function which is just continuous and changes at an isolated minimum (in ) to isolated maximum when passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.

     

On translation invariant operators which preserve the B-spline recurrence

It was observed in [4] that the Hilbert transform of the univariate B-spline preserves the B-spline recurrence. Motivated by this observation, we characterize translation invariant operators that preserve the multivariate B-spline recurrence and analogous results are also provided for the multivariate cube spline. Charles A. Micchelli was supported in part by the US National Science of Foundation under grant CCR-0407476. Yuesheng Xu was supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation of China under grant 10371122, by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Scientists” and by Ministry of Education, People’s Republic of China, under the Changjian Scholarship through Zhongshan University.     

An application of the projective spectral theorem to commuting families of operators

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 4, pp. 469–481, July–August, 1988.     

An application of the Putnam-Fuglede theorem to normal products of self-adjoint operators

We prove that if we have two self-adjoint operators (bounded or not) and if their product is normal, then it is self-adjoint provided a certain condition is satisfied.

     

Weighted composition operators and the Gleason-Kahane-Zelazko theorem

The main purpose of this paper is to investigate extensions of the Banach-Stone theorem and of the Holsztynski theorem to some locally multiplicatively convex associative algebras. The proofs are based on a generalization of a theorem, due to A. Gleason, J.-P. Kahane and W. Zelazko, characterizing continuous characters of a unital associative Banach algebra among all its linear forms.     

An extension theorem to rough paths

We show that any continuous path of finite p-variation can be lifted to a geometric q  -rough path, where q>p$q>p$.     

An application of the theorem of Rudin to the Toeplitz operators on odd spheres

Summary In this note we will prove a theorem on Toeplitz operators on odd spheres, using the recent result of Rudin [3, Th. 2.3] concerning space of typeH+C. This theorem is a generalization of the analogous fact for Toeplitz operators on the unit circle, proved by Douglas [2, Th. 7.29].     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

An extension to a theorem of Himmelberg

This paper extends a widely used theorem of Himmelberg to topological vector spaces whose completion have a separating dual.     

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].