Abstract : In this talk the properties of some parameters in Banach spaces are discussed, the relationship between these parameters and normal structure are studied, and some well known results are improved.

Abstract : Let W Ì Rⁿ be a bounded convex domain with C² boundary. And given 0 < p, q £ ¥ and a normal weight function j(r), let H_{p, q, j} be the harmonic mixed norm space on W. We prove that the Gleason's problem (W, a, H_{p, q,j}) is always solvable for any reference point a Î W. Also, the Gleason's problem for the harmonic j-Bloch (little j-Bloch) space is solvable. The parallel results for the holomorphic functions on bounded convex domains in Cⁿ are obtained.

Abstract : In this talk, we consider algebraic commutants of certain analytic operator algebras. Let H be a Hilbert space and let B(H) be the algebra of all bounded linear operators. For a subset E of B(H), we denote by E^¢ the algebraic commutant, that is

Let M be a s-finite von Neumann algebra on H and let A be either a maximal subdiagonal algebra of M with respect to a faithful normal expectation F from M onto the diagonal D of A or an analytic operator algebra determined by a flow on M. We prove that the algebraic commutants A^¢ of A is self-adjoint, that is, A^¢ = M^¢. This may regard as an algebraic version of Fuglede's Theorem.

Abstract : The paper is concerned with the study of those subsets of spaces of continuous and bounded functions for which it is possible to find a nonexpansive selection of the metric projection. We first characterize those hyperplanes of these spaces with the above property. Later we use these results in order to find more general subsets with the same property.

Abstract : Let W be the Eucliden space Rⁿ, the upper half-space Rⁿ₊, a special Lipschitz domain or a bounded strongly Lipschitz domain in Rⁿ. Divergence-free Hardy spaces H¹_z,div(W, Rⁿ) on W is the space of divergence-free functions in Hardy space H¹(Rⁿ, Rⁿ) with support in W. We exhibit an divergence-free atomic decomposition and use this to characterize the dual space, and prove a ``div-curl'' type theorem on W. As applications we prove an atomic decomposition for the Hardy-Sobolev space and give coercivity properties for some polyconvex forms.

The main results are:

Atomic Decomposition: f Î H¹_z,div(W, Rⁿ) if and only if it can be written as a sum of H¹_z,div(W, Rⁿ)-atoms.

Dual Space:

Div-Curl Type Theorem on W: Let b Î L²_loc(W, Rⁿ). Then

Abstract : We will give a characterization of operator spaces amongst all Banach K(l²)-bimodules, a characterization of M_n(\BbbC)-bimodule that can be extended to an operator space structure as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. All these three characterizations depend on whether certain natural map from a Banach bimodules to its canonical bidual being isometric.

Abstract : In this talk we introduce the Hardy type inequalities on the Heisenberg group via the Picone type identities of p-sub-Laplacians. Some new results are obtained.

Abstract : To answer a question in relation to analytic signals in time-frequency analysis we prove that an analytic function is an inner function in the unit disc or in the upper-half complex plane if and only if the imaginary part of its boundary value is the Hilbert transform of its real part.

Abstract : In this paper, we obtain some results on the relationship among essential spectrum, deficiency index of M,where M is a 2N-order differential operator, and the number of solutions of My = ly in L²[a,+¥). It is proved that for every l Î I if the number of Dirichlet solution of (M-lI)f = 0 is N (their deficiency indices may be in middle case ), then continuous spectrum of H is empty in I , where H is a self -adjoint operator associated with M . Furthermore if point spectrum has not accumulation point in I , then essential spectrum s_e(H) ÇI = Æ for every self-adjoint operator H associated with M. The results extent Weidmann's result in the limit-point case ( minimal deficiency indices ) to the middle deficiency indices cases.

Abstract : Let u be a solution to a second order elliptic equation with singular potentials belonging to the Kato-FeRerman-Phong's class in Lipschitz domains, an elementary proof of the doubling property for u² over balls is presented, if the balls contain in the domain or center at some points near some open subset of the boundary on which the solution u vanishes continuously. Moreover, we found the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations.

Abstract : It is proved that linear structures can be constructed by means of metrics on translation spaces under certain conditions. The concept of sub-normed Z-linear spaces is introduced and it is clarified that a translation space is a sub-normed Z-linear space if and only if its translation group is Abelian. Analogues of the bounded linear operator theorem, the Hahn-Banach theorem and the resonance theorem are established in sub-normed Z-linear space.

Metrics can induce topologies on a given non-empty set X and different metrics may induce different topologies on X. An interesting question is: Can metrics induce linear structures on X? The aim of the present paper is to answer this question.

The concept of a translation space (X,T) is introdued in [2], where T is the translation group. A much more general form of the Scheeffer's theorem in point set topology has been established in translation spaces, while this paper stresses linear properties rather than topological properties. We prove that a linear structure with coefficients in the ring Z of integers can be endowed with X if T is an Abelian group, and the resulted space is called a sub-normed Z-linear space, briefly, a Z-space. Propositions that are analogues of the bounded linear operator theorem, the Hahn-Banach theorem, and the resonance theorem respectively are obtained in Z-spaces. In section 1, concepts related to translation spaces are briefly introduced which are necessary for the subsequent sections. In section 2, bounded, locally bounded, and continuous operators and their relations are discussed in commutative translation spaces. Section 3 deals with a generalization of the Hahn-Banach theorem and section 4 deals with that of the resonance theorem.Lastly, section 5 is conclusion.

Key words   translation space; sub-normed Z-linear space; Hahn-Banach theorem; resonance theorem

Reference

[1] J.L.Kelley, General Topology, Springer-Verlag, 1955, New York.

[2] Guo-jun Wang, Wei Wang, Generalization of the Scheeffer's theorem, Indian J. Math., 1999,41(3):407-414.

[3] A. Wilansky, Functional Aanlysis, Blaisdell Publishing Company, 1964, New York.

Abstract : We study a class of Volterra type operators on analytic functions. We characterize the boundedness and compactness of the operator in terms of its symbol function.

Abstract : We consider properties of second-order operators H = -å^d_{ij, = 1} ¶_i c_ij¶_j on \Bbb R^d with bounded real symmetric measurable coefficients. We assume that C = (c_ij) ³ 0 almost everywhere, but allow for the possibility that C is singular. We associated with H a canonical self-adjoint viscosity operator H₀ and examine properties of the viscosity semigroup S⁽⁰⁾ generated by H₀. The semigroup extends to a positive contraction semigroup on the L_p-spaces with p Î [1,¥]. It conserves probability, satisfies L₂ off-diagonal bounds and the wave equation associated with H₀ has finite speed of propagation. Nevertheless S⁽⁰⁾ is not always strictly positive, this demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Hölder continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C^2-e (\Bbb R^d) with e > 0.