Session 5: Harmonic Analysis, Functional Analysis The Normal Structure and Some Parameters in Banach Spaces

Ji Gao
Department of Mathematics
Community College of Philadelphia
1700 Spring Garden St., Philadelphia, PA 19130-3991, U.S.A.

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.

The Gleason's Problem for Harmonic Mixed Norm Spaces

Zhangjian Hu
Department of Mathematics
Huzhou Teachers College
Huzhou, Zhejiang, 313000, P.R.China

Abstract : Let W Rn be a bounded convex domain with C2 boundary. And given 0 < p, q and a normal weight function j(r), let Hp, q, j be the harmonic mixed norm space on W. We prove that the Gleason's problem (W, a, Hp, q,j) is always solvable for any reference point a 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.

Commutants of Certain Analytic Operator Algebras

Guoxing Ji
College of Mathematics and Information Science
Shaanxi Normal University
Xi'an, 710062, P. R. China

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

E = {X B(H): AX = XA, "A E}.
If T B(H), we call {T} the algebraic commutant of T. The well-known Fuglede's Theorem says that if N is normal and X commutes with N, so does X*. That is, the algebraic commutant {N} of N is self-adjoint. Note that {N} is the same as the commutant of the algebra generated by N and I, which is non-self-adjoint in general. Thus it may be asked that the algebraic commutant of which subalgebra is self-adjoint. For example, if a subalgebra either is self-adjoint or every element in it is normal, then its algebraic commutant is self-adjoint. In general, this problem is not particularly interesting. However special cases of this problem are interesting. F. Gilfeather and D.R. Larson in 1982 showed that the algebraic commutant of a nest subalgebra of a von Neumann algebra is self-adjoint. We note that a nest subalgebra of a von Neumann algebra is a kind of analytic operator subalgebras. Thus it is interesting to consider this problem for general analytic operator algebras.

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.

Metric Projection on C(K) Spaces

Genaro López
Department of Mathematics
Universidad de Sevilla
Seville 41080, Spain

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.

Divergence-free Hardy Space on Lipschitz Domains

Zengjian Lou
Department of Mathematics
Shantou University
Guangdong Shantou 515063
P. R. China

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 H1z,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 H1z,div(W, Rn) if and only if it can be written as a sum of H1z,div(W, Rn)-atoms.

Dual Space:

H1z,div(W, Rn)* BMOr,curl(W,Rn).

Div-Curl Type Theorem on W: Let b L2loc(W, Rn). Then

u, v 

b . ( u ×v )  dx ~ || b||BMOr,curl(W, Rn),
where the supremum is taken over all u,  v W1,20(W, Rn) with ||u||L2, ||v||L2 1, W1,20(W, Rn) is the Sobolev space.

Some Special Banach Bimodules

Chi-Keung Ng
Department of Mathematics
Nankai University

Abstract : We will give a characterization of operator spaces amongst all Banach K(l2)-bimodules, a characterization of Mn(\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.

Hardy Type Inequalities on Heisenberg Group

Pengcheng Niu
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an, Shanxi, P.R.China

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.

Hilbert Transform Characterization of Boundary Values of Inner Functions

Tao Qian
Department of Mathematics
University of Macau
P.O. Box 3001, Macao

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.

The Number of Dirichlet Solutions and Discreteness
of Spectrum of Differential Operators
with Middle Deficiency Indices

Jiong Sun*
Department of Mathematics
Inner Mongolia university
Huhhot 010021 P.R.China

D.E. Edmunds
Department of Mathematics
Mantell Building
University of Sussex
Brighton BN1 9RF, U.K.

Aiping Wang
Department of Mathematics
Inner Mongolia university
Huhhot 010021 P.R.China

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 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 se(H) I = 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.

On the Unique Continuation Properties for Elliptic Operators with Singular Potentials

Xiangxing Tao
Faculty of Science
Ningbo University
Ningbo, Zhejiang, 315211, P.R. China

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.

Intrinsic linear Structure on Translation Spaces-metric Induced Linear Structure

Guojun Wang
Institute of Mathematics
Shaanxi Normal University
Xi'an 710062, China

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


[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.

Volterra Type Operators on Analytic Function Spaces

Zhijian Wu
Department of Mathematics
The University of Alabama
Tuscaloosa, Alabama 35487, U.S.A.

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.

Second-Order Operators with Degenerate Coefficients

Yueping Zhu
Department of Mathematics
Nantong University
Nantong City, Jiangsu Province 226007, P.R. China

Joint work with Tom ter Elst, Derek W Robinson and Adam Sikora

Abstract : We consider properties of second-order operators H = -dij, = 1 i cijj on \Bbb Rd with bounded real symmetric measurable coefficients. We assume that C = (cij) 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 Lp-spaces with p [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 C2-e (\Bbb Rd) with e > 0.

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On 21 Dec 2004, 12:18.