HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 30, No. 1, 2004

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics

Contents

Ismail M. Idris, Department of Mathematics, Faculty of Science, Ain-Shams University, Cairo 11566, Egypt; current address: Mathematics Department, Faculty of Science, UAE university, P.O.Box 17551, Al-Ain, United Arab Emirates (ismail.idris@uaeu.ac.ae).
*-Value Functions of *-Rings, pp. 1-9.
ABSTRACT. The notion of a *-value function on a noncommutative ring with involution is studied. The results obtained generalize those of valuations in the case of a commutative ring.

Thomas M. McCall, Charles J. Parry, Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 (Parry@math.vt.edu), and Romona R. Ranalli, Department of Mathematics, 3900 University Boulevard, University of Texas at Tyler, Tyler, TX 757999.
The 2-Class Group of Certain Number Fields, pp. 11-26.
ABSTRACT. In this article we describe a method for determining the structure of the 2 - class group of a bicyclic biquadratic extension of an arbitrary number field with odd class number. We also determine all imaginary octic fields of type (2,2,2) having class number less than or equal 16 or prime class number.

Vadim Ponomarenko, Department of Mathematics, Trinity University, 715 Stadium Drive, San Antonio, Texas 78212-7200 (vadim@trinity.edu).
Reduction of Jump Systems, pp. 27-33.
ABSTRACT. A jump system is a set of integer lattice points satisfying an exchange axiom. We discuss an operation on lattice points, called reduction, that preserves the jump system two-step axiom. We use reduction to prove a weakened version of a matroid conjecture by Rota.

Young Suk Choi and Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu, 702-701, KOREA (yjsuh@bh.knu.ac.kr) and Jung-Hwan Kwon, Department of Mathematics Eduacation, Taegu University, 702-701, Korea (jhkwon@biho.daegu.ac.kr).
On Chern Type Problems in Space-Like Complex Submanifolds of an Indefinite Complex Hyperbolic Space, pp. 35-54.
ABSTRACT. In this paper, we introduce a kind of Chern type problem in an n-dimensional complete space-like submanifold of an (n+p)-dimensional indefinite complex hyperbolic space of constant holomorphic sectional curvature c with signature (2p, 2(n+p)). Moreover, we give a best possible estimation for the norm of the second fundamental form of a complex quadric Q^n immersed in indefinite complex hyperbolic space with signature (2,2(n+1)).

W. Gu, 1250 N. Dartmouth Ave., Claremont, CA 91711} (gu@math.hmc.edu), and Christopher Pries, 340 E. FootHill Blvd., Claremont, CA 91711 (cpries@hmc.edu).
Examples of Cayley 4-Manifolds, pp. 55-87.
ABSTRACT. We determine several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds and Cayley 4-cycles in Calabi-Yau 4-folds and Spin(7) holonomy manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry.
We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself. We consider graphs which are invariant under the action of three dimensional subgroups of Spin(7) which fix the quaternions as a subgroup of the Cayley numbers. Spin(7) is a subgroup of SO(8) which preserves the Cayley form. Systems of ODEs and PDEs are derived and solved, some special cases of a classic theorem of Harvey and Lawson are investigated, and theorems aiding in the classification of all such manifolds described here are proven. Several families of interesting Cayley 4-dimensional manifolds are discovered. Some of them are novel.

Sophia Zafiridou, Department of Mathematics, University of Patras, 26500 Patras, Greece (zafeirid@math.upatras.gr).
Planar Rim-Scattered Compactifications of Planar Spaces, pp. 89-97.
ABSTRACT. We prove that a space X admits a planar rim-scattered compactification iff there exists a homeomorphism h of X into the plane such that h(X) is nowhere dense in the plane and each point of the closure of h(X) in the plane is contained in the interior of an arbitrarily small disk, whose boundary intersects h(X) in a set with a scattered closure in the plane.

Gerardo Acosta, Instituto de Matematicas, Circuito Exterior, Ciudad Universitaria, Area de la Investigacion cientifica, Mexico, D. F., 04510, MEXICO (gacosta@math.unam.mx).
On Smooth Fans and Unique Hyperspaces, pp. 99-115.
ABSTRACT. In this paper we show that if X is a smooth fan and Y is a fan such that the hyperspaces of subcontinua C(X) and C(Y) are homeomorphic, then X and Y are homeomorphic. This is a generalization of a result by C. Eberhart and S. B. Nadler, Jr.

Michael E. Taylor, Math. Dept., Univ. of North Carolina Chapel Hill, NC 27599-3250 (met@math.unc.edu).
Fourier Series and Lattice Point Problems, pp. 117-135.
ABSTRACT. Here we study functions on the torus satisfying a thin-shell Fourier series estimate, a phenomenon that arose in the study of pointwise convergence of Fourier series in previous work of the author. We consider classes of functions with conormal singularities, and see that the geometry of the singular set influences the nature of thin-shell estimates. This analysis brings in certain non-isotropic lattice point estimates, and these estimates in turn lead to variants of stationary phase methods, investigated by studying Schrodinger equations with singular initial data.

Steven M. Seubert, Bowling Green State University, Bowling Green, OH, 43403-0221 (sseuber@bgnet.bgsu.edu).
Semigroups of Analytic Toeplitz Operators on H², pp. 137-145.
ABSTRACT. Using a result of D. Suarez, we show that a closed operator R having domain D(R) dense in the Hardy space H² of the open unit disc and commuting with the standard unilateral shift S on D(R) is given by an unbounded analytic Toeplitz operator R = T_C having symbol C from the Nevanlinna class N⁺.
Using this result, we show that any collection of bounded linear operators R_t on H² commuting with S defines a C₀-semigroup if and only if there exists a function C analytic and having real part bounded above on the open unit disc for which R_t = T_e^tC for all nonnegative numbers t.

David W. Kribs, Department of Mathematics and Statistics, University of Guelph, Guelph, ON, CANADA N1G 1M8 (dkribs@uoguelph.ca).
Non-Selfadjoint operator algebras generated by weighted shifts on Fock space, pp. 147-169.
ABSTRACT. Non-commutative multi-variable versions of weighted shifts arise naturally as `weighted' left creation operators acting on Fock space. We investigate the weak operator topology closed algebras they generate. The unweighted case yields non-commutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the non-commutative setting. We prove these algebras are reflexive when the eigenvalues for the adjoint algebra include an open set in complex n-space, and provide a new elementary proof of reflexivity for the unweighted case. We compute eigenvalues for the adjoint algebras in general, finding geometry not present in the single variable setting. Motivated by this work, we obtain general information on the spectral theory for non-commuting n-tuples of operators.

Taskinen, Jari , University of Joensuu, FIN 80101 Joensuu, Finland (jari.taskinen@joensuu.fi).
On the Continuity of Bergman and Szego Projections, pp. 171-190.
ABSTRACT. We study the triplet of function spaces, call them H, h, and L, of analytic, harmonic and measurable functions on the open unit disk of the complex place. The following facts hold: the Bergman projection is continuous from L onto H, the Szego projection is continuous from h onto H, and harmonic conjugation is an isomorphism on h. We show that these spaces are in a sense the smallest extensions of the classical Banach space of bounded analytic functions (and related spaces) which have the above mentioned property.

J. Pecaric, Faculty of Textile Technology, University of Zagreb, Pierottieva 6, 10000 Zagreb, Croatia (pecaric@hazu.hr), J. Micic, Electrical Engineering Department, Polytechnic of Zagreb, Konavoska 2, 10000 Zagreb, Croatia (jmicic@public.srce.hr), and Y. Seo, Tennoji Branch, Senior Highschool, Osaka Kyoiku University, Tennoji, Osaka 543-0054, Japan (yukis@cc.osaka-kyoiku.ac.jp).
Inequalities Between Operator Means Based on the Mond-Pecaric Method, pp. 191-207.
ABSTRACT. As a continuation of the previous paper  by  J. Micic, J. Pecaric , and Y. Seo, Complementary inequalities to inequalities of Jensen and Ando based on Mond- Pecaric method, Linear Alg. Appl., 318 (2000), 87--107, we show further general complementary inequalities to operator inequalities on a positive linear map associated with two operator means.

Raymond Mortini, Université de Metz, F-57045 Metz, France ( mortini@poncelet.sciences.univ-metz.fr).
Maximal Gleason parts and support sets for trivial points, pp. 209-218.
ABSTRACT. The paper is concerned with the topological space S of trivial points in the algebra of bounded analytic functions in the open unit disk. It is shown that  every point in the closure of the set of trivial points outside the fiber  M₁ has a maximal Gleason part within M₁.  Also, every point  in  M₁ which lies in the closure of E\ M₁ for an  interpolation set E of trivial points  has maximal support.

Alec Matheson,Department of Mathematics, Lamar University, Beaumont TX 77710 (matheson@math.lamar.edu).
Isometries into Function Algebras, pp. 219-230.
ABSTRACT. The isometries from a uniform algebra into another come in two distinct types. The Type 1 isometries are associated with a set of uniqueness in the Shilov boundary of the range, and are completely described for certain algebras of analytic functions. Some more or less concrete examples of Type 2 isometries are also given.

Yiqiang Liu and Yifeng Xue (xyf63071@public9.sta.net.cn), Department of Mathematics, East China University of Science and Technology, Shanghai 200237, P.R. China and Fahui Zhai, Qingdao Chemical Technology College, Qingdao 266042, P.R. China.
The closures of (U+K)-orbits of certain essentially normal models, pp. 231-244.
ABSTRACT. Let Ω be a simply connected analytic Cauchy domain and μ be a measure on ∂Ω equivalent to the arc length measure on ∂ Ω. Let M(Ω, μ) be the multiplication operator on the Hilbert space of all μ--integrable functions on ∂Ω which are analytic in Ω. Let M(i) be the direct sum of i M(Ω, μ) s' and M[j] be the direct sum of j M(Ω*, μ) s'. In this paper, we determine the closure of (U+K)-orbit of the operator M(i)⊕ M[j] with i and j finte. This solves the problem presented by M. Dost\'al in his Ph.D thesis.

Espínola, Rafael, Departamento de Analisis Matematico, Universidad de Sevilla, Sevilla, 41-080 Spain (espinola@us.es), and Wisnicki, Andrzej, Department of Mathematics, Maria Curie - Sklodowska University, 20-031 Lublin, Poland (awisnic@golem.umcs.lublin.pl),and Wosko, Jacek, Department of Mathematics, Maria Curie - Sklodowska University, 20-031 Lublin, Poland (jwosko@golem.umcs.lublin.pl).
On a Unified Study of Relative Chebyshev Radii and Hausdorff Measures of Noncompactness, pp. 245-257.
ABSTRACT. The paper is concerned with the notion of the Lifschitz modulus introduced by Wisnicki and Wosko (1996) and its relationship with both relative Chebyshev radii and Hausdorff measures of noncompactness.

Metcalfe, Jason, Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160 (metcalfe@math.gatech.edu ).
Global Existence for Semilinear Wave Equations Exterior to Nontrapping Obstacles, pp. 259-281.
ABSTRACT. In this paper, we prove the existence of global small amplitude solutions to semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle.  This generalizes the work of Hayashi in a domain exterior to a ball and of Shibata and Tsutsumi in spatial dimensions greater than or equal to 6.