Some Characterizations of Commutative Subspace Lattices

Let H be a not necessarily separable Hilbert space, and letBH denote the space of all bounded linear operators on H. Itis proved that a commutative lattice D of self-adjoint projectionsin H that contains 0 and I is spatially complete if and onlyif it is a closed subset of BH in the strong operator topology.Some related results are obtained concerning commutative lattice-orderedcones of self-adjoint operators that contain D. 2000 MathematicsSubject Classification 47D03, 47L35, 47L07, 46L10, 54F05, 54G05,46E05.     

Arens Regularity of the Algebra of Operators on a Banach Space

A short proof is given that if E is a super-reflexive Banachspace, then B(E), the Banach algebra of operators on E withcomposition product, is Arens regular. Some remarks are madeon necessary conditions on E for B(E) to be Arens regular. 2000Mathematics Subject Classification 47L10 (primary), 46H05 (secondary).     

On an Extension Problem for Polynomials

Consider the following problem: given complex numbers a₁, ...,a_n, find an L function f of minimum norm whose Fourier coefficientsc_k(f) are equal to a_k for k between 0 and n. We show the uniquenessof this function, and we estimate its norm. The operator-valuedcase is also discussed. 2000 Mathematics Subject Classification30E05, 47A20, 47A56, 47A57.     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Corrigendum

A UNIQUENESS THEOREM IN THE INVERSE SPECTRAL THEORY OF A CERTAINHIGHER-ORDER ORDINARY DIFFERENTIAL EQUATION USING PALEY–WIENERMETHODS E. ANDERSSON erika{at}maths.lth.se Footnotes 2000 Mathematics Subject Classification 30D15, 34L40, 47E05(primary), 30D10, 34A55, 34B20, 34L05 (secondary). Received September 21, 2006; published online July 8, 2007.     

Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes

This paper concerns a Markov operator T on a space L₁, and aMarkov process P which defines a Markov operator on a spaceM of finite signed measures. For T, the paper presents necessaryand sufficient conditions for:

a the existence of invariant probabilitydensities (IPDs)

b the existence of strictly positive IPDs,and

c the existence and uniqueness of IPDs.

Similar resultson invariant probability measures for P are presented. The basicapproach is to pose a fixed-point problem as the problem ofsolving a certain linear equation in a suitable Banach space,and then obtain necessary and sufficient conditions for thisequation to have a solution. 1991 Mathematics Subject Classification:60J05, 47B65, 47N30.     

Transformations Between Sturm-Liouville Problems with Eigenvalue Dependent and Independent Boundary Conditions

Explicit relationships are given connecting ‘almost’isospectral Sturm-Liouville problems with eigen-value dependent,and independent, boundary conditions, respectively. Applicationis made to various direct and inverse spectral questions. 2000Mathematics Subject Classification 34B24, 34L05, 47E05.     

On the High-Energy Asymptotics of the Integrated Density of States

Assuming that the integrated density of states of a Schrödingeroperator admits a high-energy asymptotic expansion, the authorsgive explicit formulae for the coefficients of this expansionin terms of the heat invariants. 2000 Mathematics Subject Classification35J10, 47F05, 35P20.     

How and Why to Solve the Operator Equation AX-XB = Y

The entities A, B, X, Y in the title are operators, by whichwe mean either linear transformations on a finite-dimensionalvector space (matrices) or bounded (= continuous) linear transformationson a Banach space. (All scalars will be complex numbers.) Thedefinitions and statements below are valid in both the finite-dimensionaland the infinite-dimensional cases, unless the contrary is stated.1991 Mathematics Subject Classification 15A24, 47A10, 47A62,47B47, 47B49, 65F15, 65F30.     

A Note on Positivity of Elementary Operators

We show that operators on n x n matrices which are representablein the form (where a_i andb_i are n x n matrices) and are k-positive for must be completely positive. As a consequence, elementaryoperators on a C^*-algebra with minimal length l which are k-positivefor must be completely positive. 1991 Mathematics Subject Classification 47B47, 46L05, 47B65.     

Disintegration-of-measure techniques for commuting multivariable weighted shifts

We employ techniques from the theory of disintegration of measuresto study the Lifting Problem for commuting n-tuples of subnormalweighted shifts. We obtain a new necessary condition for theexistence of a lifting, and generate new pathology associatedwith bringing together the Berger measures associated to eachindividual weighted shift. For subnormal 2-variable weightedshifts, we then find the precise relation between the Bergermeasure of the pair and the Berger measures of the shifts associatedto horizontal rows and vertical columns of weights. 2000 MathematicsSubject Classification 47B20, 47B37, 47A13, 28A50 (primary),44A60, 47-04, 47A20 (secondary).     

Finitary Orbit Equivalence of Odometers

A finitary orbit equivalence mapping between the binary andternary odometers is constructed. 2000 Mathematics Subject Classification28D05, 37A05, 37A20.     

On Holomorphic Principal Bundles Over a Compact Riemann Surface Admitting a Flat Connection, II

Holomorphic principal bundles over a compact Riemann surfaceX that admits a flat connection are considered. A holomorphicG-bundle over X, where G is a connected semisimple linear algebraicgroup over C, admits a flat connection if and only if the adjointvector bundle admits one. More generally, for a complex reductivegroup G, the necessary and sufficient condition on a G-bundleto admit a flat connection is described. This simplifies thecriterion obtained by the authors and given in Math. Ann. 322(2002) 333–346. 2000 Mathematics Subject Classification53C05, 32L05.     

Composition Operators with Closed Image on Spaces of Real Analytic Functions

We characterize composition operators on spaces of real analyticfunctions which at the same time have closed image and are openonto their images. Under some mild assumptions, we also characterizecomposition operators with closed range and composition operatorsopen onto their images. 2000 Mathematics Subject Classification46E10, 47B33, 32C07 (primary), 26E05, 32E30, 32D15 (secondary).     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).     

On the Approximation Numbers for the Finite Sections of Block Toeplitz Matrices

In this paper we discuss the asymptotic distribution of theapproximation numbers of the finite sections for a Toeplitzoperator and µ R, witha continuous matrix-valued generating function a. We prove thatthe approximation numbers of the finite sections T_n(a) = P_nT(a)P_nhave the k-splitting property, provided T(a) is a Fredholm operatoron . 2000 Mathematics SubjectClassification 47B35 (primary), 15A18, 47A58, 47A75, 65F20 (secondary).     

Self-Similarity and Probability: Parameters Describing the Geometry of Cantor Sets

A deterministic self-similar Cantor set F defines a naturalprobability space, the elements of which are neighbourhoodsof various sizes of all points in F. Considering random variableson this probability space, some porosity and density parametersare explicitly calculated. 1991 Mathematics Subject Classification28A80, 60D05.     

On Suborbital Graphs for the Modular Group

This paper proves a conjecture of G. A. Jones, D. Singermanand K. Wicks, that a suborbital graph for the modular groupis a forest if and only if it contains no triangles. 2000 MathematicsSubject Classification 05C05, 05C20, 20H35.     

Hurwitz Groups with Given Centre

A Hurwitz group is any non-trivial finite group that can be(2,3,7)-generated; that is, generated by elements x and y satisfyingthe relations x² = y³ = (xy)⁷ = 1. In this short paper a completeanswer is given to a 1965 question by John Leech, showing thatthe centre of a Hurwitz group can be any given finite abeliangroup. The proof is based on a recent theorem of Lucchini, Tamburiniand Wilson, which states that the special linear group SL_n(q)is a Hurwitz group for every integer n 287 and every prime-powerq. 2000 Mathematics Subject Classification 20F05 (primary);57M05 (secondary).