Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Some Compactness Results in Real Interpolation for Families of Banach Spaces

The paper shows that, if the operator T:A()B() is compact foralmost every , then is compact when or is the interpolation functor constructed for infinitefamilies of Banach spaces and S satisfies certain conditions.     

Multipliers on Banach spaces of functions on a locally compact Abelian group

Let E be a Banach space of functions on a locally compact Abeliangroup G satisfying certain conditions. It has been proved thatfor every bounded operator M on E commuting with translationsthere exists such that , where is a suitable subset of the group of the continuous morphismsfrom G into * and is a generalized Fourier transform of g defined on .     

Total Chromatic Number of Graphs of Order 2n + l having Maximum Degree 2n 1

Let G be a graph of order 2n + l having maximum degree 2n –1. We prove that the total chromatic number of G is 2n if andonly if e + ' n, where w is a vertex of minimum degree in G, is the complement of G —w, e is the size of , and ' is the edge independence number of .     

Codeterminants for the Symplectic Schur Algebra

Standard codeterminants for Donkin's symplectic Schur algebra are defined. It is shown that they form a basis of . They are then used to give a purely combinatorial proof that is quasi-hereditary.     

On Ramanujan's Inequalities for exp(k)

Ramanujan claimed in his first letter to Hardy (16 January 1913)that where (k) lies between 2/21 and 8/45. This conjecture was provedin 1995 by Flajolet et al. The paper establishes the followingrefinement. where Both bounds for ^*(k) are sharp.     

Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy), , where det = > 0, c 0, det > 0, t 0 and (cs + dr –au – tb)² < 4(cr – ta)(ds – ub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let w_n denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions w_n(0)= sin and pw'_n(0) = –cos and let _n = –aw_n(1)+cpw'_n(1).Define _n and _n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /.     

Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.     

The Sector of Analyticity of the Ornstein-Uhlenbeck Semigroup on Lp Spaces with Respect to Invariant Measure

The sector of analyticity of the Ornstein–Uhlenbeck semigroupis computed on the space := L^p (R^N; µ) with respect to its invariant measure µ.If A= + Bx· denotes the generator of the Ornstein–Uhlenbecksemigroup, then the angle ₂ of the sector of analyticity in is /2 minus the spectral angleof BQ, Q being the matrix determining the Gaussian measure µ.The angle of analyticity in is then given by the formula      

A New Simple Class of Rational Functions whose Julia Set is the whole Riemann Sphere

The paper first gives sufficient conditions on the criticalpoints and the Schwarzian derivative of a real rational functionR such that the Julia set of R is . Further, it is shown that under mild conditions on another realrational function with possibly non-empty Fatou set, the Julia set of R is the whole Riemann sphere again. Then familiesof rational functions are given whose Julia set is and whose critical points are not necessarily preperiodic.Concrete examples were previously available only for the preperiodiccase. Finally, it is demonstrated that the methods presentedalso apply to the construction of polynomials whose Julia setsare dendrites and whose critical points in the Julia set arenot necessarily preperiodic.     

One-Sided BMO Spaces

In this paper we introduce the one-sided sharp functions dennedby and where z⁺ = max(z, 0). We study the BMO spaces associatedto and and their relation with the good weights for theone-sided Hardy-Littlewood maximal functions. Finally, as anapplication of our results, we characterize the weights forone-sided fractional integrals and one-sided fractional maximaloperators.     

An Entire Function Defined by a Nonlinear Recurrence Relation

For the nonlinear recurrence relation it is proved that the limit exists and defines an entire function of ₂ = ₁(1-₁).     

Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

Weighted Norm Inequalities of Hardy Type for a Class of Integral Operators

Necessary and sufficient conditions for the boundedness from to of Volterra integral operators of the form where k(x,y) is a non-negative kernel under suitablemonotone conditions, are given. The cases 1 <p q < ,1 < q < p < and 0 < q < l < p < are considered.The results extend the well-known weighted norm Hardy inequality.     

The Moduli Spaces of Bielliptic Curves of Genus 4 with more Bielliptic Structures

Let C be an irreducible, smooth, projective curve of genus g 3 over the complex field C. The curve C is called biellipticif it admits a degree-two morphism : C E onto an ellipticcurve E such a morphism is called a bielliptic structure onC. If C is bielliptic and g6, then the bielliptic structureon C is unique, but if g=3,4,5, then this holds true only genericallyand there are curves carrying n>>1 bielliptic structures.The sharp bounds n 21,10,5 exist for g=3,4,5 respectively.Let M_g be the coarse moduli space of irreducible, smooth, projectivecurves of genus g=3,4,5. Denote by the locus of points in M_g $ representing curves carrying atleast n bielliptic structures. It is then natural to ask thefollowing questions. Clearly does hold? What are the irreducible components of ? Are the irreducible components of rational? How do the irreducible components of intersect each other? Let how many non-isomorphic elliptic quotients can it have? Completeanswers are given to the above questions in the case g=4.     

The Boundary Harnack Principle for Non-Divergence form Elliptic Operators

If L is a uniformly elliptic operator in non-divergence form,the boundary Harnack principle for the ratio of positive L-harmonicfunctions holds in Hölder domains of order if > . A counterexample shows that is sharp. For Hölder domainsof order with (0,1], the boundary Harnack principle holds providedthe domain also satisfies a strong uniform regularity condition.     

The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let be a bounded analytic semigroup withgenerator –A on some Banach space X. It is proved thatif A^1/2 is admissible for A, that is, if there is an estimate then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate .for , Re()<0. This holds in particular if is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given.     

Link Polynomials of Higher Order

In this paper, we study certain polynomial invariants of links(singular or non-singular) that are related to the Homfly polynomialand Vassiliev's invariants. The Homfly polynomial H_L [3] (alsoknown as the Flypmoth polynomial) satisfies the well-known skeinrelation The Vassiliev invariants [1, 2] (of order 1) satisfy the relations and The invariants that we study satisfy the skein relations      

Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities ofthe form in the set of alltrigonometric polynomials of order n which have only real zeros.The function is assumed to be convex and increasing on [0,). As a corollary of the main result we get Turan's inequalities with the exact constantc(n, k, q) for each 1 q , n and k.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.