On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

Hypersurfaces through higher-codimensional submanifolds of ℂ n with preserved Levi-kernelwith preserved Levi-kernel

For a “generic” submanifoldS of a complex manifoldX, we show that there exists a hypersurfaceM⊃S which has the same number of negative (or positive) Levi-eigenvalues asS at one prescribed conormal (cf. also [9]). When ranksL _S is constant, thenM may be found such thatL _M andL _S have the same number of negative eigenvalues at any common conormal. Assuming the existence of a hypersurfaceM with the above property, we then discuss the link between complex submanifolds ofS whose tangent plane belongs to the null-space of the Levi-formL _S ofS (of all complex submanifolds whenL _S is semi-definite), and complex submanifolds ofT _S ^* X. As an application we give a simple result on propagation of microanalyticity for CR-hyperfunctions along complex,L _S -null, curves (cf. [3]).     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

The rational loewy series and nilpotent ideals of endomorphism rings

Sufficient conditions are given, in module-theoretic terms, for the idealN(S) of the endomorphism ringS of a moduleM consisting of the endomorphisms with essential kernel to be nilpotent. This extends in a natural way several known results on the nilpotency ofN(S). WhenM is a quasi-injective module such thatS is right noetherian, it is shown thatS is right artinian if and only ifM has a finite rational Loewy series whose length is, in this case, equal to the index of nilpotency ofN(S). The author has been partially supported by the CAICYT.     

The ring of endomorphisms of a finite dimensional module

LetS denote the ring of endomorphisms of a finite dimensional moduleM _R. Necessary and sufficient conditions for a nil subring ofS to be nilpotent are given. We place conditions onM _R so that every nil subring ofS will be nilpotent.     

A characterization of convexity and central symmetry for planar polygonal sets

LetS be a compact set inR ² with nonempty interior,L(u,k) be a line 〈u, x〉 =k, and ζ _u (k) be the linear Lebesgue measure ofS∩L(u,k). It is well known that for a convexS, ζ _u (k) is unimodal, that is, as a function ofk, it is first non-decreasing and then nonincreasing for everyu∈R ². Further, ifS is centrally symmetric with respect toM, ζ _u (k) achieves maximum whenL(u, k) passes throughM. Converse propositions are considered in this paper for polygonalS with Jordan boundary. It is shown that unimodality alone does not suffice for convexity. However, if ζ _u (k) achieves maximum wheneverL(u, k) passes through some fixed pointM then unimodality yields convexity as well as central symmetry. It is also shown that continuity of ζ _u (k) in the interior of its support implies convexity ofS. This last result, however, is false for non-polygonal sets. Research supported by National Science Foundation Grant GP-28154.     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

On the endomorphism ring of a module noetherian with respect to a torsion theory

IfM is a module torsionfree and noetherian with respect to a torsion theory, ifS is the endomorphism ring ofM, and ifL(S) is the ideal ofS consisting of all endomorphisms with large kernels, thenL(S) is nilpotent and a bound on the index of nilpotency ofL(S) is given.     

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, ifM is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection ofM is quasi-Einstein, too, provided thatM is tangent to the Lee field ofM. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ _m (of a complex Hopf manifoldS ^2m+1 ×S ¹).     

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM ₀#M ₁ unlessM ₀ orM ₁ is homeomorphic toS ⁴. LetN be the nucleus of the minimal elliptic Enrique surfaceV ₁(2, 2) and putM=N∪ _∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S ²×S²) is diffeomorphic toM ₀#M ₁ for non-simply connected closed smooth four-manifoldsM ₀ andM ₁ if and only ifk≥8. On the other hand we show thatM is homeomorphic toM ₀#M ₁ for closed topological four-manifoldsM ₀ andM ₁ withπ ₁(M_i)=ℤ/2.     

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS ⁿ ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM ₁#M ₂ with certain metrics is close to the union of the spectra ofM ₁ and ofM ₂.Partially supported by SFB 256 and by the GADGET program of the EU     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Konstruktion regulärer Hüllen konstanter Breite

LetM be a compact, convex set of diameter 2 inE ^d. There exists a bodyK of constant width 2 containingM such that every symmetry ofM is one ofK and every singular boundary point ofK is a boundary point ofM, for which the set of antipodes inK is the convex hull of the antipodes, which are already inM.

---

---

Mit 1 Abbildung     

M-harmonic Besovp-spaces and Hankel operators in the Bergman space on the ball in ℂ n

In this paper, we characterize the Hardy class ofM-harmonic functions on the unit ballB in ℂ ⁿ in terms of the Berezin transform. We define and study the Besovp-spaces ofM-harmonic functions. For anM-harmonic symbolf, we give various criteria for the Hankel operatorsH _f andH _f to be bounded, compact or in the Schatten-von-Neumann classS _p . These criteria establish a close relationship among Besovp-spaces, Berezin transform, the invariant Laplacian, and Hankel operators on the unit ballB.     

Conormal bundles with vanishing Maslov form

IfM is a Riemannian manifold, andL is a Lagrangian submanifold ofT ^* M, the Maslov class ofL has a canonical representative 1-form which we call theMaslov form ofL. We prove that ifL =v ^* N = conormal bundle of a submanifoldN ofM, its Maslov form vanishes iffN is a minimal submanifold. Particularly, ifM is locally flatv ^* N is a minimal Lagrangian submanifold ofT ^* M iffN is a minimal submanifold ofM. This strengthens a result of Harvey and Lawson [H L].     

Retracts and monomorphism monoids in semigroup varieties

If a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for anyA∈V and any left cancellative monoidM, there is a semigroupS∈V such thatA is a retract ofS andM is isomorphic to the monoid of all injective endomorphisms ofS.