A pointwise spectrum and representation of operators

For a self-adjoint operator commuting with an increasing family of projections we study the multifunction an open set of the topology containing , where is the spectrum of on . Let be the measure of maximal spectral type. We study the condition that is essentially a singleton, is not a singleton. We show that if is the density topology and if satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that is a Borel function of . If is the usual topology then the condition is equivalent to the fact that is approched in norm by step functions , where the set of intervals covers the set where is a singleton.

     

Subnormal Subgroups of Group Ring Units

Let be an arbitrary group. If satisfies , , then the units , generate a nonabelian free subgroup of units. As an application we show that if is contained in an almost subnormal subgroup of units in then either contains a nonabelian free subgroup or all finite subgroups of are normal. This was known before to be true for finite groups only.

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

A remark on Gelfand-Kirillov dimension

Let be a finitely generated non-PI Ore domain and the quotient division algebra of . If is the center of , then .

     

Cardinal invariants concerning bounded families of extendable and almost continuous functions

In this paper we introduce and examine a cardinal invariant closely connected to the addition of bounded functions from to . It is analogous to the invariant defined earlier for arbitrary functions by T. Natkaniec. In particular, it is proved that each bounded function can be written as the sum of two bounded almost continuous functions, and an example is given that there is a bounded function which cannot be expressed as the sum of two bounded extendable functions.

     

An estimate on the distortion of the logarithmic capacity

Let be a Fuchsian group. We show that the existence of a set on with no -equivalent points and positive logarithmic capacity does not imply that is of convergence type.

     

Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

Angular derivatives at boundary fixed points for self-maps of the disk

Let be a one-to-one analytic function of the unit disk into itself, with . The origin is an attracting fixed point for , if is not a rotation. In addition, there can be fixed points on where has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of is a one-to-one analytic function defined on such that , where . If is the first iterate of that does have b.r.f.p., we compute the Hardy number of , , in terms of the smallest angular derivative of at its b.r.f.p.. In the case when no iterate of has b.r.f.p., then , and vice versa. This has applications to composition operators, since is a formal eigenfunction of the operator . When acts on , by a result of C. Cowen and B. MacCluer, the spectrum of is determined by and the essential spectral radius of , . Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, can be computed in terms of . Hence, our result implies that the spectrum of is determined by the derivative of at the fixed point and the angular derivatives at b.r.f.p. of or some iterate of .

     

Inradius and integral means for green's functions and conformal mappings

Let be a convex planar domain of finite inradius . Fix the point and suppose the disk centered at and radius is contained in . Under these assumptions we prove that the symmetric decreasing rearrangement in of the Green's function , for fixed , is dominated by the corresponding quantity for the strip of width . From this, sharp integral mean inequalities for the Green's function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison estimates for the hyperbolic metric of with that of the strip and a careful analysis of geodesics.

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

Properties of subgenerators of regularized semigroups

We introduce two operations , in the set of subgenerators of a given - regularized semigroup and prove that is a complete partially ordered lattice with respect to , and the operator inclusion . Also presented are some other properties and examples for

     

Closure ordering and the Kostant-Sekiguchi correspondence

Let be a real semisimple Lie group with Lie algebra . The Kostant-Sekiguchi correspondence is a bijection between nilpotent orbits on and nilpotent orbits on . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant-Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

     

A result on the Gelfand-Kirillov dimension of representations of classical groups

Let be the reductive dual pair . We show that if is a representation of (respectively ) obtained from duality correspondence with some representation of (respectively ), then its Gelfand-Kirillov dimension is less than or equal to
(respectively ).

     

A note on Kamenev type theorems for second order matrix differential systems

Some oscillation criteria are given for the second order matrix differential system , where and are real continuous matrix functions with symmetric, . These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation , where is the identity matrix, is a given function on and .

     

Stability of weakly almost conformal mappings

We prove a stability of weakly almost conformal mappings in for not too far below the dimension by studying the -quasiconvex hull of the set of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Hölder inequalities from the Ekeland variational principle.