Spectra of composition operators on the bloch and bergman spaces

Whenϕ is an analytic map of the unit diskU into itself, andX is a Banach space of analytic functions onU, define the composition operatorC _ϕ byC _ϕ (f)=f o ϕ, forf∈X. In this paper we show how to use the Calderón theory of complex interpolation to obtain information on the spectrum ofC _ϕ (under suitable hypotheses onϕ) acting on the Bloch spaceB and BMOA, the space of analytic functions in BMO. To do this we first obtain some results on the essential spectral radius and spectrum ofC _ϕ on the Bergman spacesA ^pand Hardy spacesH ^p,spaces which are connected toB and BMOA by the interpolation relationships [A ¹,B] _t =A ^pand [H ¹,BMOA] _t =H ^pfor 1=p(1−t).     

Elementary abelian operator groups

Letp be a prime,n a positive integer. Suppose thatG is a finite solvablep'-group acted on by an elementary abelianp-groupA. We prove that ifC _G (ϕ) is of nilpotent length at mostn for every nontrivial element ϕ ofA and |A|≥p ⁿ⁺¹ thenG is of nilpotent length at mostn+1.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Approximation order provided by refinable function vectors

In this paper we considerL _p-approximation by integer translates of a finite set of functionsϕ _v (v=0, ...,r − 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vectorϕ=(ϕ ^=0/ r−1 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates ofϕ _v, then a factorization of the refinement mask ofϕ can be given. This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e ^−iu )/2) ^m if approximation orderm is achieved. Dedicated to Professor L. Berg on the occasion of his 65th birthday     

Finite representability ofl p(X) in Orlicz function spaces

We show that ifl _p(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceL _ϕ (which does not containc ₀) then the Banach spaceX is isomorphic to a subspace ofL _p. The same remains true forp = 2 whenL _ϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

Una nota sui gruppi policiclici che ammettono un automorfismo con numero di Reidemeister finito

LetG be a group and ϕ an automorphism ofG. Two elementsx, y ∈ G are called ϕ-conjugate if there existsg ∈ G such thatx=g ⁻¹ yg ^θ. It is easily verified that the ϕ-conjugation is an equivalence relation; the numberR(ϕ) of ϕ-classes ofG is called the Reidemeister number of the automorphism ϕ. In this paper we prove that if a polycyclic groupsG admits an automorphism ϕ of ordern such thatR(ϕ)<∞, thenG contains a subgroup of finite index with derived length at most 2 ⁿ⁻¹ .

     

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

Dual space,carleson measure and sequence interpolation onA p (ϕ)(O<p<1)

LetD={z∈Σ:|z|<1} and ϕ be a normal function on [0,1). Forp∈(0,1) such a function ϕ is used to define a Bergman spaceA ^p (ϕ) onD with weight ϕ ^p (|·|)/(1-|·|²). In this paper, the dual space ofA ^p (ϕ) is given, four characteristics of Carleson measure onA ^p (ϕ) are obtained. Moreover, as an application, three sequence interpolation theorems inA ^p (ϕ) are derived. Supported by the Doctoral Program Foundation of Institute of Higher Education, P.R. China.     

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL ^p , each of mean zero, is shown to be isomorphic tol ²,l ^p ,l ²⊕l ^p , or a new spaceX _p , and the linear topological properties ofX _p are investigated. It is proved thatX _p is isomorphic to a complemented subspace ofL ^p and another uncomplemented subspace ofL ^p , whence there exists an uncomplemented subspace ofl ^p isomorphic tol ^p . It is also proved thatX _p is not isomorphic to the previously knownℒ _p spaces. The work for this research was partially supported by the National Science Foundation GP-12997.     

Daniell integral represented by Radon measures

LetL be a sublattice of the space of real continuous functions defined on a Suslin spaceX, such that at no point all the functions inL vanish. Then it is shown that every Daniell integrall μ:L → IR is representable by a Radon measurem onX: μ(ϕ)=∫ϕdm ∀ϕ∈L. The measurem may be uniquely determined by constraining it to be concentrated on a certain type of subset ofX. The relation betweenL ¹(μ) andL ¹(m) is examined in detail.     

Parameter estimation for a misspecified arma model with infinite variance innovations

We consider a general linear model , where the innovations Z_t belong to the domain of attraction of an α-stable law for α<2, so that neither Z_t nor X_t have a finite variance. We do not assume that (X_t) is a standardARMA process of the form φ(B)X_t=ϕ(B)Z_t, but we fit anARMA process of a given order to the data X₁,...,X_n by estimating the coefficients of φ and ϕ. Given that (X_t) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)^1/α, which compares favorably with the L² situation with rate . In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

On the comparison theorem for étale cohomology of non-Archimedean analytic spaces

Let ϕ:Y →X be a morphism of finite type between schemes of locally finite type over a non-Archimedean fieldk, and letF be an étale constructible sheaf onY. In [Ber2] we proved that if the torsion orders ofF are prime to the characteristic of the residue field ofk then the canonical homomorphisms (R ^Q ϱ*F)^an →R ^q ϱ _* ^an F ^an are isomorphisms. In this paper we extend the above result to the class of sheavesF with torsion orders prime to the characteristic ofk.     

Simultaneous estimation of location parameters of the distribution with finite support

Summary LetX _i ,i=1,..., p be theith component of thep×1 vectorX=(X ₁,X ₂,...,X _p )′. Suppose thatX ₁,X ₂,...,X _p are independent and thatX _i has a probability density which is positive on a finite interval, is symmetric about θ _i and has the same variance. In estimation of the location vector θ=(θ₁, θ₂,...,θ _p )′ under the squared error loss function explicit estimators which dominateX are obtained by using integration by parts to evaluate the risk function. Further, explicit dominating estimators are given when the distributions ofX _i ′s are mixture of two uniform distributions. For the loss function such an estimator is also given when the distributions ofX _i ′s are uniform distributions.     

Entropy bounds and second cohomology

When X is a finite complex and p₁X\pi_{1}X acts on \mathbbR²{\mathbb{R}}^2 by translations we give criteria involving H²X for an equivariant map F : [(X)\tilde] ? \mathbbR²F : \tilde{X} \rightarrow {\mathbb{R}}^2 to be onto. Following work of Manning and Shub, this leads to entropy bounds related to Shub’s entropy conjecture.     

l p n Superspaces of spans of independent random variables

We show that for 1 ≦p < ∞,p ≠ 2, ifɛ > 0 is small enough andX ≦L _p is the span ofn independent Rademacher functions orn independent Gaussian random variables, then any superspaceY ofX satisfyingd(Y,L _p ^m ) ≦ 1 +ɛ has dimension larger thanr ⁿ, wherer =r(ɛ, p) > 1. This forms part of the author’s doctoral dissertation prepared at Texas A&M University under the direction of Professor W. B. Johnson. Supported in part by NSF DMS-85 00764.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

Existence d’un generateur suivant une sous-suite de densite superieure nulle

Ergodic theory: for every dynamical system (X,A,T, μ), totally ergodic and of finite entropy, there exist a sequenceS of integers, of upper density zero, and a partitionQ ofX, such that V _i∈S T ⁻ⁱ Q is the whole σ-algebraA. Furthermore, there is a “universal” sequenceS ⁰ for which this property is true if we restrict ourselves to the class of strongly mixing systems.      

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

The minimal cardinality where the Reznichenko property fails

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood ofx. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a setX of real numbers such thatC _p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space^Nℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC _p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ. The author is partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).