Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ⊆R₊×C which is not of zero three-dimensional Lebesgue measure, the family has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if and φ∈H^∞(D) is non-constant, then the family has a common hypercyclic vector, where Mφ:H²(D)→H²(D), Mφf=φf, and , providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family has a common hypercyclic vector, where Tbf(z)=f(z−b) acts on the Fréchet space H(C) of entire functions on one complex variable.     

Dual disjoint hypercyclic operators

Let E be a separable Fréchet space. The operators T₁,…,Tm are disjoint hypercyclic if there exists x∈E such that the orbit of (x,…,x) under (T₁,…,Tm) is dense in E×?×E. We show that every separable Banach space E admits an m-tuple of bounded linear operators which are disjoint hypercyclic. If, in addition, its dual E^∗ is separable, then they can be constructed such that are also disjoint hypercyclic.     

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T(a)∈L(?p), 1<p<∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections Tn(a)=PnT(a)Pn depends heavily on the Fredholm properties of the operators T(a) and . In particular, if the operators T(a) and are Fredholm on ?p, then the approximation numbers of Tn(a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than .     

Cores for Feller semigroups with an invariant measure

Let be an elliptic differential operator with unbounded coefficients on RN and assume that the associated Feller semigroup (T(t))_t?0 has an invariant measure μ. Then (T(t))_t?0 extends to a strongly continuous semigroup (Tp(t))_t?0 on Lp(μ)=Lp(RN,μ) for every 1?p<∞. We prove that, under mild conditions on the coefficients of A, the space of test functions is a core for the generator (Ap,Dp) of (Tp(t))_t?0 in Lp(μ) for 1?p<∞.     

Some Zygmund type L inequalities for polynomials

Let p(z) be a polynomial of degree n which does not vanish in |z|<k. It is known that for each q>0 and k?1,

     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Hyponormal operators with rank-two self-commutators

In this paper it is shown that if T∈L(H) satisfies

(i)

T is a pure hyponormal operator;

(ii)

[T^∗,T] is of rank two; and

(iii)

ker[T^∗,T] is invariant for T,

then T is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T_|ker[T^∗,T] has a rank-one self-commutator then T is subnormal and if instead T_|ker[T^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.     

On the ideal structure of some Banach algebras related to convolution operators on L(G)

Let G be a locally compact group and let p∈(1,∞). Let be any of the Banach spaces C_δ,p(G), PF_p(G), M_p(G), AP_p(G), WAP_p(G), UC_p(G), PM_p(G), of convolution operators on L^p(G). It is shown that PF_p(G)′ can be isometrically embedded into UC_p(G)′. The structure of maximal regular ideals of (and of MA_p(G)″, B_p(G)″, W_p(G)″) is studied. Among other things it is shown that every maximal regular left (right, two sided) ideal in is either dense or is the annihilator of a unique element in the spectrum of A_p(G). Minimal ideals of is also studied. It is shown that a left ideal M in is minimal if and only if , where Ψ is either a right annihilator of or is a topologically x-invariant element (for some x∈G). Some results on minimal right ideals are also given.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and be nonexpansive mappings satisfying the weakly inward condition and F(T)≠∅, and be a fixed contractive mapping. The implicit iterative sequence {xt} is defined by for t∈(0,1)

xt=P(tf(xt)+(1−t)Txt).     

Inequalities for a polynomial and its derivative

Let , 1?μ?n, be a polynomial of degree n such that p(z)≠0 in |z|<k, k>0, then for 0<r?R?k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved

     

Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights

We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap(w)⊂Lq(μ), 0<p,q<∞, where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights , α>0, and some double exponential type weights.As an application of that result, we characterize the boundedness and the compactness of Tg:Ap(w)→Aq(w), 0<p,q<∞, w∈W, where Tg is the integration operator

     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

Classifying spaces with virtually cyclic stabilisers for certain infinite cyclic extensions

Let G?B?Z be an infinite cyclic extension of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain strictly descending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|≥2. We construct a model for , the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimal dimension of . Finally we determine the geometric dimension when G is a soluble Baumslag-Solitar group.     

Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

Operator Hölder-Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q−Qf(A) and quasicommutators f(A)Q−Qf(B). Finally, we estimate the norms of finite differences for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.     

On zero varieties of holomorphic functions in Hardy spaces

In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball B_n in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no f∈H^p(B_n), 0<p<∞, with . Here N_g(ζ,1) denotes the integrated zero counting function associated with the slice function g_ζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces H^p(B_n), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner.     

On the Weyl-Heisenberg frames generated by simple functions

Let with , and let (?,a,1), 0<a?1 be a Weyl-Heisenberg system {e2πimx?(x−na):m,n∈Z}. We show that if E=[0,1] (and some modulo extension of E), then (?,a,1) is a frame for each 0<a?1 (for certain a, respectively) if and only if the analytic function has no zero on the unit circle {z:|z|=1}. These results extend the case of Casazza and Kalton (2002) [6] that and a=1, which brought together the frame theory and the function theory on the closed unit disk. Our techniques of proofs are based on the Zak transform and the distribution of fractional parts of {na}_n∈Z.     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.