The characteristic of noncompact convexity and random fixed point theorem for set-valued operators

Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: C → KC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × C → KC(C) has a random fixed point.     

On the supercyclicity and hypercyclicity of the operator algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

On weak positive supercyclicity

A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT ⁿ x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ _p (T ^*) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT ⁿ x: n ∈ ℂ, r ∈ ℝ₊} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions. Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225. Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Faber-hypercyclic operators

Let Ω be a bounded domain of the complex plane whose boundary is a closed Jordan curve and (F _n ) _n≥0 the sequence of Faber polynomials of Ω. We say that a bounded linear operator T on a separable Banach space X is Ω-hypercyclic if there exists a vector x of X such that {F _n (T)x: n ≥ 0} is dense in X. We show that many of the results in the spectral theory of hypercyclic operators involving the unit disk or its boundary have Ω-hypercyclic counterparts which involve the domain Ω or its boundary. The influence of the geometry of Ω or the smoothness of its boundary on Faber-hypercyclicity is also discussed.     

On the existence of automorphisms with simple Lebesgue spectrum

It is shown that ifT is a measure preserving automorphism on a probability space (Ω,B, m) which admits a random variable X₀ with mean zero such that the stochastic sequence X₀ o T_n,n ε ℤ is orthonormal and spans L₀ ²(Ω,B,m), then for any integerk ≠ 0, the random variablesX o T^nk,n ε ℤ generateB modulom.     

The Kato-type spectrum and local spectral theory

Let T ∈ ℒ(X) be a bounded operator on a complex Banach space X. If V is an open subset of the complex plane such that λ-T is of Kato-type for each λ ∈ V, then the induced mapping f(z) ↦ (z-T)f(z) has closed range in the Fréchet space of analytic X-valued functions on V. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of T. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves.     

Higher order Poincaré inequalities associated with linear operators on stratified groups and applications

 This paper considers the dual of anisotropic Sobolev spaces on any stratified groups 𝔾. For 0≤k<m and every linear bounded functional T on anisotropic Sobolev space W ^m−k,p (Ω) on Ω⊂𝔾, we derive a projection operator L from W ^m,p (Ω) to the collection 𝒫 _k+1 of polynomials of degree less than k+1 such that T(X ^I (Lu))=T(X ^I u) for all uW ^m,p (Ω) and multi-index I with d(I)≤k. We then prove a general Poincaré inequality involving this operator L and the linear functional T. As applications, we often choose a linear functional T such that the associated L is zero and consequently we can prove Poincaré inequalities of special interests. In particular, we obtain Poincaré inequalities for functions vanishing on tiny sets of positive Bessel capacity on stratified groups. Finally, we derive a Hedberg-Wolff type characterization of measures belonging to the dual of the fractional anisotropic Sobolev spaces W ^α,p 𝔾. Received: 25 March 2002; in final form: 10 September 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 46E35, 41A10, 22E25 The second author was supported partly by U.S NSF grant DMS99-70352 and the third author was supported partly by NNSF grant of China.     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

On a partially isometric transform of divergence-free vector fields

The paper deals with the so-called M-transform, which maps divergence-free vector fields in Ω ^T := {x ∈ Ω| dist(x, ∂Ω) < T}, Ω ⊂⊂ \mathbbR \mathbb{R} ³, to the space of transversal fields. The latter space consists of vector fields in Ω ^T tangential to the equidistant surfaces of the boundary ∂Ω. In papers devoted to the dynamical inverse problem for the Maxwell system, in the framework of the BC-method, the operator M ^T was defined for T < T _ω, where T _ω depends on the geometry of Ω. This paper provides a generalization for arbitrary T. It is proved that M ^T is partially isometric, and its intertwining properties are established. Bibliography: 6 titles.     

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

Entropy and<Emphasis Type="Italic">σ</Emphasis>-algebra equivalence of certain random walks on random sceneries

LetX=X ₀,X ₁,…be a stationary sequence of random variables defining a sequence space Σ with shift mapσ and let (T _t, Ω) be an ergodic flow. Then the endomorphismT _X(x, ω)=(σ(x),T _{x 0}(ω)) is known as a random walk on a random scenery. In [4], Heicklen, Hoffman and Rudolph proved that within the class of random walks on random sceneries whereX is an i.i.d. sequence of Bernoulli-(1/2, 1/2) random variables, the entropy ofT _t is an isomorphism invariant. This paper extends this result to a more general class of random walks, which proves the existence of an uncountable family of smooth maps on a single manifold, no two of which are measurably isomorphic. This research was sustained in part by fellowship support from the National Physical Science Consortium and the National Security Agency.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a...     

Extremely non-complex Banach spaces

A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T ²∥ = 1+∥T ²∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis and that the infimum of diameters of the slices of its unit ball is positive.     

Exact and explicit analytic solutions of an extended Jabotinsky functional differential equation

A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T: E → X for which dens X/E = dens X/TE can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c ₀ is automorphic and conjectured that c ₀ and ℓ₂ are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c ₀(Γ), for Γ uncountable. That c ₀(Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of ℓ _n ^p , 1 ≤ p < ∞, p ≠ 2. We derive that infinite dimensional spaces such as L _p (μ), p ≠ 2, C(K) spaces not isomorphic to c ₀ for K metric compact, subspaces of c ₀ which are not isomorphic to c ₀, the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic. The work of the first author has been supported in part by project MTM2004-02635     

A structure theorem for boundary-transitive graphs with infinitely many ends

We prove a structure theorem for locally finite connected graphsX with infinitely many ends admitting a non-compact group of automorphisms which is transitive in its action on the space of ends, Ω _X . For such a graphX, there is a uniquely determined biregular treeT (with both valencies finite), a continuous representationφ : Aut(X) → Aut(T) with compact kernel, an equivariant homeomorphism λ : Ω _X → Ω _T , and an equivariant map τ : Vert(X) → Vert(T) with finite fibers. Boundary-transitive trees are described, and some methods of constructing boundary-transitive graphs are discussed, as well as some examples.