In this note we investigate lacunarity or ‘thin’ subsets in the dual object of a compact group via different classes of summing operators between Banach spaces. In particular, we give characterisations of Sidon and ∧ (p) sets, 2 < p < ∞

In this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

This paper gives a complete classification of essentially commutative C*-algebras whose essential spectrum is homeomorphic to S2n−1 by their characteristic numbers. Let 1, 2 be such two C*-algebras; then they are C*-isomorphic if and only if they have the same n-th characteristic number. Furthermore, let γn() = m then is C*-isomorphic to C*(Mzl, …, Mzn) if m = 0, is C*-isomorphic C*(Tz1, …, Tzn−1, Tznm) if m ≠ 0. Some examples are given to show applications of the classfication theorem. We finally remark that the proof of the theorem depends on a construction of a complete system of representatives of Ext(S2n−1).

A holomorphic map ψ of the unit disk ito itself induces an operator Cψ on holomorphic functions by composition. We characterize bounded and compact composition operators Cψ on Qp spaces, which coincide with the BMOA for p = 1 and Bloch spaces for p > 1. We also give boundedness and compactness characterizations of Cψ from analytic function space X to Qp spaces, X = Dirichlet space D, Bloch space B or B0 = {f: f′ ∈ H∞}.

One of the useful features of spectral measures which happen to be equicontinuous is that their associated integration maps are bicontinuous isomorphisms of the corresponding L1-space onto their ranges. It is shown here that equicontinuity is not necessary for this to be the case; a somewhat weaker property suffices. This is of some interest in practice since there are many natural examples of spectral measures which fail to be equiconontinuous.

We characterize the boundedness and compactness of weighted composition operators between weighted Banach spaces of analytic functions and . we estimate the essential norm of a weighted composition operator and compute it for those Banach spaces which are isomorphic to c0. We also show that, when such an operator is not compact, it is an isomorphism on a subspace isomorphic to c0 or l∞. Finally, we apply these results to study composition operators between Bloch type spaces and little Bloch type spaces.

In this paper, from several commutative self-adjoint operators on a Hilbert space, we define a class of spaces of fundamental functions and generalized functions, which are characterized completely by selfadjoint operators. Specially, using the common eigenvectors of these self-adjoint operators, we give the general form of expansion in series of generalized functions

Let A and B be (not necessarily bounded) linear operators on a Banach lattice E such that |(s – B)-1x|≤ (s – A)-1|x| for all x in E and sufficiently large s ∈ R. The main purpose of this paper is to investigate the relation between the spectra σ(B) and σ(A) of B and A, respectively. We apply our results to study asymptotic properties of dominated C0-semigroups.

In this note, a characterization of the Möbius invariant space Qp for the range 1 - 1/n lt; p ≤ 1 is given. As a special case p = 1, we get the Möbius boundedness of BMOA in the space H2. This extends the corresponding result for 1-dimension.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

In this paper we show that, for analytic composition operators between weighted Bergman spaces (including Hardy spaces) and as far as boundedness, compactness, order boundedness and certain summing properties of the adjoint are concerned, it is possible to modify domain spaces in a systematic fashion: there is a space of analytic functions which embeds continuously into each of the spaces under consideration and on which the above properties of the operator are decided.

A remarkable consequence is that, in the setting of composition operators between weighted Bergman spaces, the properties in question can be identified as properties of the operator as a map between appropriately chosen Hilbert spaces.

We consider the space L1 (ν, X) of all real functions that are integrable with respect to a measure v with values in a real Fréchet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l∞ in the space of all linear continuous operators from a complete DF-space Y to a Fréchet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure ν and the space X that imply that L1 (ν, X) has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given.

We characterize those analytic self-maps ϕ of the unit disc which generate bounded or compact composition operators Cϕ between given weighted Banach spaces H∞v or H0v of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v.

We consider the Bergman spaces consisting of harmonic functions on the unit ball in Rn that are squareintegrable with respect to radial weights. We will describe compactness for certain classes of Toeplitz operators on these harmonic Bergman spaces.

We investigate the relationship between the peripheral spectrum of a positive operator T on a Banach lattice E and the peripheral spectrum of the operators S dominated by T, that is, ]Sx] ≤ T]x] for all x ε E. This can be applied to obtain inheritance results for asymptotic properties of dominated operators.

Let G1, G2 be locally compact real-compact spaces. A linear map T defined from C(G1) into C(G2) is said to be separating or disjointness preserving if f = g ≡ 0 implies Tf = Tg ≡ 0 f or all f, g ∈ C(G1). In this paper we prove that both a separating map which preserves non-vanishing functions and a separating bijection which satisfies condition (M) (see Definition 4) are automatically continuous and can be written as weighted composition maps. We also study the effect of separating surjections (respectively injections) on the underlying spaces G1 and G2.

Next we apply the above results to give an algebraic characterization of locally compact Abelian groups, similar to the one given in [7] for compact Abelian groups in the presence of ring isomorphisms.

Finally, locally compact (not necessarily Abelian) groups are considered. We provide a sharpening of a result of Edwards and study the effect of onto (respectively injective) weighted composition maps on the groups G1 and G2.

In this paper we discuss the asymptotic behaviour, as t → ∞, of the integral solution u(t) of the non-linear evolution equation where {A(t)}t≥0 is a family of m-dissipative operators in a Hilbert space H, and g ∈ Lloc (0, ∞ H).We give some sufficient conditions and some sufficient and necessary conditions to ensure that are weakly convergent.

We describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.

We show that the Kato conjecture is true for m-accretive operators with highly singular coefficients. For operators of the form A = *F, where formally corresponds to d/dx + zδ on L2 (R), we prove that Dom (A1/2) = Dom() = e-zHH1(R) where H is the Heavysied function. By adapting recent methods of Auscher and Tchamitchian, we characterize Dom (A) in terms of an unconditional wavelet basis for L2(R).

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.