Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

On the Cohen strongly p-summing multilinear operators

In this paper, we introduce and study a new concept of summability in the category of multilinear operators, which is the Cohen strongly p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem and we compare the notion of p-dominated multilinear operators with this class by generalizing a theorem of Bu-Cohen.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

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.     

The p-approximation property in terms of density of finite rank operators

We characterize the p-approximation property (p-AP) introduced by Sinha and Karn [D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ?p, Studia Math. 150 (2002) 17-33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi-p-nuclear operators. This relates the p-AP to Saphar's approximation property APp^′. As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.     

A surjection theorem and a fixed point theorem for a class of positive operators

This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay−Ax?M(y−x) for θ?x?y?v₀, where θ denotes the zero element and v₀ is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations.     

Gaps of operators

We construct examples which distinguish clearly the classes of p-hyponormal operators for 0<p?∞. In addition, we show that those examples classify the classes of w-hyponormal, absolute-p-paranormal, and normaloid operators on the complex Hilbert space. Only a few examples of p-hyponormal operators have been examined. Our technique can provide many examples related to the above operators.     

Transference principles for semigroups and a theorem of Peller

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows one to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to Lp-spaces and—involving the concept of γ-boundedness—to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.     

Landau’s theorem for log-p-harmonic mappings

The main aim of this paper is to prove the existence of Landau-Bloch constant for log-p-harmonic mappings.     

Reverse inclusions for multiple summing operators

Multiple summing operators have been proven to be useful in several areas of analysis and mathematical physics. In this paper we prove reverse inclusions for this class of operators, completing the work already initiated in [D. Perez-Garcia, The inclusion theorem for multiple summing operators, Studia Math. 165 (3) (2004) 275-290].     

Bounded linear non-absolutely summing operators

We show that, in certain situations, we have lineability in the set of bounded linear and non-absolutely summing operators. Examples on lineability of the set Πp(E,F)?Ip(E,F) are also presented and some open questions are proposed.     

Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients

We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in . If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t^(n−θ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order <m.     

Disjointly improjective operators and domination problem

In this work we introduce the disjointly improjective operators between Banach lattices. We investigate this class of operators. Also, we extend the Flores–Hernández’s theorem on the domination problem by disjoint strictly singular operator.     

Gaps of operators via rank-one perturbations

In this note we consider rank-one perturbations of weighted shifts to examine distinctions between various sorts of weak hyponormalities, including p-hyponormality, p-paranormality, and absolute-p-paranormality. Our examples enable us to add to the small collection of examples that exhibit the gaps between these classes.     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

Weyl's theorem and hypercyclic/supercyclic operators

Necessary and sufficient conditions for hypercyclic/supercyclic Banach space operators T to satisfy are proved.     

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . This generalizes some known results obtained recently.

     

A Maurey type result for operator spaces

The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C(K) and ?₂ is 2-summing. However, it is shown in [M. Junge, Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2) (2005) 225-286] that the operator space analogue fails. Not every cb-map is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem: every cb-map is (q,cb)-summing for any q>2 and hence admits a factorization ‖v(x)‖?c(q)‖vcb_‖‖axbq_‖ with a,b in the unit ball of the Schatten class S2q.