Isometries of certain operator spaces

Let and be Banach spaces, and be the spaces of bounded linear operators from into In this paper we give full characterization of isometric onto operators of for a certain class of Banach spaces, that includes We also characterize the isometric onto operators of and the compact operators on Furthermore, the multiplicative isometric onto operators of , when multiplication on is taken to be the Schur product, are characterized.

     

Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions

Some Hardy-type integral inequalities in general measure spaces, where the corresponding Hardy operator is replaced by a more general Volterra type integral operator with kernel , are considered. The equivalence of such inequalities on the cones of non-negative respective non-increasing functions are established and applied.

     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

A representation theorem for maxitive measures

A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(U_j A_j ) = sup_j η(A_j) for all countable disjoint families of sets (A_j) in Σ. A representation theorem for such measures is established, and next applied to represent Köthe function M-spaces as L_∞-spaces.     

Refined limiting imbeddings for Sobolev spaces of vector-valued functions

We prove a refined limiting imbedding theorem of the Brézis-Wainger type in the first critical case, i.e. , for Sobolev spaces and Bessel potential spaces of functions with values in a general Banach space E. In particular, the space E may lack the UMD property.     

Isometries of Lipschitz type function spaces

In this article, we describe isometries over the Lipschitz spaces under certain conditions. Indeed, we provide a unified proof for the main results of 3 and 5 in a more general setting. Finally, we extend our results for some other functions spaces like the space of vector‐valued little Lipschitz maps and pointwise Lipschitz maps.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

Weighted reverse weak type inequality for general maximal functions

Necessary and sufficient conditions are found to be imposed on a pair of weights, for which a weak type two-weighted reverse inequality holds, in the case of general maximal functions defined in homogenous type spaces.     

Maximally singular functions in Besov spaces

Assuming that 0 < α p < N, p, q ∈(1,∞), we construct a class of functions in the Besov space such that the Hausdorff dimension of their singular set is equal to N − α p. We show that these functions are maximally singular, that is, the Hausdorff dimension of the singular set of any other Besov function in is ≦ N − α p. Similar results are obtained for Lizorkin-Triebel spaces and for the Hardy space . Some open problems are listed. Received: 5 July 2005; revised: 18 October 2005     

Self-duality of bounded monotone boolean functions and related problems

In this paper we examine the problem of determining the self-duality of a monotone boolean function in disjunctive normal form (DNF). We show that the self-duality of monotone boolean functions with n disjuncts such that each disjunct has at most k literals can be determined in O(2^k2k²n) time. This implies an O(n²logn) algorithm for determining the self-duality of -DNF functions. We also consider the version where any two disjuncts have at most c literals in common. For this case we give an O(n^4(c+1)) algorithm for determining self-duality.     

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

Capacity for potentials of functions in Musielak-Orlicz spaces

We define a capacity for potentials of functions in Musielak-Orlicz spaces. Basic properties of such capacity are studied. We also estimate the capacity of balls and give some applications of the estimates.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB _p and the Dirichlet spaceD ^p . In the case ofB _p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Double piling structure of matrix monotone functions and of matrix convex functions

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f(0)0 and f is n-convex in [0,α),

(ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra M_n,

f(cac)cf(a)c,