Certain Norm Related Quantities of Unbounded Linear Operators

Certain norm related functions of linear operators are considered in the very general setting of not necessarily continuous linear operators in normed spaces. These are shown to be closely related to the theory of precompact, strictly cosingular and a class of Φ_? type operators in addition to having applications to perturbation theory. We also obtain some basic continuity and precompactness properties of linear operators in normed spaces which are expressed in terms of the functions under consideration.     

Chebyshev polynomials for operators

The problem of uniqueness of the Chebyshev polynomials for bounded linear operators on normed linear spaces is investigated. Herrn Professor Dr. Dr. h.c. Heinz K?nig zu seinem achtzigsten Geburtstag gewidmet     

Esistenza,unicità e dipendenza continua per una classe di problemi ai limiti non lineari

Summary The following abstract boundary value problem is considered: Lx=Hx, Qx=0, where L, H, Q denote operators, L linear, mapping normed spaces into normed spaces. Existence and uniqueness criteria are given and the dependence of the solution on L, H, Q is studied.

---

---

Lavoro eseguito nell'ambito dell'attività dei Gruppi di ricerca matematica del C.N.R. I numeri 1–5 sono dovuti aG. Pulvirenti e i numeri 6–10 aG. Santagati.     

Paracomplete normed spaces and Fredholm theory

A normed space is paracomplete if it admits a new norm, stronger than the initial one, that makes it complete. Here we give a characterization of paracomplete normed spaces. As a consequence, we show that operators on paracomplete spaces have compact spectrum in the algebra of all operators, and that the class of paracomplete spaces is not stable under ℓ₂-sums. Moreover, we give characterizations for the closed Fredholm operators on paracomplete spaces and for the almost semi-Fredholm operators of Harte on normed spaces.     

Tractability of Tensor Product Linear Operators

This paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not.     

Normes des extensions quaternionique d'opérateurs réels

We consider bounded linear operators defined on real normed spaces, and with range in quaternionic spaces. We study the norms of the quaternionic extensions of such operators. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

PROJECTIONS ON OPERATOR DUALS

Abstract

We discuss the existence of a projection with kernel K_b(E,F) ¹ (the annihilator of the quasi-compact operators) on the dual space of the space L _b,(E, F) of continous linear operators. Our results are proved in the context of Hausdorff locally convex spaces, but also provide extensions of recent results in the context of Banach spaces.     

On the jackson theorem for periodic functions in spaces with integral metric

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L _p , 0 < p < 1, and L ₀. In particular, we prove the multidimensional Jackson theorem in L _p (T ^m ), 0 < p < 1.     

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear, and rotated Q₁ polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q₁ IFE shape functions are always guaranteed regardless of the Lamé parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi‐point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.     

Multiple summing operators on<Emphasis Type="Italic">C(K)</Emphasis> spaces

In this paper, we characterize, for 1≤p<∞, the multiple (p, 1)-summing multilinear operators on the product ofC(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators onC(K) in terms of their representing measures and a new multilinear characterization ofL _∞ spaces. We also solve a problem stated by M.S. Ramanujan and E. Schock, improve a result of H. P. Rosenthal and S. J. Szarek, and give new results about polymeasures. Both authors were partially supported by DGICYT grant BMF2001-1284.     

Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

On approximation properties of Balázs-Szabados operators their Kantorovich extension

In this paper we deal with a sequence of positive linear operatorsR _n^[β] approximating functions on the unbounded interval [0, t8) which were firstly used by K. Balázs and J. Szabados. We give pointwise estimates in the framework of polynomial weighted function spaces. Also we establish a Voronovskaja type theorem in the same weighted spaces for K_n^[β] operators, representing the integral generalization in Kantorovich sense of the R_n^[β].     

A-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceL _n f of positive linear operators.     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

K -- Convex Operators and Walsh Type Norms

A celebrated result of G. Pisier states that the notions of B-convexity and K-convexity coincide for Banach spaces. We complement this in the setting of linear and bounded operators between Banach spaces. Our approach is local and even yields inequalities between gradations of K-convexity norms and Walsh type norms of operators. Our method combines G. Pisier's original ideas and the main steps in the proof of the Beurling-Kato theorem on extensions of C₀-semigroups of operators to holomorphic semigroups with the technique of ideal norms.     

Decomposable symmetric mappings between infinite-dimensional spaces

Decomposable mappings from the space of symmetric k-fold tensors over E, , to the space of k-fold tensors over F, , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.     

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·) _i and (·, ·) _S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.     

Norm attaining operators fromL 1 intoL ∞

We show that the set of norm attaining operators is dense in the space of all bounded linear operators fromL ₁ intoL _∞. Partially supported by Human Capital and Mobility. Project No. ERB4050Pl922420, Geometry of Banach spaces. Supported by D.G.I.C.Y.T., Project No. PB93-1142.