Norm attaining operators

Every Banach space is isomorphic to a space with the property that the norm-attaining operators are dense in the space of all operators into it, for any given domain space. A super-reflexive space is arbitrarily nearly isometric to a space with this property.     

Bidual renormings of Banach spaces

Given a separable Banach space X with no isomorphic copies of ℓ₁ and a separable subspace Y of its bidual, we provide a sufficient condition on Y to ensure that X admits an equivalent norm such that the restriction to Y of the corresponding bidual norm is midpoint locally uniformly rotund. This result applies to the separable subspaces of the bidual of a Banach space with a shrinking unconditional Schauder basis and to the bidual of the James space.     

ALUR dual renormings of Banach spaces

We give a covering type characterization for the class of dual Banach spaces with an equivalent ALUR dual norm.     

Asymptotic properties of Banach spaces under renormings

It is shown that a separable Banach space can be given an equivalent norm with the following properties: If is relatively weakly compact and , then converges in norm. This yields a characterization of reflexivity once proposed by V.D. Milman. In addition it is shown that some spreading model of a sequence in is 1-equivalent to the unit vector basis of (respectively, ) implies that contains an isomorph of (respectively, ).

     

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1 < p < ∞ and 1 ≤ q < ∞, a linear operator T: ℓ _p → ℓ _q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p = 1). For 1 < p ≠ q < ∞, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: ℓ _p → ℓ _q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p = q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.     

On equivalent strictly G-convex renormings of Banach spaces

We study strictly G-convex renormings and extensions of strictly G-convex norms on Banach spaces. We prove that ℓ_ω(Γ) space cannot be strictly G-convex renormed given Γ is uncountable and G is bounded and separable.     

Uniformly smooth renormings of uniformly convex Banach spaces

In this note we study the quantitative side of the famous Enflo-Pisier theorem on the possibility of equivalent uniformly smooth renormings of superreflexive Banach spaces (in particular, uniformly convex and uniformly nonsquare ones). Typical re result: let the modulus of convexity of the space X, which has a locally unconditional structure, satisfy the condition _x() C·^p. Then the space X admits an equivalent q-smooth renorming for any q, satisfying the inequality q<1n 2/1n (2× (1–C·2^–p/2)).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 120–134, 1984.     

A note on subsymmetric renormings of Banach spaces

We revisit the concept of a subsymmetric norm and construct a subsymmetric renorming of a Banach space with a subsymmetric basis. As a by-product of our work we introduce the concept of a lower symmetric basis and investigate its connection with subsymmetric bases and subsymmetric renormings.     

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.     

Superstable operators on Banach spaces

It has been shown by W. Arendt—C.J.K. Batty and Yu.I. Lyubich— V.Q. Phong that the powers of a linear contraction on a reflexive Banach space converge strongly to zero if the boundary spectrum is countable and contains no eigenvalues. In this paper we characterize the countability of the boundary spectrum through a stronger convergence property in terms of ultrapower extensions. This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG     

Dissipative operators on Banach spaces

A bounded linear operator T on a Banach space is said to be dissipative if ‖etT‖?1 for all t?0. We show that if T is a dissipative operator on a Banach space, then:

(a)

.

(b)

If σ(T)∩iR is contained in [−iπ/2,iπ/2], then

     

Norm attaining polynomials

If L 0 is a continuous symmetric n-linear form on a Banachspace and is the associatedcontinuous n-homogeneous polynomial, the ratio always lies between 1 and nⁿ/n!. At one extreme,if L is defined on Hilbert space, then . If L attains norm on Hilbert space, then also attains norm; in this case, we give anexplicit construction to provide a unit vector x₀ with . At the other extreme, if and L attains norm, then attains norm. We prove that in general the converseis not true.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Zeros ofm-accretive operators in Banach spaces

An operatorT defined on a subsetD(T) of a Banach spaceE and taking values in 2 ^E is said to bem-accretive if for each λ>0, the mappingJ _λ=(I+λT)⁻¹ is a single-valued nonexpansive mapping defined on all ofE. In this paper, facts derived from an elementary analysis of the bahavior of {J _λ x}_λ≧0 for fixedxεD(T) are applied to obtain theorems concerning existence of zeros and surjectivity. Research supported in part by National Science Foundation grant: MCS 76-03945-A01.