Norm-attaining Operators Research Articles

Rank-one perturbations and norm-attaining operators

Rank-one perturbations and norm-attaining operators

On Numerical Ranges and Spectra of Norm Attaining Operators in C*-Algebras

In this paper, we study norm attaining operators in C∗-algebras. We characterize their numerical ranges and spectra. In particular, we show that if a norm-attaining operator S is self-adjoint, then its spectrum lies in the interval [−||S||, ||S||].

On a Set of Norm Attaining Operators and the Strong Birkhoff–James Orthogonality

Continuing the study of recent results on the Birkhoff–James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia–Šemrl property for operators which is weaker than the Bhatia–Šemrl property. The set of operators with the adjusted Bhatia–Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia–Šemrl property. It is known that the set of operators with the Bhatia–Šemrl property is norm-dense if the domain space X of the operators has the Radon–Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $$c_0$$ , $$L_1[0,1]$$ and C[0, 1]. In contrast with the Bhatia–Šemrl property, we show that the set of operators with the adjusted Bhatia–Šemrl property is norm-dense when the domain space is $$c_0$$ or $$L_1[0,1]$$ . Moreover, we show that the set of functionals having the adjusted Bhatia–Šemrl property on C[0, 1] is not norm-dense but such a set is weak- $$*$$ -dense in $$C(K)^*$$ for any compact Hausdorff K.

Group Invariant Operators and Some Applications to Norm-Attaining Theory

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties \(\alpha \) of Schachermayer and \(\beta \) of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, which says that if X has the Radon–Nikodým property, then X has the G-Bishop–Phelps property for G-invariant operators whenever \(G \subseteq \mathcal {L}(X)\) is a compact group of isometries on X.

Residuality in the set of norm attaining operators between Banach spaces

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if C is a bounded subset of a Banach space X which admit an LUR renorming satisfying that, for every Banach space Y, the operators T from X to Y for which the supremum of ‖Tx‖ with x∈C is attained are dense, then the Gδ set of those functionals which strongly exposes C is dense in X⁎. This extends previous results by J. Bourgain and K.-S. Lau. The particular case in which C is the unit ball of X, in which we get that the norm of X⁎ is Fréchet differentiable at a dense subset, improves a result by J. Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the Gδ set of absolutely strongly exposing operators from X to Y by requiring that the set of strongly exposing functionals on X is dense and conditions on Y or Y⁎ involving RNP and discreteness on the set of strongly exposed points of Y or Y⁎. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications to the classical theory of norm-attaining operators, to the theory of norm-attaining bilinear forms, to the geometry of the preduals of spaces of Lipschitz functions, and to the theory of strongly norm-attaining Lipschitz maps. In particular, we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.

Norm-Attaining Tensors and Nuclear Operators

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators \(\mathcal N(X,Y)\) and in the projective tensor product space \(X\widehat{\otimes }_\pi Y\). We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in \(\mathcal N(X,Y)\) and in \(X\widehat{\otimes }_\pi Y\) is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in \(X\widehat{\otimes }_\pi Y\) which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

AbstractIn this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ;(ii)Every operator from E into F attains its norm;(iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ ,where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

On the weak maximizing properties

Quite recently, a new property related to norm-attaining operators has been introduced: the weak maximizing property (WMP). In this note, we define a generalised version of it considering other topologies than the weak one (mainly the \(\hbox {weak}^*\) topology). We provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair (X, Y) enjoys a certain maximizing property. This approach not only allows us to (re)obtain as a direct consequence that the pair \((\ell _p,\ell _q)\) has the WMP but also provides many more natural examples of pairs having a given maximizing property.

Norm-attaining operators which satisfy a Bollobás type theorem

In this paper, we are interested in studying the set $$\mathcal {A}_{\Vert \cdot \Vert }(X, Y)$$ of all norm-attaining operators T from X into Y satisfying the following: given $$\varepsilon >0$$ , there exists $$\eta $$ such that if $$\Vert Tx\Vert > 1 - \eta $$ , then there is $$x_0$$ such that $$\Vert x_0 - x\Vert < \varepsilon $$ and T itself attains its norm at $$x_0$$ . We show that every norm one functional on $$c_0$$ which attains its norm belongs to $$\mathcal {A}_{\Vert \cdot \Vert }(c_0, \mathbb {K})$$ . Also, we prove that the analogous result holds neither for $$\mathcal {A}_{\Vert \cdot \Vert }(\ell _1, \mathbb {K})$$ nor $$\mathcal {A}_{\Vert \cdot \Vert }(\ell _{\infty }, \mathbb {K})$$ . Under some assumptions, we show that the sphere of the compact operators belongs to $$\mathcal {A}_{\Vert \cdot \Vert }(X, Y)$$ and that this is no longer true when some of these hypotheses are dropped. The analogous set $$\mathcal {A}_{{{\,\mathrm{nu}\,}}}(X)$$ for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets $$\mathcal {A}_{\Vert \cdot \Vert }(X, X)$$ and $$\mathcal {A}_{\text {nu}}(X)$$ when $$X=c_0$$ or $$\ell _{p}$$ . As a consequence, we get that the canonical projections $$P_N$$ on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to $$\mathcal {A}_{\Vert \cdot \Vert }(X, X)$$ but not to $$\mathcal {A}_{{{\,\mathrm{nu}\,}}}(X)$$ and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.

Spaceability in norm-attaining sets

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set {1,…,n}, there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on ℓ1 such that, for all nonzero elements B of such a subspace, the Arens extension associated to the permutation σ of B is norm-attaining if and only if σ is an element of P. We also study the structure of the set of norm-attaining n-linear forms on c0.

Norm Attaining Operators and Pseudospectrum

It is shown that if 1 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥ > 1 and I + T does not attain its norm.

Norm-attaining operators into Lorentz sequence spaces

We prove that the Lorentz sequence spaces do not have the property B of Lindenstrauss. In fact, for any admissible sequences w, v ∈ c0 \ l1, the set of norm-attaining operators from the Orlicz space hϕ(w) (ϕ is a certain Orlicz function) into d(v, 1) is not dense in the corresponding space of operators. We also characterize the spaces such that the subset of norm-attaining operators from the Marcinkiewicz sequence space into its dual is dense in the space of all bounded and linear operators between them.

Norm-attaining operators between Marcinkiewicz and Lorentz spaces

Bishop and Phelps proved that the set of norm-attaining functionals on any Banach space is dense in the topological dual. After that, the study of the same kind of problems for operators was initiated by Lindenstrauss, and several general positive results were proved. It was then consistently continued for different classes of spaces including L 1 (μ) or C(K). Here a similar problem is studied in the context of classical interpolation Marcinkiewicz and Lorentz spaces, M W 0 and Λ 1, v , in both the real and the complex cases. We show that if wv ∈ L 1 then the identity operator between these spaces is bounded, but it is not possible to approximate it by norm-attaining operators. We also prove that every compact operator from M W 0 to Λ 1, v can be approximated by finite-rank norm-attaining operators.

Norm Attaining Operators on Some Classical Banach Spaces

We show that for the Köthe space X = c0 + 𝓁1(w), equipped with the Luxemburg norm, the set of norm attaining operators from X into any infinite-dimensional strictly convex Banach space Y is not dense in the space of all bounded operators. The same assertion holds for any infinite-dimensional L1(μ). This gives the first example of a classical space X satisfying the previous property. We also prove that all the spaces c0 + 𝓁1(w) are isomorphic for a large class of weights w.

Denseness of norm-attaining operators into strictly convex spaces

We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 &lt; p &lt; ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

Norm-Attaining Operators into Strictly Convex Banach Spaces

If a strictly convex Banach spaceYcontains either a symmetric basic sequence which is not equivalent to thel1-basis or a normalized sequence with an upperp-estimate, then there is a Banach spaceXsuch that the set of norm-attaining operators is not dense in the Banach space of all bounded linear operators fromXintoY. We deduce that no infinite-dimensional uniformly convex Banach space has Lindenstrauss' property B.

Symmetric block bases of sequences with large average growth

We show that if 0<e≦1, 1≦p<2 andx 1, …,x n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ l n ei x l‖≧n 1/p, then one can find a block basisy 1, …,y m ofx 1, …,x n which is (1+e)-symmetric and has cardinality at leastγn 2/p-1(logn)−1, where γ depends on e only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex 1, …,x n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n 2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn 2/p-1(logn)4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.

Norm Attaining Operators and Simultaneously Continuous Retractions

A compact metric space $S$ is constructed and it is shown that there is a bounded linear operator $T:{L^1}[0,1] \to C(S)$ which cannot be approximated by a norm attaining operator. Also it is established that there does not exist a retract of ${L^\infty }[0,1]$ onto its unit ball which is simultaneously weak* continuous and norm uniformly continuous.

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.

Norm Attaining Operators and Norming Functionals

The question of whether a countably additive measure with values in a Banach space attains the diameter of its range was unresolved. In this paper an example is given of a countably additive vector measure, taking values in a $C(K)$ space, for which the diameter of the range is not attained. A property stronger than the attainment of the diameter, but which is possessed by many measures taking values in $L$-spaces, is shown to fail for infinite-dimensional measures into a space having smooth dual. As an application of the concept of norming functional (the existence of which is equivalent to the attainment of diameter), a characterization is given of the countably additive measures into space having smooth dual.