Functionals that do not attain their norm

Acosta, Marı́a D.; Aizpuru, A.; Aron, Richard M.; García-Pacheco, Francisco Javier

doi:10.36045/bbms/1190994202

Cited by 27 publications

(23 citation statements)

References 14 publications

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“…Let y P RpT q " RpT 2 q. Then y " T 2 v for some v P N pT 2 q K " N pT q K (see (1) of Remark (4.2)). Now T : y " T : T 2 v " P N pT q K T v " T v, as RpT q Ď N pT q K .…”

Section: Normal Am-operatorsmentioning

confidence: 99%

Spectral properties of absolutely minimum attaining operators

Bala

Ramesh

2020

Banach J. Math. Anal.

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Let T : H 1 Ñ H 2 be a bounded linear operator defined between complex Hilbert spaces H 1 and H 2 . We say T to be minimum attaining if there exists a unit vector x P H 1 such that }T x} " mpT q, where mpT q :" inf t}T x} : x P H 1 , }x} " 1u is the minimum modulus of T . We say T to be absolutely minimum attaining (AM-operators in short), if for any closed subspace M of H 1 the restriction operator T | M : M Ñ H 2 is minimum attaining. In this paper, we give a new characterization of positive absolutely minimum attaining operators (AM-operators, in short), in terms of its essential spectrum. Using this we obtain a sufficient condition under which the adjoint of an AM-operator is AM. We show that a paranormal absolutely minimum attaining operator is hyponormal. Finally, we establish a spectral decomposition of normal absolutely minimum attaining operators. In proving all these results we prove several spectral results for paranormal operators. We illustrate our main result with an example.

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“…Let y P RpT q " RpT 2 q. Then y " T 2 v for some v P N pT 2 q K " N pT q K (see (1) of Remark (4.2)). Now T : y " T : T 2 v " P N pT q K T v " T v, as RpT q Ď N pT q K .…”

Section: Normal Am-operatorsmentioning

confidence: 99%

Spectral properties of absolutely minimum attaining operators

Bala

Ramesh

2020

Banach J. Math. Anal.

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“…The canonical basis of c 0 under this equivalent renorming is still a Schauder basis, but it is not monotone, since in (Theorem 3.1(1), [14]), it was proved that, if a Banach space X has a monotone Schauder basis, then NA(X) contains an infinite dimensional vector subspace.…”

Section: Renormings Concerning Schauder Basesmentioning

confidence: 99%

Pre-Schauder Bases in Topological Vector Spaces

García-Pacheco

Fernández

2019

Symmetry

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A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

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“…When is N A(X) spaceable? These questions have been deeply studied in [35] by Bandyopadhyay and Godefroy, who provided, among other results, conditions that ensure that N A(X) is not spaceable; see also the recent works [1,153] for more results on the linear structure of N A(X). Very recently, García-Pacheco and Puglisi [158] showed that every Banach space admitting an infinite dimensional separable quotient can be equivalently renormed in such a way that the set of its norm attaining functionals contains an infinite dimensional linear subspace.…”

Section: Some Remarks and Conclusion General Techniquesmentioning

confidence: 99%

“…2 (4) At times we shall say that B is, simply, κ-algebrable if there exists a κgenerated subalgebra C of A with C ⊂ B ∪ {0}. 1 Indeed, this λ(M ) might not exist. It is not difficult to provide natural examples of sets which are n-lineable for every n ∈ N but which are not infinitely lineable.…”

Section: Introduction "Strange" Mathematical Objects Throughout Historymentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

197

149

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For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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Functionals that do not attain their norm

Cited by 27 publications

References 14 publications

Spectral properties of absolutely minimum attaining operators

Spectral properties of absolutely minimum attaining operators

Pre-Schauder Bases in Topological Vector Spaces

Linear subsets of nonlinear sets in topological vector spaces

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