Extremal properties of the set
of vector-valued Banach limits

García-Pacheco, Francisco Javier

doi:10.1515/math-2015-0067

Cited by 2 publications

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The class of normed spaces holding vector-valued Banach limits is indeed pretty large. It is closed under arbitrary ℓ ∞ -sums (see [40,Theorem 3]) and contains the Banach spaces 1-complemented in their bidual ([12, Corollary 1]) and the 1-injective Banach spaces (see [38,Theorem 2.3]).…”

Section: Existence Of Vector-valued Banach Limitsmentioning

confidence: 99%

Vector-Valued Banach Limits and Vector-Valued Almost Convergence

Garcia-Pacheco,

Perez-Fernandez

2018

Preprint

View full text Add to dashboard Cite

This book was intended to be Rafael Armario's Ph.D. thesis dissertation for obtaining his doctoral degree in Pure Mathematics at the University of Cádiz (Spain, EU). Unfortunately he passed away on January 28 th , 2013, right after he started writing this manuscript. His two Ph.D. advisors at the time of his death (the authors of this book) decided then to finish his work in book format and this is how this manuscript was given birth. Rafael Armario's lifeRafael Armario, better known as "Rafa" by his friends, was a joyful and warm person, a beloved son and friend, a very hard-working graduate student, an excellent highschool teacher, and an extremely brilliant mathematician. Childhood and early lifeRafa was born and raised in Cadiz, the oldest European city, within a humble middleclass family. He quickly developed a strong interest in the Cadiz culture and folklore, in particular, in the worldwide famous Cadiz carnival. Other local feasts such as the El Puerto fair also grew on him in a very profound way. University of Cadiz undergraduate periodRafa decided to register as a math major in the university of Cadiz after graduating from his neighborhood high school. He met a lot of friends there among students and i ii PREFACE professors, including the one who left the strongest imprint on him, Prof. Antonio Aizpuru, who eventually became his Ph.D. advisor. He also met a bunch of friends a buddies like Fernandito, Mari, Sole, Marina, Marimar, el Gafas, etc. Rafael Armario's workDuring his working period as a top-level mathematician, Rafa accomplished many objectives and obtained many different results. Indeed, all the theorems, definitions, examples, etc., which are anonymous in this book (including those of the framework) are due to his joint work with Prof. Aizpuru and the authors of this manuscript. University of Cadiz graduate periodRafa's graduate period was undoubtedly conditioned by the death of his Ph.D. advisor at the time, Prof. Antonio Aizpuru, who passed away in March 2008 after Rafa and him published their first paper (see [5]), which was also co-authored by his Ph.D. co-advisor at the time, Prof. Francisco Javier Pérez-Fernández.In September 2010, Prof. García-Pacheco, better known as Paquito, was hired at the level of Assistant Professor by the Mathematics Department of the University of Cadiz after spending nine semesters (three American academic years) at Kent State University (Ohio, USA) and ten semesters at Texas A&M University (Texas, USA). One of the first assignments of Prof. García-Pacheco was to co-advice Rafael Armario together with Prof. Pérez-Fernández, continuing this way the work once started by Prof. Antonio Aizpuru. The fruit of this co-advising was the co-authorship of the papers [3, 4,12,13,40]. The present book consists of those papers.Many anecdotes occurred during his graduate school period. For instance, in one occasion, Rafa and Paquito shared a hotel room, due to a lack of research funding, while attending a conference held in the university of Almeria (Spain, EU). The hotel wor...

show abstract

Section: Existence Of Vector-valued Banach Limitsmentioning

confidence: 99%

Vector-Valued Banach Limits and Vector-Valued Almost Convergence

Garcia-Pacheco,

Perez-Fernandez

2018

Preprint

View full text Add to dashboard Cite

show abstract

General methods of convergence and summability

2021

View full text Add to dashboard Cite

This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ N ; and that if X is not complete, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\mathbb{N} $ N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ c ( X ) is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ c ( X ) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$ HB ( lim ) : = { T ∈ B ( ℓ ∞ ( X ) , X ) : T | c ( X ) = lim and ∥ T ∥ = 1 } and prove that $\mathcal{HB}(\lim )$ HB ( lim ) is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$ B L X 0 if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $ L X 0 : = { T ∈ B ( ℓ ∞ ( X ) , X ) : c 0 ( X ) ⊆ ker ( T ) } . Finally, we study the multipliers associated with series for the above methods of convergence.

show abstract

Extremal properties of the set of vector-valued Banach limits

Cited by 2 publications

References 16 publications

Vector-Valued Banach Limits and Vector-Valued Almost Convergence

Vector-Valued Banach Limits and Vector-Valued Almost Convergence

General methods of convergence and summability

Contact Info

Product

Resources

About