Related papers: On density and Bishop-Phelps-Bollob\'as type prope…

The $p$-Gelfand Phillips property ($1\le p<\infty$) is studied in spaces of operators. Dunford - Pettis type like sets are studied in Banach spaces. We discuss Banach spaces $X$ with the property that every $p$-convergent operator $T:X\to…

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let $A_b(B_X:X)$ be the Banach space of all bounded continuous functions $f$ on the…

G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this…

Let $H^\infty$ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by $\mathscr{B}(H^\infty)$ the Banach space of all bounded linear operators from $H^\infty$ to itself. We prove that the…

We prove that the class of positive operators from $L_\infty (\mu)$ to $L_1 (\nu)$ has the Bishop-Phelps-Bollob\'as property for any positive measures $\mu$ and $\nu$. The same result also holds for the pair $(c_0, \ell_1)$. We also provide…

We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is octahedral for every non-zero Banach space $Y$. They are those satisfying that, for every finite dimensional subspace $Z$, $\ell_\infty$ can be finitely-representable in…

Let X be a separable Banach space and Y a space which has the Radon-Nikodym property. In this work, we show that L(X, Y) has the Radon-Nikodym property, if L(X, Y) is weakly locally uniformly convex or if L(X, Y) is a weakly compactly gen-…

Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of continuous quasinorms $\mathcal{Q}_0=\{\|\cdot\|:E\to\mathbb{R}\}$ as $$d(\|\cdot\|_X,\|\cdot\|_Y)=\min\{\mu:\|\cdot\|_X…

In this paper, we investigate the relationship between the weak min-max property and the diameter uniformity of domains in Banach spaces with dimensions at least $2$. As an application, we show that diameter uniform domains are invariant…

We characterize $k-$smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize $k-$smoothness…

This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an…

We study complex distribution spaces given over a bounded Lipschitz domain $\Omega$ and associated with an elliptic differential operator $A$ with $C^{\infty}$-coefficients on $\overline{\Omega}$. If $X$ and $Y$ are quasi-Banach…

We introduce a Bochner integral approach to projective norm attainment in tensor products of Banach spaces by defining the class of integral projective norm-attaining tensors. This framework provides a broader, measure-theoretic approach to…

In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff-James orthogonality techniques, we give a necessary condition for a bounded linear operator…

We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikod\'ym property to a "less linear" frame. We note that a certain part of the theory can be developed in rather…

We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is…

In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in \cite{B3},…

We prove that for a given Banach space $X$, the subset of norm attaining Lipschitz functionals in $\mathrm{Lip}_0(X)$ is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate…

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain…

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we…