Let $(\Omega , \mathfrak {M}, \mu )$ be a measure space. We show that if a surjective map $f: S_{L^1(\mu )} \to S_Y$ between the unit spheres of real $L^1(\mu )$ and of an arbitrary real normed space Y satisfies

$$ \begin{align*} \min\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\min\{\|x+y\|, \|x-y\|\},\quad x,y\in S_{L^1(\mu)}, \end{align*} $$

then there exists a phase function $\varepsilon : S_{L^1(\mu )} \to \{-1, 1\}$ such that $\varepsilon \cdot f$ is a surjective isometry from $S_{L^1(\mu )}$ onto $S_Y$, and furthermore, this isometry can be extended to a linear isometry on the whole space $L^1(\mu )$.

We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.

Building on a recent construction of Plebanek and Salguero-Alarcón, which solved the Complemented Subspace Problem for $C(K)$-spaces, and the subsequent work of De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete solving the Complemented Subspace Problem for Banach lattices, we show that the class of Banach lattices is not primary. Specifically, we exhibit a compact Hausdorff space L such that $C(L) \simeq X \oplus \tilde {X}$ and neither X nor $\tilde {X}$ is isomorphic to a Banach lattice. In particular, it also follows that the class of $C(K)$-spaces is not primary.

Let T be a bounded linear operator on a separable Banach space that satisfies geometric properties similar to those of $\ell ^p,\, p>1$. We prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for T can only take the values $0$ or $1$. The three classes of bounded linear operators emerging from the dichotomy of these extremal norm values coincide with the partition, created by considering the norm-attaining property and if the essential norm equals the norm.

This article studies the stability of the $L_p$ torsional measure in dimension $n \geq 3$. We show that the ball is the unique domain for the solution to the related overdetermined boundary value problem. Two concepts of relative asymmetry distance and $L^c$-distance have been used to estimate the stability of the $L_p$ torsional measure, based on different stability results of the $L_p$-width functionals. The result has been uniformly given via relative asymmetry distance for all $1<p \neq n+2$. In terms of $L^c$-distance, the result has been split into two cases: $p\in (1,n)$, and $n< p\neq n+2$.

In this article, we define a new class of operators called weak U-Dunford–Pettis, which generalizes the U-Dunford–Pettis, weak Dunford–Pettis, and order Dunford–Pettis classes, then we also give a characterization for this class, which we compare with some lattice properties, we then set out the conditions under which this class coincides with the U-Dunford–Pettis class, the weak Dunford–Pettis class, and the order Dunford–Pettis class.

For bounded operators on Kreĭn spaces, isometric or unitary dilations always exist. We prove that any minimal isometric or unitary dilation has a precise geometrical structure. Moreover, a bounded operator T has a unique minimal unitary dilation if and only if T and $T^*$ have unique minimal isometric dilation if and only if T is either contractive or expansive and $T^*$ is either contractive or expansive. Passing to the bi-dimensional case, a minimal unitary extension (in short, m.u.e.) $U=(U_1, U_2)$ is obtained for a pair $V=(V_1, V_2)$ of commuting bounded isometries on a Kreĭn space. There is a link with the one-dimensional case: if U is an m.u.e. for $V,$ then $U_1U_2$ is an m.u.e. for $V_1V_2$. Also, if $(V_1V_2)^*$ is either contractive or expansive, then V has a unique minimal unitary extension. A minimal regular isometric dilation is obtained for a commuting pair $T=(T_1, T_2)$ of bounded operators on a Kreĭn space such that $T_1,\ T_2$ are contractions and T is a bidisc contraction or $T_1,\ T_2$ are expansions and T is a bidisc expansion. The existence of a minimal unitary extension is used to provide a minimal regular unitary dilation for T. Discussions about uniqueness and geometric structure conclude the article.

We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly n-point homogeneous metric space for any (or every) $n \geqslant 3$. The counterpart for $n=2$ is the Banach–Mazur problem.

Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space E. The relations between these lattices allow us to give multiple characterizations of when the underlying Banach space E is finite-dimensional and when it is reflexive. Furthermore, we show that lattice homomorphisms between free Banach lattices are always composition operators, and study how these operators behave on the scale of lattices of positively homogeneous functions.

We introduce the super Alternative Daugavet property (super ADP), which lies strictly between the Daugavet property (DP) and the ADP. A Banach space X has the super ADP if for every element x in the unit sphere and every relatively weakly open subset W of the unit ball intersecting the unit sphere, there are an element $y\in W$ and a modulus one scalar θ such that $\|x+\theta y\|$ is almost two. Spaces with the DP satisfy this condition, and it implies the ADP. We first provide examples of super ADP spaces that fail the DP. We show that the norm of a super ADP space is rough, hence the space cannot be Asplund, and we also prove that the space fails the point of continuity property (particularly, the Radon–Nikodým property). In particular, we get examples of spaces with the ADP that fail the super ADP. For a better understanding of the differences between the super ADP, the DP and the ADP, we consider the localizations of these properties and prove that they behave rather differently. As a consequence, we provide characterizations of the super ADP for spaces of vector-valued continuous functions and of vector-valued integrable functions.

In this article, we study various properties and characterizations of weaving frames in Krein spaces. In support of our findings, several examples and counter examples are provided, illustrating the applicability of the theoretical results. Additionally, we extend the discussion to an important application in probabilistic erasure, highlighting how weaving frames can be used to mitigate data loss in such scenarios.

We give some characterizations of $\sigma $-unbounded Dunford–Pettis operators (whenever $x_{n}\overset {uaw}{\longrightarrow } 0$ implies that $T(x_{n})\overset {un}{\longrightarrow } 0$). In addition, we study some properties of this operator, some other interesting results are also obtained.

We investigate the notion of ideal (equivalently: filter) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell _p$, $c_0$, and James’ space). Those examples lead to distinguishing and characterizing ideals (equivalently: filters) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and ideals (equivalently: filters) on the set of natural numbers.

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

Let C be a convex body (i.e., a proper closed convex subset with nonempty interior) in a normed space X. We consider four moduli of local uniform rotundity for C at a given point $x\in \partial C$, that extend in a natural way the corresponding notions for the unit ball $B_X$, and we prove that they all coincide. This extends a known result of J. Daneš from 1976 concerning the particular case when $C=B_X$.

Chan and Seceleanu have shown that if a weighted shift operator on $\ell^p(\mathbb{Z})$, $1\leq p \lt \infty$, admits an orbit with a non-zero limit point then it is hypercyclic. We present a new proof of this result that allows to extend it to very general sequence spaces. In a similar vein, we show that, in many sequence spaces, a weighted shift with a non-zero weakly sequentially recurrent vector has a dense set of such vectors, but an example on $c_0(\mathbb{Z})$ shows that such an operator is not necessarily hypercyclic. On the other hand, we obtain that weakly sequentially hypercyclic weighted shifts are hypercyclic. Chan and Seceleanu have, moreover, shown that if an adjoint multiplication operator on a Bergman space admits an orbit with a non-zero limit point then it is hypercyclic. We extend this result to very general spaces of analytic functions, including the Hardy spaces.

In this paper, we prove a new uncertainty principle for functions with radial symmetry by differentiating a radial version of the Stein–Weiss inequality. The difficulty is to prove the differentiability in the limit of the best constant that unlike the general case it is not known. We provide also an integral alternative formula for the logarithmic weight $(\log|\xi|)$ in Fourier domain.

Let $\varphi : B_d\to \mathbb {D}$, $d\ge 1$, be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb {C}^d$ and $\mathbb {D}= B_1$. Let $\Theta : \mathbb {D} \to \mathbb {D}$ be an inner function, and let $K^p_\Theta $ denote the corresponding model space. For $p>1$, we characterize the compact composition operators $C_\varphi : K^p_\Theta \to H^p(B_d)$, where $H^p(B_d)$ denotes the Hardy space.

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim toward applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into $\mathbb {R}$-trees have Lipschitz free spaces isomorphic to $L^1$-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into $\mathbb {R}$-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to $\ell ^1$, (2) the Lipschitz free space over hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-space and (3) every infinite, finitely generated hyperbolic group stochastically embeds into an $\mathbb {R}$-tree, has Lipschitz free space isomorphic to $\ell ^1$, and admits a proper, uniformly Lipschitz affine action on $\ell ^1$.