We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\ell_1$.
We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional Banach space with the additional condition that $T$ attains norm at each extreme point of $B_{\ell_{\infty}^{n}}.$ We also characterize $k-$smoothness of an operator defined between $\ell_{\infty}^3$ and $\ell_{1}^3.$
We prove the existence of free objects in certain subcategories of Banach lattices, including $p$-convex Banach lattices, Banach lattices with upper $p$-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (\emph{Israel J.\ Math.}~2019). In the $p$-con\-vex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear $p$-summing maps.
For $1<p\leqslant \infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\textsf{A}_p$ and $\textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.
We study the smoothness and the norm attainment of bounded bilinear operators between Banach spaces, using the concepts of Birkhoff-James orthogonality and semi-inner-products. In the finite-dimensional case, we characterize Birkhoff-James orthogonality of bilinear operators in terms of the norm attainment set. This yields a nice characterization of smoothness of bilinear operators between Banach spaces, in the finite-dimensional case. Without any restriction on the dimension of the space, we obtain a complete characterization of the norm attainment set of a bounded bilinear operator using sem…
Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property
University of Agriculture in Krakow（克拉科夫农业大学）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）；Lviv Polytechnic National University（利沃夫国立理工大学）；Comenius University in Bratislava（布拉迪斯拉发康斯坦丁尼查尔大学）
We study four asymptotic smoothness properties of Banach spaces, denoted $\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their description by proving the missing renorming theorem for $\textsf{A}_p$. We prove that asymptotic uniform flattenability (property $\textsf{T}_\infty$) and summable Szlenk index (property $\textsf{A}_\infty$) are three space properties. Combined with the positive results of the first named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three space problem for this family of properties. We also …
Free Banach lattices over pre-ordered Banach spaces
Mathematical Institute, Leiden University（莱顿大学数学研究所）；Department of Mathematics and Applied Mathematics, University of Pretoria（比勒陀利亚大学数学与应用数学系）；College of Mathematics, Sichuan University（四川大学数学学院）
We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We demonstrate an important class of \emph{real} Banach space $ \mathbb{X} $ for which the numerical radius defines a norm on $ \mathbb{L}(\mathbb{X}), $ the space of all bounded linear operators on $ \mathbb{X} $. We characterize Birkhoff-James orthogonality in the space of bounded linear operators on a finite-dimensional Banach space, endowed with the numerical radius norm. Some examples are a…
We introduce a family of quasi-Banach spaces - which we call wave packet smoothness spaces - that includes those function spaces which can be characterised by the sparsity of their expansions in Gabor frames, wave atoms, and many other frame constructions. We construct Banach frames for and atomic decompositions of the wave packet smoothness spaces and study their embeddings in each other and in a few more classical function spaces such as Besov and Sobolev spaces.
We study $k-$smoothness of bounded linear operators defined between arbitrary Banach spaces. As an application, we characterize $k-$smooth operators defined from $\ell_1^n$ to an arbitrary Banach space. We also completely characterize $k-$smooth operators defined between arbitrary two-dimensional Banach spaces.
Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $Γ$, a free set is called $Γ$-convex if it closed under isometric conjugation by isometries intertwining $Γ$. We establish an Effros-Winkler Hahn-Banach separation theorem for $Γ$-convex sets; they are delineated by linear pencils in the coordinates of $Γ$ and the variables $x$.
Motivated by the Lyapunov convexity theorem in infinite dimensions, we extend the convexity of the integral of a decomposable set to separable Banach spaces under the strengthened notion of nonatomicity of measure spaces, called "saturation", and provide a complete characterization of decomposability in terms of saturation.
In this paper we investigate some reflexivity-type properties of separable measurable Banach bundles over a $σ$-finite measure space. Our two main results are the following: - The fibers of a bundle are uniformly convex (with a common modulus of convexity) if and only if the space of its $L^p$-sections is uniformly convex for every $p\in(1,\infty)$. - The fibers of a bundle are reflexive if and only if the space of its $L^p$-sections is reflexive. These results generalise the well-known corresponding ones for Lebesgue-Bochner spaces.
Working in infinite dimensional linear spaces, we deal with support for closed sets without interior. We generalize the Convexity Theorem for closed sets without interior. Finally we study the infinite dimensional version of Jordan hypersurfaces. Our whole work never assumes smoothness and is based exclusively on non-differential Convex Analysis tools and, in particular, on theory of convex cones. A crucial mathematical tool for our results is obtained solving the decomposition problem for non-closed non pointed cones.
