Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-liń\-ski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff-James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff-James orthogonality.
Associated to Birkhoff orthogonality, we study Birkhoff angles in a normed space and present some of their basic properties. We also discuss how to decide whether an angle is more acute or more obtuse than another. In addition, given two vectors $x$ and $y$ in a normed space, we study the formula for Birkhoff `cosine' of the angle from $x$ to $y$ from which we can, in principal, compute the angle. Some examples will be presented.
In this article, the relationship between Birkhoff-James orthogonality of elementary tensors in certain tensor product spaces with the Birkhoff-James orthogonality of individual elements in their respective spaces is studied.
In this paper, we extend the investigations regarding Birkhoff-James orthogonality of linear operators to bounded continuous functions on metric spaces. We introduce Birkhoff-James extensions of continuous functions and study them in detail, in the separate contexts of compact and non-compact metric spaces. We conclude by discussing an application of our ideas to the study of Birkhoff-James orthogonality in $C(X)$ with the supremum norm, where $X$ is a compact metric space.
It is shown that Birkhoff-James orthogonality knows everything about the smooth norms in reflexive Banach spaces and can also compute the dimensions of the underlying normed spaces.
Real smooth three-dimensional or higher Banach spaces are isomorphic with respect to the nonlinear structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic. Moreover, using smooth Radon planes and non-smooth direct sums, in arbitrary dimensions, we construct examples of non-isometric pairs of real Banach spaces that admit norm-preserving homogeneous bicontinuous Birkhoff-James orthogonality preservers among them.
We study Birkhoff-James orthogonality and its local symmetry in some sequence spaces namely $\ell_p,$ for $1\leq p\leq\infty$, $p\neq2$, $c$, $c_0$ and $c_{00}$. Using the characterization of the local symmetry of Birkhoff-James orthogonality, we characterize isometries of each of these spaces onto itself and obtain the Banach-Lamperti theorem for onto operators on the sequence spaces.
We study Birkhoff-James orthogonality and its pointwise symmetry in commutative $C^*$ algebras, i.e., the space of all continuous functions defined on a locally compact Hausdorff space that vanish at infinity. We use this characterization to obtain the characterization of Birkhoff-James orthogonality on $L_\infty$ space defined on any arbitrary measure space. We also do the same for the $L_p$ spaces for $1\leq p<\infty$.
In this paper we characterize Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Banach space $ \mathbb{X}. $ We also explore the symmetry of Birkhoff-James orthogonality of linear operators defined on $ \mathbb{X}. $ Using some of the related results proved in this paper, we finally prove that $ T \in \mathbb{L}(l_{p}^2) (p \geq 2, p \neq \infty) $ is left symmetric with respect to Birkhoff-James orthogonality if and only if $ T $ is the zero operator. We conjecture that the result holds for any finite dimensional strictly convex and smooth real Banach space…
James-type constants for Birkhoff--James orthogonality and approximate isosceles
School of Mathematics and Statistics, Anqing Normal University（安庆师范大学数学与统计学院）；School of Mathematics and Statistics, Guangdong University of Foreign Studies（广东外语外贸大学数学与统计学院）；Department of Mathematics, Sun Yat-sen University（中山大学数学系）
We study Birkhoff-James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff-James orthogonality set. We also present some best approximation type results in the space of bounded linear opera…
We explore Birkhoff-James orthogonality of two elements in a complex Banach space by using the directional approach. Our investigation illustrates the geometric distinctions between a smooth point and a non-smooth point in a complex Banach space. As a concrete outcome of our study, we obtain a new proof of the Bhatia-Šemrl Theorem on orthogonality of linear operators.
We study the relationship between the point-wise symmetry of Birkhoff-James orthogonality and the geometry of the space of operators $\mathbb{B}(\ell_\infty^n,\ell_1^m)$. We show that any non-zero left-symmetric point in this space is a smooth point. We also show that for $n\geq4$, any unit norm right-symmetric point of this space is an extreme point of the closed unit ball. This marks the first step towards characterizing the extreme points of these unit balls and finding the Grothendieck constants $G(m,n)$ using Birkhoff-James orthogonality techniques.
In this paper, we establish several characterizations of the $A$-parallelism of bounded linear operators with respect to the seminorm induced by a positive operator $A$ acting on a complex Hilbert space. Among other things, we investigate the relationship between $A$-seminorm-parallelism and $A$-Birkhoff-James orthogonality of $A$-bounded operators. In particular, we characterize $A$-bounded operators which satisfy the $A$-Daugavet equation. In addition, we relate the $A$-Birkhoff-James orthogonality of operators and distance formulas and we give an explicit formula of the center mass for $A$-…
This paper deals with study of Birkhoff-James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For arbitrary Hilbert space $ \mathbb{H},$ we also study orthogonality to subspace of the space of linear operators $L(\mathbb{H}), $ both with respect to operator norm as well as numerical radius norm.
We study the norm derivatives in the context of Birkhoff-James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff-James orthogonality in $\ell_1^n$ and $\ell_\infty^n$ spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff-James orthogonality. We further study …
Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-Šemrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show th…
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.
Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given a positive operator $A\in\B(\h)$, and a number $λ\in [0,1]$, a seminorm ${\|\cdot\|}_{(A,λ)}$ is defined on the set $\B_{A^{1/2}}(\h)$ of all operators in $\B(\h)$ having an $A^{1/2}$-adjoint. The seminorm ${\|\cdot\|}_{(A,λ)}$ is a combination of the sesquilinear form ${\langle \cdot, \cdot\rangle}_A$ and its induced seminorm ${\|\cdot\|}_A$. A characterization of Birkhoff--James orthogonality for operators with respect to th…
In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T\perp^B_A S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T + γS\|}_A\geq {\|T\|}_A$ for all $γ\in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T\perp^B_A S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\math…
We study Birkhoff-James orthogonality and isosceles orthogonality of bounded linear operators between Hilbert spaces and Banach spaces. We explore Birkhoff-James orthogonality of bounded linear operators in light of a new notion introduced by us and also discuss some of the possible applications in this regard. We also study isosceles orthogonality of bounded (positive) linear operators on a Hilbert space and some of the related properties, including that of operators having disjoint support. We further explore the relations between Birkhoff-James orthogonality and isosceles orthogonality in a…
Let $x$ and $y$ be two unit vectors in a normed plane $\mathbb{R}^2$. We say that $x$ is Birkhoff orthogonal to $y$ if the line through $x$ in the direction $y$ supports the unit disc. A B-measure (Fankhänel 2011) is an angular measure $μ$ on the unit circle for which $μ(C)=π/2$ whenever $C$ is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.
In this paper, we obtain some characterizations of the (strong) Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules and certain elements of $\mathbb{B}(\mathscr{H})$. Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the norm attaining set $\mathbb{M}_T$ is a unit sphere of some finite dimensional subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{\mathscr{H}_0}^\perp} < \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong Birkhoff--James orthogonal to $S$ if a…
We establish variant Khintchine inequalities on normed spaces of Hanner type and cotype, in which the Rademacher distribution corresponding to classical Khintchine inequality is replaced by general symmetric distributions. The proof involves the $p$-barycenter and Birkhoff's ergodic theorem. More importantly, by employing these Khintchine inequalities, we get some lower bounds for Banach-Mazur distance between $l^p$-ball and a general centrally symmetric convex body.
We obtain necessary and sufficient conditions for a matrix $A$ to be Birkhoff-James orthogonal to another matrix $B$ in the Ky Fan $k$-norms. A characterization for $A$ to be Birkhoff-James orthogonal to any subspace $\mathscr W$ of $\mathbb M(n)$ is also obtained.
We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson (1977) : "If a normed space $E$ does not contain any asymptotically isometric copy of $\ell^1(\IN)$, then every bounded sequence of $E'$ has a normalized block sequence pointwise converging to 0".
We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance formulae are presented. As an application of our study, we obtain some crucial inequalities, which also strengthen the classical Hölder's inequality. The relevance of the algorithms and the inequalities are discussed through concrete examples.
We study the geometry of $\mathcal{L}(X)_w,$ the space of all bounded linear operators on a Banach space $X,$ endowed with the numerical radius norm, whenever the numerical radius defines a norm. We obtain the form of the extreme points of the unit ball of the dual space of $\mathcal{L}(X)_w.$ Using this structure, we explore Birkhoff-James orthogonality, best approximation and deduce distance formula in $\mathcal{L}(X)_w.$ A special attention is given to the case of operators satisfying a notion of smoothness. Finally, we obtain an equivalence between Birkhoff-James orthogonality in $\mathcal…
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 attaining norm at a particular point of the unit sphere. We prove a number of corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff-James orthogonality. Restricting our attention to $ l_{p}^{2} (p \in \mathbb{N}\setminus \{…
