James Orthogonality Research Articles

Geometric nonlinear classification of Hilbert $$C^*$$-modules based on strong Birkhoff–James orthogonality

Geometric nonlinear classification of Hilbert $$C^*$$-modules based on strong Birkhoff–James orthogonality

Diameter of the BJ-orthograph in finite-dimensional C*-algebras

We determine the exact diameter of the orthograph related to mutual strong Birkhoff–James orthogonality in arbitrary finite-dimensional C ∗ -algebra. Additionally, we will estimate the distance between vertices in an arbitrary C ∗ -algebra that can be represented as a direct sum.

Birkhoff–James orthogonality in certain tensor products of Banach spaces II

Birkhoff–James orthogonality in certain tensor products of Banach spaces II

Characterization of a-Birkhoff–James orthogonality in $$C^*$$-algebras and its applications

Characterization of a-Birkhoff–James orthogonality in $$C^*$$-algebras and its applications

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

Convex functions and approximate Birkhoff–James orthogonality

AbstractFor the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.

ON REAL NONISOMORPHIC BANACH SPACES WITH HOMEOMORPHIC GEOMETRIC STRUCTURE SPACES

AbstractWe give an example of a pair of real Banach spaces such that they are neither linearly isomorphic nor isomorphic with respect to the structure of Birkhoff–James orthogonality, but have mutually homeomorphic geometric structure spaces.

Approximate smoothness in normed linear spaces

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.

A study of local symmetry of Birkhoff–James orthogonality in Banach spaces

We present a complete characterization of the right-symmetric points in the one sum of two Banach spaces. We also obtain some basic properties of the left-symmetric (right-symmetric) points in the p sum, 1 ≤ p ≠ 2 < ∞ ( 1 < p ≠ 2 < ∞ ), of two Banach spaces. Using these properties we (a) give examples of Banach spaces which do not have any non-zero left-symmetric points and (b) prove a complete characterization of those left-symmetric and right-symmetric points in the p sum, 1 < p ≠ 2 < ∞ , of two Banach spaces, whose components satisfy an additional norm assumption. We give examples of Banach spaces where all non-zero left-symmetric or right-symmetric points satisfy this additional norm assumption. We also present an alternative proof of the recently obtained characterization of the left-symmetric and the right-symmetric points in ℓ p n , n ≥ 3 , and ℓ p , 1 ≤ p ≠ 2 < ∞ .

On a norming property of subspaces of a Banach space and some applications

We introduce and study m-heavy subspaces of a Banach space and obtain a complete analytic characterization of the same. The importance of m-heavy subspaces in studying the geometry of a Banach space is illustrated by establishing its various applications, including a strengthening of the classical Birkhoff–James orthogonality, duality theory and linear Hahn–Banach extension operators.

On a Set of Norm Attaining Operators and the Strong Birkhoff–James Orthogonality

Continuing the study of recent results on the Birkhoff–James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia–Šemrl property for operators which is weaker than the Bhatia–Šemrl property. The set of operators with the adjusted Bhatia–Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia–Šemrl property. It is known that the set of operators with the Bhatia–Šemrl property is norm-dense if the domain space X of the operators has the Radon–Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $$c_0$$ , $$L_1[0,1]$$ and C[0, 1]. In contrast with the Bhatia–Šemrl property, we show that the set of operators with the adjusted Bhatia–Šemrl property is norm-dense when the domain space is $$c_0$$ or $$L_1[0,1]$$ . Moreover, we show that the set of functionals having the adjusted Bhatia–Šemrl property on C[0, 1] is not norm-dense but such a set is weak- $$*$$ -dense in $$C(K)^*$$ for any compact Hausdorff K.

The dual of the sequence spaces with mixed norms l^{(p, infty )}

We identify a norm-dense subspace of the dual of the sequence space l^{(p,infty )}, thus closing the existing gap in the literature. We based our approach on the notion of James orthogonality, absolutely continuous norms and on the uniform convexity and the uniform smoothness of the underlying subspaces.

What does Birkhoff--James orthogonality know about the norm?

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.

On the best coapproximation problem in

We study the best coapproximation problem in the Banach space by using Birkhoff–James orthogonality techniques. Given a subspace of , we completely identify the elements x in for which best coapproximations to x out of exist. The methods developed in this article are computationally effective and it allows us to present an algorithmic approach to the concerned problem. We also identify the coproximinal subspaces and co-Chebyshev subspaces of .

An Extension of Birkhoff–James Orthogonality Relations in Semi-Hilbertian Space Operators

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 $\lambda\in [0,1]$, a seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined on the set $\B_{A^{1/2}}(\h)$ of all operators in $\B(\h)$ having an $A^{1/2}$-adjoint. The seminorm ${\|\cdot\|}_{(A,\lambda)}$ 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 the discussed seminorm is given. Moving $\lambda$ along the interval $[0,1]$, a wide spectrum of seminorms are obtained, having the $A$-numerical radius $w_A(\cdot)$ at the beginning (associated with $\lambda=0$) and the $A$-operator seminorm ${\|\cdot\|}_A$ at the end (associated with $\lambda=1$). Moreover, if $A=I$ the identity operator, the classical operator norm and numerical radius are obtained. Therefore, the results in this paper are significant extensions and generalizations of known results in this area.

On Birkhoff–James orthogonality preservers between real non-isometric Banach spaces

We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results of Koldobsky and of Wójcik. Moreover, in an arbitrary dimension, we construct examples of non-isometric pairs of non-smooth real Banach spaces that admit norm preserving homogeneous bicontinuous Birkhoff–James orthogonality preservers among them.

On best approximations in Banach spaces from the perspective of orthogonality

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.

A nonlinear characterization of type I factors based on strong Birkhoff–James orthogonality

A nonlinear characterization of type I factors based on strong Birkhoff–James orthogonality

From norm derivatives to orthogonalities in Hilbert C*-modules

Let be a Hilbert -module over a -algebra and let be the set of states on . In this paper, we first compute the norm derivative for nonzero elements x and y of as follows: We then apply it to characterize different concepts of orthogonality in . In particular, we present a simpler proof of the classical characterization of Birkhoff–James orthogonality in Hilbert -modules. Moreover, some generalized Daugavet equation in the -algebra of all bounded linear operators acting on a Hilbert space is solved.

Approximate orthogonality in complex normed spaces: a directional approach

We introduce and study the notion of approximate directional orthogonality in a complex normed space. We further explore the relevant scenario in the topological setting, which allows us to characterize approximate Birkhoff–James orthogonality of operators on a finite-dimensional complex Banach space. Our investigation extends some existing results in the literature to a broader scope.