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We explore Birkhoff–James orthogonality of two elements in a complex Banach space 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...
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Citations
... Observe that lim Ax n , y n : x n , y n ∈ S H ∀n ∈ N, lim T x n , y n = T = lim Ax n , T x n : x n ∈ S H ∀n ∈ N, lim T x n = T and that the latter set is convex (this was first stated without proof in [ In the particular case when H is finite-dimensional, Bhatia and Šemrl were the first to write down the characterization of BJ-orthogonality of two matrices in terms of the elements of H [5, Theorem 1.1]. An alternative proof of this characterization was given by Roy, Bagchi, and Sain in [32]. We obtain this result as a consequence of Corollary 3.12. ...
... However, it is natural to study for which operators T it is possible to have a Bhatia-Šemrl theorem for all operators A: conditions on T such that whenever T ⊥ B A, one has that there is a norm-one x such that T x = T and T x ⊥ B Ax (that is, whether we may remove the convex hull in Corollary 3.11). This has been done in [26,37,40] for the real case and in [27,32] for the complex case. Our aim in what follows is to give a unified approach that allows to recover some of these results and to obtain an improvement in the complex setting. ...
... Given elements x, y of a Banach space Z , we say that x is orthogonal to y in the direction of γ ∈ T, which we denote by x ⊥ γ y, if x + tγ y x for every t ∈ R. Obviously, x ⊥ B y if and only if x ⊥ γ y for every γ ∈ T. In the real case, it is obvious that x ⊥ B y if and only if x ⊥ 1 y if and only if x ⊥ −1 y. In the complex case, there are easy examples showing that x ⊥ B y while x ⊥ γ y for some γ ∈ T is possible, see [32,Example 1]. It is shown in [32,Theorem 4] that ...
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.
... Observe that lim⟨Ax n , y n ⟩ : x n , y n ∈ S H ∀n ∈ N, lim⟨T x n , y n ⟩ = ∥T ∥ = {lim⟨Ax n , T x n ⟩ : x n ∈ S H ∀n ∈ N, lim ∥T x n ∥ = ∥T ∥ and that the latter set is convex (this was first stated without proof in [ An alternative proof of this characterization was given by Roy, Bagchi, and Sain in [30]. We obtain this result as a consequence of Corollary 3.12. ...
... However, it is natural to study for which operators T it is possible to have a Bhatia-Šemrl theorem for all operators A: conditions on T such that whenever T ⊥ B A, one has that there is a norm-one x such that ∥T x∥ = ∥T ∥ and T x ⊥ B Ax (that is, whether we may remove the convex hull in Corollary 3.11). This has been done in [25,35,38] for the real case and in [26,30] for the complex case. Our aim in what follows is to give a unified approach that allows to recover some of these results and to obtain an improvement in the complex setting. ...
... Given x, y elements of a Banach space Z, we say that x is orthogonal to y in the direction of γ ∈ T, which we denote by x ⊥ γ y, if ∥x + tγy∥ ⩾ ∥x∥ for every t ∈ R. Obviously, x ⊥ B y if and only if x ⊥ γ y for every γ ∈ T. In the real case, it is obvious that x ⊥ B y if and only if x ⊥ 1 y if and only if x ⊥ −1 y. In the complex case, there are easy examples showing that x ̸ ⊥ B y while x ⊥ γ y for some γ ∈ T is possible, see [30,Example 1]. It is shown in [30,Theorem 4] that (4.1) ...
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-\v{S}emrl 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.
... In order to carry forward the study of operator orthogonality in Hilbert spaces to the setting of complex normed spaces, a weaker notion of Birkhoff-James orthogonality, namely, the directional orthogonality, has been introduced and studied in [1,2]. In the present article, we extend the notion of directional orthogonality to that of approximate directional orthogonality. ...
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.
Without a shadow of doubt, Birkhoff–James (B–J) orthogonality is the principal tool used in this monograph in studying the geometry of operator spaces. Therefore, understanding B–J orthogonality of operators is of paramount importance to us. In this chapter, we gradually build the theory of characterizing B–J orthogonality of operators between Banach (Hilbert) spaces, up to its fullest generality.
We present Birkhoff–James orthogonality from historical perspectives to the current development. We compare it with some other orthogonalities, present its properties and its applications, and review the characterizations of Birkhoff–James orthogonality in classical Banach spaces like \(\mathbb B(\mathcal {H})\), C
∗-algebras, Hilbert C
∗-modules, or the space of rectangular matrices normed with Schatten norms. We also present the results on characterizations of preservers of Birkhoff–James orthogonality and, by devising a directed graph of the relation, show that in smooth spaces it can completely determine the norm up to (conjugate) linear isometry.Most, though not all, of the results that we state are supplied with (sketches of) the proof.KeywordsNormed vector spaceBirkhoff–James orthogonality
C
∗-algebraHilbert C
∗-modulePreserversGraphClique
