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# Birkhoff–James Orthogonality and Applications: A Survey

01 Jan 2021-Vol. 282, pp 293-315
Abstract: In the last few decades, the concept of Birkhoff–James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff–James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff–James orthogonality, which are also mentioned. In the end, some new results are obtained.

Topics: , Banach space (55%), Subderivative (54%) ...read more
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Open accessPosted Content
Abstract: For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained.

Topics: Orthogonality (50%)

3 Citations

Journal Article
Abstract: Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.

Topics: Normed vector space (64%), Vector space (57%),  ...read more

1 Citations

Open accessJournal Article
Sushil Singla1Institutions (1)
Abstract: We find an expression for the Gateaux derivative of the C ⁎ -algebra norm. Using this, we obtain a characterization of orthogonality of an operator A ∈ B ( H , K ) to a subspace, under the assumption dist ( A , K ( H , K ) ) ‖ A ‖ . We obtain an expression for the subdifferential set of the norm function at A ∈ B ( H ) when dist ( A , K ( H ) ) ‖ A ‖ . We also give new proofs of known results on the closely related notions of smooth points and Birkhoff-James orthogonality for the spaces B ( H ) and C b ( Ω ) , respectively.

Topics: Gâteaux derivative (50%)

Open accessPosted Content
Sushil Singla1Institutions (1)
Abstract: We find an expression for Gateaux derivative of the $C^*$-algebra norm. This gives us alternative proofs or generalizations of various known results on the closely related notions of subdifferential sets, smooth points and Birkhoff-James orthogonality for spaces $\mathscr B(\mathcal H)$ and $C_b(\Omega)$. We also obtain an expression for subdifferential sets of the norm function at $A\in\mathscr B(\mathcal H)$ and a characterization of orthogonality of an operator $A\in\mathscr B(\mathcal H, \mathcal K)$ to a subspace, under the condition $dist(A, \mathscr K(\mathcal H))< \|A\|$ and $dist(A, \mathscr K(\mathcal H, \mathcal K))< \|A\|$ respectively.

##### References
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Open accessBook
25 Sep 2001-
Abstract: Introduction: Notation, Elementary Results.- Convex Sets: Generalities Convex Sets Attached to a Convex Set Projection onto Closed Convex Sets Separation and Applications Conical Approximations of Convex Sets.- Convex Functions: Basic Definitions and Examples Functional Operations Preserving Convexity Local and Global Behaviour of a Convex Function First- and Second-Order Differentiation.- Sublinearity and Support Functions: Sublinear Functions The Support Function of a Nonempty Set Correspondence Between Convex Sets and Sublinear Functions.- Subdifferentials of Finite Convex Functions: The Subdifferential: Definitions and Interpretations Local Properties of the Subdifferential First Examples Calculus Rules with Subdifferentials Further Examples The Subdifferential as a Multifunction.- Conjugacy in Convex Analysis: The Convex Conjugate of a Function Calculus Rules on the Conjugacy Operation Various Examples Differentiability of a Conjugate Function.

Topics: Subderivative (80%), Convex analysis (78%),  ...read more

1,167 Citations

Open accessBook
01 Nov 1970-
Abstract: espace lineaire norme # espace metrique # meilleure approximation # sous-espace lineaire # sous-espace lineaire de dimension finie # sous-espace lineaire ferme de codimension finie # element d'ensemble # element d'ensemble non-lineaire

Topics: Linear subspace (50%)

728 Citations

Open accessBook
08 Dec 2005-
Abstract: Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Topics: Affiliated operator (71%), Nest algebra (71%), Von Neumann algebra (70%) ...read more

583 Citations

Open accessJournal Article
Abstract: The natural definition of orthogonality of elements of an abstract Euclidean space is that x ly if and only if the inner product (x, y) is zero. Two definitions have been given [11](2) which are equivalent to this and can be generalized to normed linear spaces, preserving the property that every twodimensional linear subset contain nonzero orthogonal elements. The definition which will be used here (Definition 1.2) has the added advantage of being closely related to the theories of linear functionals and hyperplanes. The theory and applications of this orthogonality have been organized in the following sections, which are briefly outlined: 1. Fundamental definitions. An element x is orthogonal to an element y if and only if j|x+kyjj > ||x|l for all k. This orthogonality is homogeneous, but is neither symmetric nor additive. 2. Existence of orthogonal elements. An element x of a normed linear space is orthogonal to at least one hyperplane through the origin, while for elements x and y there is at least one number a for which ax+y Ix or ||ax+yj| is minimum (Theorems 2.2-2.3). 3. Orthogonality in general normed linear spaces. The limits N?(x; y) =+lim,+.?|lnx+yil jjnxjj =liml0?o [||x+hyjj -lixil ]/hexistandsatisfyweakened linearity conditions. Also, x Iax+y if and only if N_(x; y) < -a||x||

Topics: Linear subspace (56%), , Reflexive space (55%) ...read more

437 Citations

Open accessJournal Article
G.A. Watson1Institutions (1)
Abstract: A characterization is given of the subdifferential of matrix norms from two classes, orthogonally invariant norms and operator (or subordinate) norms. Specific results are derived for some special cases.

Topics: , Subderivative (54%), Matrix norm (50%)

377 Citations

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