Some remarks on Birkhoff–James orthogonality of linear operators

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 operators.     

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.     

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.     

Characterization of Day–James Spaces and Generalized Day–James Spaces

In this paper, we introduce the definition of generalized Day–James space on R~n(n ≥2) and give a characterization of it, which extend some known results. In addition, we provide a sufficient and necessary condition for Day–James space, which reappeared Day's construction for any two-dimensional normed space to make Birkhoff orthogonality symmetry.     

Isosceles-orthogonality preserving property and its stability

For real normed spaces, we consider the class of linear operators, preserving approximately the relation of isosceles-orthogonality. We show some general properties of such mappings. Next, we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one. In regard to this, we generalize some results obtained earlier for inner product spaces with standard orthogonality relation.     

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·) _i and (·, ·) _S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.     

Mappings approximately preserving orthogonality in normed spaces

We answer many open questions regarding approximately orthogonality preserving mappings (in Birkhoff-James sense) in normed spaces. In particular, we show that every approximately orthogonality preserving linear mapping (in Chmieliński sense) is necessarily a scalar multiple of an ε-isometry. Thus, whenever ε-isometries are close to isometries we obtain stability. An example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries, that is, stability does not hold.     

Orthogonalities,linear operators,and characterization of inner product spaces

It is proved that a normed space, whose dimension is at least three, admitting a nonzero linear operator reversing Birkhoff orthogonality is an inner product space, which releases the smoothness condition in one of J. Chmieliński’s results. Further characterizations of inner product spaces are obtained by studying properties of linear operators related to Birkhoff orthogonality and isosceles orthogonality.     

Area orthogonality in normed linear spaces

A new concept of orthogonality in real normed linear spaces is introduced. Typical properties of orthogonality (homogeneity, symmetry, additivity, ...) and relations between this orthogonality and other known orthogonalities (Birkhoff, Boussouis, Unitary-Boussouis and Diminnie) are studied. In particular, some characterizations of inner product spaces are obtained.     

Complements on Diminnie Orthogonality

A new orthogonality relation for normed linear spaces is introduced by C. R. DIMINNIE in [10]. Some interesting properties of such orthogonality and its relationship with Birkhoff orthogonality are studied in the above paper. The first part of this paper begins with a geometrical interpretation of Diminnie-orthogonality which allows us to obtain some other properties of such orthogonality. The second part deals with relationships between Diminnie orthogonality and some other known orthogonalities.     

Spectral Theory for Riemannian Manifolds with Cusps and a Related Trace Formula

A new orthogonality relation for normed linear spaces is introduced using a concept of area of a parallelogram given by E. Silverman. Comparisons are drawn between this relation and an earlier relation used by G. Birkhoff. In addition, this new relation is utilized to obtain new characterizations of inner-product spaces.     

Linear approximation of approximately biorthogonality preserving maps

We investigate linear properties of mappings from a bounded domain of an n-dimensional normed space into another n-dimensional normed space such that the image of some almost biorthogonal system is almost biorthogonal. In this way we generalize a result of the author on stability of orthogonality in Euclidean spaces.     

Approximate Roberts orthogonality sets and {(\delta, \varepsilon)}-(a,b)-isosceles-orthogonality preserving mappings

We introduce the concept of approximate Roberts orthogonality set and investigate the geometric properties of such sets. In addition, we introduce the notion of approximate a-isosceles-orthogonality and consider a class of mappings, which approximately preserve a-isosceles-orthogonality.     

A new type of orthogonality for normed planes

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ⩾ 3.     

An orthogonality in normed linear spaces based on angular distance inequality

In this paper, we present a new orthogonality in a normed linear space which is based on an angular distance inequality. Some properties of this orthogonality are discussed. We also find a new approach to the Singer orthogonality in terms of an angular distance inequality. Some related geometric properties of normed linear spaces are discussed. Finally a characterization of inner product spaces is obtained.     

Linear mappings preserving ρ-orthogonality

We define a ρ-orthogonality in a real normed space and we consider the class of linear mappings preserving this relation. We show that a linear mapping preserving ρ-orthogonality has to be a similarity, i.e., a scalar multiple of an isometry. As a result, we give a characterization of smooth spaces in terms of this orthogonality.     

Pythagorean orthogonality and additive mappings

Summary In this note we consider a real normed vector spaceX equipped with the isosceles orthogonality or the Pythagorean orthogonality, both of them defined by R. C. James. It is known that any odd, isosceles orthogonally additive mapping fromX into an Abelian group is unconditionally additive whenever dimX 3. Also, it is worth mentioning that this result was the first of this sort based on a non-homogeneous relation. In this context, we derive here the same for the other non-homogeneous orthogonality, the Pythagorean one, answering in part a pretty old and famous question. The proof uses the corresponding result for isosceles orthogonality and a detailed analysis of the geometry of normed spaces.Dedicated to Professor Jürg Rätz on the occasion of his 60th birthday     

On a ρ-orthogonality

In a normed space we introduce an exact and approximate orthogonality relation connected with “norm derivatives” ${\rho^{\prime}_{\pm}}$ . We also consider classes of linear mappings preserving (exactly and approximately) this kind of orthogonality.     

Tangent segments and orthogonality types in normed planes

The following characterizations of the Euclidean plane are obtained: the two tangent segments of the unit circle of a normed plane from each point of a disc centered at the origin with sufficiently large diameter have equal lengths; the lengths of the tangent segments from each point of a fixed circle centered at the origin are determined only by the radius of this circle. Three further characterizations of the Euclidean plane are obtained by considering properties of certain points related to an exterior point of the unit disc and the two tangent segments corresponding to it. To obtain one of these characterizations, the notion of arc-length orthogonality is introduced, and the Euclidean plane is also characterized via a relation between arc-length orthogonality and Birkhoff orthogonality.     

Some fixed point theorems in fuzzy reflexive Banach spaces

In this paper, we first show that there are some gaps in the fixed point theorems for fuzzy non-expansive mappings which are proved by Bag and Samanta, in [Bag T, Samanta SK. Fixed point theorems on fuzzy normed linear spaces. Inf Sci 2006;176:2910–31; Bag T, Samanta SK. Some fixed point theorems in fuzzy normed linear spaces. Inform Sci 2007;177(3):3271–89]. By introducing the notion of fuzzy and α- fuzzy reflexive Banach spaces, we obtain some results which help us to establish the correct version of fuzzy fixed point theorems. Second, by applying Theorem 3.3 of Sadeqi and Solati kia [Sadeqi I, Solati kia F. Fuzzy normed linear space and it’s topological structure. Chaos, Solitons & Fractals, in press] which says that any fuzzy normed linear space is also a topological vector space, we show that all topological version of fixed point theorems do hold in fuzzy normed linear spaces.