An extension of pythagorean and isosceles orthogonality and a characterization of inner-product spaces

A new orthogonality relation in normed linear spaces which generalizes pythagorean orthogonality and isosceles orthogonality is defined, and it is shown that the new orthogonality is homogeneous (additive) if and only if the space is a real inner-product space.     

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.     

ε-近似保正交映射的稳定性与扰动

对于Hilbert空间之间的ε-近似保正交线性映射T,给出了|〈T(x),T(y)〉-||T||~2〈x,y〉|的一个估计,得到了ε-近似保正交线性映射的充分条件,研究了ε-近似保正交线性映射的稳定性,并获得了ε-近似保正交线性映射的扰动定理.     

Orthogonality in $$C^{*}$$-algebras

The aim in this paper is to study algebraic orthogonality between positive elements of a \(C^{*}\)-algebra in the context of geometric orthogonality. It has been shown that the algebraic orthogonality in certain classes of \(C^{*}\)-algebras is equivalent to geometric orthogonality when supported with some order-theoretic conditions. Further more, algebraic orthogonality between positive elements in a \(C^{*}\)-algebra is also characterized in terms of positive linear functionals.     

Characterizations of connected orthogonality graphs of projections of Rickart ∗-rings

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected.     

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.     

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.     

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.     

Characterization of orthomaps on the Cayley plane

We classify maps which preserve orthogonality on the Cayley projective plane over octonions. In addition, we also classify orthogonality preserving maps on finite dimensional projective spaces over reals, complexes, or quaternions. Unlike similar results which extend Uhlhorns’s theorem we assume neither injectivity/surjectivity nor that orthogonality is preserved in both directions.     

Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.     

ε-近似保正交映射的稳定性与扰动

对于Hilbert空间之间的ε-近似保正交线性映射T,给出了的一个估计,得到了ε-近似保正交线性映射的充分条件,研究了ε-近似保正交线性映射的稳定性,并获得了ε-近似保正交线性映射的扰动定理.       

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.     

Vector-Orthogonality and Lanczos-Type Methods

A method for solving a linear system is defined. It is a Lanczos-type method, but it uses formal vector orthogonality instead of scalar orthogonality. Moreover, the dimension of vector orthogonality may vary which gives a large freedom in leading the algorithm, and controlling the numerical problems. The ideas of truncated and restarted methods are revisited. The obtained residuals are exactly orthogonal to a space of increasing dimension. Some experiments are done, the problem of finding automaticaly good directions of projection remains partly open.     

Various Notions of Orthogonality in Normed Spaces

In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also considered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.     

On mappings approximately preserving orthogonality

We answer a question posed by Chmieliński, whether a linear map which approximately preserves orthogonality must be close to an orthogonality preserving one. Furthermore, we give a short proof of the stability of the orthogonality equation on finite dimensional Hilbert spaces.     

Generalized n-gons and Chebychev polynomials

Two sequences of orthogonal polynomials are given whose weight functions consist of an absolutely continuous part and two point masses. Combinatorial proofs of the orthogonality relations are given. The polynomials include natural q-analogs of the Chebychev polynomials. The technique uses association schemes of generalized n-gons to find approximating discrete orthogonality relations. The Feit-Higman Theorem is a corollary of these orthogonality relations for the polynomials.     

广义正交表正交性的等价条件

广义正交表是一种类似于正交表的新设计.正交平衡性是广义正交表必须满足的基本要求之一,它是正交表正交性的推广,它能够使得试验因子在方差分析中保持柯赫伦定理成立,因而可以像正交表一样进行试验设计和方差分析,从而不但保证其数据分析模型符合"不自生"逻辑,而且也可以保证试验因子的各种关系比较的数据分析结论具有客观一致性和可重复再现性,但试验次数大幅减少.利用矩阵象技术,提出并证明了广义正交表的组合正交性不但等价于其矩阵象的正交性,而且也等价于其广义关联矩阵的正交性.借助于SAS软件可以方便快速的验证某些区组设计相应的行列设计是否为广义正交表.     

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.     

A NEW LOOK AT SOME CLASSIC PROBLEMS IN ORTHOGONAL POLYNOMIALS

<正> Until recently there were two kinds of orthogonality commonly accepted:that generatedby a real weight function and that generated by a complex weight function with integrationtaken along a path in the complex plane.The authors following R.D.Morton and A.M.Krall [10] research work are discussing the orthogonality from the point of view of distribu-tion theory.In doing so a third and new kind of orthogonality is pointed out.In some of the authors' forthcoming papers vrious kinds of orthogonality for polynomialsin several variables will be discussed [14]—[16],while the appropriate papers will deal withC.Brezinski's comprehensive very recent results [17] and with M.Picone's“total derivativeoperators”systems in the framework of D.Mangeron's polyvibrating or polywave equations,respectively.     

Strong additivity of metric isosceles orthogonality

It is known that the property of additivity of isosceles orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper it is shown that suitably metrized concepts of additivity of metric isosceles orthogonality characterize euclidean or hyperbolic spaces among complete, convex, externally convex metric spaces.