scholarly journals Numerical range and orthogonality in normed spaces

Filomat ◽  
2009 ◽  
Vol 23 (1) ◽  
pp. 21-41 ◽  
Author(s):  
A. Bachir ◽  
A. Segres
Keyword(s):  
Normed Linear Space ◽  
Numerical Range ◽  
Positive Answer ◽  
Duality Mapping ◽  
Normed Algebra ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normalized Duality Mapping ◽  
James Orthogonality ◽  
Orthogonality In Normed Spaces

Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V (T)A which generalize those given by J.P. William and J.P. Stamplfli. Finally, we give a positive answer of the Mathieu's question. .

scholarly journals On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality

2018 ◽  
Vol 11 (3) ◽  
pp. 793-802
Author(s):  
Mahdi Iranmanesh ◽  
M. Saeedi Khojasteh ◽  
M. K. Anwary
Keyword(s):  
Numerical Range ◽  
Alternative Definition ◽  
Normed Spaces ◽  
Operator Approach ◽  
Linear Spaces ◽  
Continuity Properties ◽  
Lower Semi

In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {μ : μ ∈ C, p(x − μy, y) = 1} is upper and lower semi continuous, where p(x, y) = sup{pz1,...,zn−2 (x, y) : z1, . . . , zn−2 ∈ X} and pz1,...,zn−2 (x, y) = kPx,z1,...,zn−2,yk−1 where Px,z1,...,zn−2,y denotes the projection parallel to y from X to the subspace generated by {x, z1, . . . , zn−2}. This can be considered as an alternative definition for numerical range in linear spaces.

scholarly journals Orthogonality in smooth countably normed spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Tawfeek ◽  
Nashat Faried ◽  
H. A. El-Sharkawy
Keyword(s):  
Normed Space ◽  
Nonempty Subset ◽  
Metric Projection ◽  
Orthogonal Complement ◽  
Duality Mapping ◽  
Dual Cone ◽  
Birkhoff Orthogonality ◽  
Uniformly Convex ◽  
Normed Spaces ◽  
Normalized Duality Mapping

AbstractWe generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

2-HH NORM AND BIRKHOFF-JAMES ORTHOGONALITY IN NORMED SPACES

2019 ◽  
Vol 8 (3) ◽  
pp. 1
Author(s):  
PRASAD OJHA BHUWAN ◽  
MUNI BAJRACHARYA PRAKASH ◽  
◽  
Keyword(s):  
Normed Spaces ◽  
James Orthogonality ◽  
Orthogonality In Normed Spaces

scholarly journals Some geometric characterizations of inear product spaces

1981 ◽  
Vol 24 (2) ◽  
pp. 239-246 ◽  
Author(s):  
O. P. Kapoor ◽  
S. B. Mathur
Keyword(s):  
Linear Space ◽  
Product Space ◽  
Normed Linear Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Isosceles Orthogonality ◽  
Linear Spaces ◽  
Von Neumann ◽  
James Orthogonality

There are several geometric characterizations of inner product spaces amongst the normed linear spaces. Mahlon M. Day's refinement “rhombi suffice as well as parallelograms”, of P. Jordan and J. von Neumann parallelogram law is well known. There are some characterizations which employ various notions of orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality then the space is an inner product space; geometrically it means that if the diagonals of a rectangle, with sides perpendicular in Birkhoff-James sense, are equal then the space is an inner product space. In the main result of this note we improve upon this characterization and show that here unit squares suffice as well as rectangles.

scholarly journals Orthogonality and characterizations of inner product spaces

1978 ◽  
Vol 19 (3) ◽  
pp. 403-416 ◽  
Author(s):  
O.P. Kapoor ◽  
Jagadish Prasad
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Product Spaces ◽  
Isosceles Orthogonality ◽  
Linear Spaces ◽  
Strictly Convex ◽  
Inner Product Spaces ◽  
James Orthogonality ◽  
Pythagorean Orthogonality

Using the notions of orthogonality in normed linear spaces such as isosceles, pythagorean, and Birkhoff-James orthogonality, in this paper we provide some new characterizations of inner product spaces besides giving simpler proofs of existing similar characterizations. In addition we prove that in a normed linear space pythagorean orthogonality is unique and that isosceles orthogonality is unique if and only if the space is strictly convex.

scholarly journals The numerical range of an element of a normed algebra

1969 ◽  
Vol 10 (1) ◽  
pp. 68-72 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Linear Space ◽  
Unit Sphere ◽  
Dual Space ◽  
Normed Linear Space ◽  
Numerical Range ◽  
Normed Algebra ◽  
Bounded Linear ◽  
Image Position

Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by

Symmetrized Birkhoff–James orthogonality in arbitrary normed spaces

2021 ◽  
pp. 125203
Author(s):  
Ljiljana Arambašić ◽  
Alexander Guterman ◽  
Bojan Kuzma ◽  
Rajna Rajić ◽  
Svetlana Zhilina
Keyword(s):  
Normed Spaces ◽  
James Orthogonality

scholarly journals Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Songnian He ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Duality Mapping ◽  
Normalized Duality Mapping ◽  
Uniformly Smooth ◽  
Computational Work

LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceX,{Tk}k=1∞:C→Can infinite family of nonexpansive mappings with the nonempty set of common fixed points⋂k=1∞Fix⁡(Tk), andf:C→Ca contraction. We introduce an explicit iterative algorithmxn+1=αnf(xn)+(1-αn)Lnxn, whereLn=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk,  andwk>0with∑k=1∞ωk=1. Under certain appropriate conditions on{αn}, we prove that{xn}converges strongly to a common fixed pointx*of{Tk}k=1∞, which solves the following variational inequality:〈x*-f(x*),J(x*-p)〉≤0,    p∈⋂k=1∞Fix(Tk), whereJis the (normalized) duality mapping ofX. This algorithm is brief and needs less computational work, since it does not involveW-mapping.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 27-36
Author(s):  
M. M. GUEYE ◽  
M. SENE ◽  
M. NDIAYE ◽  
N. DJITTE
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Maximal Monotone ◽  
Point Theory ◽  
Duality Mapping ◽  
Monotone Mappings ◽  
Normalized Duality Mapping ◽  
Maximal Monotone Mappings ◽  
Pseudocontractive Mappings

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

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