In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ϵ > 0; the notion of Birkhoff-James approximate orthogonality sets for ϵ-higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural genera...

In the last few decades, concept of Birkhoff–James orthogonality has been used in several applications. this survey article, results known on necessary and sufficient conditions for certain Banach spaces are mentioned. Their applications studying geometry normed given. The connections between orthogonality, Gateaux derivative subdifferential set norm function provided. Several interesting dista...

It is shown that Birkhoff-James orthogonality knows everything about the smooth norms in reflexive Banach spaces and can also compute dimensions of underlying normed spaces.

Abstract We give an example of a pair real Banach spaces such that they are neither linearly isomorphic nor with respect to the structure Birkhoff–James orthogonality, but have mutually homeomorphic geometric spaces.

We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

An extension of the metric space in which distance same point is not always zero called a partial space. Orthogonality relation two perpendicular lines at one intersection forming right angle. There are several ways to define orthogonality, including Pythagorean Orthogonality, Isosceles and Birkhoff-James Orthogonality. The purpose this research study consistency definition orthogonality Based ...