Various Notions of Orthogonality in Normed Spaces

On Orthogonality of Elementary Operators in Norm-attainable Classes

“…If V (A) ⊆ R, then A is called a Hermitian element. Given two Hermitian elements S and R such that SR = RS, then D = S + Ri is called normal [32].…”

“…Others who have also worked on orthogonality include: Kittaneh [21], Mecheri [29] among others. For details see [30][31][32][33][34][35]. We shall investigate the orthogonality of the range and the kernel of several types of important elementary operators in Banach spaces.…”

On Norm-Attainable Operators in Banach Spaces

“…Furthermore, there exists a unit vector η ∈ H such that ∥Zη∥ = ∥Z∥ with hZη, ηi = β, where W 0 ðSÞ denotes the maximal numerical range of the operator S: Moreover, norm-attainability conditions for elementary operators and generalized derivations have been given. For orthogonality of elementary operators in norm-attainable classes, a detailed exposition has been given in [4,8]. A superclass of Hilbert space operators has also been considered in norm-attainable classes.…”

Norm Inequalities for Inner Product Type Integral Transformers

“…Remark 1.1. When µ is the counting measure on N then such transformers are known as elementary operators whose some of the properties have been studied in details(see [7] and the references therein on orthogonality property). This work is organized as follows: Section 1: Introduction; Section 2: Unitarily invariant norms; Section 3: Operator valued functions; Section 4: Norm inequalities for inner product type integral transformers and lastly; Section 5: Applications in quantum theory.…”