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Abstract

This paper aims to study the approximate orthogonality (Chmielinski orthogonality) of Birkhoff–James techniques in real Banach space (X,‖·‖)—even if the concepts symbolised by⊥εBJCorthogonality showno ambiguity—and provide some new geometric characterisations that serve as the basis of our main definitions.This paper also explores the relation between two types of⊥εBJCorthogonalities, namely,⊥εBJCorthogonality ina real Banach space (X,‖·‖) and⊥εBJCorthogonality in the space of the bounded linear operatorB(X,Y). Weobtain complete characterisations of these two⊥εBJCorthogonalities in some types of Banach spaces, such as strictlyconvex, smooth and reflexive spaces. This study provides different results about the concept symmetry of Chmielinskiorthogonality for a compact linear operator on a reflexive, strictly convex Banach space having a Kadets–Kleeproperty by exploring new generalised results with Birkhoff–James orthogonality in the space of bounded linearoperators. We also obtain a smooth compact linear operator with a spectral value that is defined on a reflexive, strictlyconvex Banach space having a Kadets–Klee property that is either having zero nullity or not⊥εBJC-right-symmetric

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