Approximate Birkhoff-James orthogonality of analytic functions defined by a generalized operator

The aim of this thesis is to study new results of an approximate orthogonality of Birkhoff-James techniques in a real Banach 𝗌𝗉𝖺𝖼e (X,∥∙∥), namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by ⊥_BJC^ϵorthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. In this thesis, we explore relation between two different types of ⊥_BJC^ϵorthogonalities. First of them ⊥_BJC^ϵorthogonality in a real Banach space (X,∥∙∥) and the other ⊥_BJC^ϵorthogonality in the space of bounded linear transformation B(X ,Y). We obtain a complete characterizations of these two ⊥_BJC^ϵorthogonalities such as left symmetric and right symmetric in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. Keywords: Approximate birkhoff-james orthogonality, Roots of polynomials, Zeros of analytical functions, Riemann zeta function, Chebyshev polynomials, Cofficients bound