Dear Dr Abeer Alkhawaldeh
I think the papers
http://www.m-hikari.com/ijma/ijma-password-2007/ijma-password9-12-2007/mazaheriIJMA9-12-2007-1.pdf
&
https://math.stackexchange.com/questions/860622/orthogonality-on-banach-spaces
&
https://www.hindawi.com/journals/jmath/2017/1073589/
&
Article An Extension of the Notion of Orthogonality to Banach Spaces
&
https://www.researchgate.net/publication/266281658_Orthogonality_in_Banach_spaces
http://raiith.iith.ac.in/1602/
are very important for the question
Dear Abeer,
I happen to share the same field of interest as yours. You may have a look at my ResearchGate publications, and more importantly, the references therein, to find some recent publications on orthogonality in Banach spaces.
I work mostly with Birkhoff-James orthogonality only, but there are excellent works on other kind of orthogonality types in Banach spaces, e.g., isosceles orthogonality.
It might also be a good idea to begin with the three famous papers by R. C. James on Birkhoff-James orthogonality in normed linear spaces. You will find all these references, and more, in our work, and recent works by several other mathematicians.
Recall that in any vector lattice X, two elements x,y are said to be orthogonal if inf{|x|,|y|}=0. There are many function spaces or spaces of measures which have a natural structure of a vector lattice, but have not a natural structure of a prehilbertian space. Most of them are Banach lattices. Besides, an order complete Banach lattice (and a commutative algebra) of self-adjoint operators associated to a self-adjoint operator A acting on a Hilbert space H is defined and discussed in the monograph of Romulus Cristescu, "Ordered Vector Spaces and Linear Operators".
I also recommend you:
Article Orthogonalities and functional equations
Article On Birkhoff orthogonality and isosceles orthogonality in nor...
You may check www.sciencedirect.com/science/article/pii/S0022247X01977340
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