Orthogonality in Banach Spaces via projective tensor product

2020/10/02 by Dhara, Kousik, Rakshit, Narayan, Sarkar, Jaydeb +1
#46B20 #46B22 #46B28 #47A30 #47B01 #47L05 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2010.00978

Abstract

Let X be a complex Banach space and x,y∈ X. By definition, we say that x is Birkhoff-James orthogonal to y if ‖x+λy‖_X ≥ ‖x‖_X for all λ∈ ℂ. We prove that x is Birkhoff-James orthogonal to y if and only if there exists a semi-inner product φ on X such that ‖φ‖ = 1, φ(x,x)=‖x‖² and φ(x,y)=0. A similar result holds for C^*-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.

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