Birkhoff–James ε-orthogonality sets of vectors and vector-valued polynomials

Abstract

Consider a complex normed linear space (X,‖⋅‖), and let χ,ψ∈X with ψ≠0. Motivated by recent works on rectangular matrices and on normed linear spaces, we study the Birkhoff–James ε-orthogonality set of χ with respect to ψ, give an alternative definition for this set, and explore its rich structure. We also introduce the Birkhoff–James ε-orthogonality set of polynomials in one complex variable whose coefficients are members of X, and survey and record extensions of results on matrix polynomials to these vector-valued polynomials.