Algebraic degree in spatial matricial numerical ranges of linear operators

Abstract

We study the maximal algebraic degree of principal ortho-compressions of linear operators that constitute spatial matricial numerical ranges of higher order. We demonstrate (amongst other things) that for a (possibly unbounded) operator L L on a Hilbert space, every principal m m -dimensional ortho-compression of L L has algebraic degree less than m m if and only if r a n k ( L − λ I ) ≤ m − 2 rank(L-\lambda I)\le m-2 for some λ ∈ C \lambda \in \mathbb {C} .