Chebyshev polynomials for operators (Q1024181)

Let \(P^1_n\) denote the set of all monic polynomials of degree \(n\), where \(n\in\mathbb{N}_0\). Let \(({\mathcal R},\|\cdot \|)\) be a unital Banach algebra. If \(A\in{\mathcal R}\) and \(n\in\mathbb{N}_0\) then \(p_n\in{\mathcal P}^1_n\) is a Chebyshev polynomial for \(A\) if \[ \| p_n(A)\|= \underset{p\in{\mathcal P}^1_n}{}{\text{inf}}\,\| p(A)\|. \] If \(X\) is a normed space, let \({\mathcal L}(X)\) denote the algebra of bounded linear operators on \(X\). The paper is concerned with the question of uniqueness of Chebyshev polynomials and proves Theorem 2. If \((X,\|\cdot \|)\) is a uniformly convex normed space then for each \(A\in{\mathcal L}(X)\) and each \(n\in\mathbb{N}_0\) such that \(n< m\) if \(p\in{\mathcal P}^1_m\) and \(p(A)= 0\), then there exists a unique Chebyshev polynomial of degree \(n\) for \(A\). The result is placed in context by a racy discussion of the history of Chebyshev polynomials, by the suggestion that the supremum norm in Chebyshev's original problem (\({\mathcal R}= C([-1,1])\), \(A(t)= t\) for all \(t\in[-1, 1]\)) might be replaced by other `reasonable' norms, and by the exhibition of one `reasonable' and one `unreasonable' norm.