A note on Söderlind’s conjecture

Abstract

Let f be a Lipschitz operator from a path-connected set D⊆ C m into C m , with the lub-Lipschitz constant L( f) and the so-called “Gerschgorim range radius” r( f) subordinate to a given vector norm ∥·∥ of C m . In 1986 [Numer. Math. 50 (1986) 27], Söderlind’s conjectured that if r( f)< L( f), then there exists a new vector norm ∥·∥ * of C m such that the induced lub-Lipschitz constant L *( f)⩽ r( f). In this paper, we affirmatively prove Söderlind’s conjecture for several class of Lipschitz operators f, whilst we construct a counterexample to disprove Söderlind’s conjecture.