Krein space numerical ranges: compressions and dilations

Abstract

A criterion for the numerical range of a linear operator acting in a Krein space to be a two-component hyperbolical disc is given, using the concept of support function. A characterization of the Krein space numerical range as a union of hyperbolical discs is obtained by a reduction to the two-dimensional case. We revisit a famous result of Ando concerning the inclusion relation $W(A)\subseteq W(B)$ of the numerical ranges of two operators $A$ and $B$ acting in (possibly different) Hilbert spaces, and the condition that $A$ can be dilated to an operator of the form $B\otimes I$. The extension of this result to operators acting in Krein spaces is investigated.