Numerical range of s(φ)

Abstract

We make a detailed study of the numerical ranges W(T) of completely nonunitary contractions T with the property rank (1-T^∗T)=1 on a finite-dimensional Hilbert space. We show that such operators are completely characterized by the Poncelet property of their numerical ranges, namely, an n-dimensional contraction T is in the above class if and only if for any point λ on the unit circle there is an (n+l)-gon which is inscribed in the unit circle, circumscribed about W(T) and has λ as a vertex. We also obtain a dual form of this property and the information on the inradii of numerical ranges of arbitrary finite-dimensional operators.

Keywords:

• Numerical range
• compression of the shift
• Poncelet property
• Jordan block
• numerical radius
• inradius

Mathematics Subject Classification:

• 15A60
• 47A12

^∗Corresponding author. e-mail:[email protected]

^∗Corresponding author. e-mail:[email protected]

Notes

^∗Corresponding author. e-mail:[email protected]