Convex domains and K-spectral sets

Abstract

the introduction was changed and some remarks have been added. 26 pages ; to appear in Math. ZInternational audienceLet $\Omega$ be an open convex domain of the complex plane. We study constants K such that $\Omega$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $\Omega$. Several approaches are discussed

Resolvent characterisation of generators of cosine functions and C 0-groups

“…We refer the reader also to the series of papers [3,5,[11][12][13]43], where the relationships between the notions of dilation, boundedness of functional calculus and numerical range are studied for general convex domains.…”

Operators with numerical range in a conic domain

“…holomorphic in Ω, and continuous and bounded on the closure Ω. Also, in this paper, we do not consider the completely bounded version C cb (Ω) of our constants (see for instance [1] for the definition), but the reader familiar with this notion will easily notice that all our estimates are still valid with C cb (Ω) in place of C(Ω).…”

“…The general bound C(Ω) ≤ 11.1 has been established in [5]. This bound is considered to be pessimistic: for instance, for a disk D it is known [1] that C(D) = 2, and Crouzeix conjectures in [3] that C(Ω) ≤ 2 for any open convex set Ω.…”

“…Here one often requires sharp estimates for C(Ω) for particular sets Ω. Using techniques different from those in [5], it is shown in [4] that C(Ω) ≤ 4.75 for a parabolic domain Ω, and in [1,2] that…”

“…Finally we compare in Section 4 our findings for a sector with those from [1,2]. holomorphic in Ω, and continuous and bounded on the closure Ω.…”