On a problem of Bernard Chevreau concerning the 𝜌-contractions

Abstract

We prove new results for the operators of class C ρ ( ρ > 0 ) C_{\rho } (\rho >0) on Hilbert spaces defined by B. Sz.-Nagy and C. Foiaş. The main result is an answer to a problem posed in 2006 by B. Chevreau: Let p ≥ 2 p\geq 2 be a natural number and T ∈ L ( H ) T \in \mathrm {L}(\mathcal {H}) ; if there exists ρ 0 > 0 \rho _{0} >0 such that T p ∈ C ρ 0 T^{p} \in C_{\rho _0} , then necessarily is T ∈ ⋃ ρ > 0 C ρ T \in \bigcup _{\rho >0}C_{\rho } ?