arXiv subjects
Publications and source records attributed to Jens de Vries.
Let $T$ be a $ρ$-contraction on a Hilbert space. We establish sharp bounds for the functional calculus of $T$ for matrix-valued analytic functions on the unit disc. These are complete versions of bounds established by Drury and by Schwenninger and the second author.
We investigate when the algebraic numerical range is a $C$-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex $2\times2$-matrices with the induced $1$-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.
Let $A$ be a $ρ$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $ρ$-contractions we prove that \begin{align*} \big\|f(A)\big\|\leq \fracρ{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{ρ^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for $ρ=2$, is consistent with a result by Drury.
We prove bounds for a class of unital homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix--Palencia, Okubo--Ando and von Neumann in a unified way and to refine a recent result by Crouzeix--Greenbaum.