We study different operator radii of homomorphisms from an operator algebra into $B(H)$ and show that these can be computed explicitly in terms of the usual norm. As an application, we show that if $Ω$ is a $K$-spectral set for a Hilbert space operator, then it is a $M$-numerical radius set, where $M=\frac{1}{2}(K+K^{-1})$. This is a counterpart of a recent result of Davidson, Paulsen and Woerdeman. More general results for operator radii associated with the class of operators having $ρ$-dilations in the sense of Sz.-Nagy and Foias are given. A version of a result of Drury concerning the joint numerical radius of non-commuting $n$-tuples of operators is also obtained.
Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces
众所周知，具有法锥$X_+$的有序赋范向量空间$X$具有与原始范数等价的保序范数。\begin{equation} \sharp x \sharp = \max \{ d(x, X_+), d(x, - X_+)\}, \qquad x \in X. \end{equation}给出了这样一个等价的保序范数。本文讨论了这个范数和半范数$ψ(x) = d(x,-X_+)$的性质，与锥是否正规无关。我们使用$ψ$推导抽象积分方程解的比较原理，推导非线性正映射点耗散性的条件，比较有界齐次映射的Collatz-Wielandt数、界和序谱半径，并给出局部上Collatz-Wielandt半径具有下正特征向量的条件.
Inequalities and equalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators
Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators
