当前位置: X-MOL 学术Linear Multilinear Algebra › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
When does ‖f(A)‖ = f(‖A‖) hold true?
Linear and Multilinear Algebra ( IF 1.1 ) Pub Date : 2021-10-19 , DOI: 10.1080/03081087.2021.1985054
Florian Bünger 1 , Siegfried M. Rump 1, 2
Affiliation  

Let a strongly stable norm on the set Mn of complex n-by-n matrices be given, which means that AkAk for all A ∈ Mn and all k=1,2,. Furthermore, let f(x)=k=0ckxk be a power series with nonnegative coefficients ck ≥ 0 and radius of convergence R>0. If I>1, we additionally suppose that c0 = f(0) = 0. We aim to characterize those A with A<R, which fulfil f(A)=f(A). We first show how to reduce the discussion of f to Neumann series. For matrix norms induced by uniformly convex vector norms, like the ℓp-norms, p ∈ (1, ∞), it follows from known results of the Daugavet equation I+A=1+A that f(A)=f(A) holds true if, and only if, A is an eigenvalue of A, provided that ckck+1 ≠ 0 for some k ≥ 0. Under adapted assumptions on the ck we prove that this equivalence remains true for the ℓ1- and the ℓ-norm, for unitarily invariant matrix norms and for the numerical radius. We conjecture this equivalence to be valid for all strongly stable norms if ck > 0 for all k ≥ 1.



中文翻译:

‖f(A)‖ = f(‖A‖) 什么时候成立?

让一个强大稳定的规范在复数n × n矩阵的集合M n上给出,这意味着AkAk对于所有A  ∈  M n和所有k=1个,2个,……. 此外,让F(X)=k=0CkXk是非负系数c k  ≥ 0 且收敛半径R > 0 的幂级数。如果>1个,我们还假设c 0  =  f (0) = 0。我们的目标A<R, 满足F(A)=F(A). 我们首先展示如何将f的讨论减少到 Neumann 级数。对于由一致凸向量范数导出的矩阵范数,如 ℓ p -范数,p  ∈ (1, ∞),它遵循 Daugavet 方程的已知结果+A=1个+AF(A)=F(A)成立当且仅当A是A的特征值,前提是 对于某些k ≥ 0, c k c k +1 ≠ 0。根据对c k的适应性 假设,我们证明对于 ℓ 1 - 和 ℓ ∞ -范数,对于单一不变矩阵范数和数值半径。如果对于所有k ≥ 1, c k  > 0,我们推测这种等价性对所有强稳定范数都有效 。

更新日期:2021-10-19
down
wechat
bug