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
Let a strongly stable norm on the set Mn of complex n-by-n matrices be given, which means that for all A ∈ Mn and all . Furthermore, let be a power series with nonnegative coefficients ck ≥ 0 and radius of convergence R>0. If , we additionally suppose that c0 = f(0) = 0. We aim to characterize those A with , which fulfil . 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 that holds true if, and only if, 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,这意味着对于所有A ∈ M n和所有. 此外,让是非负系数c k ≥ 0 且收敛半径R > 0 的幂级数。如果,我们还假设c 0 = f (0) = 0。我们的目标是用, 满足. 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 方程的已知结果那成立当且仅当是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 。