Resolvent bounds and spectral variation
The spectral variation between linear operators A, B mapping an n- dimensional space over \({\mathbb{C}}\) is given by \(v(A,B)=\min_{\pi}\max_{i}| \lambda_ i-\mu_{\pi (i)}|\) where \(\pi\) varies over all permutations of \(\{\) 1,...,n\(\}\). Here \(\{\lambda_ 1,...,\lambda_ n\}\) is the spectrum of A and \(\{\mu_ 1,...,\mu_ n\}\) is the spectrum of B; \(\| \cdot \|\) is a norm when applied to a vector and the inherited operator norm when applied to a linear operator. It is shown that \(v(A,B)\leq 8\times 4^{-1/n}M^{1- 1/n}\| A-B\|^{1/n}\times [\sum^{n- 1}_{i=0}(\frac{v(A,B)}{2M})^ i]^{1/n},\) where \(M=\max \{\| A\|,\| B\| \}\). This compares to a known result \(v(A,B)\leq 8\times 4^{-1/n}n^{1/n}M^{1-1/n}\| A-B\|^{1/n}\).
scientific article; zbMATH DE number 4181533