%0 Unpublished work
%T Maximum of the resolvent over matrices with given spectrum
%+ Department of Applied Mathematics and Theoretical Physics (DAMTP)
%+ Mathématiques Fondamentales
%A Szehr, Oleg
%A Zarouf, Rachid
%8 2015-01-27
%D 2015
%Z 1501.07007
%R 10.1016/j.jfa.2016.07.005
%K Toeplitz matrix
%K Resolvent
%K Model matrix
%K Condition number
%K Blaschke product 2010 Mathematics Subject Classification: Primary: 15A60
%K Secondary:30D55
%Z Mathematics [math]/Numerical Analysis [math.NA]
%Z Mathematics [math]/Spectral Theory [math.SP]
%Z Mathematics [math]/Functional Analysis [math.FA]Preprints, Working Papers, ...
%X In numerical analysis it is often necessary to estimate the condition number $CN(T)=\left|\!\left|T\right|\!\right|_{} \cdot\left|\!\left|T^{-1}\right|\!\right|_{}$ and the norm of the resolvent $\left|\!\left|(\zeta-T)^{-1}\right|\!\right|_{}$ of a given $n\times n$ matrix $T$. We derive new spectral estimates for these quantities and compute explicit matrices that achieve our bounds. We recover the well-known fact that the supremum of $CN(T)$ over all matrices with $\left|\!\left|T\right|\!\right|_{} \leq1$ and minimal absolute eigenvalue $r=\min_{i=1,...,n}\left|\lambda_{i}\right|>0$ is the Kronecker bound $\frac{1}{r^{n}}$. This result is subsequently generalized by computing the corresponding supremum of $\left|\!\left|(\zeta-T)^{-1}\right|\!\right|_{}$ for any $\left|\zeta\right| \leq1$. We find that the supremum is attained by a triangular Toeplitz matrix. This provides a simple class of structured matrices on which condition numbers and resolvent norm bounds can be studied numerically. The occuring Toeplitz matrices are so-called model matrices, i.e.~matrix representations of the compressed backward shift operator on the Hardy space $H_2$ to a finite-dimensional invariant subspace.
%G English
%2 https://hal.science/hal-01110346/document
%2 https://hal.science/hal-01110346/file/Resolvent_estimates_27_01_15_22h44.pdf
%L hal-01110346
%U https://hal.science/hal-01110346
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ TDS-MACS