LAUSR

This note surveys recent strategies to estimate the condition number $$CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $$CN(T)=‖T‖·‖T-1‖ of complex $$n\times n$$n×n matrices T with given spectrum. More precisely, we present a proof of the fact that if T acts on the Hilbert space $$\mathbb {C}^{n}$$Cn, then the supremum of CN(T) over all contractions T with smallest eigenvalues of modulus $$r>0$$r>0, is equal to $$1/r^{n}$$1/rn, and is achieved by an analytic Toeplitz matrix. The same question is treated for n-dimensional Banach spaces. These strategies provide with explicit and constructive solutions to the so-called Halmos and Schäffer’s problems, and are also shown to be effective in a closely related situation, namely considering Kreiss matrices instead of contractions.