Condition numbers of large matrices, and analytic capacities

Abstract:

Given an operator $T:X\longrightarrow X$ on a Banach space $X$, we compare the condition number of $T$, $\operatorname {CN}(T)= \Vert T\Vert \cdot \Vert T^{-1}\Vert$, and the spectral condition number defined as $\operatorname {SCN}(T)= \Vert T\Vert \cdot r(T^{-1})$, where $r(\cdot )$ stands for the spectral radius. For a set $\Upsilon$ of operators, we put $\Phi (\Delta ) = \sup \{\operatorname {CN}(T): T\in \Upsilon , \operatorname {SCN}(T) \le \Delta \}$, $\Delta \in [1,\infty )$, and say that $\Upsilon$ is spectrally $\Phi$-conditioned. As $\Upsilon$ we consider certain sets of $(n\times n)$-matrices or, more generally, algebraic operators with $\deg (T)\le n$ that admit a specific functional calculus. In particular, the following sets are included: Hilbert (Banach) space power bounded matrices (operators), polynomially bounded matrices, Kreiss type matrices, Tadmor–Ritt type matrices, and matrices (operators) admitting a Besov class $B^{s}_{p,q}$-functional calculus. The above function $\Phi$ is estimated in terms of the analytic capacity $\operatorname {cap}_{A}(\cdot )$ related to the corresponding function class $A$. In particular, for $A= B^{s}_{p,q}$, the quantity $\Phi (\Delta )$ is equivalent to $\Delta ^{n}n^{s}$ as $\Delta \longrightarrow \infty$ (or as $n\longrightarrow \infty$) for $s>0$, and is bounded by $\Delta ^{n}(\log (n))^{1/q}$ for $s=0$.