Condition numbers of large matrices, and analytic capacities

Given an operator $T\colon X\to X$ on a Banach space $X$, we compare the condition number of $T$, $\mathrm{CN}(T)=\|T\|\cdot\|T^{-1}\|$, and the spectral condition number defined as $\mathrm{SCN}(T)=\|T\|\cdot r(T^{-1}\|$, where $r(\cdot)$ stands for the spectral radius. For a set $\Upsilon T$ of operators, we put $\Phi(\Delta)=\sup\{\mathrm{CN}(T):T\in\Upsilon Y,\mathrm{SCN}(T)\leq\Delta\}$, $\Delta\in[1,\infty)$, and say that $\Upsilon Y$ is spectrally $\Phi$-conditioned. As $\Upsilon Y$ we consider certain sets of $(n\times n)$-matrices or, more generally, algebraic operators with $\deg(T)\leq n$ 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 $\Delta\to\infty$ (or as $n\to\infty$) for $s.0$, and is bounded by $\Delta^n(\log(n))^{1/q}$ for $s=0$.