Эта публикация цитируется в
25 статьях
Статьи
Condition numbers of large matrices, and analytic capacities
N. K. Nikolskiab a St. Petersburg Branch Steklov Mathematical Institute, Russian Academy of Sciences, Russia
b Département de Mathématiques, Université de Bordeaux 1, France
Аннотация:
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
$s=0$.
Поступила в редакцию: 15.04.2005
Язык публикации: английский