zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Simultaneous contractibility. (English) Zbl 0912.15033
Authors’ summary: Let $C$ be a set of $n \times n$ complex matrices. For $m = 1,2,\dots,$ $C^{m}$ is the set of all products of matrices in $C$ of length $m$. Denote by $\widehat{r}(C)$ the joint spectral radius of $C$, that is, $$\widehat{r}(C) := \limsup_{m\to\infty}[\sup_{A\in C^m}\| A\| ]^{\frac{1}{m}}.$$ We call $C$ simultaneously contractible if there is an invertible matrix S such that $$\sup\{\| S^{-1}AS\| ;\ A\in C\}<1,$$ where $\|\cdot\|$ is the spectral norm. This paper is primarily devoted to determining the optimal joint spectral radius range for simultaneous contractibility of bounded sets of $n \times n$ complex matrices, that is, the maximum subset $J$ of $[0,1)$ such that if $C$ is a bounded set of $n \times n$ complex matrices and $\widehat{r}(C)\in J$, then $C$ is simultaneously contractible. The central result proved in this paper is that this maximum subset is $[0,\frac{1}{\sqrt{n}}).$ Our method of proof is based on a matrix-theoretic version of complex John’s ellipsoid theorem and the generalized Gelfand spectral radius formula.

15A60Applications of functional analysis to matrix theory
15A18Eigenvalues, singular values, and eigenvectors
47A10Spectrum and resolvent of linear operators
Full Text: DOI