Contractibility of compact contractions in Hilbert space. (English) Zbl 0999.15010
A necessary and sufficient condition for the spectral radius of a compact contraction being less than one is stated.
Let \(\Sigma\) denote a finite set of compact contractions on a complex Hilbert space. Then the spectral radius of \(A\) is less than one for all \(A\) in the multiplicative semigroup generated by \(\Sigma\) if and only if there exists a positive integer \(N\) such that norm of \(A\) is less than one for all \(A\) in the set of all products of operators in \(\Sigma\) of length \(N\).
Related extensions and examples are considered.
Let \(\Sigma\) denote a finite set of compact contractions on a complex Hilbert space. Then the spectral radius of \(A\) is less than one for all \(A\) in the multiplicative semigroup generated by \(\Sigma\) if and only if there exists a positive integer \(N\) such that norm of \(A\) is less than one for all \(A\) in the set of all products of operators in \(\Sigma\) of length \(N\).
Related extensions and examples are considered.
Reviewer: Jitka Drkošova (Praha)
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 47A10 | Spectrum, resolvent |
| 47B07 | Linear operators defined by compactness properties |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
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