Abstract
For a finite set Σ of compact contractions in a complex Hilbert space ( H·∥·∥), it is shown that r( A)<1 for all A in the multiplicative semigroup generated by Σ if and only if there exists a positive integer N such that ∥ A∥<1 for all A in the multiplicative semigroup generated by Σ with length greater than N. Here r( A) denotes the spectral radius of A. As an application, an answer is given to an infinite-dimensional case of the finiteness conjecture for the generalized spectral radius attributed to J.C. Lagarias and Y. Wang [Linear Algebra Appl. 214 (1995) 17].
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