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Analysis demonstrates bounds of rho-contraction norms in Hilbert spaces, indicating critical thresholds for vector behavior.
For Ο > 1 Ο >1 , let A A be a Ο Ο -contraction on a Hilbert space H H . Letting n β₯ 1 nβ₯ 1 and x x be a unit vector in H H , we show that (a) if β A x β = β A 2 x β = β― = β A n x β β‘ c \|Ax\|=\|A^2x\|=β― =\|A^nx\|β‘ c , then c β€ Ο ( Ο + n β 1 ) / n cβ€ βΟ (Ο +n-1)/n , and (b) β A x β = β A
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Gau et al. (2026) studied this question.
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