A Class of Contractions in Hilbert Space and Applications

Nick Dungey¹

¹ Department of Mathematics Macquarie University Sydney, NSW 2109 Australia

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) no. 4, pp. 347-355

Résumé

We characterize the bounded linear operators $T$ in Hilbert space which satisfy $T = \beta I + (1-\beta)S$ where $\beta\in (0,1)$ and $S$ is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup $(T^n)_{n=1, 2, \ldots}$ by the continuous semigroup $(e^{-t(I-T)})_{t\geq 0}$. Moreover, we give a stronger quadratic form inequality which ensures that $\sup \{ n \| T^n - T^{n+1} \| \colon n = 1, 2, \ldots \} \infty$. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

Détail

DOI : 10.4064/ba55-4-6

Keywords: characterize bounded linear operators hilbert space which satisfy beta beta where beta contraction characterizations include quadratic form inequality domination condition discrete semigroup ldots continuous semigroup t i t geq moreover stronger quadratic form inequality which ensures sup colon ldots infty results apply large classes markov operators countable spaces locally compact groups