## A Class of Contractions in Hilbert Space and Applications

### Volume 55 / 2007

Bulletin Polish Acad. Sci. Math. 55 (2007), 347-355
MSC: 47A30, 47A10, 60G50, 60G15.
DOI: 10.4064/ba55-4-6

#### Abstract

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.