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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Dungey N.

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A class of contractions in Hilbert space and applications

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Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom Vol. 55, no 2

347-355

We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1 - β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup (T[sup]n)n=1,2,... by the continuous semigroup (e[sup]-t(I-T)t≥o. Moreover, we give a stronger quadratic form inequality which ensures that sup{n||T[sup]n - T[sup]n+1 ||: n = 1, 2,...} < ∞. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

Some gradient estimates on covering manifolds

100%

Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

tom Vol. 52, nr 4

437-443

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.