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A note on the differences of the consecutive powers of operators

Andrzej Święch (1997)

Banach Center Publications

We present two examples. One of an operator T such that T n ( T - I ) n = 1 is precompact in the operator norm and the spectrum of T on the unit circle consists of an infinite number of points accumulating at 1, and the other of an operator T such that T n ( T - I ) n = 1 is convergent to zero but T is not power bounded.

A note on the Hyers-Ulam problem

Yunbai Dong (2015)

Colloquium Mathematicae

Let X,Y be real Banach spaces and ε > 0. Suppose that f:X → Y is a surjective map satisfying | ∥f(x)-f(y)∥ - ∥x-y∥ | ≤ ε for all x,y ∈ X. Hyers and Ulam asked whether there exists an isometry U and a constant K such that ∥f(x) - Ux∥ ≤ Kε for all x ∈ X. It is well-known that the answer to the Hyers-Ulam problem is positive and K = 2 is the best possible solution with assumption f(0) = U0 = 0. In this paper, using the idea of Figiel's theorem on nonsurjective isometries, we give a new proof of...

A note on the Poincaré inequality

Alireza Ranjbar-Motlagh (2003)

Studia Mathematica

The Poincaré inequality is extended to uniformly doubling metric-measure spaces which satisfy a version of the triangle comparison property. The proof is based on a generalization of the change of variables formula.

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