Even the first iterate of a Markov operator is contracting in anL2 norm
A weightedL2 norm is introduced in which Markov operators, e.g., associated with noisy maps, are contracting provided the kernel (i.e., the transitional distribution) is smooth enough. This results in strong relaxational properties of noisy maps. Similar to this norm, integral functionals appear useful when studying spatiotemporal chaos and random fields.
Let X and Y be two inflnite dimensional real or complex Banach spaces. In this note we determine the forms of surjective additive maps ` : L(X) ! L(Y ) preserving the kernel's dimension or the range's codimension. As consequence, we establish that ` : L(X) ! L(X) preserves the kernel (respectively, the range) if and only if there exists an invertible operator A 2 L(X) such that `(T) = AT (respectively, `(T) = TA) for all T 2 L(X).