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Some orthogonal decompositions of Sobolev spaces and applications

H. Begehr, Yu. Dubinskiĭ (2001)

Colloquium Mathematicae

Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of W - 1 for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the Δ k -solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form. In the second kind decomposition the Δ k -solenoidal...

Some remarks on the asymptotic behaviour of the iterates of a bounded linear operator

Anatoliĭ Antonevich, Jürgen Appell, Petr Zabreĭko (1994)

Studia Mathematica

We discuss the problem of characterizing the possible asymptotic behaviour of the norm of the iterates of a bounded linear operator between two Banach spaces. In particular, given an increasing sequence of positive numbers tending to infinity, we construct Banach spaces such that the norm of the iterates of a suitable multiplication operator between these spaces assumes (or exceeds) the values of this sequence.

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