The celebrated Kreiss matrix theorem is one of several results relating the norms of the powers of a matrix to its pseudospectra (i.e. the level curves of the norm of the resolvent). But to what extent do the pseudospectra actually determine the norms of the powers? Specifically, let A, B be square matrices such that, with respect to the usual operator norm ‖ · ‖, (∗) ‖(zI − A)−1‖ = ‖(zI − B)−1...

Let L be the space of complex-valued functions f on the set of vertices T of an infinite tree rooted at o such that the difference of the values of f at neighboring vertices remains bounded throughout the tree, and let Lw be the set of functions f ∈ L such that |f v − f v− | O |v|−1 , where |v| is the distance between o and v and v− is the neighbor of v closest to o. In this paper, we character...

We study the optimal approximation of the solution of an operator equation A(u) = f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain Hs-norm where s was given by the operator since we assumed that A : Hs 0(Ω) → H−s(Ω) is an isomorphism. The most typical case here is s = 1. It is well known that for certain regular ...

Weighted norm inequalities are proved for a rough homogeneous singular integral operator and its corresponding maximal truncated singular operator. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals.