We show that the knowledge of the set of the Cauchy data on the boundary of a C bounded open set in R, n ≥ 3, for the Schrödinger operator with continuous magnetic and bounded electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.

Johnson and Zippin recently showed that ifX is a weak∗-closed subspace of l1 and T : X → C(K) is any bounded operator then T can extended to a bounded operator T̃ : l1 → C(K). We give a converse result: if X is a subspace of l1 so that l1/X has a (UFDD) and every operator T : X → C(K) can be extended to l1 then there is an automorphism τ of l1 so that τ(X) is weak∗-closed. This result is proved ...

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown th...

1.1 (Fredholm operators). Let X, Y be real Banach spaces and denote their dual spaces by X, Y . A bounded linear operator D : X → Y is called Fredholm if it has a closed image and if its kernel and cokernel (the quotient space Y/imD) are finite dimensional. Equivalently, there exists a bounded linear operator T : Y → X such that the operators TD− idX and DT − idY are compact. The Fredholm index...

Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator D ∈ B(X) is a Drazin inverse of T if TD = DT , D = TD and T k = T D for some nonnegative integer k. In this paper we look at the Jörgens algebra, an algebra of operators on a dual system, and characterise when an operator in that algebra has a Drazin inverse that is also in the algebra. This resu...

I fhI is the respective Haar coefficient, and σ(I) = ±1. This operator, which we denote by Tσ, is a dyadic martingale transform. The martingale transform is bounded as an operator on L(R,C). We want to find a condition on matrix weights, U and V , that implies that all martingale transforms are uniformly bounded as operators from L(R,C, V ) to L(R,C, U) where L(R,C, V ) is the space of function...

For an R-bounded families of operators on L1 we associate a family of representing measures and show that they form a weakly compact set. We consider a sectorial operator A which generates an R-bounded semigroup on L1. We show that given 2 > 0 there is an invertible operator U : L1 → L1 with ‖U − I‖ < 2 such that for some positive Borel function w we have U(D(A)) ⊃ L1(w).

In this paper we get the formula for the condition number of the W -weighted Drazin inverse solution of a linear system WAWx = b, where A is a bounded linear operator between Hilbert spaces X and Y , W is a bounded linear operator between Hilbert spaces Y and X, x is an unknown vector in the range of (AW ) and b is a vector in the range of (WA). AMS Mathematics Subject Classification (2000): 47...

Normality of bounded and unbounded adjointable operators is discussed. If T is an adjointable operator on a Hilbert C*-module which has polar decomposition, then T is normal if and only if there exists a unitary operator U which commutes with T and T ∗ such that T = U T ∗. Kaplansky’s theorem for normality of the product of bounded operators is also reformulated in the framework of Hilbert C*-m...

In this paper, we prove that every multiplier M (i.e. every bounded operator commuting whit the shift operator S) on a large class of Banach spaces of sequences on Z is associated to a function essentially bounded by ‖M‖ on spec(S). This function is holomorphic on ◦ spec(S), if ◦ spec(S) 6= ∅. Moreover, we give a simple description of spec(S). We also obtain similar results for Toeplitz operato...