Sufficient spectral conditions for the existence of a spectral decomposition of an operator T defined on a Banach space X, with countable spectrum, are given. We apply the results to obtain the West decomposition of certain Riesz operators.
The asymptotic limit of a bicontraction T (that is, a pair of commuting contractions) on a Hilbert space is used to describe a Nagy-Foiaş-Langer type decomposition of T. This decomposition is refined in the case when the asymptotic limit of T is an orthogonal projection. The case of a bicontraction T consisting of hyponormal (even quasinormal) contractions is also considered, where we have .
We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces for which is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces can be divided into subsets in such a way that if and are in different subsets,...
We show that if (Tₙ) is a hypercyclic sequence of linear operators on a locally convex space and (Sₙ) is a sequence of linear operators such that the image of each orbit under every linear functional is non-dense then the sequence (Tₙ + Sₙ) has dense range. Furthermore, it is proved that if T,S are commuting linear operators in such a way that T is hypercyclic and all orbits under S satisfy the above non-denseness property then T - S has dense range. Corresponding statements for operators and sequences...
In a former paper we describe the geometric properties of the space of continuous functions with values in the space of operators acting on a Hilbert space. In particular we show that dent B(L(H)) = ext B(L(H)) if dim H < 8 and card K < 8 and dent B(L(H)) = 0 if dim H < 8 or card K = 8, and x-ext C(K,L(H)) = ext C(K,L(H)).
The realization of an elliptic operator A under suitable boundary conditions is considered and the dependence of the square-root of A from the various conditions is studied.
In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.