Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Structure of spectra of linear operators in Banach spaces

Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separable Banach space are obtained.

Time operators and shift representation of dynamical systems

We introduce and characterise time operators for unilateral shifts and exact endomorphisms. The associated shift representation of evolution is related to the spectral representation by a generalized Fourier transform. We illustrate the results for a simple exact system, namely the Renyi map.

Semi-formal Models to Support Program Development: Autonomic Management within Component Based Parallel and Distributed Programming

Functional and non-functional concerns require different programming effort, different techniques and different methodologies when attempting to program efficient parallel/distributed applications. In this work we present a "programmer oriented" methodology based on formal tools that permits reasoning about parallel/distributed program development and refinement. The proposed methodology is semi-formal in that it does not require the exploitation of highly formal tools and techniques, while providing a palatable and effective support to programmers developing parallel/distributed applications, in particular when handling non-functional concerns.