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.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Behavior of the heterogeneous electron-transfer rate constants of arenes and substituted anthracenes in room-temperature ionic liquids

We report the anodic oxidation of several arenes and anthracenes within room-temperature ionic liquids (RTILs). In particular, the heterogeneous electron-transfer rates (k(0)) for substituted anthracenes and arenes are also investigated in 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([C(2)mim][NTf2]) and found not to obey the outer-sphere Marcus-type behavior of these compounds in contrast to the behavior in traditional organic solvents,in particular the predictions for k(0) with molecular size and solvent static dielectric constant. To obtain the electron-transfer rate for 9-phenylanthracene, the dimerization and heterogeneous electron-transfer kinetics of its electrogenerated radical cations is studied in [C(2)mim][NTf2] and eight other RTILs and are both found to be largely independent of the solution viscosity.

Bimolecular rate constants for diffusion in ionic liquids

The temperature dependence of the bimolecular rate constants for a diffusion controlled reaction involving neutral reactants have been directly determined in five commonly used ionic liquids over the temperature range 5-70 degreesC.