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.

New ways to break an old bond: the bacterial carbon-phosphorus hydrolases and their role in biogeochemical phosphorus cycling

Phosphonates are organophosphorus molecules that contain the highly stable C-P bond, rather than the more common, and more labile, C-O-P phosphate ester bond. They have ancient origins but their biosynthesis is widespread among more primitive organisms and their importance in the contemporary biosphere is increasingly recognized; for example phosphonate-P is believed to play a particularly significant role in the productivity of the oceans. The microbial degradation of phosphonates was originally thought to occur only under conditions of phosphate limitation, mediated exclusively by the poorly characterized C-P lyase multienzyme system, under Pho regulon control. However, more recent studies have demonstrated the Pho-independent mineralization by environmental bacteria of three of the most widely distributed biogenic phosphonates: 2-aminoethylphosphonic acid (ciliatine), phosphonoacetic acid, and 2-amino-3-phosphonopropionic acid (phosphonoalanine). The three phosphonohydrolases responsible have unique specificities and are members of separate enzyme superfamilies; their expression is regulated by distinct members of the LysR family of bacterial transcriptional regulators, for each of which the phosphonate substrate of the respective degradative operon serves as coinducer. Previously no organophosphorus compound was known to induce the enzymes required for its own degradation. Whole-genome and metagenome sequence analysis indicates that the genes encoding these newly described C-P hydrolases are distributed widely among prokaryotes. As they are able to function under conditions in which C-P lyases are inactive, the three enzymes may play a hitherto-unrecognized role in phosphonate breakdown in the environment and hence make a significant contribution to global biogeochemical P-cycling.

Isolation and characterisation of bacterial strains containing enantioselective DMSO reductase activity: application to the kinetic resolution of racemic sulfoxides

The kinetic resolution of racemic sulfoxides by dimethyl sulfoxide (DMSO) reductases was investigated with a range of microorganisms. Three bacterial isolates (provisionally identified as Citrobacter braakii, Klebsiella sp. and Serratia sp.) expressing DMSO reductase activity were isolated from environmental samples by anaerobic enrichment with DMSO as terminal electron acceptor. The organisms reduced a diverse range of racemic sulfoxides to yield either residual enantiomer depending upon the strain used. C. braakii DMSO-11 exhibited wide substrate specificity that included dialkyl, diaryl and alkylaryl sulfoxides, and was unique in its ability to reduce the thiosulfinate 1,4-dihydrobenzo-2, 3-dithian-2-oxide. DMSO reductase was purified from the periplasmic fraction of C. braakii DMSO-11 and was used to demonstrate unequivocally that the DMSO reductase was responsible for enantiospecific reductive resolution of racemic sulfoxides.