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.

Src and ADAM-17-Mediated Shedding of Transforming Growth Factor-alpha Is a Mechanism of Acute Resistance to TRAIL

Tumor necrosis factor-related apoptosis-inducing ligand (TRAIL/Apo-2L) has emerged as a promising anticancer agent. However, resistance to TRAIL is likely to be a major problem, and sensitization of cancer cells to TRAIL may therefore be an important anticancer strategy. In this study, we examined the effect of the epidermal growth factor receptor (EGFR)tyrosine kinase inhibitor (TKI) gefitinib and a human epidermal receptor 2 (HER2)-TKI (M578440) on the sensitivity of human colorectal cancer (CRC) cell lines to recombinant human TRAIL (rhTRAIL). A synergistic interaction between rhTRAIL and gefitinib and rhTRAIL and M578440 was observed in both rhTRAIL-sensitive and resistant CRC cells. This synergy correlated with an increase in EGFR and HER2 activation after rhTRAIL treatment. Furthermore, treatment of CRC cells with rhTRAIL resulted in activation of the Src family kinases (SFK). Importantly, we found that rhTRAIL treatment induced shedding of transforming growth factor-alpha (TGF-alpha) that was dependent on SFK activity and the protease ADAM-17. Moreover, this shedding of TGF-alpha was critical for rhTRAIL-induced activation of EGFR. In support of this, SFK inhibitors and small interfering RNAs targeting ADAM-17 and TGF-alpha also sensitized CRC cells to rhTRAIL-mediated apoptosis. Taken together, our findings indicate that both rhTRAIL-sensitive and resistant CRC cells respond to rhTRAIL treatment by activating an EGFR/HER2-mediated survival response and that these cells can be sensitized to rhTRAIL using EGFR/HER2-targeted therapies. Furthermore, this acute response to rhTRAIL is regulated by SFK-mediated and ADAM-17-mediated shedding of TGF-alpha, such that targeting SFKs or inhibiting ADAM-17, in combination with rhTRAIL, may enhance the response of CRC tumors to rhTRAIL. [Cancer Res 2008;68(20):8312-21]

Inhibitor profiling of the Pseudomonas aeruginosa virulence factor LasB using N-alpha mercaptoamide template-based inhibitors

We report on the synthesis and biological evaluation of a focussed library of N-alpha mercaptoamide containing dipeptides as inhibitors of the zinc metallopeptidase Pseudomonas aeruginosa elastase (LasB, EC 3.4.24.26). The aim of the study was to derive an inhibitor profile for LasB with regard to mapping the S´1 binding site of the enzyme. Consequently, a focussed library of 160 members has been synthesised, using standard Fmoc-solid phase methods (on a Rink-amide resin), in which a subset of amino acids including examples of those with basic (Lys, Arg), aromatic (Phe, Trp), large aliphatic (Val, Leu) and acidic (Asp, Glu) side-chains populated the P´2 position of the inhibitor sequence and all 20 natural amino acids were incorporated, in turn, at the P´1 position. The study has revealed a preference for aromatic and/or large aliphatic amino acids at P´1 and a distinct bias against acidic residues at P´2. Ten inhibitor sequences were discovered that exhibited sub to low micromolar Ki values.

Enhanced sensitivity of protein kinase B/Akt to insulin in hypoxia is independent of HIF1 alpha and promotes cell viability

Maintenance of oxygen homeostasis is a key requirement to ensure normal mammalian cell growth and differentiation. Hypoxia arises when oxygen demand exceeds supply, and is a feature of multiple human diseases including stroke, cancer and renal fibrosis. We have investigated the effect of hypoxia on kidney cells, and observed that insulin-induced cell viability is increased in hypoxia. We have characterized the role of protein kinase B (PKB/ Akt) in these cells as a potential mediator of this effect. PKB/Akt activity was increased by low oxygen concentrations in kidney cells, and insulin-stimulated activation of PKB/Akt was stronger, more rapid and more sustained in hypoxia. Reduction of HIF1 alpha levels using antimycin-A or siRNA targeting HlF1 alpha did not affect PKB/Akt activation in hypoxia. Pharmacologic stabilization of HIF1 alpha independent of hypoxia did not increase insulin-stimulated PKB/Akt activation. Although increased insulin-stimulated cell viability was observed in hypoxia, no differences in the degree of insulin-stimulated glucose uptake were observed in L6 muscle cells in hypoxia compared to normoxia. Thus, PKB/Akt may regulate specific cellular responses to growth factors such as insulin under adverse conditions such as hypoxia. alpha 2007 Elsevier GmbH. All rights reserved.

SN 2008S: an electron-capture SN from a super-AGB progenitor?

We present comprehensive photometric and spectroscopic observations of the faint transient SN 2008S discovered in the nearby galaxy NGC 6946. SN 2008S exhibited slow photometric evolution and almost no spectral variability during the first nine months, implying a long photon diffusion time and a high-density circumstellar medium. Its bolometric luminosity (similar or equal to 10(41) erg s(-1) at peak) is low with respect to most core-collapse supernovae but is comparable to the faintest Type II-P events. Our quasi-bolometric light curve extends to 300 d and shows a tail phase decay rate consistent with that of Co-56. We propose that this is evidence for an explosion and formation of Ni-56 (0.0014 +/- 0.0003 M-circle dot). Spectra of SN 2008S show intense emission lines of H alpha, [Ca II] doublet and Ca II near-infrared (NIR) triplet, all without obvious P-Cygni absorption troughs. The large mid-infrared (MIR) flux detected shortly after explosion can be explained by a light echo from pre-existing dust. The late NIR flux excess is plausibly due to a combination of warm newly formed ejecta dust together with shock-heated dust in the circumstellar environment. We reassess the progenitor object detected previously in Spitzer archive images, supplementing this discussion with a model of the MIR spectral energy distribution. This supports the idea of a dusty, optically thick shell around SN 2008S with an inner radius of nearly 90 AU and outer radius of 450 AU, and an inferred heating source of 3000 K. The luminosity of the central star is L similar or equal to 10(4.6) L-circle dot. All the nearby progenitor dust was likely evaporated in the explosion leaving only the much older dust lying further out in the circumstellar environment. The combination of our long-term multiwavelength monitoring data and the evidence from the progenitor analysis leads us to support the scenario of a weak electron-capture supernova explosion in a super-asymptotic giant branch progenitor star (of initial mass 6-8 M-circle dot) embedded within a thick circumstellar gaseous envelope. We suggest that all of main properties of the electron-capture SN phenomenon are observed in SN 2008S and future observations may allow a definitive answer.