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.

Time operator for diffusion

Volume: 11 Issue: 4 Pages: 465-477 Published: MAR 2000 Times Cited: 9 References: 15 Citation MapCitation Map beta Abstract: We extend the concept of time operator for general semigroups and construct a non-self-adjoint time operator for the diffusion equation which is intertwined with the unilateral shift. We obtain the spectral resolution, the age eigenstates and a new shift representation of the solution of the diffusion equation. Based on previous work we obtain similarly a self-adjoint time operator for Relativistic Diffusion. (C) 2000 Elsevier Science Ltd. All rights reserved.

Derivation of clinically compliant MSCs from CD105+, CD24-differentiated human ESCs

Adult tissue-derived mesenchymal stem cells ( MSCs) have demonstrated therapeutic efficacy in treating diseases or repairing damaged tissues through mechanisms thought to be mediated by either cell replacement or secretion of paracrine factors. Characterized, self- renewing human ESCs could potentially be an invariable source of consistently uniform MSCs for therapeutic applications. Here we describe a clinically relevant and reproducible manner of generating identical batches of hESC- derived MSC ( hESC- MSC) cultures that circumvents exposure to virus, mouse cells, or serum. Trypsinization and propagation of HuES9 or H1 hESCs in feeder- and serum-free selection media generated three polyclonal, karyotypically stable, and phenotypically MSC-like cultures that do not express pluripotency- associated markers but displayed MSC- like surface antigens and gene expression profile. They differentiate into adipocytes, osteocytes, and chondrocytes in vitro. Gene expression and fluorescence- activated cell sorter analysis identified CD105 and CD24 as highly expressed antigens on hESC- MSCs and hESCs, respectively. CD105+, CD24- monoclonal isolates have a typical MSC gene expression profiles and were identical to each other with a highly correlated gene expression profile ( r(2) >.90). We have developed a protocol to reproducibly generate clinically compliant and identical hESC- MSC cultures.