330 research outputs found

    And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models

    Full text link
    One of the most important results in operator theory is And\^o's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an And\^o dilation for a commuting pair (T1,T2)(T_1,T_2) of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of And\^o's dilation theorem by giving both Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction TT into the product of two commuting contractions. Unlike the one-variable case, two minimal And\^o dilations need not be unitarily equivalent. However, we show that the compressions of the two And\^o dilations constructed in this paper to the minimal dilation spaces of the contraction T1T2T_1T_2, are unitarily equivalent. In the special case when the product T=T1T2T=T_1T_2 is pure, i.e., if Tn0T^{* n}\to 0 strongly, an And\^o dilation was constructed recently in \cite{D-S-S}, which, as this paper will show, is a corollary to the Douglas-type construction. We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair (T1,T2)(T_1,T_2) of commuting contractions such that T1T2T_1T_2 is a pure contraction.Comment: 24 page

    Federal Recovery Funds Allocated

    Get PDF
    Figure S1. Overview of the informatics pipeline used to the identify lincRNAs in goat. (PDF 216 kb

    Silencing integrin β3 expression diminishes migration and invasion-modulating properties of the HMF-Sdc1-derived ECM.

    No full text
    <p>A) Quantitative assessment of migration directionality of MDA-MB-231 and MCF10DCIS.com cells within the ECMs produced by HMF-mock or HMF-Sdc1 cells treated with control or β3 siRNA using time-lapse microscopy. Different letters above columns indicate significant difference (p<0.001). B) Invasion of MDA-MB-231 and MCF10DCIS.com cells through ECMs produced by HMF-mock or HMF-Sdc1 cells treated with control or β3 siRNA. Different letters above columns indicate significant difference (at least p<0.05). Ctrl si, control siRNA; β3 si, β3 siRNA.</p

    Generation of stable HMF cell lines expressing mutant forms of Sdc1.

    No full text
    <p>A) Schematic representation of Sdc1 mutant constructs introduced into HMF cells. B) Flow cytometry analysis of exogenous Sdc1 in stably transfected HMF reveals that wildtype Sdc1 and Sdc1 mutants are expressed at comparable levels.</p

    Integrin αvβ3 activity is involved in Sdc1-induced morphological changes during 3D ECM production.

    No full text
    <p>A-B, representative phase contrast images of HMF mock and Sdc1 cells cultured on gelatin-coated plates under semiconfluent states (A; original magnification: 100x) and the corresponding mean elongation indices (ratio of cell length/width) of HMF mock and Sdc1 cells (B). C, representative phase contrast images of HMF mock and Sdc1 cells after 7 d of culture under conditions permissive for 3D ECM production (Original magnification: 100x). D-E, representative confocal images of the nuclei of HMF mock and Sdc1 cells cultured on gelatin-coated plates under semiconfluent states (D; original magnification: 200x) and the corresponding mean nuclear elongation indices (ratio of nuclear length/width) of HMF mock and Sdc1 cells (E). The nuclei were labeled with Hoechst 33342. F-G, representative confocal images of the nuclei of HMF mock and Sdc1 cells after 7 d of culture under conditions permissive for 3D ECM production (F; Original magnification: 200x) and the corresponding mean nuclear elongation indices of HMF mock and Sdc1 cells (G). The nuclei were labeled with Hoechst 33342. *indicates that the difference was statistically significant (p<0.0001). H-I, representative confocal images of the nuclei of control siRNA or β3 siRNA treated HMF mock and Sdc1 cells after 7 d of culture under conditions permissive for 3D ECM production (H; original magnification: 200x) and the corresponding mean nuclear elongation indices (I). SiRNA treatment was performed as described in Material and Methods. The nuclei were labeled with Hoechst 33342. Letters above the columns indicate the results of statistical comparisons by ANOVA. Columns sharing the same letter are not significantly different; columns labeled with different letters are significantly different (at least p<0.05).</p

    Activation of integrin αvβ3 alone is not sufficient for generating a migration and invasion-permissive ECM.

    No full text
    <p>A) Quantitative assessment of migration directionality of MDA-MB-231 and MCF10DCIS.com cells within the ECMs from HMF mock and Sdc1 cells treated with control or β3 clasp peptide using time-lapse microscopy. Columns labeled with different letters are significantly different (p<0.001). B) Invasion of MDA-MB-231 and MCF10DCIS.com cells through ECMs from HMF mock and Sdc1 cells treated with control or β3 clasp peptide. Columns labeled with different letters are significantly different (at least p<0.05). Ctrl pep, control peptide; β3 pep, β3 clasp peptide.</p

    Sdc1 ectodomain and HS chains are required for producing an ECM that promotes directional migration and invasion.

    No full text
    <p>A) Time-lapse migration study of MDA-MB-231 and MCF10DCIS.com cells in 3D ECMs derived from HMF-mock cells, HMF-Sdc1 cells, HMF cells expressing the HS-deficient Sdc1 (Sdc1-ΔHS) and HMF cells expressing the ectodomain-truncated Sdc1 (Sdc1-ΔEcto). Cell movements were recorded every 30 minutes for a period of 5–6 hours. The directional persistence of cell migration was determined as the ratio of the migration distance (net distance in a direct line from start to end point) to the total distance traveled. Columns labeled with different letters are significantly different (p<0.001). B) Invasion of MDA-MB-231 and MCF10DCIS.com cells through the indicated HMF-derived ECMs deposited in the inserts of Matrigel invasion chambers. Invasion is reported as the number of invading cells per field. Columns labeled with different letters are significantly different (p<0.001).</p

    Forced activation of integrin αvβ3 leads to partial ECM fiber alignment.

    No full text
    <p>A) Activation of the αvβ3 integrin can be achieved independent of Sdc1 by β3 clasp peptide. Integrin αvβ3 activities were accessed by measuring the attachment of HMF cells, including untreated HMF cells, HMF cells treated with 250μM control peptide (ctrl pep), HMF cells treated with 250μM β3 clasp peptide (β3 pep) and HMF mock cells treated with 2 mM MgCl<sub>2</sub>, to culture dishes pre-coated with vitronectin after 15min incubation at 37°C. Columns labeled with different letters are significantly different (at least p<0.05). B) Representative confocal images of immunofluorescently labeled FN fibers of ECMs from HMF mock and Sdc1 cells treated with control or β3 clasp peptide. Original magnification: 200x. C) Mean fiber-to-fiber angles of the indicated HMF ECMs. Columns labeled with different letters are significantly different (at least p<0.05). Ctrl pep, control peptide; β3 pep, β3 clasp peptide.</p

    Potential molecular mechanisms by which stromal Sdc1 regulates ECM fiber alignment.

    No full text
    <p>A) In Sdc1-negative normal fibroblasts, FN fibrillogenesis is likely initiated by integrin α5β1, which gives rise to an ECM with haphazard fiber arrangement. B) In the stromal fibroblasts of some invasive breast carcinomas, Sdc1 is aberrantly expressed. Sdc1 induces morphological changes of the fibroblasts, which may contribute to the production of aligned ECM. Sdc1 may also modulate FN fibrillogenesis by directly regulating the unfolding of dimeric FNs and the interactions between cell-associated FN dimers through its HS chains. In addition, Sdc1 may modulate the initiation of FN fibrillogenesis by activating integrin αvβ3 through its ectodomain. The combined effects of Sdc1 affect the architecture of the multimeric FN fibrils, resulting in an ECM with an organized, aligned fiber arrangement.</p
    corecore