Flat extensions of positive moment matrices: recursively generated relations

We develop new computational tests for existence and uniqueness of representing measures in the Truncated Complex Moment Problem: γij = Z zizj d (0 i + j 2n):(TCMP) We characterize the existence of nitely atomic representing measures in terms of positivity and extension properties of the moment matrix M(n)(γ) associated with γ γ(2n): γ00; : : : ; γ0;2n; : : : ; γ2n;0, γ00> 0 (Theorem 1.5). We study conditions for flat (i.e., rank-preserving) extensions M(n + 1) of M(n) 0; each such extension corresponds to a distinct rank M(n)-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consis-tency, and normal consistency, we reduce the existence problem for minimal representing measures to the solubility of small systems of multivariable alge

n-Tuples of operators satisfying σT(AB)=σT(BA)

AbstractFor “criss-cross commuting” tuples A and B of Banach space operators we give two sufficient conditions for the spectral equality σT(AB)=σT(BA)

Subnormality for arbitrary powers of 2-variable weighted shifts whose restrictions to a large invariant subspace are tensor products

The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. \ We study LPCS within the class of commuting 2-variable weighted shifts $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invariant subspace generated by all vectors $e_{(k_1,k_2)}$ with $k_1,k_2 \geq 1$; we say that $c(\mathbf{T})$ is of \textit{tensor form} if it is unitarily equivalent to a shift of the form $(I \otimes W_\alpha, W_\beta \otimes I)$, where $W_\alpha$ and $W_\beta$ are subnormal unilateral weighted shifts. \ Given a 2-variable weighted shift $\mathbf{T}$ whose core is of tensor form, we prove that LPCS is solvable for $\mathbf{T}$ if and only if LPCS is solvable for any power $\mathbf{T}^{(m,n)}:=(T^m_1,T^n_2)$ ($m,n\geq 1$). \Comment: article in pres

Toral and spherical Aluthge transforms of 2-variable weighted shifts

We introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties. Next, we study the class of spherically quasinormal $2$-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. Finally, we briefly discuss the relation between spherically quasinormal and spherically isometric 2-variable weighted shifts.Comment: Accepted for publication by C.R. Acad. Sci. Paris, Ser. I (2016

Time-dependent moments from partial differential equations and the time-dependent set of atoms

We study the time-dependent moments $s_\alpha(t) = \int x^\alpha\cdot f(x,t)~\mathrm{d} x$ of the solution $f$ of the partial differential equation $\partial_t f = \nu\Delta f + g\cdot\nabla f + h\cdot f$ with initial Schwartz function data $f_0\in\mathcal{S}(\mathbb{R}^n)$. At first we describe the dual action on the polynomials, i.e., the time-evolution of $f$ is completely moved to the polynomial side $s_\alpha(t) = \int p(x,t)\cdot f_0(x)~\mathrm{d} x$. We investigate the special case of the heat equation. We find that several non-negative polynomials which are not sums of squares become sums of squares under the heat equation in finite time. Finally, we solve the problem of moving atoms under the equation $\partial_t f = g\cdot\nabla f + h\cdot f$ with $f_0 = \mu_0$ being a finitely atomic measure. We find that in the time evolution $\mu_t = \sum_{i=1}^k c_i(t)\cdot \delta_{x_i(t)}$ the atom positions $x_i(t)$ are governed only by the transport term $g\cdot\nabla$ and that the time-dependent coefficients $c_i(t)$ have an analytic solution depending on $x_i(t)$

Subnormality of Bergman-like weighted shifts

Abstract For a, b, c, d 0 with ad − bc > 0, we consider the unilateral weighted shift S (a, b, c, d) with weights α n := an+b cn+d (n 0). Using Schur product techniques, we prove that S (a, b, c, d) is always subnormal; more generally, we establish that for every p 1, all p-subshifts of S (a, b, c, d

N-terminal mutants of human apolipoprotein A-I : Structural perturbations associated to protein misfolding

Since the early description of different human apolipoprotein A-I variants associated to amyloidosis, the reason that determines its deposition inducing organ failure has been under research. To shed light into the events associated to protein aggregation, we studied the effect of the structural perturbations induced by the replacement of a Leucine in position 60 by an Arginine as it occurs in the natural amyloidogenic variant (L60R). Circular dichroism, intrinsic fluorescence measurements and assays of binding to ligands indicate that L60R is more unstable, more sensitive to proteolysis and interacts with sodium dodecyl sulfate (a model of negative lipids) more than the protein with the native sequence and other natural variant tested, involving a replacement of a Trytophan by and Arginine in the amino acid 50 (W50R). In addition, the small structural rearrangement observed under physiological pH leads to the release of tumor necrosis factor α and interleukin-lβ from a model of macrophages. Our results strongly suggest that the chronic disease may be a consequence of the loss in the native conformation which alters the equilibrium among native and cytotoxic proteins conformation.Instituto de Investigaciones Bioquímicas de La PlataFacultad de Ciencias MédicasInstituto de Investigaciones Fisicoquímicas Teóricas y Aplicada