arXiv subjects

Raúl E. Curto

Publications and source records attributed to Raúl E. Curto.

8 recordsLinked to original sources

arXiv2026

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

We study the time-dependent moments and associated polynomials arising from the partial differential equation $\partial_t f = νΔf + g\cdot\nabla f + h\cdot f$, and consider in detail the dual equation. For 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. We show that every non-negative polynomial in $\mathbb{R}[x,y,z]_{\leq 4}$ becomes a sum of squares in finite time under the heat equation. We solve the problem of moving atoms under the equation $\partial_t f = g\cdot\nabla f + h\cdot f$ with $f_0 = μ_0$ being a finitely atomic measure. The time evolution $μ_t = \sum_{i=1}^k c_i(t)\cdot δ_{x_i(t)}$ of the atom positions $x_i(t)$ are described by the transport term $g\cdot\nabla$ and the time-dependent coefficients $c_i(t)$ have an explicit solution depending on $x_i(t)$, $h$, and $\mathrm{div}\, g$.

math.FA↗

arXiv2026

Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts

In the study of the geometrically regular weighted shifts (GRWS) -- see [5] -- signed power representing measures (which we call Berger-type charges) played an important role. Motivated by their utility in that context, we establish a general theory for Berger-type charges. We give the first result of which we are aware showing that k-hyponormality alone (as opposed to subnormality) yields measure/charge-related information. More precisely, for signed countably atomic measures with a decreasing sequence of atoms we prove that k-hyponormality of the associated shift forces positivity of the densities of the largest k+1 atoms. Further, for certain completely hyperexpansive weighed shifts, we exhibit a Berger-type charge representation, in contrast (but related) to the classical Lévy-Khinchin representation. We use Berger-type charges to investigate when a non-subnormal GRWS weighted shift may be scaled to become conditionally positive definite, and close with an example indicating a distinction between the study of moment sequences and the study of weighted shifts.

math.FA↗

arXiv2026

Classes of operators related to subnormal operators

In this paper we attempt to lay the foundations for a theory encompassing some natural extensions of the class of subnormal operators, namely the $n$--subnormal operators and the sub-$n$--normal operators. We discuss inclusion relations among the above mentioned classes and other related classes, e.g., $n$--quasinormal and quasi-$n$--normal operators. We show that sub-$n$--normality is stronger than $n$--subnormality, and produce a concrete example of a $3$--subnormal operator which is not sub-$2$--normal. In \cite{CU1}, R.E. Curto, S.H. Lee and J. Yoon proved that if an operator $T$ is subnormal, left-invertible, and such that $T^n$ is quasinormal for some $n \le 2$, then $T$ is quasinormal. in \cite{JS}, P.Pietrzycki and J. Stochel improved this result by removing the assumption of left invertibility. In this paper we consider suitable analogs of this result for the case of operators in the above-mentioned classes. In particular, we prove that the weight sequence of an $n$--quasinormal unilateral weighted shift must be periodic with period at most $n$.

math.FA↗

arXiv2026

The Role of Gröbner Bases in the Study of Extremal Truncated Moment Problems

In a 2014 paper, R.E. Curto and S. Yoo proved that a moment matrix $M(3)$ with specific harmonic polynomials as column relations admits a representing measure if and only if a condition at the level of moments holds. \ In this paper, we generalize the 2014 result to arbitrary moment matrices $M(k)$ ($k \in \mathbb{Z}_{+}$), with column relations given by general harmonic polynomials. \ We accomplish this by proving that the Gröbner basis for the ideal generated by a finite variety associated with the moment matrix provides all the necessary column relations for the matrix as well as a suitable condition on the moments, which is equivalent to the existence of a representing measure.

math.FA↗

arXiv2023

$\mathcal{MID}$ and safe quotients for GRWS

Geometrically regular weighted shifts (in short, GRWS) are those with weights $α(N,D)$ given by $α_n (N,D) = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $(N,D)$ is fixed in the open unit square $ (-1, 1)\times (-1, 1)$. We study here the zone of pairs $ (M,P)$ for which the weight $\frac{α(N,D) }{ α(M,P) }$ gives rise to a moment infinitely divisible ($ \mathcal {MID}$) or a subnormal weighted shift, and deduce immediately the analogous results for product weights $α(N,D) α(M,P)$, instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.

math.FA↗

arXiv2023

The realizability problem as a special case of the infinite-dimensional truncated moment problem

The realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated $K-$moment problem where $K$ is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited in \cite{KuLeSp11}, where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained in \cite{CGIK2022} for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results in \cite{KuLeSp11}, allowing to finally embed them in the unified framework for the infinite-dimensional truncated moment problem presented in \cite{CGIK2022}.

math.PR↗

arXiv2022

Jörg Eschmeier's mathematical work

An outline of Jörg Eschmeier's main mathematical contributions is organized both on a historical perspective, as well as on a few distinct topics. The reader can grasp from our essay the dynamics of spectral theory of commutative tuples of linear operators during the last half century. Some clear directions of future research are also underlined.

math.HO↗

arXiv2016

Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory

In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. \ We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. \ We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. \ We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fej\' er Interpolation Problem for matrix rational functions. \ We then extend the $H^\infty$-functional calculus to an $\overline{H^\infty}+H^\infty$-functional calculus for the compressions of the shift. \ Next, we consider the subnormality of Toeplitz operators with matrix-valued bounded type symbols and, in particular, the matrix-valued version of Halmos's Problem 5; we then establish a matrix-valued version of Abrahamse's Theorem. \ We also solve a subnormal Toeplitz completion problem of $2\times 2$ partial block Toeplitz matrices. \ Further, we establish a characterization of hyponormal Toeplitz pairs with matrix-valued bounded type symbols, and then derive rank formulae for the self-commutators of hyponormal Toeplitz pairs.

math.FA↗