arXiv subjects

Raul E. Curto

Publications and source records attributed to Raul E. Curto.

At least 19 recordsLinked to original sources

arXiv2026

Model Theory for Operators Related to Square Roots of Normal Operators

In this paper we prove that a square root of a cyclic normal operator is unitarily equivalent to a block multiplication operator on a vector-valued Lebesgue space with a $2$--normal symbol. In addition, we show that a cyclic operator which admits a cyclic $2$--normal extension is unitarily equivalent to a block multiplication operator on a corresponding vector-valued Hardy space with $2$--normal symbol. We consider the existence of bounded point evaluations in the vector-valued Hardy space setting; as an application, we prove that an operator admitting a cyclic $2$--normal extension has nontrivial invariant subspaces. We also study uniqueness for minimal $n$--normal extensions of operators that have cyclic $2$--normal extensions.

math.FA↗

arXiv2026

Operator-valued rational functions

In this paper we show that every inner divisor of the operator-valued coordinate function, $zI_E$, is a Blaschke-Potapov factor. We also introduce a notion of operator-valued "rational" function and then show that $Δ$ is two-sided inner and rational if and only if it can be represented as a finite Blaschke-Potapov product; this extends to operator-valued functions the well-known result proved by V.P. Potapov for matrix-valued functions.

math.FA↗

arXiv2026

The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ({\bf SP}) (related to the Aluthge transform) and the Square Root Problem ({\bf SRP}) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For $p$--atomic measures with $p \le 6$, our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a $7$--atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by G.R. Exner in 2009, and to a recent conjecture formulated by R.E. Curto et al in 2019.

math.FA↗

arXiv2026

Propagation Phenomena for Operator-Valued Weighted Shifts

This paper is devoted to the study of propagation phenomena for $2$--hyponormal, quadratically hyponormal, and cubically hyponormal operator-valued weighted shifts. \ First, we show that every {\it quadratically} hyponormal matrix-valued weighted shift with two equal weights ({\it excluding the initial weight}) is flat. \ Second, we show that a {\it cubically} hyponormal operator-valued weighted shift with two equal weights ({\it possibly including the initial weight}) is flat. \ Next, we introduce a {\it local flatness} notion for matrix-valued weighted shifts. \ We prove that $2$--hyponormal (in particular, subnormal) matrix-valued weighted shifts satisfy this stronger propagation phenomenon. \ As a result, we prove a {\it structural decomposition theorem} for $2$--hyponormal matrix-valued weighted shifts.

math.FA↗

arXiv2026

The Local Operator Moment Problem on $\mathbb{R}$

We study the connections between operator moment sequences ${\mathcal T}=\displaystyle(T_n)_{n\in\mathbb{Z}_+}$ of self-adjoint operators on a complex Hilbert space $\mathcal{H}$ and the local moment sequences $\langle{\mathcal T}x,x\rangle = (\langle T_nx,x\rangle)_{n\in\mathbb{Z}_+}$ for arbitrary $x\in \mathcal{H}$. We provide necessary and sufficient conditions for solving the operator moment problem on $\mathbb{R}$, and we show that these criteria are automatically valid on compact subsets of $\mathbb{R}$. Applications of the compact case are used to study subnormal operator weighted shifts. A Stampfli-type propagation theorem for subnormal operator weighted shifts is also established. In addition, we discuss the validity of Tchakaloff's Theorem for operator moment sequences with compact support. In the case of a recursively generated sequence of self-adjoint operators, necessary and sufficient conditions for an affirmative answer to the operator recursive moment problem are provided, and the support of the associated representing operator-valued measure is described.

math.FA↗

arXiv2026

Geometrically regular weighted shifts

We study a general class of weighted shifts whose weights $α$ are given by $α_n = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $N$ and $D$ are parameters so that $(N,D) \in (-1, 1)\times (-1, 1)$. Some few examples of these shifts have appeared previously, usually as examples in connection with some property related to subnormality. In sectors nicely arranged in the unit square in $(N,D)$, we prove that these geometrically regular weighted shifts exhibit a wide variety of properties: moment infinitely divisible, subnormal, $k$- but not $(k+1)$-hyponormal, or completely hyperexpansive, and with a variety of well-known functions (such as Bernstein functions) interpolating their weights squared or their moment sequences. They provide subshifts of the Bergman shift with geometric, not linear, spacing in the weights which are moment infinitely divisible. This new family of weighted shifts provides a useful addition to the library of shifts with which to explore new definitions and properties.

math.FA↗

arXiv2026

Semi-hyponormality of commuting pairs of Hilbert space operators

We first find an explicit formula for the square root of positive $2 \times 2$ operator matrices with commuting entries, and then use it to define and study semi-hyponormality for commuting pairs of Hilbert space operators. \ For the well-known $3$--parameter family $W_{(α,β)}(a,x,y)$ of $2$--variable weighted shifts, we completely identify the parametric regions in the open unit cube where $W_{(α,β)}(a,x,y)$ is subnormal, hyponormal, semi-hyponormal, and weakly hyponormal. As a result, we describe in detail concrete sub-regions where each property holds. For instance, we identify the specific sub-region where weak hyponormality holds but semi-hyponormality does not hold, and vice versa. \ To accomplish this, we employ a new technique emanating from the homogeneous orthogonal decomposition of $\ell^2(\mathbb{Z}_+^2)$. The technique allows us to reduce the study of semi-hyponormality to positivity considerations of a sequence of $2 \times 2$ scalar matrices. It also requires a specific formula for the square root of $2 \times 2$ scalar and operator matrices, and we obtain that along the way. As an application of our main results, we show that the Drury-Arveson shift is {\it not} semi-hyponormal. Taken together, the new results offer a sharp contrast between the above-mentioned properties for unilateral weighted shifts and their $2$--variable counterparts.

math.FA↗

arXiv2026

Circle companions of Hardy spaces of the unit disk

This paper gives a complete answer to the following problem: Find the circle companion of the Hardy space of the unit disk with values in the space of all bounded linear operators between two separable Hilbert spaces. Classically, the problem asks whether for each function $h$ on the unit {\it disk}, there exists a ``boundary function" $bh$ on the unit {\it circle} such that the mapping $bh\mapsto h$ is an isometric isomorphism between Hardy spaces of the unit circle and the unit disk with values in some Banach space. For the case of bounded linear operator-valued functions, we construct a Hardy space of the unit circle such that its elements are SOT measurable, and their norms are integrable: indeed, this new space is isometrically isomorphic to the Hardy space of the unit disk via a ``strong Poisson integral."

math.CV↗

arXiv2026

A Combinatorial Formula for Recursive Operator Sequences and Applications

We study sequences of bounded operators $(T_n)_{n \ge 0}$ on a complex separable Hilbert space $\mathcal{H}$ that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients $A_0, A_1, \dots, A_{r-1}$ are pairwise commuting bounded operators on $\mathcal{H}$. \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for $T_n$. As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

math.FA↗

arXiv2026

Hyponormal block Toeplitz operators with finite rank self-commutators

In this paper, we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. \ Recall that an operator $T_φ$ is hyponormal and $[T_φ^{*}, T_φ]$ is a finite rank operator if and only if there exists a finite Blaschke product $b$ in $\mathcal{E}(φ)$, where $$ \mathcal{E}(φ) := \{k \in H^\infty(\mathbb{T}): \left\|k\right\|_\infty \le 1 \textrm{ and } φ-k\cdot \barφ \in H^\infty(\mathbb{T})\}. $$ An analogous set $\mathcal{E}(Φ)$ can be defined for a matrix-valued symbol $Φ$. \ In the block Toeplitz operator case, we first establish that if a symbol $Φ$ is in $L^\infty(\mathbb{T}, M_n)$ and if $\mathcal{E}(Φ)$ contains a constant unitary matrix $U$, then $T_Φ$ is normal. \ We then obtain a suitable converse, under a mild assumption on the symbol. \ Next, we provide a partial answer to a conjecture recently posed by R.E. Curto, I.S. Hwang, and W.Y. Lee. \ Concretely, assume that $Φ\in H^{\infty}(\mathbb{T}, M_n)$ is such that $Φ^{\ast}$ is of bounded type and $T_Φ$ is hyponormal. \ Then $[T_Φ^{\ast}, T_Φ]$ is a finite rank operator if and only if there exists a finite Blaschke-Potapov product in $\mathcal{E}(\widetildeΦ)$, where $\widetildeΦ:=\breveΦ^*$ and $\breveΦ(e^{iθ}):=Φ(e^{-iθ})$.

math.FA↗

arXiv2026

Subnormal block Toeplitz operators

In this paper we consider the subnormality of block Toeplitz operators $T_Φ$, where $Φ$ is an $n\times n$ matrix-valued function on the unit circle $\mathbb T$ of the form $$ Φ=QΦ^* \quad \hbox{($Q$ is a finite Blaschke--Potapov product).} $$ This is related to a matrix-valued version of Halmos's Problem 5 and Nakazi-Takahashi Theorem. We ask whether $T_Φ$ is either normal or analytic if $T_Φ$ is subnormal, where $Φ$ is of the above form. We give answers to this problem for different cases of the symbol. Moreover, we provide a sufficient condition for the answer to be affirmative when $Φ^*$ is not of bounded type.

math.FA↗

arXiv2024

The Truncated Moment Problem for Unital Commutative R-Algebras

We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation w.r.t. a positive Radon measure supported on a closed subset $K$ of the character space of $A$. We provide a criterion for the existence of such a representation for $L$ when $A$ is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for $A$ not necessarily equipped with a topology. This allows us to extend well-known classical results on truncated moment problems.

math.FA↗

arXiv2021

Time-dependent moments from the heat equation and a transport equation

We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation $\partial_t u = νΔu$, with initial data $u_0 \in\mathcal{S}(\mathbb{R}^n)$, we first compute the moments $s_α(t)$ of the unique solution $u \in \mathcal{S}(\mathbb{R}^n)$. These moments are polynomials in the time variable, of degree comparable to $α$, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve) which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone which are also moment sequences. \ We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation $\partial_t u = ax \cdot \nabla u$, and conduct a similar analysis. Along the way we incorporate several illustrating examples.

math.AP↗

arXiv2021

Conditional positive definiteness as a bridge between k-hyponormality and n-contractivity

For sequences $α\equiv \{α_n\}_{n=0}^{\infty}$ of positive real numbers, called weights, we study the weighted shift operators $W_α$ having the property of moment infinite divisibility ($\mathcal{MID}$); that is, for any $p > 0$, the Schur power $W_α^p$ is subnormal. We first prove that $W_α$ is $\mathcal{MID}$ if and only if certain infinite matrices $\log M_γ(0)$ and $\log M_γ(1)$ are conditionally positive definite (CPD). Here $γ$ is the sequence of moments associated with $α$, $M_γ(0),M_γ(1)$ are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of $W_α$, and $\log$ is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between $k$--hyponormality and $n$--contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift $W_α$ is $\mathcal{MID}$ if and only if for all $p>0$, $M_γ^p(0)$ and $M_γ^p(1)$ are CPD.

math.FA↗

arXiv2020

Moment Infinite Divisibility of Weighted Shifts: Sequence Conditions

We consider weighted shift operators having the property of moment infinite divisibility; that is, for any $p > 0$, the shift is subnormal when every weight (equivalently, every moment) is raised to the $p$-th power. By reconsidering sequence conditions for the weights or moments of the shift, we obtain a new characterization for such shifts, and we prove that such shifts are, under mild conditions, robust under a variety of operations and also rigid in certain senses. In particular, a weighted shift whose weight sequence has a limit is moment infinitely divisible if and only if its Aluthge transform is. We also consider back-step extensions, subshifts, and completions.

math.FA↗

arXiv2020

Polynomial embeddings of unilateral weighted shifts into $2$-variable weighted shifts

Given a bounded sequence ωof positive numbers and its associated unilateral weighted shift W_ω acting on the Hilbert space \ell^2(\mathbb{Z}_+), we consider natural representations of W_ω as a 2-variable weighted shift, acting on \ell^2(\mathbb{Z}_+^2). Alternatively, we seek to examine the various ways in which the sequence ωcan give rise to a 2-variable weight diagram. Our best (and more general) embedding arises from looking at two polynomials p and q nonnegative on a closed interval I in R_+ and the double-indexed moment sequence \{\int p(r)^k q(r)^{\ell} dσ(r)\}_{k,\ell \in \mathbb{Z}_+}, where W_ω is assumed to be subnormal with Berger measure σsuch that \supp \; σ\subseteq I; we call such an embedding a (p,q)-embedding of W_ω. We prove that every (p,q)-embedding of a subnormal weighted shift W_ω is (jointly) subnormal, and we explicitly compute its Berger measure. We apply this result to answer three outstanding questions: (i) Can the Bergman shift A_2 be embedded in a subnormal 2-variable spherically isometric weighted shift W_{(α,β)}? If so, what is the Berger measure of W_{(α,β)}? (ii) Can a contractive subnormal unilateral weighted shift be always embedded in a spherically isometric 2-variable weighted shift? (iii) Does there exist a hyponormal 2-variable weighted shift Θ(W_ω) (where Θ(W_ω) denotes the classical embedding of a hyponormal unilateral weighted shift W_ω) such that some integer power of Θ(W_ω) is not hyponormal? As another application, we find an alternative way to compute the Berger measure of the Agler j-th shift A_{j} (j\geq 2). Our research uses techniques from the theory of disintegration of measures, Riesz functionals, and the functional calculus for the columns of the moment matrix associated to a polynomial embedding.

math.FA↗

arXiv2020

Solution of the Reconstruction-of-the-Measure Problem for Canonical Invariant Subspaces

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts $W_{(α,β)}$, when the initial data are given as the Berger measure of the restriction of $W_{(α,β)}$ to a canonical invariant subspace, together with the marginal measures for the 0-th row and 0-th column in the weight diagram for $W_{(α,β)}$. We prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, we give a concrete formula for the Berger measure of $W_{(α,β)}$. Our strategy is to build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspaces of $\ell^2(\mathbb{Z}_+^2)$.

math.FA↗

arXiv2020

The Spectral Picture and Joint Spectral Radius of the Generalized Spherical Aluthge Transform

For an arbitrary commuting $d$--tuple $\bT$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\dbT$ and we prove that the spectral radius of $\bT$ can be calculated from the norms of the iterates of $\dbT$. \ Let $\bm{T} \equiv (T_1,\cdots,T_d)$ be a commuting $d$--tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let $P:=\sqrt{T_1^*T_1+\cdots+T_d^*T_d}$, and let $$ \left( \begin{array}{c} T_1 \\ \vdots \\ T_d \end{array} \right) = \left( \begin{array}{c} V_1 \\ \vdots \\ V_d \end{array} \right) P $$ be the canonical polar decomposition, with $(V_1,\cdots,V_d)$ a (joint) partial isometry and $$ \bigcap_{i=1}^d \ker T_i=\bigcap_{i=1}^d \ker V_i=\ker P. $$ \medskip For $0 \le t \le 1$, we define the generalized spherical Aluthge transform of $\bm{T}$ by $$ Δ_t(\bm{T}):=(P^t V_1P^{1-t}, \cdots, P^t V_dP^{1-t}). $$ We also let $\left\|\bm{T}\right\|_2:=\left\|P\right\|$. \ We first determine the spectral picture of $Δ_t(\bm{T})$ in terms of the spectral picture of $\bm{T}$; in particular, we prove that, for any $0 \le t \le 1$, $Δ_t(\bm{T})$ and $\bm{T}$ have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. \ We then study the joint spectral radius $r(\bm{T})$, and prove that $r(\bm{T})=\lim_n\left\|Δ_t^{(n)}(\bm{T})\right\|_2 \,\, (0 < t < 1)$, where $Δ_t^{(n)}$ denotes the $n$--th iterate of $Δ_t$. \ For $d=t=1$, we give an example where the above formula fails.

math.FA↗