arXiv subjects

Raul E. Curto

Publications and source records attributed to Raul E. Curto.

At least 37 records · Page 2Linked to original sources

arXiv2019

Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators

Let $\mathbf{T} \equiv (T_1,\cdots,T_n)$ be a commuting $n$-tuple of operators on a Hilbert space $\mathcal{H}$, and let $T_i \equiv V_i P \; (1 \le i \le n)$ be its canonical joint polar decomposition (i.e., $P:=\sqrt{T_1^*T_1+\cdots+T_n^*T_n}$, $(V_1,\cdots,V_n)$ a joint partial isometry, and $\bigcap_{i=1}^n \ker T_i = \bigcap_{i=1}^n \ker V_i = \ker P)$. \ The spherical Aluthge transform of $\mathbf{T}$ is the (necessarily commuting) $n$-tuple $\hat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\cdots,\sqrt{P}V_n\sqrt{P})$. \ We prove that $σ_T(\hat{\mathbf{T}})=σ_T(\mathbf{T})$, where $σ_T$ denotes Taylor spectrum. \ We do this in two stages: away from the origin we use tools and techniques from criss-cross commutativity; at the origin we show that the left invertibility of $\mathbf{T}$ or $\hat{\mathbf{T}}$ implies the invertibility of $P$. \ As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions.

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arXiv2019

Quasinormality of powers of commuting pairs of bounded operators

We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal $2$-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator $T$ whose square $T^{2}$ is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal $n$-tuples. Fourth, we show that if a $2$-variable weighted shift $W_{\left(α,β\right) }$ and its powers $W_{\left(α,β\right)}^{(2,1)}$ and $W_{\left(α,β\right)}^{(1,2)}$ are all spherically quasinormal, then $W_{\left( α,β\right)}$ may not necessarily be jointly quasinormal. Moreover, it is possible for both $W_{\left(α,β\right)}^{(2,1)}$ and $W_{\left(α,β\right)}^{(1,2)}$ to be spherically quasinormal without $W_{\left(α,β\right)}$ being spherically quasinormal. Finally, we prove that, for $2$-variable weighted shifts, the common fixed points of the toral and spherical Aluthge transforms are jointly quasinormal.

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arXiv2019

The Beurling-Lax-Halmos Theorem for Infinite Multiplicity

In this paper, we consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes the shift-invariant subspaces of vector-valued Hardy spaces. The Beurling-Lax-Halmos Theorem states that a backward shift-invariant subspace is a model space $\mathcal{H}(Δ) \equiv H_E^2 \ominus ΔH_{E}^2$, for some inner function $Δ$. Our first question calls for a description of the set $F$ in $H_E^2$ such that $\mathcal{H}(Δ)=E_F^*$, where $E_F^*$ denotes the smallest backward shift-invariant subspace containing the set $F$. In our pursuit of a general solution to this question, we are naturally led to take into account a canonical decomposition of operator-valued strong $L^2$-functions. Next, we ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? As we know, the kernel of a Hankel operator is shift-invariant, so the above question is equivalent to seeking a solution to the equation $\ker H_Φ^*=ΔH_{E^{\prime}}^2$, where $Δ$ is an inner function satisfying $Δ^* Δ=I_{E^{\prime}}$ almost everywhere on the unit circle $\mathbb{T}$ and $H_Φ$ denotes the Hankel operator with symbol $Φ$. Consideration of the above question on the structure of shift-invariant subspaces leads us to study and coin a new notion of "Beurling degree" for an inner function. We then establish a deep connection between the spectral multiplicity of the model operator and the Beurling degree of the corresponding characteristic function. At the same time, we consider the notion of meromorphic pseudo-continuations of bounded type for operator-valued functions, and then use this notion to study the spectral multiplicity of model operators (truncated backward shifts) between separable complex Hilbert spaces. In particular, we consider the multiplicity-free case.

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arXiv2017

Moment infinitely divisible weighted shifts

We say that a weighted shift $W_α$ with (positive) weight sequence $α: α_0, α_1, \ldots$ is {\it moment infinitely divisible} (MID) if, for every $t > 0$, the shift with weight sequence $α^t: α_0^t, α_1^t, \ldots$ is subnormal. \ Assume that $W_α$ is a contraction, i.e., $0 < α_i \le 1$ for all $i \ge 0$. \ We show that such a shift $W_α$ is MID if and only if the sequence $α$ is log completely alternating. \ This enables the recapture or improvement of some previous results proved rather differently. \ We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.

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arXiv2017

Weyl's Theorem for pairs of commuting hyponormal operators

Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$ \textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbolλ) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbolλ})^*), $$ for every $\boldsymbolλ$ in the Taylor spectrum $σ(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $ω(\mathbf{T})$, satisfies the identity $$ ω(\mathbf{T})=σ(\mathbf{T}) \setminus π_{00}(\mathbf{T}), $$ where $π_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of finite multiplicity. Our method of proof relies on a (strictly $2$-variable) fact about the topological boundary of the Taylor spectrum; as a result, our proof does not hold for $d$-tuples of commuting hyponormal operators with $d>2$.

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arXiv2017

A new approach to the nonsingular cubic binary moment problem

We present an alternative solution to nonsingular cubic moment problems, using techniques that are expected to be useful for higher-degree truncated moment problems. In particular, we apply the theory of recursively determinate moment matrices to deal with a case of rank-increasing moment matrix extensions.

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arXiv2017

Aluthge transforms of 2-variable weighted shifts

We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is preserved under both transforms. Third, we consider whether these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor and Taylor essential spectra of 2-variable weighted shifts behave under the toral and spherical Aluthge transforms; as a special case, we consider the Aluthge transforms of the Drury-Arveson 2-shift. Finally, we briefly discuss the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points for the spherical Aluthge transform.

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arXiv2016

The Division Algorithm in Sextic Truncated Moment Problems

For a degree 2n finite sequence of real numbers $β\equiv β^{(2n)}= \{ β_{00},β_{10}, β_{01},\cdots, β_{2n,0}, β_{2n-1,1},\cdots, β_{1,2n-1},β_{0,2n} \}$ to have a representing measure $μ$, it is necessary for the associated moment matrix $\mathcal{M}(n)$ to be positive semidefinite, and for the algebraic variety associated to $β$, $\mathcal{V}_β \equiv \mathcal{V}(\mathcal{M}(n))$, to satisfy $\operatorname{rank} \mathcal{M}(n)\leq \operatorname{card} \mathcal{V}_β$ as well as the following consistency} condition: if a polynomial $p(x,y)\equiv \sum_{ij}a_{ij}x^{i}y^j$ of degree at most 2n vanishes on $\mathcal{V}_β$, then the Riesz functional $Λ(p) \equiv p(β):=\sum_{ij}a_{ij}β_{ij}=0$. Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic ($n=1$) and quartic ($n=2$) moment problems. Also, positive semidefiniteness, combined with consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the Division Algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.

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arXiv2016

Non-extremal sextic moment problems

Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another necessary condition, consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. However, these conditions are not sufficient for non-extremal sextic or higher-order truncated moment problems. In this paper we settle three key instances of the non-extremal (i.e., r<v) sextic moment problem, as follows: when r=7, positive semidefiniteness, consistency and the variety condition guarantee the existence of a 7-atomic representing measure; when r=8 we construct two determining algorithms, corresponding to the cases v=9 and $v=+ \infty$. \ To accomplish this, we generalize the rank-reduction technique developed in previous work, where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure.

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arXiv2016

A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space

A well known result of C. Cowen states that, for a symbol $φ\in L^{\infty }, \; φ\equiv \bar{f}+g \;\;(f,g\in H^{2})$, the Toeplitz operator $T_{φ}$ acting on the Hardy space of the unit circle is hyponormal if and only if $f=c+T_{\bar{h}}g,$ for some $c\in {\mathbb C}$, $h\in H^{\infty }$, $\left\| h\right\| _{\infty}\leq 1.$ \ In this note we consider possible versions of this result in the {\it Bergman} space case. \ Concretely, we consider Toeplitz operators on the Bergman space of the unit disk, with symbols of the form $$φ\equiv αz^n+βz^m +γ\overline z ^p + δ\overline z ^q,$$ where $α, β, γ, δ\in \mathbb{C}$ and $m,n,p,q \in \mathbb{Z}_+$, $m < n$ and $p < q$. \ By letting $T_φ$ act on vectors of the form $$z^k+c z^{\ell}+d z^r \; \; (k<\ell<r),$$ we study the asymptotic behavior of a suitable matrix of inner products, as $k \rightarrow \infty$. \ As a result, we obtain a sharp inequality involving the above mentioned data: $$ \left|α\right|^2 n^2 + \left|β\right|^2 m^2 - \left|γ\right|^2 p^2 - \left|δ\right|^2 q^2 \ge 2 \left|\bar αβm n - \bar γδp q \right|. $$ This inequality improves a number of existing results, and it is intended to be a precursor of basic necessary conditions for joint hyponormality of tuples of Toeplitz operators acting on Bergman spaces in one or several complex variables.

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arXiv2016

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.

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arXiv2014

Concrete Solution to the Nonsingular Quartic Binary Moment Problem

Given real numbers $β\equiv β^{\left( 4\right) }\colon β_{00}$, $β_{10}$, $β_{01}$, $β_{20}$, $β_{11}$, $ β_{02}$, $β_{30}$, $β_{21}$, $β_{12}$, $β_{03}$, $β_{40}$, $β_{31}$, $β_{22}$, $β_{13}$, $β_{04}$, with $β_{00} >0$, the quartic real moment problem for $β$ entails finding conditions for the existence of a positive Borel measure $μ$, supported in $\mathbb{R}^2$, such that $β_{ij}=\int s^{i}t^{j}\,dμ\;\;(0\leq i+j\leq 4) $. Let $\mathcal{M}(2)$ be the 6 x 6 moment matrix for $β^{(4)}$, given by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=β_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$. In this note we find concrete representing measures for $β^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6-atomic.

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arXiv2014

Berger measure for some transformations of subnormal weighted shifts

A subnormal weighted shift may be transformed to another shift in various ways, such as taking the p-th power of each weight or forming the Aluthge transform. \ We determine in a number of cases whether the resulting shift is subnormal, and, if it is, find a concrete representation of the associated Berger measure, directly for finitely atomic measures, and using both Laplace transform and Fourier transform methods for more complicated measures. \ Alternatively, the problem may be viewed in purely measure-theoretic terms as the attempt to solve moment matching equations such as $(\int t^n \, dμ(t))^2 = \int t^n \, dν(t)$ ($n=0, 1, \ldots$) for one measure given the other.

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arXiv2014

Cubic column relations in truncated moment problems

For the truncated moment problem associated to a complex sequence $γ^{(2n)}=\{γ_{ij}\}_{i,j\in Z_{+},i+j \leq 2n}$ to have a representing measure $μ$, it is necessary for the moment matrix $M(n)$ to be positive semidefinite, and for the algebraic variety $\mathcal{V}_γ$ to satisfy $\operatorname{rank}\;M(n) \leq \;$ card$\;\mathcal{V}_γ$ as well as a consistency condition: the Riesz functional vanishes on every polynomial of degree at most $2n$ that vanishes on $\mathcal{V}_γ$. In previous work with L. Fialkow and M. Möller, the first-named author proved that for the extremal case (rank$\;M(n)=$ card$\;\mathcal{V}_γ$), positivity and consistency are sufficient for the existence of a representing measure. In this paper we solve the truncated moment problem for cubic column relations in $M(3)$ of the form $Z^{3}=itZ+u\bar{Z}$ ($u,t \in \mathbb{R}$); we do this by checking consistency. For $(u,t)$ in the open cone determined by $0 < \left|u\right| < t < 2 \left|u\right|$, we first prove that the algebraic variety has exactly $7$ points and $\operatorname{rank}\;M(3)=7$; we then apply the above mentioned result to obtain a concrete, computable, necessary and sufficient condition for the existence of a representing measure.

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arXiv2013

Abrahamse's Theorem for matrix-valued symbols and subnormal Toeplitz completions

This paper deals with subnormality of Toeplitz operators with matrix-valued symbols and, in particular, with an appropriate reformulation of Halmos's Problem 5: Which subnormal Toeplitz operators with matrix-valued symbols are either normal or analytic? In 1976, M. Abrahamse showed that if $φ\in L^\infty$ is such that $φ$ or $\overlineφ$ is of bounded type and if $T_φ$ is subnormal, then $T_φ$ is either normal or analytic. In this paper we establish a matrix-valued version of Abrahamse's Theorem and then apply this result to solve the following Toeplitz completion problem: Find the unspecified Toeplitz entries of the partial block Toeplitz matrix $$ A:=\begin{bmatrix} T_{\overline b_α} & ?\\?& T_{\overline b_β}\end{bmatrix}\quad\hbox{($α,β\in\mathbb D$)} $$ so that $A$ becomes subnormal, where $b_λ$ is a Blaschke factor of the form $b_λ(z):=\frac{z-λ}{1-\overline λz}$ ($λ\in \mathbb D$).

math.FA↗

arXiv2012

Recursively determined representing measures for bivariate truncated moment sequences

A theorem of Bayer and Teichmann implies that if a finite real multisequence β= β^(2d) has a representing measure, then the associated moment matrix M_d admits positive, recursively generated moment matrix extensions M_(d+1), M_(d+2),... For a bivariate recursively determinate M_d, we show that the existence of positive, recursively generated extensions M_(d+1),...,M_(2d-1) is sufficient for a measure. Examples illustrate that all of these extensions may be required to show that βhas a measure. We describe in detail a constructive procedure for determining whether such extensions exist. Under mild additional hypotheses, we show that M_d admits an extension M_(d+1) which has many of the properties of a positive, recursively generated extension.

math.FA↗

arXiv2011

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 $(I \otimes W_α, W_β\otimes I)$, where $W_α$ and $W_β$ 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 $\mathbf{T}^{(m,n)}:=(T^m_1,T^n_2)$ ($m,n\geq 1$). \

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arXiv2010

A new approach to the 2-variable subnormal completion problem

We study the Subnormal Completion Problem (SCP) for 2-variable weighted shifts. We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion.

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