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Raul E. Curto

Publications and source records attributed to Raul E. Curto.

50 records · Page 3Linked to original sources
arXiv2007

Reconstruction of the Berger measure when the core is of tensor form

Let $\mathfrak{H}_{0}$ denote the class of commuting pairs of subnormal operators on Hilbert space, and let $\mathcal{TC}:=\{\mathbf{T}\in \mathfrak{% H}_{0}:c(\mathbf{T)}$ is of tensor form$\}$, where $c(\mathbf{T})$ is the core of $\mathbf{T}$. We obtain a concrete necessary and sufficient condition for the subnormality of $\mathbf{T}\equiv (T_{1},T_{2})\in \mathcal{TC}$ in terms of $c(\mathbf{T})$, the marginal measures of $T_{1}$ and $T_{2}$, and the weight $α_{01}$.

math.FA↗
arXiv2006

Spectral pictures of 2-variable weighted shifts

We study the spectral pictures of (jointly) hyponormal 2-variable weighted shifts with commuting subnormal components. By contrast with all known results in the theory of subnormal single and 2-variable weighted shifts, we show that the Taylor essential spectrum can be disconnected. We do this by obtaining a simple sufficient condition that guarantees disconnectedness, based on the norms of the horizontal slices of the shift. We also show that for every k >= 1 there exists a k-hyponormal 2-variable weighted shift whose horizontal and vertical slices have 1- or 2-atomic Berger measures, and whose Taylor essential spectrum is disconnected.

math.FA↗
arXiv2006

The extremal truncated moment problem

For a degree 2n real d-dimensional multisequence β^(2n) to have a representing measure, it is necessary for the associated moment matrix M(n) to be positive semidefinite and for the algebraic variety V = V(β) associated to βto satisfy rank M(n) <= card V as well as the following consistency condition: if a polynomial p vanishes on V, then p(β) = 0. We prove that for the extremal case (rank M(n) = card V), positivity of M(n) and consistency are sufficient for the existence of a (unique, rank M(n)-atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of M(n).

math.FA↗
arXiv2006

Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not. Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\infty; this further stretches the gap between the classes H_1 and H_\infty. Finally, we prove that there exists a large class of 2-variable weighted shifts (T_1,T_2) (namely those pairs in H_0 whose cores are of tensor form) for which the subnormality of (T_1^2,T_2) and (T_1,T_2^2) does imply the subnormality of (T_1,T_2).

math.FA↗
arXiv2006

Propagation phenomena for hyponormal 2-variable weighted shifts

We study the class of hyponormal 2-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights leads to horizontal or vertical flatness, in a way that resembles the situation for 1-variable weighted shifts. In 1-variable, it is well known that flat weighted shifts are necessarily subnormal (with finitely atomic Berger measures). By contrast, we exhibit a large collection of flat (i.e., horizontally and vertically flat) 2-variable weighted shifts which are hyponormal but not subnormal. Moreover we completely characterize the hyponormality and subnormality of symmetrically flat contractive 2-variable weighted shifts.

math.FA↗
arXiv2006

Generalized Browder's and Weyl's Theorems for Banach space operators

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T. We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of σ(T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f \in H(T), the space of functions analytic on an open neighborhood of σ(T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f \in H(σ(T)).

math.FA↗
arXiv2005

k-hyponormality of finite rank perturbations of unilateral weighted shifts

In this paper we explore finite rank perturbations of unilateral weighted shifts $W_α$. First, we prove that the subnormality of $W_α$ is never stable under nonzero finite rank pertrubations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients of $D_n(s):=\text{det} P_n [(W_α+sW_α^2)^*, W_α+s W_α^2] P_n$ are nonnegative, for every $n\ge 0$, where $P_n$ denotes the orthogonal projection onto the basis vectors $\{e_0,...,e_n\}$. Finally, for $α$ strictly increasing and $W_α$ 2-hyponormal, we show that for a small finite-rank perturbation $α^\prime$ of $α$, the shift $W_{α^\prime}$ remains quadratically hyponormal.

math.FA↗
arXiv2005

Truncated K-moment problems in several variables

Let $β\equivβ^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(β)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure μ, which is r-atomic, with supp μ$ equal to $\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of $\mathcal{M}(n+1)$. Further, βhas an r-atomic (minimal) representing measure supported in a semi-algebraic set $K_{\mathcal{Q}}$ subordinate to a family $\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ if and only if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving extension $\mathcal{M}(n+1)$ for which the associated localizing matrices $\mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ are positive semidefinite $(1\leq i\leq m)$; in this case, μ(as above) satisfies supp μ\subseteq K_{\mathcal{Q}}$, and μhas precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ atoms in $\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$, $1\leq i\leq m$.

math.FA↗
arXiv2005

Solution of the truncated hyperbolic moment problem

Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $β\equivβ^{2n)}=\{β_{ij}\}_{i,j\geq0,i+j\leq2n}$, with $β_{00}>0$, the truncated Q-hyperbolic moment problem for βentails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x,y)=0, such that $β_{ij}=\int y^{i}x^{j} dμ(0\leq i+j\leq2n)$. We prove that βadmits a Q-representing measure μ(as above) if and only if the associated moment matrix $\mathcal{M}(n)(β)$ is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety $\mathcal{V}(β)$ associated to βsatisfies $card\mathcal{V}(β)\geq\rank\mathcal{M}(n)(β)$. In this case, $rank\mathcal{M}(n)\leq2n+1$; if $rank\mathcal{M}(n)\leq2n$, then βadmits a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if $rank\mathcal{M}(n)=2n+1$, then βadmits a Q-representing measure μsatisfying $2n+1\leqcard suppμ\leq2n+2$.

math.FA↗
arXiv2005

k-hyponormality of multivariable weighted shifts

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k>=1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.

math.FA↗
arXiv2002

Weyl's theorem, a-Weyl's theorem, and local spectral theory

We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl's theorem and $a$-Weyl's theorem. We show that if $T$ or $T^{\ast}$ has SVEP and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(σ(T))$. When $T^{\ast}$ has SVEP, $T$ is transaloid and $T$ is $a$-isoloid, then $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(σ(T))$. We also prove that if $T$ or $T^{\ast}$ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.

math.FA↗
arXiv2002

A duality proof of Tchakaloff's theorem

Tchakaloff's Theorem establishes the existence of a quadrature rule of prescribed degree relative to a positive, compactly supported measure that is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}^{d}$. Subsequent extensions were obtained by Mysovskikh and by Putinar. We provide new proofs and partial extensions of these results, based on duality techniques utilized by Stochel. We also obtain new uniqueness criteria in the Truncated Complex Moment Problem.

math.FA↗