Raúl E Curto

We study the connections between operator moment sequences T = (Tn)n∈Z+ of self-adjoint operators on a complex Hilbert space H and the local moment sequences ⟨T x, x⟩ = (⟨Tnx, x⟩)n∈Z+ for arbitrary x ∈ H. We provide necessary and sufficient conditions for solving the operator moment problem on R, and we show that these criteria are au- tomatically valid on compact subsets of 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 Theo- rem 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 recur- sive moment problem are provided, and the support of the associated representing operator-valued measure is described.

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 functionhon the unit ıt disk, there exists a ``boundary function"bhon the unit ıt circle such that the mappingbh↦ his 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."

In this paper we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. Recall that an operator Tφ is hyponormal and [T ∗ φ , Tφ] is a finite rank operator if and only if there exists a finite Blaschke product b in E(φ), where E(φ) := {k ∈ H∞(T) : ∥k∥∞ ≤ 1 and φ − k · ¯φ ∈ H∞(T)}. An analogous set 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∞(T, Mn) and if E(Φ) contains a constant unitary matrix U , then TΦ 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 [10, Conjecture 6.1]. Concretely, assume that Φ ∈ H∞(T, Mn) is such that Φ∗ is of bounded type and TΦ is hyponormal. Then [T ∗ Φ, TΦ] is a finite rank operator if and only if there exists a finite Blaschke–Potapov product in E(eΦ), where eΦ := ˘Φ∗ and ˘Φ(eiθ ) := Φ(e−iθ ).

In this paper we consider the subnormality of block Toeplitz operators TΦ, where Φ is an n × n matrix-valued function on the unit circle T of the form Φ = QΦ∗ (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 TΦ is either normal or analytic if TΦ 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

We first find an explicit formula for the square root of positive 2×2 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 ℓ2(Z2 +). The technique allows us to reduce the study of semi-hyponormality to positivity considerations of a sequence of 2 × 2 scalar matrices. It also requires a specific formula for the square root of 2 × 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 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.

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 quadratically hyponormal matrix-valued weighted shift with two equal weights (excluding the initial weight) is flat. Second, we show that a cubically hyponormal operator-valued weighted shift with two equal weights (possibly including the initial weight) is flat. Next, we introduce a 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 structural decomposition theorem for 2–hyponormal matrix-valued weighted shifts.

We study sequences of bounded operators (Tn)n≥0 on a complex separable Hilbert space H that satisfy a linear recurrence relation of the form Tn+r = A0Tn + A1Tn+1 + · · · + Ar−1Tn+r−1 (for all n ≥ 0), where the coefficients A0, A1, . . . , Ar−1 are pairwise commuting bounded operators on 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 Tn. 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 Ak = ak IH, with ak ∈ 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.

In a 2014 paper, R.E. Curto and S. Yoo proved that a moment matrix 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 ( ), 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.

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 Tn is quasinormal for some n≤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.

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 L\'{e}vy-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.