Raúl E Curto

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 this paper we lay the foundations for a new theory encompassing two 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 (Curto et al., J. Funct. Anal. 278, 108342 2020), R.E. Curto, S.H. Lee and J. Yoon proved that if an operator T is subnormal, left-invertible, and such that
is quasinormal, then T is quasinormal. In subsequent work, (Putnam, C.R., Proc. Amer. Math. Soc. 8, 768–769 1957), 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 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 the study ([5]) of the geometrically regular weighted shifts (GRWS), signed 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 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.

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(Z+2). 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.

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 ⟨Tx,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 automatically 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 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.

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-Fejer Interpolation Problem for matrix rational functions. We then extend the H-infinity-functional calculus to an (H) over bar (infinity) + H-infinity-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' Problem 5; we then establish a matrix-valued version of Abrahamse's Theorem. We also solve a subnormal Toeplitz completion problem of 2 x 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.