Raúl E Curto

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.

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.
for 2–hyponormal matrix-valued weighted shifts.

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.

In this paper we consider the subnormality of block Toeplitz operators T, where is an n × n matrix-valued function on the unit circle T of the form = Q∗ (Q is a finite Blaschke–Potapov product).
This is related to a matrix-valued version of Halmos’ Problem 5 and the Nakazi–Takahashi Theorem. We ask whether T is either normal or analytic if T is subnormal, where 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 ∗ is not of bounded type

We study a general class of weighted shifts whose weights α are given by αn =√ p n +N p n +D , where p>1 and N and D are parameters so that (N , D) ∈ (−1, 1) × (−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.

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.

Geometrically regular weighted shifts (in short, GRWS) are those with weights alpha(N, D) given by alpha(n)(N, D) = root p(n)+D/p(n)+D, where p > 1 and (N, D) is fixed in the open unit square (-1, 1) x (-1, 1). We study here the zone of pairs (M, P) for which the weight alpha(N,D)/alpha(M,P ) gives rise to a moment infinitely divisible (MID) or a subnormal weighted shift, and deduce immediately the analogous results for product weights alpha(N, D)alpha(M, P), instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS. (c) 2024 Elsevier Inc .All rights reserved.

We consider Hankel and Toeplitz operators on H2(Tn), the Hardy space of the n-torus Tn. Given symbols φ and ψ in L∞(Tn) with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator Hψ,n and the Toeplitz operator Tφ,n to commute. We then extend the study to the more general situation where no assumptions are imposed on φ, and provide new, non-trivial necessary conditions for the commutativity of Hψ,n and Tφ,n. We also show that certain well known commutativity results between Hankel and Toeplitz operators in the one-variable case do not extend to the multivariable setting.

The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics.