We study the connections between operator moment sequences 𝒯= (T_n)_n∈ℤ_+ of self-adjoint operators on a complex Hilbert space ℋ and the local moment sequences ⟨𝒯x,x⟩ = (⟨ T_nx,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.

We study a general class of weighted shifts whose weights α are given by α _n = √(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.

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’ Problem 5 and the 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.

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.

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 $$k+1$$ k + 1 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.

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.

The realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated $K-$moment problem where $K$ is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited in \cite{KuLeSp11}, where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained in \cite{CGIK2022} for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results in \cite{KuLeSp11}, allowing to finally embed them in the unified framework for the infinite-dimensional truncated moment problem presented in \cite{CGIK2022}.

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 Krakow 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.

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ({\bf SP}) (related to the Aluthge transform) and the Square Root Problem ({\bf SRP}) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For $p$--atomic measures with $p \le 6$, our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a $7$--atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by G.R. Exner in 2009, and to a recent conjecture formulated by R.E. Curto et al in 2019.

We consider Hankel and Toeplitz operators on H 2 ( T n ), the Hardy space of the n- torus T n . Given symbols phi and psi in L infinity ( T n ) with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator H psi,n and the Toeplitz operator T phi,n to commute. We then extend the study to the more general situation where no assumptions are imposed on phi, and provide new, non-trivial necessary conditions for the commutativity of H psi,n and T phi,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. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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 function h on the unit disk, there exists a "boundary function" bh on the unit circle such that the mapping bh↦h is 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 the notion of slantification of a Hankel operator on the space H^2(𝕋^n) , the Hardy space of n -torus, is introduced. Various properties including hyponormality, isometric behaviour, co-isometric behaviour and compactness of these operators are also studied.

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

We investigate when a linear functional L defined on a linear sub-space B of a unital commutative real algebra A admits an integral representa-tion with respect to a positive Radon measure supported on a closed subset K of the character space of A. We provide a criterion for the existence of such a representation for L when A is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for A not necessarily equipped with a topology.This allows us to extend well-known results on truncated moment problems.

We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation partial derivative(t)u = v Delta u, with initial data u(0) is an element of S(R-n), we first compute the moments s(alpha)(t) of the unique solution u is an element of S(R-n). These moments are polynomials in the time variable, of degree comparable to alpha, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve), which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone, which are also moment sequences. We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation partial derivative(t)u = ax.del u and conduct a similar analysis. Along the way we incorporate several illustrating examples. We show that while partial derivative(t)u = nu Delta u + ax . del u has no explicit solution, the time-dependent moments can be explicitly calculated.

An outline of Jörg Eschmeier’s main mathematical contributions is organized both on a historical perspective, as well as on a few distinct topics. The reader can grasp from our essay the dynamics of spectral theory of commutative tuples of linear operators during the last half century. Some clear directions of future research are also underlined.

We study the time-dependent moments and associated polynomials arising from the partial differential equation $\partial_t f = \nu\Delta f + g\cdot\nabla f + h\cdot f$, and consider in detail the dual equation. For the heat equation we find that several non-negative polynomials which are not sums of squares become sums of squares under the heat equation in finite time. We show that every non-negative polynomial in $\mathbb{R}[x,y,z]_{\leq 4}$ becomes a sum of squares in finite time under the heat equation. We solve the problem of moving atoms under the equation $\partial_t f = g\cdot\nabla f + h\cdot f$ with $f_0 = \mu_0$ being a finitely atomic measure. The time evolution $\mu_t = \sum_{i=1}^k c_i(t)\cdot \delta_{x_i(t)}$ of the atom positions $x_i(t)$ are described by the transport term $g\cdot\nabla$ and the time-dependent coefficients $c_i(t)$ have an explicit solution depending on $x_i(t)$, $h$, and $\mathrm{div}\, g$.

. For 𝑛 ∈ ℕ , we consider the algebraic variety 𝒱 obtained by inter-secting 𝑛+1 algebraic curves of degree 𝑛 in ℝ 2 , when the leading terms of the associated bivariate polynomials are all different. We provide a new proof, based on the Flat Extension Theorem from the theory of truncated moment problems, that the cardinality of 𝒱 cannot exceed ( 𝑛+1 2 ) . In some instances, this provides a slightly better estimate than the one given by Bézout’s Theorem. Our main result contributes to the growing literature on the interplay between linear algebra, operator theory, and real algebraic geometry.

We show that every inner divisor of the operator-valued coordinate function, zIE, is a Blaschke-Potapov factor. We also introduce a notion of operator-valued "rational" function and then show that Δ is two-sided inner and rational if and only if it can be represented as a finite Blaschke-Potapov product; this extends to operator-valued functions the well-known result proved by V.P. Potapov for matrix-valued functions.