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 birth and death processes with rates lambda(n) = q(n)(1-aq(n+1)) and mu (n) = aqn(1-qn), for n >= 0 and 0 < a, q < 1. Using the associated generating functions, we show that the corresponding orthogonal polynomials generalize the little q-Laguerre polynomials. We also provide the minimal solution of the three-term recurrence relation and derive explicit formulas for the convergents of the continued fractions associated with the little q-Laguerre orthogonal polynomials.

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ 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 \(T_n\). 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 $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, 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.

The main goal of this paper is to study some local spectral properties of the generalized derivation operator and the elementary multiplication operator. More precisely, let A, B be bounded operators on a Banach space E, delta(A,B) : X is an element of L(E) -> AX - XB, and Delta(A,B) : X is an element of L(E) -> AXB be the generalized derivation and the elementary multiplication operator, respectively. We investigate the relation between point spectrum of A, B and those of delta(A,B) and Delta(A,B). We also consider the set where delta(A,B) and Delta(A,B) fail to have (SVEP), and Bishop property (beta). Some recent results from the literature are also recaptured.

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.

The main goal of this paper is to study some local spectral properties of the generalized derivation operator and the elementary multiplication operator. More precisely, let A, B be bounded operators on a Banach space E, δ _A, B: X∈ℒ(E)→ AX-XB , and Δ _A, B: X ∈ℒ(E) → AXB be the generalized derivation and the elementary multiplication operator, respectively. We investigate the relation between point spectrum of A, B and those of δ _A, B and Δ _A, B . We also consider the set where δ _A, B and Δ _A, B fail to have (SVEP), and Bishop property (β ) . Some recent results from the literature are also recaptured.

The main goal of this article is to study the class of doubly commuting n-tuples satisfying the wandering subspace property and the Beurling-type theorem or admitting a Wold-type decomposition. It is shown that a doubly commuting n-tuple T=(T-1,& mldr;,T-n) satisfies the wandering subspace property if and only if T(i )does for every 1 <= i <= n. Applications are given in the case of Hilbert spaces of analytic functions, and various recent results are extended. Several results concerning the Beurling-type theorem for doubly commuting n-tuples are also provided. Finally, in the case where Ti admits a Wold-type decomposition for every i, we show a Wold-type decomposition for the doubly commuting tuples (T-1,& mldr;,T-n). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we generalize and refine several operator inequalities involving the joint numerical radius and the joint operator norm of spherical Aluthge transform to generalized spherical Aluthge transforms.Moreover, we investigate the link between nontrivial joint invariant subspaces of the generalized spherical Aluthge transform and the original commuting d -tuples of bounded operators.

In this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in \(\mathbb{R}_+\).

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.

This article is an introductory work to a larger research project devoted to pure, applied and philosophical aspects of dimension theory. It concerns a novel approach toward an alternate dimension theory foundation: the point-dimension theory. For this purpose, historical research on this notion and related concepts, combined with critical analysis and philosophical development proved necessary. Hence, our main objective is to challenge the conventional zero dimension assigned to the point. This reconsideration allows us to propose two new ways of conceiving the notion of dimension, which are the two sides of the same coin. First as an organization; accordingly, we suggest the existence of the Dimensionad, an elementary particle conferring dimension to objects and space-time. The idea of the existence of this particle could possibly adopted as a projection to create an alternative way to unify quantum mechanics and Einstein's general relativity. Secondly, in connection with Boltzmann and Shannon entropies, dimension appears essentially as a comparison between entropies of sets. Thus, we started from the point and succeeded in constructing a point-dimension notion allowing us to extend the principle of box dimension in many directions. More precisely, we introduce the notion of point-extended box dimension in the large framework of topological vector spaces, freeing it from the notion of metric. This general setting permits us to treat the case of finite, infinite and invisible dimensions. This first part of our research project focuses essentially on general properties and is particularly oriented towards establishing a well founded framework for infinite dimension. Among others, one prospect is to test the possibility of using other types of spaces as a setting for quantum mechanics, instead of limiting it to the exclusive Hilbertian framework.

In this paper, we consider the matrix-valued truncated complex moment problem. We notice first that if a truncated complex matrix-valued sequence admits a representing measure, then it is the initial data of an infinite complex matrix-valued sequence verifying some suitable finite-dimensional property. We show that finite-dimensional completion of a truncated data provides a necessary and sufficient condition, and hence a solution, for the matrix-valued truncated complex moment problem. As a consequence, we obtain a matrix generalization of Curto–Fialkow’s result on flat positive extensions of moment matrices.

We devote this paper to Hamburger-type weighted shifts. We give first an affirmative answer to a problem concerning subnormality of the Aluthge transform of Hamburger moment measures with finite support. We also extend the notion of “the jumping flatness property” introduced recently in [11] for Hamburger-type weighted shifts and provide several results related to the representing measure of such weighted shifts.

In this paper, we pursue our investigations on $$\rho _n-$$ contractions on a complex Hilbert space. We associate with a $$\rho _n-$$ contraction an adequate kernel and use it to provide several additional results extending those known for $$\rho -$$ contractions. We also study the behavior of orbits under such operators.

For a non-negative measure μ with p atoms, we study the relation between the Square Root Problem of μ and the problem of subnormality of W̃_μ the Aluthge transform of the associated unilateral weighted shift. We use an approach based on uniquely represented elements in the support of μ *μ . We first show that if W̃_μ is subnormal, then 2p-1≤ card(supp(μ *μ ))≤ [(p-1)^2+6/2] . We rewrite several results known for finitely atomic measure having at most five atoms and give a complete solution for measures six atoms.

This survey aims to give a brief introduction to Wold-type decomposition for some closed range operators satisfying some operator inequalities. As a cornerstone in the theory of the Hardy space, Beurling theorem for unweighted shift is our starting point that we try to transfer to regular operators. Also, several results on left invertible operators close to isometries, as extensions of the Hardy shifts, are listed and extended to the case of regular operators. We define and study the Cauchy dual for such operators by using the Moore-Penrose inverse of closed range operators. The Cauchy dual plays the role of the left inverse in our approach for this general setting.

In this paper, we show that multidimensional K-moment problems are closely related to multi-sequences of Fibonacci type. This motivated us to investigate such sequences. Interesting and useful results are obtained. Consequently, new necessary and sufficient conditions for the truncated multidimensional K-moment problem are provided. In addition, we exploit our results to get an abstract solution to the multidimensional subnormal completion problem.

We show in this paper that a Wold-type decomposition holds for the class of regular operators with regular Moore–Penrose inverse. We also give several examples and investigate various properties of such class of operators.

We investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly.