Operators In Spaces Research Articles

A Note on the Differential Operator on Weighted Fock Spaces

The differential operator Df = f′ is a canonical example of unbounded operators in many Banach spaces. In this study, we identify a class of weighted Fock spaces in which the differential operator is bounded. Then, we obtain some properties of the differential operator on these spaces in terms of its C*-algebraic structure in the one and two-dimensional cases.

Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

\( \newcommand\norm[1]{\left\lVert#1\right\rVert}\newcommand\normx[1]{\left\Vert#1\right\Vert} \)In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping $f$$$\norm{(1\otimes B-A\otimes 1)^{-1}[\operatorname{exp}(1\otimes B)-\operatorname{exp}(A\otimes 1)]- \operatorname{exp}\left(\frac{A\otimes 1+1\otimes B}{2}\right)}$$$$\leqslant \norm{1\otimes B-A\otimes 1}^{2}\frac{\norm{f''}_{I,+\infty}}{24}.$$

A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities

Abstract The primary goal of this research is to investigate the approximate numerical solution of variational inequalities using quasimonotone operators in infinite-dimensional real Hilbert spaces. In this study, the sequence obtained by the proposed iterative technique for solving quasimonotone variational inequalities converges strongly toward a solution due to the viscosity-type iterative scheme. Furthermore, a new technique is proposed that uses an inertial mechanism to obtain strong convergence iteratively without the requirement for a hybrid version. The fundamental benefit of the suggested iterative strategy is that it substitutes a monotone and non-monotone step size rule based on mapping (operator) information for its Lipschitz constant or another line search method. This article also provides a numerical example to demonstrate how each method works.

Generalized Cauchy–Schwarz Inequalities and A-Numerical Radius Applications

The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes a positive semidefinite operator in a complex Hilbert space. Some of our novel A-numerical radius inequalities expand upon the existing literature on numerical radius inequalities with Hilbert space operators, which are important tools in functional analysis. We use techniques from semi-Hilbert space theory to prove our results and highlight some applications of our findings.

Maneuvering into the future: Open-Architecture Data Repository (OADR) prototype: Towards civil and commercial space traffic coordination

In 2018, the US National Space Council released Space Policy Directive-3 (SPD-3), which instructed relevant US government agencies to begin re-assigning many aspects of space traffic management (STM) and space traffic coordination (STC) serving non-military US space operators. This was identified as a ‘whole of government’ approach, and the organization identified to lead many of the efforts was the National Oceanic & Atmospheric Administration (NOAA) Office of Space Commerce (OSC), in the US Department of Commerce (DOC). The US Congress, in Public Law 116–93, Consolidated Appropriations Act, 2020, legislation directed that the Secretary of Commerce contract with the National Academy of Public Administration (NAPA) to study the Trump Administration's proposal. The NAPA study affirmed OSC would be best suited. Congress also mandated that OSC accomplish an important first step: assemble an Open Architecture Data Repository11Since this paper was first presented, the name for the effort, the Open Architecture Data Repository (OADR), has been changed to TraCSS (Traffic Coordination System for Space). As the focus of this work is on the prototype demonstration system, we will retain the use of OADR in this paper. (OADR) prototype to demonstrate the efficacy of the approach and architecture.OSC, along with The Aerospace Corporation, MITER, MIT Lincoln Laboratories and the University of Texas-Austin, architected and assembled a working prototype in 2021. The team obtained two-months of data from the US Space Surveillance Network (SSN), along with data from commercial space surveillance data providers and satellite operator ephemerides. Using the prototype, the authors analyzed the data, identified potential satellite conjunctions, calculated probabilities of collision, and generated Conjunction Data Messages (CDMs). The team compared the results to existing operational systems, with good agreement. OSC and their partners demonstrated the OADR prototype results to the Congress, as funded and directed by Public Law 116–260, the Consolidated Appropriations Act of 2021. This paper describes some of the background and rationale for the prototype, goals and derived requirements of the prototype, the architecture, software deployed, data utilized, conjunction analysis results, comparison of the prototype's results with existing operational US government systems, and possible next steps for the OADR.

Several inequalities for an integral transform of positive operators in Hilbert spaces with applications

For a continuous and positive function \(w\left( \lambda \right) ,\) \(\lambda>0\) and \(\mu \) a positive measure on \((0,\infty )\) we consider the following Integral Transform \[ \begin{equation*} \mathcal{D}\left( w,\mu \right) \left( T\right) :=\int_{0}^{\infty }w\left(\lambda \right) \left( \lambda +T\right)^{-1}d\mu \left( \lambda \right) , \end{equation*} \] where the integral is assumed to exist for \(T\) a postive operator on a complex Hilbert space \(H\). We show among others that, if \( \beta \geq A \geq \alpha > 0, \, B > 0 \) with \( M \geq B-A \geq m > 0 \) for some constants \( \alpha, \beta, m, M \), then \[ \begin{align*} 0 & \leq \frac{m^{2}}{M^{2}}\left[ \mathcal{D}\left( w,\mu \right) \left(\beta\right) - \mathcal{D}\left( w,\mu \right) \left(M+\beta\right) \right] \\ & \leq \frac{m^{2}}{M}\left[ \mathcal{D}\left( w,\mu \right) \left(\beta\right) - \mathcal{D}\left( w,\mu \right) \left(M+\beta\right) \right] \left( B-A\right)^{-1} \\ & \leq \mathcal{D}\left( w,\mu \right) \left(A\right) - \mathcal{D}\left(w,\mu\right) \left(B\right) \\ & \leq \frac{M^{2}}{m}\left[ \mathcal{D}\left( w,\mu \right) \left(\alpha\right) - \mathcal{D}\left( w,\mu \right) \left(m+\alpha\right) \right] \left(B-A\right)^{-1} \\ & \leq \frac{M^{2}}{m^{2}}\left[ \mathcal{D}\left( w,\mu \right) \left(\alpha\right) - \mathcal{D}\left( w,\mu \right) \left(m+\alpha\right) \right]. \end{align*} \] Some examples for operator monotone and operator convex functions as well as for integral transforms \(\mathcal{D}\left( \cdot ,\cdot \right) \) related to the exponential and logarithmic functions are also provided.

Well-posedness of fractional differential equations with finite delay in complex Banach spaces

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems.

Upper bounds for the numerical radii of powers of Hilbert space operators

We present several upper bounds for the numerical radii of 2 × 2 operator matrices. We employ these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then for every r ≥ 1 and α ∈ [0, 1]. This substantially improves on the existing inequality . Here w(.) and ||.|| denote the numerical radius and the usual operator norm, respectively.

Study on Birkhoff orthogonality and symmetry of matrix operators

Abstract We focus on the problem of generalized orthogonality of matrix operators in operator spaces. Especially, on ℬ ( l 1 n , l p n ) ( 1 ≤ p ≤ ∞ ) {\mathcal{ {\mathcal B} }}\left({l}_{1}^{n},{l}_{p}^{n})\left(1\le p\le \infty ) , we characterize Birkhoff orthogonal elements of a certain class of matrix operators and point out the conditions for matrix operators which satisfy the Bhatia-Šemrl property. Furthermore, we give some conclusions which are related to the Bhatia-Šemrl property. In a certain class of matrix operator space, such as ℬ ( l ∞ n ) {\mathcal{ {\mathcal B} }}\left({l}_{\infty }^{n}) , the properties of the left and right symmetry are discussed. Moreover, the equivalence condition for the left symmetry of Birkhoff orthogonality of matrix operators on ℬ ( l p n ) ( 1 &lt; p &lt; ∞ ) {\mathcal{ {\mathcal B} }}\left({l}_{p}^{n})\left(1\lt p\lt \infty ) is obtained.

Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces

AbstractThe main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

Model-Based Simultaneous Multi-Slice (SMS) Reconstruction with Hankel Subspace Learning for Accelerated MR T1 Mapping

Herein, we propose a novel model-based simultaneous multi-slice (SMS) reconstruction method by exploiting data-driven parameter modeling for highly accelerated T1 parameter quantification. We assume that the predefined slice-specific null space operator remains invariant along the parameter dimension. We incorporate the parameter dimension into SMS-HSL to exploit Hankel-structured and Casorati matrices. Given this consideration, the SMS signal is reformulated in k-p space as a constrained optimization problem that exploits rank deficiency for the Hankel-structured matrix and a finite-dimensional basis for a subspace containing slowly evolving signals in the parameter direction. The proposed model-based SMS reconstruction method is validated on in vivo data and compared with state-of-the-art methods with slice acceleration factors of 3 and 5, including an in-plane acceleration factor of 2. The experimental results demonstrate that the proposed method performs effective slice unfolding and signal recovery in reconstructed images and T1 maps with high precision as compared to the state-of-the-art methods.

Davis–Wielandt–Berezin radius inequalities of reproducing kernel Hilbert space operators

Davis–Wielandt–Berezin radius inequalities of reproducing kernel Hilbert space operators

On a paradox in quantum mechanics and its resolution

Consider a free Schrödinger particle inside an interval with walls characterized by the Dirichlet boundary condition. Choose a parabola as the normalized state of the particle that satisfies this boundary condition. To calculate the variance of the Hamiltonian in that state, one needs to calculate the mean value of the Hamiltonian and that of its square. If one uses the standard formula to calculate these mean values, one obtains both results without difficulty, but the variance unexpectedly takes an imaginary value. If one uses the same expression to calculate these mean values but first writes the Hamiltonian and its square in terms of their respective eigenfunctions and eigenvalues, one obtains the same result as above for the mean value of the Hamiltonian but a different value for its square (in fact, it is not zero); hence, the variance takes an acceptable value. From whence do these contradictory results arise? The latter paradox has been presented in the literature as an example of a problem that can only be properly solved by making use of certain fundamental concepts within the general theory of linear operators in Hilbert spaces. Here, we carefully review those concepts and apply them in a detailed way to resolve the paradox. Our results are formulated within the natural framework of wave mechanics, and to avoid inconveniences that the use of Dirac’s symbolic formalism could bring, we avoid the use of that formalism throughout the article. In addition, we obtain a resolution of the paradox in an entirely formal way without addressing the restrictions imposed by the domains of the operators involved. We think that the content of this paper will be useful to undergraduate and graduate students as well as to their instructors.

Coarse geometry of operator spaces and complete isomorphic embeddings into $$\ell _1$$ and $$c_0$$-sums of operator spaces

Coarse geometry of operator spaces and complete isomorphic embeddings into $$\ell _1$$ and $$c_0$$-sums of operator spaces

Topological transitivity of translation operators in a non-separable Hilbert space

We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.

On wavelet type Bernstein operators

This paper deals with construction and studying wavelet type Bernstein operators by using the compactly supported Daubechies wavelets of the given function $f$. The basis used in this construction is the wavelet expansion of the function $f$ instead of its rational sampling values $f\big( \frac{k}{n}\big)$. After that, we investigate some properties of these operators in some function spaces.

Using deep LSD to build operators in GANs latent space with meaning in real space.

Generative models rely on the idea that data can be represented in terms of latent variables which are uncorrelated by definition. Lack of correlation among the latent variable support is important because it suggests that the latent-space manifold is simpler to understand and manipulate than the real-space representation. Many types of generative model are used in deep learning, e.g., variational autoencoders (VAEs) and generative adversarial networks (GANs). Based on the idea that the latent space behaves like a vector space Radford et al. (2015), we ask whether we can expand the latent space representation of our data elements in terms of an orthonormal basis set. Here we propose a method to build a set of linearly independent vectors in the latent space of a trained GAN, which we call quasi-eigenvectors. These quasi-eigenvectors have two key properties: i) They span the latent space, ii) A set of these quasi-eigenvectors map to each of the labeled features one-to-one. We show that in the case of the MNIST image data set, while the number of dimensions in latent space is large by design, 98% of the data in real space map to a sub-domain of latent space of dimensionality equal to the number of labels. We then show how the quasi-eigenvectors can be used for Latent Spectral Decomposition (LSD). We apply LSD to denoise MNIST images. Finally, using the quasi-eigenvectors, we construct rotation matrices in latent space which map to feature transformations in real space. Overall, from quasi-eigenvectors we gain insight regarding the latent space topology.

On an efficient iterative scheme for a class of generalized nonexpansive operators in Banach spaces

The main purpose of this work is to connect the recently introduced iterative approximation scheme known as [Formula: see text]-iterative scheme with the class of generalized [Formula: see text]-nonexpansive mappings. We consider the setting of a Banach space to prove our weak and strong convergence results. Once again, we prove that the class of [Formula: see text]-nonexpansive mapping includes properly the classes of Suzuki nonexpansive, generalized [Formula: see text]-nonexpansive and as well Reich–Suzuki type nonexpansive mappings by providing a numerical example. Eventually, we prove that [Formula: see text]-iterative scheme of this example is essentially more efficient than the other well-known iterative scheme of the literature. The presented work is new and extends some other well-known results in the iterations theory.

Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators C_t, for tin [0,1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}, such as ell ^p, c_0, c, bv_0, bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesàro spaces ces(p) and their (isomorphic) dual spaces d_p. In most cases C_t (tnot =1) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of C_t, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}. Besides {{mathbb {C}}}^{{{mathbb {N}}}_0} itself, the Fréchet spaces considered are ell (p+), ces(p+) and d(p+), for 1le p<infty , as well as the (LB)-spaces ell (p-), ces(p-) and d(p-), for 1<ple infty .

New Results on Boas–Bellman-Type Inequalities in Semi-Hilbert Spaces with Applications

In this article, we investigate new findings on Boas–Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal significant properties of such spaces and apply these results to the field of multivariable operator theory. Specifically, we derive new inequalities that relate to the joint A-numerical radius, the joint operator A-seminorm, and the Euclidean A-seminorm of tuples of semi-Hilbert space operators. We assume that A is a nonzero positive operator. Our discoveries provide insights into the structure of semi-Hilbert spaces and have implications for a broad range of mathematical applications and beyond.