Multiplication Operators Research Articles

Surjective Isometries of Multiplication Operators and Weighted Composition Operators on nth Weighted-Type Banach Spaces of Analytic Functions

This paper is to characterize isometries of multiplication operators and weighted composition operators on the nth weighted-type Banach spaces {Vn : n ∈ N} of analytic functions on the open unit disk of which the Bloch space and the Zygmund space are particular cases at n = 1, 2. We give characterizations of the symbols ψ and φ for which the multiplication operator Mψ and the weighted composition operator Wψ,φ are surjective isometries. Moreover, we show that generalized weighted composition operators are not isometric on Vn.

Conjugations on L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} Space on the Real Line

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{L}}^\ extbf{2}$$\\end{document} space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{H}}^\ extbf{2}$$\\end{document} space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators.

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) H of entire functions associated with operator valued kernel functions. de Branges operators E=(E−,E+) analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on X, where X is a complex separable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS B(E) based on the de Branges operator E=(E−,E+) has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces B(E) has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces B(E) has been given.

Mapping Groundwater Resource using Multispectral Sentinel 2 and Fuzzy Logic method, Case Study: Salafchegan, Qom, Iran

Groundwater is one of the essential freshwater sources for human consumption, with the highest reserves of fresh water on earth after glaciers and glaciers. Conservation and maintenance of groundwater quality in a large area require an overview of the status and potential of groundwater resources in that area, which can be applied to potential areas using remote sensing technology. In this study, after extracting the factors influencing the formation of groundwater aquifers from the Sentinel satellite image, appropriate information layers were prepared and integrated into the ArcGIS using different fuzzy operators and potential maps prepared with the location of groundwater wells. The area was validated. The results of combining slope layers, slope direction, lithology, drainage length density, lineament length density, lineament buffer, drainage buffer, and vegetation in the area showed that fuzzy multiplication and gamma operators could be used as suitable operators for Introducing information layers to identify groundwater potential in the area. Also, using the gamma numbers 0.1 gave better results than larger gamma numbers. The research results showed that 15.9% of the studied area has good and very good potential for the presence of underground water in the production map using the fuzzy gamma with gamma 0.1 method. Also, this map was validated by 70.1% of water wells in the region. The normalized ratio of accuracy to validity in the final production model with this method was estimated to be 54%, which is entirely acceptable compared to other methods.

Weighted inequalities for Fourier multiplier operators of Bochner–Riesz type on $$ \mathbb{R} ^2$$

Weighted inequalities for Fourier multiplier operators of Bochner–Riesz type on $$ \mathbb{R} ^2$$

Semigrupos de clase Co en l2(Z)

In this work we begin by studying the generalized multiplication operator M on the l2(Z). We prove that this operator is not bounded, is densely defined and symmetric and therefore does not admit a symmetric linear extension to the entire space. We introduce a family of operators on the l2(Z) space with n even and demonstrate that it forms a contraction semigroup of class Co, having −M as its infinitesimal generator. We also prove that if we restrict the domains of that family of operators, they still remain a contraction semigroup. Finally, we give results of existence of solution of the associated abstract Cauchy problem and properties of continuous dependence of the solution in connection to other norms.

Oversampling on a class of symmetric regular de Branges spaces

A de Branges space is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map . Let be the reproducing kernel in and be the operator of multiplication by the independent variable with maximal domain in . Loosely speaking, we say that has the -oversampling property relative to a proper subspace of it, with , if there exists such that for all , for all and almost every self-adjoint extension of . This definition is motivated by the well-known oversampling property of Paley-Wiener spaces. In this paper, we provide sufficient conditions for a symmetric, regular de Branges space to have the -oversampling property relative to a chain of de Branges subspaces of it.

Generalisation of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schrödinger type

The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schrödinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schrödinger case.Numerical illustrations for the time-dependent Gross–Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods.The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross–Pitaevskii systems in real and imaginary time.

Integrable hierarchies, Frölicher–Nijenhuis bicomplexes and Lauricella bi-flat F-manifolds

Given the Frölicher-Nijenhuis bicomplex associated with a -tensor field L with vanishing Nijenhuis torsion, we define a multi-parameter family of bi-flat structures . This result is obtained by combining the construction of integrable hierarchies of hydrodynamic type starting from Frölicher-Nijenhuis bicomplexes with the construction of flat F-manifold structures from integrable systems of hydrodynamic type. By construction L is the operator of multiplication by the Euler vector field E and the number of parameters coincides with the number of Jordan blocks appearing in its Jordan normal form. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case flat coordinates of are Lauricella functions. The -tensor fields defining the corresponding integrable hierarchies have a similar block diagonal structure.

Conditional-mean multiplicative operator models for count time series

Multiplicative error models (MEMs) are commonly used for real-valued time series, but they cannot be applied to discrete-valued count time series as the involved multiplication would not preserve the integer nature of the data. Thus, the concept of a multiplicative operator for counts is proposed (as well as several specific instances thereof), which are then used to develop a kind of MEMs for count time series (CMEMs). If equipped with a linear conditional mean, the resulting CMEMs are closely related to the class of so-called integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models and might be used as a semi-parametric extension thereof. Important stochastic properties of different types of INGARCH-CMEM as well as relevant estimation approaches are derived, namely types of quasi-maximum likelihood and weighted least squares estimation. The performance and application are demonstrated with simulations as well as with two real-world data examples.

Gates joint locally connected network for accurate and robust reconstruction in optical molecular tomography

Optical molecular tomography (OMT) is a potential pre-clinical molecular imaging technique with applications in a variety of biomedical areas, which can provide non-invasive quantitative three-dimensional (3D) information regarding tumor distribution in living animals. The construction of optical transmission models and the application of reconstruction algorithms in traditional model-based reconstruction processes have affected the reconstruction results, resulting in problems such as low accuracy, poor robustness, and long-time consumption. Here, a gates joint locally connected network (GLCN) method is proposed by establishing the mapping relationship between the inside source distribution and the photon density on surface directly, thus avoiding the extra time consumption caused by iteration and the reconstruction errors caused by model inaccuracy. Moreover, gates module was composed of the concatenation and multiplication operators of three different gates. It was embedded into the network aiming at remembering input surface photon density over a period and allowing the network to capture neurons connected to the true source selectively by controlling three different gates. To evaluate the performance of the proposed method, numerical simulations were conducted, whose results demonstrated good performance in terms of reconstruction positioning accuracy and robustness.

Holomorphic Hörmander-type functional calculus on sectors and strips

In this paper, the abstract multiplier theorems for 0 0 -sectorial and 0 0 -strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405–444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and “holomorphic Schwartz” functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937–974] to obtain an improvement—with respect to the smoothness condition—of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.

Inverse scattering problem for a third-order operator with local potential

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem.

S-Wave Accelerates Optimization-based Photoacoustic Image Reconstruction in vivo

Photoacoustic imaging has undergone rapid development in recent years. To simulate photoacoustic imaging on a computer, the most popular MATLAB toolbox currently used for the forward projection process is k-Wave. However, k-Wave suffers from significant computation time. Here we propose a straightforward simulation approach based on superposed Wave (s-Wave) to accelerate photoacoustic simulation. In this study, we consider the initial pressure distribution as a collection of individual pixels. By obtaining standard sensor data from a single pixel beforehand, we can easily manipulate the phase and amplitude of the sensor data for specific pixels using loop and multiplication operators. The effectiveness of this approach is validated through an optimization-based reconstruction algorithm. The results reveal significantly reduced computation time compared with k-Wave. Particularly in a sparse 3-D configuration, s-Wave exhibits a speed improvement >2000 times compared with k-Wave. In terms of optimization-based image reconstruction, in vivo imaging results reveal that using the s-Wave method yields images highly similar to those obtained using k-Wave, while reducing the reconstruction time by approximately 50 times. Proposed here is an accelerated optimization-based algorithm for photoacoustic image reconstruction, using the fast s-Wave forward projection simulation. Our method achieves substantial time savings, particularly in sparse system configurations. Future work will focus on further optimizing the algorithm and expanding its applicability to a broader range of photoacoustic imaging scenarios.

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg–Semmes, Lord–McDonald–Sukochev–Zanin and Fan–Lacey–Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator Mf on Heisenberg groups via homogeneous Sobolev norm of the symbol f. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

An enhanced arithmetic optimization algorithm for optimal control of reactive power

AbstractIn this article, an enhanced arithmetic optimization algorithm (EAOA) is utilized to resolve the optimal reactive power dispatch problem (ORPD) of power plants, which is a non‐linear, non‐smooth, complex optimization problem. Typically, it is formulated as a constrained optimal power flow problem. This paper utilizes the gamma distribution to generate initial random solutions, which improves the accuracy of the arithmetic optimization algorithm (AOA). The original mode of deviation of math optimizer accelerated (MOA) is replaced with the improved pattern of change in the prey's energy in Harris Hawk Optimization (HHO), which creates a superior balance between the exploration and exploitation of AOA. This paper implements logarithmic and exponential operators in place of division and multiplication operators, which improves the exploration ability of AOA and also utilizes the gamma distribution in the generation of random numbers, which are essential for the selection of operators in both search phase which enlarges the search area and improves the global optimum solution. A parametric analysis is performed on 10 benchmark functions for nine possible values before selecting the algorithm's initial parameters. A comparative statistical analysis along with Friedman's test is carried out on 23 benchmark functions to test the overall performance as compared to Sine‐Cosine Algorithm (SCA), Swarm‐Salp Algorithm, Grey Wolf Optimization (GWO), and basic Arithmetic Optimization Algorithm (AOA). EAOA is applied to the IEEE 57 bus system for optimal reactive power control. EAOA reports a significant reduction in a power loss of 23.2 MW, which is 18.596% smaller than the system's base case power loss of 28.5 MW as compared to Artificial Bee Colony (ABC), Firefly Algorithm (FA), Bat‐inspired Algorithm (BAT), Cuckoo Search Algorithm (CSA), and Modified Ant Lion Algorithm (MALO) which reports a reduction in a power loss of 14.73%, 15.43%, 15.43%, 16.14%, and 16.14% respectively. EAOA reduces the TVD to 0.76 pu, compared to ABC, FA, BAT, CSA, and MALO, which minimizes the TVD to 0.84, 0.82, 0.88, 0.82, and 0.79, respectively. This proves that EAOA performs better in resolving ORCP problems regarding the quality of solutions and convergence. A detailed comparative statistical analysis of EAOA with CSA, BAT, and WOA (Whale Optimization Algorithm) is conducted in terms of Probability Distribution Functions (PDF) and Cumulative Distribution Functions (CDF), along with the Wilcoxson sign rank test, which proves that EAOA gives the solution with a higher degree of precision.

Spectrum of One-Dimensional Potential Perturbed by a Small Convolution Operator: General Structure

We consider an operator of multiplication by a complex-valued potential in L2(R), to which we add a convolution operator multiplied by a small parameter. The convolution kernel is supposed to be an element of L1(R), while the potential is a Fourier image of some function from the same space. The considered operator is not supposed to be self-adjoint. We find the essential spectrum of such an operator in an explicit form. We show that the entire spectrum is located in a thin neighbourhood of the spectrum of the multiplication operator. Our main result states that in some fixed neighbourhood of a typical part of the spectrum of the non-perturbed operator, there are no eigenvalues and no points of the residual spectrum of the perturbed one. As a consequence, we conclude that the point and residual spectrum can emerge only in vicinities of certain thresholds in the spectrum of the non-perturbed operator. We also provide simple sufficient conditions ensuring that the considered operator has no residual spectrum at all.

On simultaneous similarity of families of commuting operators

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

Estimates for entropy numbers of multiplier operators of multiple series

The asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system {ϕm}m∈N0d on a probability space and bounded in L∞, is studied. The orthonormal system can be of the type ϕm(x)=ϕm1(1)(x1)⋯ϕmd(d)(xd), where each {ϕl(j)}l∈N0 is an orthonormal system, that can be different for each j, for example, it can be a Vilenkin system, a Walsh system on a real sphere or the trigonometric system on the unit circle. General upper and lower bounds for the entropy numbers are established by using Levy means of norms constructed using the orthonormal system. These results are applied to get upper and lower bounds for entropy numbers of specific multiplier operators, which generate, in particular cases, sets of finitely and infinitely differentiable functions, in the usual sense and in the dyadic sense. It is shown that these estimates have order sharp in various important cases.

Riesz transform associated with the fractional Fourier transform and applications in image edge detection

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.