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Journal articles on the topic 'Integral operator algebras'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Pavlakos, Panaiotis K. "Integral representation theorems in partially ordered vector spaces." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 51, no. 2 (1991): 187–215. http://dx.doi.org/10.1017/s1446788700034194.

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AbstractDefining a Radon-type integration process we extend the Alexandroff, Fichtengolts-KantorovichHildebrandt and Riesz integral representation theorems in partially ordered vector spaces.We also identify some classes of operators with other classes of operator-valued set functions, the correspondence between operator and operator-valued set function being given by integration.All these established results can be immediately applied in C* -algebras (especially in W* -algebras and AW* -algebras of type I), in Jordan algebras, in partially ordered involutory (O*-)algebras, in semifields, in q

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2

CASTRO, OSVALDO OSUNA, та ELMAR WAGNER. "AN OPERATOR-THEORETIC APPROACH TO INVARIANT INTEGRALS ON QUANTUM HOMOGENEOUS SLn+1(ℝ)-SPACES". International Journal of Geometric Methods in Modern Physics 09, № 01 (2012): 1250012. http://dx.doi.org/10.1142/s0219887812500120.

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We present other examples illustrating the operator-theoretic approach to invariant integrals on quantum homogeneous spaces developed by Kürsten and the second author. The quantum spaces are chosen such that their coordinate algebras do not admit bounded Hilbert space representations and their self-adjoint generators have continuous spectrum. Operator algebras of trace class operators are associated to the coordinate algebras which allow interpretations as rapidly decreasing functions and as finite functions. The invariant integral is defined as a trace functional which generalizes the well-kn

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3

Quinn, Terrance. "Products of Decomposable Positive Operators." Canadian Journal of Mathematics 46, no. 4 (1994): 854–71. http://dx.doi.org/10.4153/cjm-1994-048-4.

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AbstractIn recent years there has been a growing interest in problems of factorization for bounded linear operators. We first show that many of these problems properly belong to the category of C*-algebras. With this interpretation, it becomes evident that the problem is fundamental both to the structure of operator algebras and the elements therein. In this paper we consider the direct integral algebra with separable and infinite dimensional. We generalize a theorem of Wu (1988) and characterize those decomposable operators which are products of non-negative decomposable operators. We do this

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4

Wang, Zhongwei, Zhen Guan, Yi Zhang, and Liangyun Zhang. "Rota–Baxter Operators on Cocommutative Weak Hopf Algebras." Mathematics 10, no. 1 (2021): 95. http://dx.doi.org/10.3390/math10010095.

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In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,∗,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given.

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Kalitvin, A. S., and V. A. Kalitvin. "Linear Operators and Equations with Partial Integrals." Contemporary Mathematics. Fundamental Directions 65, no. 3 (2019): 390–433. http://dx.doi.org/10.22363/2413-3639-2019-65-3-390-433.

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We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially diﬀerent compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and constru

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6

Nikitopoulos, Evangelos A. "Multiple operator integrals in non-separable von Neumann algebras." Journal of Operator Theory 89, no. 2 (2023): 361–427. https://doi.org/10.7900/jot.2021aug19.2357.

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A multiple operator integral (MOI) is an indispensable tool in several branches of noncommutative analysis. However, there are substantial technical issues with the existing literature on the ``separation of variables'' approach to defining MOIs, especially when the underlying Hilbert spaces are not separable. In this paper, we provide a detailed development of this approach in a very general setting that resolves existing technical issues. Along the way, we characterize several kinds of ``weak'' operator valued integrals in terms of easily checkable conditions and prove a useful Minkowski-typ

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7

Casper, W. Riley, and Milen T. Yakimov. "Integral operators, bispectrality and growth of Fourier algebras." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 766 (2020): 151–94. http://dx.doi.org/10.1515/crelle-2019-0031.

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AbstractIn the mid 1980s it was conjectured that every bispectral meromorphic function {\psi(x,y)} gives rise to an integral operator {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions {\psi(x,y)} where the commuting differential operator is of order {\leq 6}. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exac

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8

Hashem, H. H. G. "Solvability of a 2 x 2 block operator matrix of chandrasekhar type on a Bananch algebra." Filomat 31, no. 16 (2017): 5169–75. http://dx.doi.org/10.2298/fil1716169h.

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In this paper, we study the existence of solutions for a system of quadratic integral equations of Chandrasekhar type by applying fixed point theorem of a 2 x 2 block operator matrix defined on a nonempty bounded closed convex subsets of Banach algebras where the entries are nonlinear operators.

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9

Neeb, Karl-Hermann, Hadi Salmasian, and Christoph Zellner. "Smoothing operators and C∗-algebras for infinite dimensional Lie groups." International Journal of Mathematics 28, no. 05 (2017): 1750042. http://dx.doi.org/10.1142/s0129167x17500422.

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A smoothing operator for a unitary representation [Formula: see text] of a (possibly infinite dimensional) Lie group [Formula: see text] is a bounded operator [Formula: see text] whose range is contained in the space [Formula: see text] of smooth vectors of [Formula: see text]. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map [Formula: see text] is smooth. For unitary representations [Formula: see text] which are semibounded, i.e. there exists an element [Formula: see text] such that all operators [Formula: see text] from the deriv

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10

Ciubotaru, Dan, and Marcelo De Martino. "Dirac Induction for Rational Cherednik Algebras." International Mathematics Research Notices 2020, no. 17 (2018): 5155–214. http://dx.doi.org/10.1093/imrn/rny153.

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Abstract We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler–Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H

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11

Tarkhanov, N. "Operator algebras related to the Bochner–Martinelli integral." Complex Variables and Elliptic Equations 51, no. 3 (2006): 197–208. http://dx.doi.org/10.1080/17476930500454001.

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12

Tsukada, Haruo. "String path integral realization of vertex operator algebras." Memoirs of the American Mathematical Society 91, no. 444 (1991): 0. http://dx.doi.org/10.1090/memo/0444.

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13

Łuczak, Andrzej, and Abdulrahman Mohammed. "Stochastic integration in finite von Neumann algebras." Studia Scientiarum Mathematicarum Hungarica 44, no. 2 (2007): 233–64. http://dx.doi.org/10.1556/sscmath.2007.1016.

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We investigate various aspects of stochastic integration in finite von Neumann algebras. For integration with respect to a bounded L2 -martingale the idea of treating the integral as a bounded operator is developed. Several classes of integrable processes are defined, it turns out that some of them form a Banach or C *-algebra. We find representations of these algebras and establish relations between the von Neumann algebras generated by these representations. Finally, we characterize the range of the stochastic integration operator.

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14

THOM, ANDREAS. "INTEGER OPERATORS IN FINITE VON NEUMANN ALGEBRAS." Journal of Topology and Analysis 03, no. 04 (2011): 433–50. http://dx.doi.org/10.1142/s1793525311000635.

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Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by Fekete and Szegö, see [3, 4, 13]. More concretely, we use results by Rumely, see [12], on equidistribution of algebraic integers to obtain a description of those integer operator which have spectrum of logarithmic capacity less than or equal to one. Finally, we relate the study of integer operators to a recent construction by Petracov

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15

IRAC-ASTAUD, MICHÈLE, and GUY RIDEAU. "DEFORMED HARMONIC OSCILLATOR ALGEBRAS DEFINED BY THEIR BARGMANN REPRESENTATIONS." Reviews in Mathematical Physics 11, no. 05 (1999): 631–51. http://dx.doi.org/10.1142/s0129055x99000222.

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Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint a and a†, and two self-adjoint N and the unity 1 such as: [Formula: see text] The Bargmann Hilbert space is defined as a space of functions, holomorphic in a ring of the complex plane, equipped with a scalar product involving a true integral. In a Bargmann representation, the operators of a Deformed Harmonic Oscillator Algebra act on a Bargmann Hilbert space and the creation (or the annihilation operator) is the multiplication by z. We discuss the conditions of existence of Deformed Harmonic Oscillator

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16

Najafi, Hamed. "Some operator inequalities for Hermitian Banach $*$-algebras." MATHEMATICA SCANDINAVICA 126, no. 1 (2020): 82–98. http://dx.doi.org/10.7146/math.scand.a-115624.

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In this paper, we extend the Kubo-Ando theory from operator means on C$^{*}$-algebras to a Hermitian Banach $*$-algebra $\mathcal {A}$ with a continuous involution. For this purpose, we show that if $a$ and $b$ are self-adjoint elements in $\mathcal {A}$ with spectra in an interval $J$ such that $a \leq b$, then $f(a) \leq f(b)$ for every operator monotone function $f$ on $J$, where $f(a)$ and $f(b)$ are defined by the Riesz-Dunford integral. Moreover, we show that some convexity properties of the usual operator convex functions are preserved in the setting of Hermitian Banach $*$-algebras. In

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17

Fahem, Amor, Aref Jeribi, and Najib Kaddachi. "Existence of solutions for a system of Chandrasekhar’s equations in Banach algebras underweak topology." Filomat 33, no. 18 (2019): 5949–57. http://dx.doi.org/10.2298/fil1918949f.

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This paper is devoted to the study of a coupled system within fractional integral equations in suitable Banach algebra. In particular, we are concerned with a quadratic integral equations of Chandrasekhar type. The existence of solutions will be proved by applying fixed point theorem of a 2 x 2 block operator matrix defined on a nonempty, closed and convex subset of Banach algebra where the entries are weakly sequentially continuous operators.

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18

Duncan, Benton. "Operator algebras associated to modules over an integral domain." Advances in Operator Theory 3, no. 2 (2018): 374–87. http://dx.doi.org/10.15352/aot.1706-1181.

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19

Mu, Qiang. "Smash product construction of modular lattice vertex algebras." Electronic Research Archive 30, no. 1 (2021): 204–20. http://dx.doi.org/10.3934/era.2022011.

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<abstract><p>Motivated by a work of Li, we study nonlocal vertex algebras and their smash products over fields of positive characteristic. Through smash products, modular vertex algebras associated with positive definite even lattices are reconstructed. This gives a different construction of the modular vertex algebras obtained from integral forms introduced by Dong and Griess in lattice vertex operator algebras over a field of characteristic zero.</p></abstract>

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20

McRae, Robert. "Intertwining operators among modules for affine Lie algebra and lattice vertex operator algebras which respect integral forms." Journal of Pure and Applied Algebra 219, no. 10 (2015): 4757–81. http://dx.doi.org/10.1016/j.jpaa.2015.03.005.

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21

Hashem, Hind, Ahmed El-Sayed, and Dumitru Baleanu. "Existence Results for Block Matrix Operator of Fractional Orders in Banach Algebras." Mathematics 7, no. 9 (2019): 856. http://dx.doi.org/10.3390/math7090856.

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This paper is concerned with proving the existence of solutions for a coupled system of quadratic integral equations of fractional order in Banach algebras. This result is a direct application of a fixed point theorem of Banach algebras. Some particular cases, examples and remarks are illustrated. Finally, the stability of solutions for that coupled system are studied.

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22

Szymanski, Waclaw. "Antisymmetry and the Direct Integral Decomposition of Unstarred Operator Algebras." Proceedings of the American Mathematical Society 96, no. 3 (1986): 497. http://dx.doi.org/10.2307/2046603.

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23

Dong, Chongying, and Robert L. Griess. "Determinants for integral forms in lattice type vertex operator algebras." Journal of Algebra 558 (September 2020): 327–35. http://dx.doi.org/10.1016/j.jalgebra.2019.09.006.

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24

Szyma{ński, Wacław. "Antisymmetry and the direct integral decomposition of unstarred operator algebras." Proceedings of the American Mathematical Society 96, no. 3 (1986): 497. http://dx.doi.org/10.1090/s0002-9939-1986-0822448-6.

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25

Garetto, Claudia, and Michael Oberguggenberger. "Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities." Proceedings of the Edinburgh Mathematical Society 57, no. 2 (2013): 426–63. http://dx.doi.org/10.1017/s0013091513000424.

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AbstractThis paper addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalized functions. We employ the recently developed theory of generalized Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalized solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalized Fourier integral oper

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26

Miana, Pedro J., Juan J. Royo, and Luis Sánchez-Lajusticia. "Convolution Algebraic Structures Defined by Hardy-Type Operators." Journal of Function Spaces and Applications 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/212465.

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The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spacesLωpℝ+forp≥1. We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

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27

Grassi, Pietro Antonio. "Novel Free Differential Algebras for Supergravity." Universe 9, no. 8 (2023): 376. http://dx.doi.org/10.3390/universe9080376.

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We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D = 11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of

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Kaddachi, Najib. "Generalized form of fixed point theorems in Banach algebras under weak topology with an application." Filomat 33, no. 13 (2019): 4281–96. http://dx.doi.org/10.2298/fil1913281k.

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In this manuscript, by means of the technique of measures of weak noncompactness, we establish a generalized form of fixed point theorems for a 2 x 2 block operator matrix involving multivalued maps acting on suitable Banach algebras. The results obtained are then applied to a coupled system of nonlinear integral equations.

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Poncini, Xavier, and Jørgen Rasmussen. "Integrability of planar-algebraic models." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 7 (2023): 073101. http://dx.doi.org/10.1088/1742-5468/acdce7.

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Abstract The quantum inverse scattering method is a scheme for solving integrable models in 1 + 1 dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion an

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30

Behrend, Kai, and Pooya Ronagh. "The inertia operator on the motivic Hall algebra." Compositio Mathematica 155, no. 3 (2019): 528–98. http://dx.doi.org/10.1112/s0010437x18007881.

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We study the action of the inertia operator on the motivic Hall algebra and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra ofvirtually indecomposableelements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion ofalgebroid.

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31

Dong, Chongying, and Robert L. Griess. "Integral forms in vertex operator algebras which are invariant under finite groups." Journal of Algebra 365 (September 2012): 184–98. http://dx.doi.org/10.1016/j.jalgebra.2012.05.006.

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32

Dong, Chongying, and Robert L. Griess. "Lattice-integrality of certain group-invariant integral forms in vertex operator algebras." Journal of Algebra 474 (March 2017): 505–16. http://dx.doi.org/10.1016/j.jalgebra.2016.12.003.

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33

PETKOVA, V. B. "TWO-DIMENSIONAL (HALF-) INTEGER SPIN CONFORMAL THEORIES WITH CENTRAL CHARGE c<1." International Journal of Modern Physics A 03, no. 12 (1988): 2945–58. http://dx.doi.org/10.1142/s0217751x88001235.

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A generalized integral representation involving two types of charges is explored to construct correlation functions on the plane for c=1–6/(m(m+1))<1 discrete unitary Virasoro series. The various local operator product algebras emerging contain integer, or half-integer, spin fields along with scalar fields. The examples also include a generalization for arbitrary m of the ℤ2-statistics of the Ising model order-disorder fields.

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Farid, Mohamed Amine, Karim Chaira, El Miloudi Marhrani, and Mohamed Aamri. "Measure of Weak Noncompactness and Fixed Point Theorems in Banach Algebras with Applications." Axioms 8, no. 4 (2019): 130. http://dx.doi.org/10.3390/axioms8040130.

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In this paper, we prove some fixed point theorems for the nonlinear operator A · B + C in Banach algebra. Our fixed point results are obtained under a weak topology and measure of weak noncompactness; and we give an example of the application of our results to a nonlinear integral equation in Banach algebra.

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Pietrzkowski, Gabriel. "Inhomogeneous Spitzer-type identities for commutative and non-commutative Rota–Baxter algebras." Journal of Algebra and Its Applications 16, no. 01 (2017): 1750016. http://dx.doi.org/10.1142/s0219498817500165.

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We consider a complete filtered Rota–Baxter (RB) algebra of weight [Formula: see text] over a commutative ring. Finding the unique solution of an inhomogeneous linear algebraic equation in this algebra, we generalize Spitzer’s identity in both commutative and non-commutative cases. As an application, considering the RB algebra of power series in one variable with q-integral as the RB operator, we show certain Eulerian identities.

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36

Momani, Shaher, and Rabha W. Ibrahim. "On a fractional integral equation of periodic functions involving Weyl–Riesz operator in Banach algebras." Journal of Mathematical Analysis and Applications 339, no. 2 (2008): 1210–19. http://dx.doi.org/10.1016/j.jmaa.2007.08.001.

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Macfarlane, A. J., and Hendryk Pfeiffer. "Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator." Journal of Physics A: Mathematical and General 36, no. 9 (2003): 2305–17. http://dx.doi.org/10.1088/0305-4470/36/9/308.

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38

Zhang, Degang. "Exact Solution for Three-Dimensional Ising Model." Symmetry 13, no. 10 (2021): 1837. http://dx.doi.org/10.3390/sym13101837.

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The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical

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Osaka, Hiroyuki, Sergei Silvestrov, and Jun Tomiyama. "Monotone operator functions, gaps and power moment problem." MATHEMATICA SCANDINAVICA 100, no. 1 (2007): 161. http://dx.doi.org/10.7146/math.scand.a-15019.

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The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps $P_{n}(I) \setminus P_{n+1}(I)$ for bounded intervals $I$. We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many

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40

Kohnen, Winfried, and Geoffrey Mason. "On Generalized Modular forms and their Applications." Nagoya Mathematical Journal 192 (2008): 119–36. http://dx.doi.org/10.1017/s0027763000026003.

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AbstractWe study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficien

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41

Kaddachi, Najib, Aref Jeribi, and Bilel Krichen. "Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations." Mathematical Methods in the Applied Sciences 36, no. 6 (2012): 659–73. http://dx.doi.org/10.1002/mma.2615.

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Bokodisa, Annel Thembinkosi, and Maggie Aphane. "Existence and Uniqueness of Fixed-Point Results in Non-Solid C⋆-Algebra-Valued Bipolar b-Metric Spaces." Mathematics 13, no. 4 (2025): 667. https://doi.org/10.3390/math13040667.

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In this monograph, motivated by the work of Aphane, Gaba, and Xu, we explore fixed-point theory within the framework of C⋆-algebra-valued bipolar b-metric spaces, characterized by a non-solid positive cone. We define and analyze (FH−GH)-contractions, utilizing positive monotone functions to extend classical contraction principles. Key contributions include the existence and uniqueness of fixed points for mappings satisfying generalized contraction conditions. The interplay between the non-solidness of the cone, the C⋆-algebra structure, and the completeness of the space is central to our resul

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43

Azamov, N. A., A. L. Carey, P. G. Dodds, and F. A. Sukochev. "Operator Integrals, Spectral Shift, and Spectral Flow." Canadian Journal of Mathematics 61, no. 2 (2009): 241–63. http://dx.doi.org/10.4153/cjm-2009-012-0.

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Abstract. We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

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Sharyn, S. V. "Algebraic and differential properties of polynomial Fourier transformation." Matematychni Studii 53, no. 1 (2020): 59–68. http://dx.doi.org/10.30970/ms.53.1.59-68.

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Methods of integral transformations of (generalized) functions are widely used in the solution of initial and boundary value problems for partial differential equations. However, many problems in applied mathematics require a nonlinear generalization of distribution spaces. Besides, an algebraic structure of a space of distributions is desirable, which is needed, for example, in quantum field theory.In the article, we use the adjoint operator method as well as technique of symmetric tensor products to extended the Fourier transformation onto the spaces of so-called polynomial rapidly decreasin

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Garetto, Claudia, Günther Hörmann, and Michael Oberguggenberger. "Generalized oscillatory integrals and Fourier integral operators." Proceedings of the Edinburgh Mathematical Society 52, no. 2 (2009): 351–86. http://dx.doi.org/10.1017/s0013091506000915.

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AbstractIn this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

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Tariq, Muhammad, Sotiris K. Ntouyas, and Asif Ali Shaikh. "A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators." Axioms 12, no. 7 (2023): 719. http://dx.doi.org/10.3390/axioms12070719.

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A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,h−m)−p-convex functions, and h-preinvex functions. Include

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Zafiris, Elias, and Albrecht von Müller. "Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras." Universe 9, no. 9 (2023): 392. http://dx.doi.org/10.3390/universe9090392.

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Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons.

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Jorgensen, Palle E. T., and James Tian. "Operator theory of multiple Ito-integrals." Journal of Operator Theory 93, no. 2 (2025): 477–509. https://doi.org/10.7900/jot.2023jun14.2428.

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We study systems of Gaussian fields indexed by families F of positive sigma-finite measures μ. For a given μ, the corresponding Gaussian field W(μ) is centered and has quadratic variation equal to μ. Our focus is the induced multi-variable case of stochastic analysis and discrete time Gaussian random walk processes. The approach is operator-theoretic with three aims: (i) explicit formulas for the operators and Hilbert spaces involved; (ii) implications for Krein-Feller diffusion processes; and (iii) a study of operator systems and algebras generated by the W(μ)-induced Ito-isometries Vμ, for μ

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Prössdorf, S. "Krupnik, N. Ya., Banach Algebras with Symbol and Singular Integral Operators. Basel, Boston, Birkhäuser Verlag 1987. X, 205 pp., sfr 88.—. ISBN 3-7643-1836-8 (Operator Theory: Advances and Applications 26)." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 68, no. 12 (1988): 624. http://dx.doi.org/10.1002/zamm.19880681208.

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Yuan, Hongfen, Guohong Shi, and Xiushen Hu. "Boundary Value Problems for the Perturbed Dirac Equation." Axioms 13, no. 4 (2024): 238. http://dx.doi.org/10.3390/axioms13040238.

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The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterati

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