Relevant bibliographies by topics / Operator theory – General theory of linear operators – Functional calculus

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Academic literature on the topic 'Operator theory – General theory of linear operators – Functional calculus'

Author: Grafiati
Published: 4 June 2021
Last updated: 7 February 2022

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Journal articles on the topic "Operator theory – General theory of linear operators – Functional calculus"

1

Gantner, Jonathan. "Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators." Memoirs of the American Mathematical Society 267, no. 1297 (September 2020): 0. http://dx.doi.org/10.1090/memo/1297.

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2

Wang, Yingxu. "Inference Algebra (IA)." International Journal of Cognitive Informatics and Natural Intelligence 5, no. 4 (October 2011): 61–82. http://dx.doi.org/10.4018/jcini.2011100105.

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Inference as the basic mechanism of thought is one of the gifted abilities of human beings. It is recognized that a coherent theory and mathematical means are needed for dealing with formal causal inferences. This paper presents a novel denotational mathematical means for formal inferences known as Inference Algebra (IA). IA is structured as a set of algebraic operators on a set of formal causations. The taxonomy and framework of formal causal inferences of IA are explored in three categories: a) Logical inferences on Boolean, fuzzy, and general logic causations; b) Analytic inferences on general functional, correlative, linear regression, and nonlinear regression causations; and c) Hybrid inferences on qualification and quantification causations. IA introduces a calculus of discrete causal differential and formal models of causations; based on them nine algebraic inference operators of IA are created for manipulating the formal causations. IA is one of the basic studies towards the next generation of intelligent computers known as cognitive computers. A wide range of applications of IA are identified and demonstrated in cognitive informatics and computational intelligence towards novel theories and technologies for machine-enabled inferences and reasoning.
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3

Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. "Perturbation of the generator of a quaternionic evolution operator." Analysis and Applications 13, no. 04 (April 28, 2015): 347–70. http://dx.doi.org/10.1142/s0219530514500249.

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The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.
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4

Li, Nan, Risto Korhonen, and Lianzhong Yang. "Good Linear Operators and Meromorphic Solutions of Functional Equations." Journal of Complex Analysis 2015 (May 14, 2015): 1–8. http://dx.doi.org/10.1155/2015/960204.

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Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, q-difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and q-difference equations are largely similar. The purpose of this paper is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. As an example case, we apply our methods to study the growth of meromorphic solutions of the functional equation M(z,f)+P(z,f)=h(z), where M(z,f) is a linear polynomial in f and L(f), where L is good linear operator, P(z,f) is a polynomial in f with degree deg P≥2, both with small meromorphic coefficients, and h(z) is a meromorphic function.
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5

Garetto, Claudia, and G. Hormann. "On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets." Bulletin: Classe des sciences mathematiques et natturalles 133, no. 31 (2006): 115–36. http://dx.doi.org/10.2298/bmat0631115g.

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Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functional and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis. AMS Mathematics Subject Classification (2000): 46F30, 46A20, 47G30.
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6

LILLO, R. E., and M. MARTIN. "Characterization Results Based on a Functional Derivative Approach." Combinatorics, Probability and Computing 10, no. 5 (September 2001): 417–34. http://dx.doi.org/10.1017/s0963548301004825.

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We introduce a notion of the derivative with respect to a function, not necessarily related to a probability distribution, which generalizes the concept of derivative as proposed by Lebesgue [14]. The differential calculus required to solve linear differential equations using this notion of the derivative is included in the paper. The definition given here may also be considered as the inverse operator of a modified notion of the Riemann–Stieltjes integral. Both this unified approach and the results of differential calculus allow us to characterize distributions in terms of three different types of conditional expectations. In applying these results, a test of goodness of fit is also indicated. Finally, two characterizations of a general Poisson process are included. Specifically, a useful result for the homogeneous Poisson process is generalized.
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7

BARBU, VIOREL. "Variational approach to nonlinear stochastic differential equations in Hilbert spaces." Carpathian Journal of Mathematics 37, no. 2 (June 9, 2021): 295–309. http://dx.doi.org/10.37193/cjm.2021.02.15.

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Here we survey a few functional methods to existence theory for infinite dimensional stochastic differential equations of the form dX+A(t)X(t)=B(t,X(t))dW(t), X(0)=X_0, where A(t) is a non\-linear maximal monotone operator in a variational couple (V,V'). The emphasis is put on a new approach of the classical existence result of N. Krylov and B. Rozovski on existence for the infinite dimensional stochastic differential equations which is given here via the theory of nonlinear maximal monotone operators in Banach spaces. A variational approach to this problem is also developed.
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8

KLOECKNER, BENOÎT R. "Effective high-temperature estimates for intermittent maps." Ergodic Theory and Dynamical Systems 39, no. 8 (December 12, 2017): 2159–75. http://dx.doi.org/10.1017/etds.2017.111.

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Using quantitative perturbation theory for linear operators, we prove a spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (‘high-temperature’ regime). Hölder and bounded $p$-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau–Manneville map, any potential with Lipschitz constant less than 0.0014 has a transfer operator acting on $\operatorname{Lip}([0,1])$ with a spectral gap; and that for any two-to-one unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on $\operatorname{BV}([0,1])$ with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in Giulietti et al [The calculus of thermodynamical formalism. J. Eur. Math. Soc., to appear. Preprint, 2015, arXiv:1508.01297], allowing all results there to be applied under the high-temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.
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9

MANZONETTO, GIULIO. "What is a categorical model of the differential and the resource λ-calculi?" Mathematical Structures in Computer Science 22, no. 3 (February 27, 2012): 451–520. http://dx.doi.org/10.1017/s0960129511000594.

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The differential λ-calculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows the application of a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows us to write the full Taylor expansion of a program. Through this expansion, every program can be decomposed into an infinite sum (representing non-deterministic choice) of ‘simpler’ programs that are strictly linear.The aim of this paper is to develop an abstract ‘model theory’ for the untyped differential λ-calculus. In particular, we investigate what form a general categorical definition of a denotational model for this calculus should take. Starting from the work of Blute, Cockett and Seely on differential categories, we develop the notion of a Cartesian closed differential category and prove that linear reflexive objects living in such categories constitute sound and complete models of the untyped differential λ-calculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This requires that every model living in such categories equates all programs having the same full Taylor expansion.We then provide a concrete example of a Cartesian closed differential category modelling the Taylor expansion, namely the category MRel of sets and relations from finite multisets to sets. We prove that the extensional model of λ-calculus we have recently built in MRel is linear, and is thus also an extensional model of the untyped differential λ-calculus. In the same category, we build a non-extensional model and prove that it is, nevertheless, extensional on its differential part.Finally, we study the relationship between the differential λ-calculus and the resource calculus, which is a functional programming language combining the ideas behind the differential λ-calculus with those behind Boudol's λ-calculus with multiplicities. We define two translation maps between these two calculi and study the properties of these translations. In particular, this analysis shows that the two calculi share the same notion of a model, and thus that the resource calculus can be interpreted by translation into every linear reflexive object living in a Cartesian closed differential category.
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10

Krasnoshchok, Mykola. "Strong solution of a hydrodinamics problem with memory." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 34 (April 24, 2021): 62–74. http://dx.doi.org/10.37069/1683-4720-2020-34-7.

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In the last few years, the concepts of fractional calculus were frequently applied to other disciplines. Recently, this subject has been extended in various directions such as signal processing, applied mathematics, bio-engineering, viscoelasticity, fluid mechanics, and fluid dynamics. In fluid dynamics, the fractional derivative models were used widely in the past for the study of viscoelastic materials such as polymers in the glass transition and in the glassy state. Recently, it has increasingly been seen as an efficient tool through which a useful generalization of physical concepts can be obtained. The fractional derivatives used most are the Riemann--Liouville fractional derivative and the Caputo fractional derivative. It is well known that these operators exhibit difficulties in applications. For example, the Riemann--Liouville derivative of a constant is not zero. We deal with so called temporal fractional derivative as a prototype of general fractional derivative. We prove the global strong solvability of a linear and quasilinear initial-boundary value problems with a singular complete monotone kernels. Our main tool is a theory of evolutionary integral equations. An abstract fractional order differential equation is studied, which contains as particular case the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. This paper concerns with an initial-boundary value problem for the Navier--Stokes--Voigt equations describing unsteady flows of an incompressible viscoelastic fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution in two-dimensional domain. We also establish an $L_2$ decay estimate for the velocity field under the assumption that the external forces field is conservative.
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Books on the topic "Operator theory – General theory of linear operators – Functional calculus"

1

I, Gohberg. Traces and determinants of linear operators. Basel: Birkhäuser Verlag, 2000.

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2

(Victor), Vinnikov V., ed. Foundations of free noncommutative function theory. Providence, Rhode Island: American Mathematical Society, 2014.

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3

1949-, Karlovich Yuri I., and Spitkovskiĭ Ilya M. 1953-, eds. Convolution operators and factorization of almost periodic matrix functions. Basel: Birkhäuser Verlag, 2002.

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Zaslavski, Alexander J., and Simeon Reich. Infinite products of operators and their applications: A research workshop of the Israel Science Foundation : May 21-24, 2012, Haifa, Israel : Israel mathematical conference proceedings. Providence, Rhode Island: American Mathematical Society, 2015.

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Simon, Barry. Operator theory. Providence, Rhode Island: American Mathematical Society, 2015.

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Roe, John. Winding around: The winding number in topology, geometry, and analysis. Providence, Rhode Island: American Mathematical Society, 2015.

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1941-, Șabac Mihai, ed. Lie algebras of bounded operators. Basel: Birkhäuser Verlag, 2001.

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International Conference on p-Adic Functional Analysis (11th 2010 Université Blaise Pascal). Advances in non-Archimedean analysis: Eleventh International Conference on p-Adic Functional Analysis, July 5-9 2010, Université Blaise Pascal, Clermont-Ferrand, France. Edited by Araujo-Gomez Jesus 1965-, Diarra B. (Bertin) 1944-, and Escassut Alain. Providence, R.I: American Mathematical Society, 2011.

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G, Samko S., ed. Equations with involutive operators. Boston: Birkhäuser, 2001.

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Abramovich, Y. A. Banach C(K)-modules and operators preserving disjointness. Harlow, Essex, England: Longman, 1992.

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