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Journal articles on the topic 'Banach spaces Linear operators Spectral theory (Mathematics)'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

ZENG, QINGPING. "Five short lemmas in Banach spaces." Carpathian Journal of Mathematics 32, no. 1 (2016): 131–40. http://dx.doi.org/10.37193/cjm.2016.01.14.

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Consider a commutative diagram of bounded linear operators between Banach spaces...with exact rows. In what ways are the spectral and local spectral properties of B related to those of the pairs of operators A and C? In this paper, we give our answers to this general question using tools from local spectral theory.

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2

Ricker, W. "Spectral operators and weakly compact homomorphisms in a class of Banach Spaces." Glasgow Mathematical Journal 28, no. 2 (July 1986): 215–22. http://dx.doi.org/10.1017/s0017089500006534.

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The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

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3

Laursen, Kjeld B., and Michael M. Neumann. "Local spectral theory and spectral inclusions." Glasgow Mathematical Journal 36, no. 3 (September 1994): 331–43. http://dx.doi.org/10.1017/s0017089500030937.

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Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

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Laursen, Kjeld B., Vivien G. Miller, and Michael M. Neumann. "Local spectral properties of commutators." Proceedings of the Edinburgh Mathematical Society 38, no. 2 (June 1995): 313–29. http://dx.doi.org/10.1017/s0013091500019106.

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For a pair of continuous linear operators T and S on complex Banach spaces X and Y, respectively, this paper studies the local spectral properties of the commutator C(S, T) given by C(S, T)(A): = SA−AT for all A∈L(X, Y). Under suitable conditions on T and S, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces of C(S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoară, Foiaş, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.

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5

Tahir, Jawad Kadhim. "Numerical Computations for One Class of Dynamical Mathematical Models in Quasi-Sobolev Space." Mathematical Modelling of Engineering Problems 8, no. 2 (April 28, 2021): 267–72. http://dx.doi.org/10.18280/mmep.080214.

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The article studies some mathematical models that represent one class of dynamical equations in quasi-Sobolev space. The analytical investigation of solvability of the Cauchy problem in the quasi-Sobolev space and theoretical results used to enhance and develop an algorithm structure of the numerical procedures to find approximate solutions for models, the steps of algorithm based on the theoretical investigation of models, new algorithm of numerical method allowing to find approximate solutions of mathematical models under study in quasi-Sobolev space. Construction a program implements an algorithm of numerical method that allow finding approximate solutions for models. To construct the theory of degenerate holomorphic semigroups of operators in quasi-Banach spaces of sequences, we used the classical methods of functional analysis, theory of linear bounded operators, spectral theory. To construct the operators of resolving semigroups we used the Laplace transform of operator-valued functions in quasi-Banach spaces of sequences. The numerical investigation for models generate some approximate solutions which are normally based on the modified projection method. The convergence of the approximate solution to the exact one theoretically is justified by the convergence of the corresponding series, the agreement of approximate computations with the theoretical solution is established.

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6

Ludkovsky, S., and B. Diarra. "Spectral integration and spectral theory for non-Archimedean Banach spaces." International Journal of Mathematics and Mathematical Sciences 31, no. 7 (2002): 421–42. http://dx.doi.org/10.1155/s016117120201150x.

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Banach algebras over arbitrary complete non-Archimedean fields are considered such that operators may be nonanalytic. There are different types of Banach spaces over non-Archimedean fields. We have determined the spectrum of some closed commutative subalgebras of the Banach algebraℒ(E)of the continuous linear operators on a free Banach spaceEgenerated by projectors. We investigate the spectral integration of non-Archimedean Banach algebras. We define a spectral measure and prove several properties. We prove the non-Archimedean analog of Stone theorem. It also contains the case ofC-algebrasC∞(X,𝕂). We prove a particular case of a representation of aC-algebra with the help of aL(Aˆ,μ,𝕂)-projection-valued measure. We consider spectral theorems for operators and families of commuting linear continuous operators on the non-Archimedean Banach space.

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7

Edmunds, D. E., W. D. Evans, and D. J. Harris. "A spectral analysis of compact linear operators in Banach spaces." Bulletin of the London Mathematical Society 42, no. 4 (May 17, 2010): 726–34. http://dx.doi.org/10.1112/blms/bdq030.

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8

Baskakov, A. G., and A. S. Zagorskii. "Spectral theory of linear relations on real Banach spaces." Mathematical Notes 81, no. 1-2 (February 2007): 15–27. http://dx.doi.org/10.1134/s0001434607010026.

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9

Zguitti, Hassane. "A note on the common spectral properties for bounded linear operators." Filomat 33, no. 14 (2019): 4575–84. http://dx.doi.org/10.2298/fil1914575z.

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Let X and Y be Banach spaces, A : X ? Y and B, C : Y ? X be bounded linear operators. We prove that if A(BA)2 = ABACA = ACABA = (AC)2A, then ?*(AC) {0} = ?*(BA)\{0} where ?+ runs over a large of spectra originated by regularities.

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10

Roman, Marcel, and Adrian Sandovici. "$B$-spectral theory of linear relations in complex Banach spaces." Publicationes Mathematicae Debrecen 91, no. 3-4 (October 1, 2017): 455–66. http://dx.doi.org/10.5486/pmd.2017.7781.

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11

Schechter, Martin. "Book Review: Spectral theory of linear operators---and spectral systems in Banach algebras." Bulletin of the American Mathematical Society 41, no. 04 (June 17, 2004): 541–44. http://dx.doi.org/10.1090/s0273-0979-04-01020-1.

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12

Manhas, J. S. "Composition Operators and Multiplication Operators on Weighted Spaces of Analytic Functions." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–21. http://dx.doi.org/10.1155/2007/92070.

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LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

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13

Ashurov, R. R., and W. N. Everitt. "Linear quasi-differential operators in locally integrable spaces on the real line." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 4 (August 2000): 671–98. http://dx.doi.org/10.1017/s0308210500000366.

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The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.

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14

TERAUDS, VENTA. "FUNCTIONAL CALCULUS EXTENSIONS ON DUAL SPACES." Bulletin of the Australian Mathematical Society 79, no. 1 (February 2009): 71–77. http://dx.doi.org/10.1017/s0004972708001032.

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AbstractIn this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.

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15

Navascués, M. A., and P. Viswanathan. "Bases of Banach and Hilbert Spaces Transformed by Linear Operators." Complex Analysis and Operator Theory 13, no. 5 (September 1, 2018): 2277–301. http://dx.doi.org/10.1007/s11785-018-0844-z.

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16

Tylli, Hans-Olav. "A spectral radius problem connected with weak compactness." Glasgow Mathematical Journal 35, no. 1 (January 1993): 85–94. http://dx.doi.org/10.1017/s0017089500009599.

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The asymptotic behaviour has been determined for several natural geometric or topological quantities related to (degrees of) compactness of bounded linear operators on Banach spaces; see for instance [24], [25] and [17]. This paper complements these results by studying the spectral properties of some quantities related to weak compactness.

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17

Omladič, Matjaž. "Spectral subspaces of operator-valued functions." Bulletin of the Australian Mathematical Society 31, no. 1 (February 1985): 61–73. http://dx.doi.org/10.1017/s0004972700002276.

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We give a generalization of the notion of spectral maximal sub-spaces, for which some of the main results are still valid. We give an application of this theory on a class of operators, defined on some reflexive Banach space.

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18

Sain, Debmalya. "On best approximations to compact operators." Proceedings of the American Mathematical Society 149, no. 10 (July 21, 2021): 4273–86. http://dx.doi.org/10.1090/proc/15494.

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We study best approximations to compact operators between Banach spaces and Hilbert spaces, from the point of view of Birkhoff-James orthogonality and semi-inner-products. As an application of the present study, some distance formulae are presented in the space of compact operators. The special case of bounded linear functionals as compact operators is treated separately and some applications to best approximations in reflexive, strictly convex and smooth Banach spaces are discussed. An explicit example is presented in ℓ p n \ell _p^{n} spaces, where 1 > p > ∞ , 1 > p > \infty , to illustrate the applicability of the methods developed in this article. A comparative analysis of the results presented in this article with the well-known classical duality principle in approximation theory is conducted to demonstrate the advantage in the former case, from a computational point of view.

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19

Wrobel, Volker. "Multi-Dimensional spectral theory of bounded linear operators in locally convex spaces." Mathematische Annalen 275, no. 3 (September 1986): 409–23. http://dx.doi.org/10.1007/bf01458614.

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20

El Kettani, Mustapha Ech-Chérif, and Aziz Lahssaini. "On the pseudospectrum preservers." Proyecciones (Antofagasta) 39, no. 6 (December 1, 2020): 1457–69. http://dx.doi.org/10.22199/issn.0717-6279-2020-06-0089.

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Let X and Y be two complex Banach spaces, and let B(X) denotes the algebra of all bounded linear operators on X. We characterize additive maps from B(X) onto B(Y ) compressing the pseudospectrum subsets Δϵ(.), where Δϵ (.) stands for any one of the spectral functions σϵ (.), σlϵ (.) and σrϵ (.) for some ϵ > 0. We also characterize the additive (resp. non-linear) maps from B(X) onto B(Y) preserving the pseudospectrum σϵ (.) of generalized products of operators for some ϵ > 0 (resp. for every ϵ > 0).

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21

Sahin, M., and M. B. Ragimov. "Spectral theory of ordered pairs of the linear operators - acting in different Banach spaces and applications." International Mathematical Forum 2 (2007): 223–36. http://dx.doi.org/10.12988/imf.2007.07021.

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22

Drnovšek, Roman, and Aljoša Peperko. "On the spectral radius of positive operators on Banach sequence spaces." Linear Algebra and its Applications 433, no. 1 (July 2010): 241–47. http://dx.doi.org/10.1016/j.laa.2010.02.020.

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23

Ye, Feng. "Toward a constructive theory of unbounded linear operators." Journal of Symbolic Logic 65, no. 1 (March 2000): 357–70. http://dx.doi.org/10.2307/2586543.

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AbstractWe show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

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24

Obukhovskii, Valeri, and Pietro Zecca. "On boundary value problems for degenerate differential inclusions in Banach spaces." Abstract and Applied Analysis 2003, no. 13 (2003): 769–84. http://dx.doi.org/10.1155/s108533750330301x.

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We consider the applications of the theory of condensing set-valued maps, the theory of set-valued linear operators, and the topological degree theory of the existence of mild solutions for a class of degenerate differential inclusions in a reflexive Banach space. Further, these techniques are used to obtain the solvability of general boundary value problems for a given class of inclusions. Some particular cases including periodic problems are considered.

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25

Benevieri, Pierluigi, and Antonio Iannizzotto. "Eigenvalue Problems for Fredholm Operators with Set-Valued Perturbations." Advanced Nonlinear Studies 20, no. 3 (August 1, 2020): 701–23. http://dx.doi.org/10.1515/ans-2020-2090.

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AbstractBy means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

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26

Aponte, Elvis, Jhixon Macías, José Sanabria, and José Soto. "Further characterizations of property (VΠ) and some applications." Proyecciones (Antofagasta) 39, no. 6 (December 1, 2020): 1435–56. http://dx.doi.org/10.22199/issn.0717-6279-2020-06-0088.

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We carry out characterizations with techniques provided by the local spectral theory of bounded linear operators T ∈ L(X), X infinite dimensional complex Banach space, which verify property (VΠ) introduced by Sanabria et al. (Open Math. 16(1) (2018), 289-297). We also carry out the study for polaroid operators and Drazin invertible operators that verify the property mentioned above.

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27

Huang, Qianglian, and Jipu Ma. "Perturbation analysis of generalized inverses of linear operators in Banach spaces." Linear Algebra and its Applications 389 (September 2004): 355–64. http://dx.doi.org/10.1016/j.laa.2004.04.011.

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28

Beanland, Kevin, and Ryan M. Causey. "Genericity and Universality for Operator Ideals." Quarterly Journal of Mathematics 71, no. 3 (June 17, 2020): 1081–129. http://dx.doi.org/10.1093/qmathj/haaa018.

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Abstract A bounded linear operator $U$ between Banach spaces is universal for the complement of some operator ideal $\mathfrak{J}$ if it is a member of the complement and it factors through every element of the complement of $\mathfrak{J}$. In the first part of this paper, we produce new universal operators for the complements of several ideals, and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use descriptive set theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal $\mathfrak{J}$ generic if whenever an operator $A$ has the property that every operator in $\mathfrak{J}$ factors through a restriction of $A$, then every operator between separable Banach spaces factors through a restriction of $A$. We prove that many classical operator ideals (such as strictly singular, weakly compact, Banach–Saks) are generic and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W. B. Johnson, which states that there is no universal operator for the complement of the ideal of completely continuous operators.

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Huang, Qianglian, Lanping Zhu, and Yueyu Jiang. "On stable perturbations for outer inverses of linear operators in Banach spaces." Linear Algebra and its Applications 437, no. 7 (October 2012): 1942–54. http://dx.doi.org/10.1016/j.laa.2012.05.004.

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30

Ding, Jiu. "New perturbation results on pseudo-inverses of linear operators in Banach spaces." Linear Algebra and its Applications 362 (March 2003): 229–35. http://dx.doi.org/10.1016/s0024-3795(02)00493-7.

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31

Ungureanu, Viorica Mariela. "Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions." Discrete Dynamics in Nature and Society 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/293930.

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We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.

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Anthoni, S. Marshal, J. H. Kim, and J. P. Dauer. "Existence of mild solutions of second-order neutral functional differential inclusions with nonlocal conditions in Banach spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 22 (2004): 1133–49. http://dx.doi.org/10.1155/s0161171204310410.

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We study the existence of mild solutions of the nonlinear second-order neutral functional differential and integrodifferential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using the theory of strongly continuous cosine families of bounded linear operators and a fixed point theorem for condensing maps due to Martelli.

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Baskakov, A. G. "Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators." Journal of Mathematical Sciences 137, no. 4 (September 2006): 4885–5036. http://dx.doi.org/10.1007/s10958-006-0286-4.

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34

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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35

CHIBA, Hayato. "A SPECTRAL THEORY OF LINEAR OPERATORS ON RIGGED HILBERT SPACES UNDER ANALYTICITY CONDITIONS II: APPLICATIONS TO SCHRÖDINGER OPERATORS." Kyushu Journal of Mathematics 72, no. 2 (2018): 375–405. http://dx.doi.org/10.2206/kyushujm.72.375.

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36

Flemming, Jens. "A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations." Journal of Inverse and Ill-posed Problems 26, no. 5 (October 1, 2018): 639–46. http://dx.doi.org/10.1515/jiip-2017-0116.

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Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

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37

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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Yang, Xiaodan, and Yuwen Wang. "Some new perturbation theorems for generalized inverses of linear operators in Banach spaces." Linear Algebra and its Applications 433, no. 11-12 (December 2010): 1939–49. http://dx.doi.org/10.1016/j.laa.2010.07.008.

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39

Eveson, Simon P., and Roger D. Nussbaum. "Applications of the Birkhoff–Hopf theorem to the spectral theory of positive linear operators." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 3 (May 1995): 491–512. http://dx.doi.org/10.1017/s0305004100073321.

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This paper may be regarded as a sequel to our earlier paper [19], where we give an elementary and self-contained proof of a very general form of the Hopf theorem on order-preserving linear operators in partially ordered vector spaces (reproduced here as Theorem 1·1).Versions of this theorem and related ideas have been used by various authors to study both linear and nonlinear integral equations (Thompson [41], Bushell [9, 11], Potter [38, 39], Eveson [16, 17], Bushell and Okrasiriski [12, 13]); the convergence properties of nonlinear maps (Nussbaum [32, 33]); so-called DAD theorems (Borwein, Lewis and Nussbaum [8]) and in the proof of weak ergodic theorems (Fujimoto and Krause [20], Nussbaum [34]).

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40

Yu, Wenhuan. "Well-posedness of determining the source term of an elliptic equation." Bulletin of the Australian Mathematical Society 50, no. 3 (December 1994): 383–98. http://dx.doi.org/10.1017/s0004972700013502.

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In this paper the inverse problem for determining the source term of a linear, uniformly elliptic equation is investigated. The uniqueness of the inverse problem is proved under mild assumptions by use of the orthogonality method and an elimination method. The existence of the inverse problem is proved by means of the theory of solvable operators between Banach spaces, moreover, the continuous dependence of the solution to the inverse problem on measurement is also obtained.

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41

Huang, Qianglian. "On perturbations for oblique projection generalized inverses of closed linear operators in Banach spaces." Linear Algebra and its Applications 434, no. 12 (June 2011): 2468–74. http://dx.doi.org/10.1016/j.laa.2010.12.033.

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42

Wang, Yuwen, and Hao Zhang. "Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces." Linear Algebra and its Applications 426, no. 1 (October 2007): 1–11. http://dx.doi.org/10.1016/j.laa.2007.02.025.

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43

Lahrech, S. "Banach-Steinhaus type theorem in locally convex spaces for linear $\Sigma$-locally Lipschitzian operators." Miskolc Mathematical Notes 6, no. 1 (2005): 43. http://dx.doi.org/10.18514/mmn.2005.95.

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44

Faried, Nashat, Mohamed S. S. Ali, and Hanan H. Sakr. "On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces." Abstract and Applied Analysis 2020 (July 31, 2020): 1–13. http://dx.doi.org/10.1155/2020/5804957.

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Due to the difficulty of representing problem parameters fuzziness using the soft set theory, the fuzzy soft set is regarded to be more general and flexible than using the soft set. In this paper, we define the fuzzy soft linear operator T~ in the fuzzy soft Hilbert space H~ based on the definition of the fuzzy soft inner product space U~,·,·~ in terms of the fuzzy soft vector v~fGe modified in our work. Moreover, it is shown that ℂnA, ℝnA and ℓ2A are suitable examples of fuzzy soft Hilbert spaces and also some related examples, properties and results of fuzzy soft linear operators are introduced with proofs. In addition, we present the definition of the fuzzy soft orthogonal family and the fuzzy soft orthonormal family and introduce examples satisfying them. Furthermore, the fuzzy soft resolvent set, the fuzzy soft spectral radius, the fuzzy soft spectrum with its different types of fuzzy soft linear operators and the relations between those types are introduced. Moreover, the fuzzy soft right shift operator and the fuzzy soft left shift operator are defined with an example of each type on ℓ2A. In addition, it is proved, on ℓ2A, that the fuzzy soft point spectrum of fuzzy soft right shift operator has no fuzzy soft eigenvalues, the fuzzy soft residual spectrum of fuzzy soft right shift operator is equal to the fuzzy soft comparison spectrum of it and the fuzzy soft point spectrum of fuzzy soft left shift operator is the fuzzy soft open disk λ~<~1~. Finally, it is shown that the fuzzy soft Hilbert space is fuzzy soft self-dual in this generalized setting.

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PROTASOV, VLADIMIR. "REFINEMENT EQUATIONS AND CORRESPONDING LINEAR OPERATORS." International Journal of Wavelets, Multiresolution and Information Processing 04, no. 03 (September 2006): 461–74. http://dx.doi.org/10.1142/s0219691306001385.

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Refinement equations of the type [Formula: see text] play an exceptional role in the theory of wavelets, subdivision algorithms and computer design. It is known that the regularity of their compactly supported solutions (refinable functions) depends on the spectral properties of special N-dimensional linear operators T0, T1 constructed by the coefficients of the equation. In particular, the structure of kernels and of common invariant subspaces of these operators have been intensively studied in the literature. In this paper, we give a complete classification of the kernels and of all the root subspaces of T0 and T1, as well as of their common invariant subspaces. This result answers several open questions stated in the literature and clarifies the structure of the space spanned by the integer translates of refinable functions. This also leads to some results on the moduli of continuity of refinable functions and wavelets in various functional spaces. In particular, it is proved that the Hölder exponent of those functions is sharp whenever it is not an integer.

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Cichoń, Mieczyław. "Differential inclusions and abstract control problems." Bulletin of the Australian Mathematical Society 53, no. 1 (February 1996): 109–22. http://dx.doi.org/10.1017/s0004972700016774.

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We prove an existence theorem for differential inclusions in Banach spaces. Here {A (t): t ∈ [0,T]} is a family of linear operators generating a continuous evolution operator K (t, s). We concentrate on maps F with F (t,·) weakly sequentially hemi-continuous.Moreover, we show a compactness of the set of all integral solutions of the above problem. These results are also applied to a semilinear optimal control problem. Some corollaries, important in the theory of optimal control, are given too. We extend in several ways theorems existing in the literature.

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Gabeleh, Moosa, Deepesh Kumar Patel, Pradip Ramesh Patle, and Manuel De La Sen. "Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems." Open Mathematics 19, no. 1 (January 1, 2021): 450–69. http://dx.doi.org/10.1515/math-2021-0033.

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Abstract This work intends to treat the existence of mild solutions for the Hilfer fractional hybrid differential equation (HFHDE) with linear perturbation of first and second type in partially ordered Banach spaces. First, we establish the results concerning the actuality of fixed point of sum and product of operators via the concepts of measure of noncompactness and simulation functions in partially ordered spaces. Then combining these fixed point theorems with the concepts in fractional calculus, new existence results for mild solutions of HFHDE’s are established. Furthermore, the presented fixed point results and existence results improve and extend the present state-of-art in the literature. Competent examples in support of theory are illustrated for better understanding.

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DANERS, DANIEL, and JOCHEN GLÜCK. "THE ROLE OF DOMINATION AND SMOOTHING CONDITIONS IN THE THEORY OF EVENTUALLY POSITIVE SEMIGROUPS." Bulletin of the Australian Mathematical Society 96, no. 2 (March 29, 2017): 286–98. http://dx.doi.org/10.1017/s0004972717000260.

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We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

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Huang, Qianglian, and Wenxiao Zhai. "Perturbations and expressions for generalized inverses in Banach spaces and Moore–Penrose inverses in Hilbert spaces of closed linear operators." Linear Algebra and its Applications 435, no. 1 (July 2011): 117–27. http://dx.doi.org/10.1016/j.laa.2011.01.008.

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MATUCCI, S. "EXISTENCE, UNIQUENESS AND ASYMPTOTIC BEHAVIOUR FOR A MULTI-STAGE EVOLUTION PROBLEM OF AN AGE-STRUCTURED POPULATION." Mathematical Models and Methods in Applied Sciences 05, no. 08 (December 1995): 1013–41. http://dx.doi.org/10.1142/s021820259500053x.

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The theory of semigroups of bounded linear operators in L1-like Banach spaces is employed to show the existence and uniqueness of a triple of non-negative functions representing the solution of a model of evolution of a population distributed in three stages of individuals, each of them being dependent on his own age and one stage also on mother age. The differential equations involved are linear and are coupled through linear boundary conditions. The detailed study of the spectrum of the linear, closed operator related to the system allows to obtain estimates for the asymptotic time behaviour of the solution; these results may be considered as a generalization of the Sharpe-Lotka theorem. Finally, the analytical structure of the solution is given.

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