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Journal articles on the topic 'Compact linear operator'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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1

Et al., Kider. "Properties of Fuzzy Compact Linear Operators on Fuzzy Normed Spaces." Baghdad Science Journal 16, no. 1 (2019): 0104. http://dx.doi.org/10.21123/bsj.2019.16.1.0104.

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In this paper the definition of fuzzy normed space is recalled and its basic properties. Then the definition of fuzzy compact operator from fuzzy normed space into another fuzzy normed space is introduced after that the proof of an operator is fuzzy compact if and only if the image of any fuzzy bounded sequence contains a convergent subsequence is given. At this point the basic properties of the vector space FC(V,U)of all fuzzy compact linear operators are investigated such as when U is complete and the sequence ( ) of fuzzy compact operators converges to an operator T then T must be fuzzy compact. Furthermore we see that when T is a fuzzy compact operator and S is a fuzzy bounded operator then the composition TS and ST are fuzzy compact operators. Finally, if T belongs to FC(V,U) and dimension of V is finite then T is fuzzy compact is proved.

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2

Bracic, Janko. "Arens regularity andweakly compact operators." Filomat 32, no. 14 (2018): 4993–5002. http://dx.doi.org/10.2298/fil1814993b.

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We explore the relation between Arens regularity of a bilinear operator and the weak compactness of the related linear operators. Since every bilinear operator has natural factorization through the projective tensor product a special attention is given to Arens regularity of the tensor operator. We consider topological centers of a bilinear operator and we present a few results related to bilinear operators which can be approximated by linear operators.

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3

Gumenchuk, A., I. Krasikova, and M. Popov. "On linear sections of orthogonally additive operators." Matematychni Studii 58, no. 1 (2022): 94–102. http://dx.doi.org/10.30970/ms.58.1.94-102.

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Our first result asserts that, for linear regular operators acting from a Riesz space with the principal projection property to a Banach lattice with an order continuous norm, the $C$-compactness is equivalent to the $AM$-compactness. Next we prove that, under mild assumptions, every linear section of a $C$-compact orthogonally additive operator is $AM$-compact, and every linear section of a narrow orthogonally additive operator is narrow.

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4

Solikhin, Solikhin, Y. D. Sumanto, Susilo Hariyanto, and Abdul Aziz. "OPERATOR PADA RUANG FUNGSI TERINTEGRAL DUNFORD." Journal of Fundamental Mathematics and Applications (JFMA) 1, no. 2 (2018): 110. http://dx.doi.org/10.14710/jfma.v1i2.17.

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An integral Dunford and an operator on Dunford integrable functional space have discussed in this article. The results were shown that the Dunford integrable functional space was a linear function. For every Dunford integrable function on a closed interval, there is an operator that is linear bounded and weak compact operator, whereas its adjoin operator is also linear bounded and weak compact. An operator is weak compact if and only if its adjoin operator is weak compact. Furthermore, the norm of this operator was equal to the norm of its adjoin operator.

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5

Sah, Nagendra Pd. "About Riesz theory of compact operators." BIBECHANA 9 (December 10, 2012): 126–29. http://dx.doi.org/10.3126/bibechana.v9i0.7186.

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In this paper, it is shown that every compact operators are bounded and continuous. The bounded and continuous properties of an operator is sufficient for a Riesz operator. For mapping T: K-?I in normed linear space with some extended [1] properties, T becomes compact. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7186 BIBECHANA 9 (2013) 126-129

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6

Gok, Omer. "On regular operators on Banach lattices." Acta et commentationes: Ştiinţe Exacte şi ale Naturii 14, no. 2 (2023): 53–56. http://dx.doi.org/10.36120/2587-3644.v14i2.53-56.

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Let $E$ and $F$ be Banach lattices and $X$ and $Y$ be Banach spaces. A linear operator $T: E \rightarrow F$ is called regular if it is the difference of two positive operators. $L_{r}(E,F)$ denotes the vector space of all regular operators from $E$ into $F$. A continuous linear operator $T: E \rightarrow X$ is called $M$-weakly compact operator if for every disjoint bounded sequence $(x_{n})$ in $E$, we have $lim_{n \rightarrow\infty} \| Tx_{n} \| =0$. $W^{r}_{M}(E,F)$ denotes the regular $M$-weakly compact operators from $E$ into $F$. This paper is devoted to the study of regular operators and $M$-weakly compact operators on Banach lattices. We show that $F$ has a b-property if and only if $L_{r}(E,F)$ has b-property. Also, $W^{r}_{M}(E,F)$ is a $KB$-space if and only if $F$ is a $KB$-space.

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7

Keten Çopur, Ayşegül, and Ramazan İnal. "Lipschitz Operators Associated with Weakly $p$-Compact and Unconditionally $p$-Compact Sets." Journal of Advanced Research in Natural and Applied Sciences 10, no. 3 (2024): 530–41. http://dx.doi.org/10.28979/jarnas.1451630.

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The study introduces different categories of Lipschitz operators linked with weakly $p$-compact and unconditionally $p$-compact sets. It explores some properties of these operator classes derived from linear operators associated with these sets and examines their interconnections. Additionally, it denotes that these classes are extensions of the related linear operators. Moreover, the study evaluates the concept of majorization by scrutinizing both newly obtained and pre-existing results and draws some conclusions based on these findings. The primary method used to obtain the results in the study is the linearization of Lipschitz operators through the Lipschitz-free space constructed over a pointed metric space.

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8

Frank, Michael. "Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules." Journal of K-theory 2, no. 3 (2008): 453–62. http://dx.doi.org/10.1017/is008001031jkt035.

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AbstractC*-algebras A of compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every bounded A-linear operator between two Hilbert A-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between Hilbert A-modules are orthogonal summands, (iii) the images of all bounded A-linear operators with closed range between Hilbert A-modules are orthogonal summands, and (iv) for every Hilbert A-module every Hilbert A-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.

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9

Garbouj, Zied. "Some generalizations of ascent and descent for linear operators." Arabian Journal of Mathematics 10, no. 2 (2021): 367–93. http://dx.doi.org/10.1007/s40065-021-00328-y.

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AbstractThe purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.

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10

Song, Xueli, and Jigen Peng. "On strong convex compactness property of spaces of nonlinear operators." Bulletin of the Australian Mathematical Society 74, no. 3 (2006): 411–18. http://dx.doi.org/10.1017/s0004972700040466.

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The strong convex compactness property is important for property persistence of operator semigroups under perturbations. It has been investigated in the linear setting. In this paper, we are concerned with the property in the nonlinear setting. We prove that the following spaces of (nonlinear) operators enjoy the strong convex compactness property: the space of compact operators, the space of completely continuous operators, the space of weakly compact operators, the space of conditionally weakly compact operators, the space of weakly completely continuous operators, the space of demicontinuous operators, the space of weakly continuous operators and the space of strongly continuous operators. Moreover, we prove the property persistence of operator semigroups under nonlinear perturbation.

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11

Zayood, Karla. "A Study on Compact Operators in Locally K -Convex Spaces." Galoitica: Journal of Mathematical Structures and Applications 5, no. 2 (2023): 08–11. http://dx.doi.org/10.54216/gjmsa.050201.

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In this paper we give an equivalent definition of continuous and compact linear operators by using orthogonal bases in non-archimedean locally K - convex spaces. We also show that if E is a space and F is a semi-Montel space, then every continuous linear operator T:E→F is compact.

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12

Chafika, Belabbaci. "New Characterizations of the Jeribi Essential Spectrum." International Journal of Analysis and Applications 21 (October 4, 2023): 109. http://dx.doi.org/10.28924/2291-8639-21-2023-109.

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In this paper, we give several characterizations of the Jeribi essential spectrum of a bounded linear operator defined on a Banach space by using the notion of almost weakly compact operators. As a consequence, we prove the stability of the Jeribi essential spectrum under compact perturbations. Furthermore, some characterizations of the Jeribi essential spectra of 3×3 upper triangular block operator matrix are also given.

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13

Latrach, Khalid, and J. Martin Paoli. "Relatively compact-like perturbations, essential spectra and application." Journal of the Australian Mathematical Society 77, no. 1 (2004): 73–90. http://dx.doi.org/10.1017/s1446788700010168.

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AbstractThe purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. IfAdenotes a closed densely defined linear operator on a Banach spaceX, our approach consists principally in considering the class ofA-closable operators which, regarded as operators in ℒ(XA,X) (whereXAdenotes the domain ofAequipped with the graph norm), are contained in the set ofA-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

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14

Cho, Chong-Man. "A Note on M-Ideals of Compact operators." Canadian Mathematical Bulletin 32, no. 4 (1989): 434–40. http://dx.doi.org/10.4153/cmb-1989-062-8.

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AbstractSuppose X and Y are closed subspaces of (ΣXn)p and (ΣYn)q (1 < p ≦ q < ∞, dim Xn < ∞, dimYn < ∞), respectively. If K(X, Y), the space of the compact linear operators from X to Y, is dense in L(X, Y), the space of the bounded linear operators from X to Y, in the strong operator topology, then K(X, Y) is an M-ideal in L(X, Y).

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15

Bajryacharya, Prakash Muni, and Keshab Raj Phulara. "Extension of Bounded Linear Operators." Journal of Advanced College of Engineering and Management 2 (November 29, 2016): 11. http://dx.doi.org/10.3126/jacem.v2i0.16094.

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<p>In this article the problem entitled when does every member of a class of operators T : E → Y admit an extension operator T : X → Y in different approaches like injective spaces, separable injective spaces, the class of compact operators and extension Into C(K ) spaces has-been studied.</p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol. 2, 2016, page: 11-13</p>

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16

Obaid, Rasha Hamzah, Aseel Ameen Harbi, and Zainab Mohammed Najm. "Compact operator of peridynamic model." Journal of Interdisciplinary Mathematics 28, no. 4 (2025): 1657–61. https://doi.org/10.47974/jim-2201.

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17

Bahreini, Manijeh, Elizabeth Bator, and Ioana Ghenciu. "Complemented Subspaces of Linear Bounded Operators." Canadian Mathematical Bulletin 55, no. 3 (2012): 449–61. http://dx.doi.org/10.4153/cmb-2011-097-2.

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AbstractWe study the complementation of the space W(X, Y) of weakly compact operators, the space K(X, Y) of compact operators, the space U(X, Y) of unconditionally converging operators, and the space CC(X, Y) of completely continuous operators in the space L(X, Y) of bounded linear operators from X to Y. Feder proved that if X is infinite-dimensional and c0 ↪ Y, then K(X, Y) is uncomplemented in L(X, Y). Emmanuele and John showed that if c0 ↪ K(X, Y), then K(X, Y) is uncomplemented in L(X, Y). Bator and Lewis showed that if X is not a Grothendieck space and c0 ↪ Y, then W(X, Y) is uncomplemented in L(X, Y). In this paper, classical results of Kalton and separably determined operator ideals with property (∗) are used to obtain complementation results that yield these theorems as corollaries.

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18

Sharma, Mami, and Debajit Hazarika. "Fuzzy Bounded Linear Operator in Fuzzy Normed Linear Spaces and its Fuzzy Compactness." New Mathematics and Natural Computation 16, no. 01 (2020): 177–93. http://dx.doi.org/10.1142/s1793005720500118.

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In this paper, we first investigate the relationship between various notions of fuzzy boundedness of linear operators in fuzzy normed linear spaces. We also discuss the fuzzy boundedness of fuzzy compact operators. Furthermore, the spaces of fuzzy compact operators have been studied.

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19

Guo, Xin, and Maofa Wang. "Compact linear combinations of composition operators over the unit ball." Journal of Operator Theory 88, no. 1 (2022): 61–84. http://dx.doi.org/10.7900/jot.2020nov28.2310.

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In this paper, we study the compactness of any finite linear combination of composition operators with general symbols on weighted Bergman spaces over the unit ball in terms of a power type criterion. The strategy of the proof involves the subtle connection of composition operator theory between weighted Bergman spaces and Korenblum spaces.

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20

Lim, Teck-Cheong. "Dynamics of a Compact Operator." ISRN Mathematical Analysis 2011 (February 16, 2011): 1–27. http://dx.doi.org/10.5402/2011/281737.

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21

Aydın, Abdullah, and Svetlana Gorokhova. "Multiplicative order compact operators between vector lattices and Riesz algebras." Filomat 38, no. 19 (2024): 6743–51. https://doi.org/10.2298/fil2419743a.

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In this paper, we present and examine the concept of multiplicative order compact operators from vector lattices to Riesz algebras. Specifically, a linear operator T from a vector lattice X to an Riesz algebra E is deemed omo-compact, if every net x? in an o-bounded subset of X possesses a subnet x?? such that Tx??mo--? y for some y ? E. Moreover, we introduce and investigate omo-M-and omo-L-weakly compact operators.

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22

Gil’, Michael. "An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components." Filomat 30, no. 13 (2016): 3415–25. http://dx.doi.org/10.2298/fil1613415g.

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Let H be a linear unbounded operator in a separable Hilbert space. It is assumed the resolvent of H is a compact operator and H ? H* is a Schatten - von Neumann operator. Various integro-differential operators satisfy these conditions. Under certain assumptions it is shown that H is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

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23

Abdulkarim, Eman. "A Study of a Basic Sufficient Condition for the Compactness of Linear Operators on Banach Spaces." Journal of Pure & Applied Sciences 24, no. 1 (2025): 67–72. https://doi.org/10.51984/jopas.v24i1.3690.

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This work examines conditions for the compactness of linear operators in Banach spaces, a key question in functional analysis with broad applications. Compactness ensures that bounded sets are mapped to relatively compact sets, making it a fundamental tool in the study of operators on infinite-dimensional spaces. This paper provides a detailed investigation of three conditions ensuring compactness: total boundedness, finite dimensionality, and completeness. It addresses a significant gap in the literature and provides a sound theoretical framework. This paper aims to (1) explain the connection between finite dimensionality and total boundedness as conditions for compactness, (2) present unified sufficient conditions for the compactness of linear operators with proofs, and (3) offer new insights into operator theory for broader mathematical applications. This study employs advanced functional analytic techniques to deduce and validate these well-founded conditions. This work addresses gaps in compact operator theory, with implications for quantum physics, differential equations, and numerical analysis. By enhancing the understanding of Banach spaces and operator theory, this study may inspire further exploration of their properties.

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24

B.,, Sanooj, and Vinodkumar P. B. "Li-Yorke Chaotic Eigen Set of Direct Sum of Linear Operators." PROOF 2 (September 13, 2022): 165–68. http://dx.doi.org/10.37394/232020.2022.2.21.

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The Li-Yorke chaotic eigen set of an operator consisting of all λ’s such that T- λI is Li-Yorke chaotic. In this paper, the Li-Yorke chaotic eigen set of the direct sum of linear operators is found to be the union of Li- Yorke chaotic sets of the corresponding operators. Also we discuss about the Li-Yorke chaotic eigen set of compact operators, normal operators and self adjoint operators.

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25

Malkowsky, E., and A. Alotaibi. "Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces." Journal of Function Spaces 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/196489.

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We study the spacesw0p,wp, andw∞pof sequences that are strongly summable to 0, summable, and bounded with indexp≥1by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaceswpinto the spacesw∞1,w1, andw01. We also give estimates for the operator norm and the Hausdorff measure of noncompactness of such operators. Finally we apply our results to characterize the classes of compact bounded linear operators fromw0pandwpintow01andw1.

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26

Kittaneh, Fuad. "Inequalities for the Schatten p-norm II." Glasgow Mathematical Journal 29, no. 1 (1987): 99–104. http://dx.doi.org/10.1017/s0017089500006716.

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This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

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27

Anastassiou, George A. "Neural Networks as Positive Linear Operators." Mathematics 13, no. 7 (2025): 1112. https://doi.org/10.3390/math13071112.

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Basic neural network operators are interpreted as positive linear operators and the related general theory applies to them. These operators are induced by a symmetrized density function deriving from the parametrized and deformed hyperbolic tangent activation function. I explore the space of continuous functions on a compact interval of the real line to the reals. I study quantitatively the rate of convergence of these neural network operators to the unit operator. The studied inequalities involve the modulus of continuity of the function under approximation or its derivative. I produce uniform and Lp, p≥1, approximation results via these inequalities. The convexity of functions is also taken into consideration.

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28

Khaleefah, Sabah A., and Buthainah A. A. Ahmed. "Spectrum of Soft Compact Linear Operator with Properties." Journal of Physics: Conference Series 1530 (May 2020): 012107. http://dx.doi.org/10.1088/1742-6596/1530/1/012107.

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29

Rakhimova, Alsu Il'darorna. "Hypercyclic and chaotic operators in space of functions analytic in domain." Ufa Mathematical Journal 16, no. 3 (2024): 84–91. https://doi.org/10.13108/2024-16-3-84.

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We consider the space $H(\Omega)$ of functions analytic in a simply connected domain $\Omega$ in the complex plane equipped with the topology of uniform convergence on compact sets. We study issues on hypercyclicity, chaoticity and frequently hypercyclic for some operators in this space. We prove that a linear continuous operator in $H(\Omega),$ which commutes with the differentiation operator, is hypercyclic. We also show that this operator is chaotic and frequently hypercyclic in $H(\Omega).$

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30

Gau, Haw-Long, Jyh-Shyang Jeang, and Nagi-Ching Wong. "Biseparating linear maps between continuous vector-valued function spaces." Journal of the Australian Mathematical Society 74, no. 1 (2003): 101–10. http://dx.doi.org/10.1017/s1446788700003153.

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AbstractLet X, Y be compact Hausdorff spaces and E, F be Banach spaces. A linear map T: C(X, E) → C(Y, F) is separating if Tf, Tg have disjoint cozeroes whenever f, g have disjoint cozeroes. We prove that a biseparating linear bijection T (that is, T and T-1 are separating) is a weighted composition operator Tf = h · f o ϕ. Here, h is a function from Y into the set of invertible linear operators from E onto F, and ϕ, is a homeomorphism from Y onto X. We also show that T is bounded if and only if h(y) is a bounded operator from E onto F for all y in Y. In this case, h is continuous with respect to the strong operator topology.

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31

Ricker, W. "Spectral operators and weakly compact homomorphisms in a class of Banach Spaces." Glasgow Mathematical Journal 28, no. 2 (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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32

Zabeti, O. "Тип теоремы Кренгеля для компактных операторов между локально плотными векторными решетками". Владикавказский математический журнал 25, № 3 (2023): 76–80. http://dx.doi.org/10.46698/g6863-7709-2981-j.

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Suppose $X$ and $Y$ are locally solid vector lattices. A linear operator $T:X\to Y$ is said to be $nb$-compact provided that there exists a zero neighborhood $U\subseteq X$, such that $\overline{T(U)}$ is compact in $Y$; $T$ is $bb$-compact if for each bounded set $B\subseteq X$, $\overline{T(B)}$ is compact. These notions are far from being equivalent, in general. In this paper, we introduce the notion of a locally solid $AM$-space as an extension for $AM$-spaces in Banach lattices. With the aid of this concept, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices. This extends [1, Theorem 5.7]} (established for compact operators between Banach lattices) to different classes of compact operators between locally solid vector lattices.

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33

Burtnyak, I., I. Chernega, V. Hladkyi, O. Labachuk, and Z. Novosad. "Application of symmetric analytic functions to spectra of linear operators." Carpathian Mathematical Publications 13, no. 3 (2021): 701–10. http://dx.doi.org/10.15330/cmp.13.3.701-710.

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The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.

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34

Brattka, Vasco. "Effective representations of the space of linear bounded operators." Applied General Topology 4, no. 1 (2003): 115. http://dx.doi.org/10.4995/agt.2003.2014.

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<p>Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.</p>

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35

Abanin, Alexander V., and Julia V. Korablina. "Compactness of Linear Operators on Quasi-Banach Spaces of Holomorphic Functions." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4-1 (216-1) (December 28, 2022): 83–89. http://dx.doi.org/10.18522/1026-2237-2022-4-1-83-89.

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We state conditions under which some classical operators acting from abstract quasi-Banach spaces of functions holomorphic in a plain domain into a weighted space of the same functions with sup-norm are compact. It is obtained abstract criteria for the compactness of a linear operator on an arbitrary quasi-Banach space which are stated in terms of delta-functions and formulate their realizations for both classical and generalized Fock spaces. The above results are applied to the weighted composition operator. It is established some conditions for the compactness of this operator which are given in terms of norms of delta-functions in the corresponding dual spaces. These results are essential generalizations of the known Zorboska’s ones. Namely, we significantly extended the class of weighted spaces of holomorphic functions with uniform norms for which one can state some conditions for the compactness of an arbitrary linear operator or the weighted composition operator.

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36

BOTTAZZI, T., C. CONDE, M. S. MOSLEHIAN, P. WÓJCIK, and A. ZAMANI. "ORTHOGONALITY AND PARALLELISM OF OPERATORS ON VARIOUS BANACH SPACES." Journal of the Australian Mathematical Society 106, no. 2 (2018): 160–83. http://dx.doi.org/10.1017/s1446788718000150.

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We present some properties of orthogonality and relate them with support disjoint and norm inequalities in $p$-Schatten ideals. In addition, we investigate the problem of characterization of norm-parallelism for bounded linear operators. We consider the characterization of the norm-parallelism problem in $p$-Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm-parallel to the identity operator. Finally, we give some equivalence assertions about the norm-parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

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37

Shahi, Mahendra. "Some special characterisations of Fredholm operators in Banach space." BIBECHANA 11 (May 10, 2014): 169–74. http://dx.doi.org/10.3126/bibechana.v11i0.10399.

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A bounded linear operator which has a finite index and which is defined on a Banach space is often referred to in the literature as a Fredholm operator. Fredholm operators are important for a variety of reasons, one being the role that their index plays in global analysis. The aim of this paper is to prove the spectral theorem for compact operators in refined form and to describe some properties of the essential spectrum of general bounded operators by the use of the theorem of Fredholm operators. For this, we have analysed the Fredholm operator which is defined in a Banach space for some special characterisations. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10399 BIBECHANA 11(1) (2014) 169-174

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38

Oikhberg, Timur. "THE OPERATOR SHIFT SPACE." Proceedings of the Edinburgh Mathematical Society 51, no. 1 (2008): 229–63. http://dx.doi.org/10.1017/s0013091505000337.

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AbstractWe construct and examine an operator space $X$, isometric to $\ell_2$, such that every completely bounded map from its subspace $Y$ into $X$ is a compact perturbation of a linear combination of multiples of a shift of given multiplicity and their adjoints. Moreover, every completely bounded map on $X$ is a Hilbert–Schmidt perturbation of such a linear combination.

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39

Kittaneh, Fuad. "Inequalities for the Schatten P-norm." Glasgow Mathematical Journal 26, no. 2 (1985): 141–43. http://dx.doi.org/10.1017/s0017089500005905.

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Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1<p<∞, si(A)p <∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).

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40

Anastassiou, George A. "Generalized Logistic Neural Networks in Positive Linear Framework." Symmetry 17, no. 5 (2025): 746. https://doi.org/10.3390/sym17050746.

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Essential neural-network operators are interpreted as positive linear operators, and the related general theory applies to them. These operators are induced by a symmetrized density function deriving from the parametrized and deformed A-generalized logistic activation function. We are acting on the space of continuous functions on a compact interval of real line to the reals. We quantitatively study the rate of convergence of these neural -network operators to the unit operator. Our inequalities involve the modulus of continuity of the function under approximation or its derivative. We produce uniform and Lp, p≥1 approximation results via these inequalities. The convexity of functions is also used to derive more refined results.

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41

GIL’, MICHAEL. "A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS." Journal of the Australian Mathematical Society 97, no. 3 (2014): 331–42. http://dx.doi.org/10.1017/s1446788714000354.

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AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

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42

Elazzouzi, Abdelhai, Khalil Ezzinbi, and Mohammed Kriche. "Periodic Solution for some Class of Linear Partial Differential Equation with infinite Delay using Semi-Fredholm perturbations." Nonautonomous Dynamical Systems 9, no. 1 (2022): 116–44. http://dx.doi.org/10.1515/msds-2022-0150.

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Abstract In this work, we study the existence of periodic solutions for a class of linear partial functional differential equations with infinite delay. Inspiring by an existing study, by applying the perturbation theory of semi-Fredholm operators, we introduce a suitable a priori estimate on the norm of the operator L to establish the periodicity of solutions in the case where the linear part is nondensely defined and satisfies the Hille-Yosida condition and without considering the exponential stability condition on the semigroup generated by the part of this operator on the closure of it’s domain. Moreover, in the special case where the linear part generates a strongly continuous semigroup and perturbed by a compact linear operator, we give some sufficient conditions to derive periodic solution from bounded ones. Finally, our theoretical results are illustrated by applications in both densely and nondensely cases.

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43

Lee, Keun Young, and Gwanghyun Jo. "The dual of a space of compact operators." AIMS Mathematics 9, no. 4 (2024): 9682–91. http://dx.doi.org/10.3934/math.2024473.

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<abstract><p>Let $ X $ and $ Y $ be Banach spaces. We provide the representation of the dual space of compact operators $ K(X, Y) $ as a subspace of bounded linear operators $ \mathcal{L}(X, Y) $. The main results are: (1) If $ Y $ is separable, then the dual forms of $ K(X, Y) $ can be represented by the integral operator and the elements of $ C[0, 1] $. (2) If $ X^{**} $ has the weak Radon-Nikodym property, then the dual forms of $ K(X, Y) $ can be represented by the trace of some tensor products.</p></abstract>

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44

Montgomery-Smith, Stephen, and Paulette Saab. "p-Summing operators on injective tensor products of spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 120, no. 3-4 (1992): 283–96. http://dx.doi.org/10.1017/s0308210500032145.

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SynopsisLet X, Y and Z be Banach spaces, and let Πp (Y, Z) (1 ≦ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a ℒ∞-space, then a bounded linear operator is 1-summing if and only if a naturally associated operator T#: X → Πl (Y, Z) is 1-summing. This result need not be true if X is not a ℒ∞-space. For p > 1, several examples are given with X = C[0, 1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on whose associated operator T# is 2-summing, but for all N ∈ N, there exists an N-dimensional subspace U of such that T restricted to U is equivalent to the identity operator on . Finally, we show that there is a compact Hausdorff space K and a bounded linear operator for which T#: C(K) → Π1 (l1, l2) is not 2-summing.

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45

Lohaj, Muhib, and Shqipe Lohaj. "Quasi-Diagonal Operators." Sarajevo Journal of Mathematics 6, no. 2 (2024): 229–35. http://dx.doi.org/10.5644/sjm.06.2.07.

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Let $H$ be a separable complex Hilbert space and let $B(H)$ denote the algebra of all bounded linear operators on $H.$ If $T$ is a quasi-normal Fredholm operator we prove that $TT^*\in (QD)(P_n)$ if and only if $T^*T\in (QD)(P_n).$ We also show that if $T$ is quasi-normal and $T(T^*T)$ is quasi-diagonal with respect to any sequence $(P_n)$ in $PF(H),$ such that $P_n\rightarrow I$ strongly, then $T=N+K,$ where $N$ is a normal operator and $K$ is a compact operator. 2000 Mathematics Subject Classification. 47Bxx, 47B20

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46

Drewnowski, Lech. "Copies of l∞ in an operator space." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (1990): 523–26. http://dx.doi.org/10.1017/s0305004100069401.

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Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the followingTheorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

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47

TALPONEN, JARNO. "OPERATORS ON C0(L,X) WHOSE RANGE DOES NOT CONTAIN c0." Bulletin of the Australian Mathematical Society 77, no. 3 (2008): 515–20. http://dx.doi.org/10.1017/s0004972708000646.

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AbstractThis paper contains two results: (a) if $\mathrm {X}\neq \{0\}$ is a Banach space and (L,τ) is a nonempty locally compact Hausdorff space without isolated points, then each linear operator T:C0(L,X)→C0(L,X) whose range does not contain an isomorphic copy of c00 satisfies the Daugavet equality $\|\mathbf {I}+T\|=1+\|T\|$; (b) if Γ is a nonempty set and X and Y are Banach spaces such that X is reflexive and Y does not contain c0 isomorphically, then any continuous linear operator T:c0(Γ,X)→Y is weakly compact.

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48

Beanland, Kevin, and Ryan M. Causey. "Genericity and Universality for Operator Ideals." Quarterly Journal of Mathematics 71, no. 3 (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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49

Redner, Oliver. "DISCRETE APPROXIMATION OF NON-COMPACT OPERATORS DESCRIBING CONTINUUM-OF-ALLELES MODELS." Proceedings of the Edinburgh Mathematical Society 47, no. 2 (2004): 449–72. http://dx.doi.org/10.1017/s0013091503000476.

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AbstractWe consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.AMS 2000 Mathematics subject classification: Primary 47A58; 45C05. Secondary 47B34

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50

He, Jia Wei, and Yong Zhou. "Stability analysis for discrete time abstract fractional differential equations." Fractional Calculus and Applied Analysis 24, no. 1 (2021): 307–23. http://dx.doi.org/10.1515/fca-2021-0013.

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Abstract In this paper, we consider a discrete-time fractional model of abstract form involving the Riemann-Liouville-like difference operator. On account of the C 0-semigroups generated by a closed linear operator A and based on a distinguished class of sequences of operators, we show the existence of stable solutions for the nonlinear Cauchy problem by means of fixed point technique and the compact method. Moreover, we also establish the Ulam-Hyers-Rassias stability of the proposed problem. Two examples are presented to explain the main results.

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