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Journal articles on the topic 'Operator spaces (= matricially normed spaces)'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Junge, Marius, Christian le Merdy, and Lahcene Mezrag. "$L^p$-matricially normed spaces and operator space valued Schatten spaces." Indiana University Mathematics Journal 56, no. 5 (2007): 2511–34. http://dx.doi.org/10.1512/iumj.2007.56.3070.

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2

Effros, Edward, and Zhong-Jin Ruan. "On matricially normed spaces." Pacific Journal of Mathematics 132, no. 2 (April 1, 1988): 243–64. http://dx.doi.org/10.2140/pjm.1988.132.243.

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3

Bekjan, Turdebek N. "ON Lp -MATRICIALLY NORMED SPACES." Acta Mathematica Scientia 25, no. 4 (October 2005): 681–86. http://dx.doi.org/10.1016/s0252-9602(17)30208-4.

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4

Ramasamy, C. T., and C. Ganesa Moorthy. "Strictly unbounded operator on normed spaces." Asian-European Journal of Mathematics 08, no. 03 (September 2015): 1550056. http://dx.doi.org/10.1142/s1793557115500564.

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5

Forouzanfar, AM, S. Khorshidvandpour, and Z. Bahmani. "Uniformly invariant normed spaces." BIBECHANA 10 (October 31, 2013): 31–33. http://dx.doi.org/10.3126/bibechana.v10i0.7555.

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In this work, we introduce the concepts of compactly invariant and uniformly invariant. Also we define sometimes C-invariant closed subspaces and then prove every m-dimensional normed space with m > 1 has a nontrivial sometimes C-invariant closed subspace. Sequentially C-invariant closed subspaces are also introduced. Next, An open problem on the connection between compactly invariant and uniformly invariant normed spaces has been posed. Finally, we prove a theorem on the existence of a positive operator on a strict uniformly invariant Hilbert space. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.7555 BIBECHANA 10 (2014) 31-33

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6

Taghavi, Ali, and Majid Mehdizadeh. "Adjoint Operator In Fuzzy Normed Linear Spaces." Journal of Mathematics and Computer Science 02, no. 03 (April 10, 2011): 453–58. http://dx.doi.org/10.22436/jmcs.02.03.08.

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7

Sukochev, Fedor. "Completeness of quasi-normed symmetric operator spaces." Indagationes Mathematicae 25, no. 2 (March 2014): 376–88. http://dx.doi.org/10.1016/j.indag.2012.05.007.

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8

ZHU, YUANGUO. "ON PARA-NORMED SPACE WITH FUZZY VARIABLES BASED ON EXPECTED VALUED OPERATOR." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 16, no. 01 (February 2008): 95–106. http://dx.doi.org/10.1142/s0218488508005066.

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Fuzzy variables are functions from credibility spaces to the set of real numbers. The set of fuzzy variables is a linear space with the classic operations of addition and multiplication by numbers. Its subspace formed by fuzzy variables with finite pth absolute moments is showed to be a complete para-normed space. The concept of para-normed space is novel, and is an extension of normed space. It is seen that most properties of normed spaces hold in para-normed spaces. Also some useful inequalities in para-normed spaces are obtained.

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9

Nakasho, Kazuhisa. "Multilinear Operator and Its Basic Properties." Formalized Mathematics 27, no. 1 (April 1, 2019): 35–45. http://dx.doi.org/10.2478/forma-2019-0004.

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Summary In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.

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10

F. Al-Mayahi, Noori, and Abbas M. Abbas. "Some Properties of Spectral Theory in Fuzzy Hilbert Spaces." Journal of Al-Qadisiyah for computer science and mathematics 8, no. 2 (August 7, 2017): 1–7. http://dx.doi.org/10.29304/jqcm.2016.8.2.27.

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In this paper we give some definitions and properties of spectral theory in fuzzy Hilbert spaces also we introduce definitions Invariant under a linear operator on fuzzy normed spaces and reduced linear operator on fuzzy Hilbert spaces and we prove theorms related to eigenvalue and eigenvectors ,eigenspace in fuzzy normed , Invariant and reduced in fuzzy Hilbert spaces and show relationship between them.

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11

Nemesh, N. T., and S. M. Shteiner. "METRIC AND TOPOLOGICAL FREEDOM FOR SEQUENTIAL OPERATOR SPACES." Vestnik of Samara University. Natural Science Series 20, no. 10 (May 29, 2017): 55–67. http://dx.doi.org/10.18287/2541-7525-2014-20-10-55-67.

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In 2002 Anselm Lambert in his PhD thesis [1] introduced the deﬁnition of sequential operator space and managed to establish a considerable amount of analogs of corresponding results in operator space theory. Informally speaking, the category of sequential operator spaces is situated ”between” the categories of normed and operator spaces. This article aims to describe free and cofree objects for diﬀerent versions of sequential operator space homology. First of all, we will show that duality theory in above-mentioned category is in many respects analogous to that in the category of normed spaces. Then, based on those results, we will give a full characterization of both metric and topological free and cofree objects.

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12

Nakasho, Kazuhisa, and Yasunari Shidama. "Continuity of Multilinear Operator on Normed Linear Spaces." Formalized Mathematics 27, no. 1 (April 1, 2019): 61–65. http://dx.doi.org/10.2478/forma-2019-0006.

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Summary In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism [4], [1] and [2]. In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space. In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its whole domain. We referred to [5], [11], [8], [9] in this formalization.

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13

Hazarika, Bipan. "On Paranormed Ideal Convergent Generalized Difference Strongly Summable Sequence Spaces Defined over n-Normed Spaces." ISRN Mathematical Analysis 2011 (June 16, 2011): 1–17. http://dx.doi.org/10.5402/2011/317423.

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We introduce a new class of ideal convergent (shortly I-convergent) sequence spaces using an Orlicz function and difference operator of order defined over the n-normed spaces. We investigate these spaces for some linear topological structures. These investigations will enhance the acceptability of the notion of n-norm by giving a way to construct different sequence spaces with elements in n-normed spaces. We also give some relations related to these sequence spaces.

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14

Bachir, A., and A. Segres. "Numerical range and orthogonality in normed spaces." Filomat 23, no. 1 (2009): 21–41. http://dx.doi.org/10.2298/fil0901021b.

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Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V (T)A which generalize those given by J.P. William and J.P. Stamplfli. Finally, we give a positive answer of the Mathieu's question. .

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15

BARNES, BRUCE A. "BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY." Glasgow Mathematical Journal 49, no. 1 (January 2007): 145–54. http://dx.doi.org/10.1017/s0017089507003503.

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Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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16

Nakasho, Kazuhisa, Yuichi Futa, and Yasunari Shidama. "Continuity of Bounded Linear Operators on Normed Linear Spaces." Formalized Mathematics 26, no. 3 (October 1, 2018): 231–37. http://dx.doi.org/10.2478/forma-2018-0021.

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Summary In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized. In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.

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17

Et al., Kider. "Properties of Fuzzy Compact Linear Operators on Fuzzy Normed Spaces." Baghdad Science Journal 16, no. 1 (March 11, 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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18

Chang, Shih-Sen, Yeol Je Cho, and Fan Wang. "On the existence and uniqueness problems of solutions for set-valued and single-valued nonlinear operator equations in probabilistic normed spaces." International Journal of Mathematics and Mathematical Sciences 17, no. 2 (1994): 389–96. http://dx.doi.org/10.1155/s0161171294000530.

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In this paper, we introduce the concept of more general probabilistic contractors in probabilistic normed spaces and show the existence and uniqueness of solutions for set-valued and single-valued nonlinear operator equations in Menger probabilistic normed spaces.

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19

Al-Saphory, Raheam A. Mansor, and Mahmood K. Jasim. "Quasi-Compactness in Quasi-Banach Spaces." JOURNAL OF ADVANCES IN MATHEMATICS 4, no. 1 (November 9, 2013): 325–41. http://dx.doi.org/10.24297/jam.v4i1.7227.

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Quasi-compactness in a quasi-Banach space for the sequence space Lp, p< 0 < p <1 has been introduced based on the important extension of Milman's reverse Brunn-Minkowiski inequality by Bastero et al. in 1995. Moreover, Many interesting results connected with quasi-compactness and quasi-completeness in a quasi-normed space, Lp for 0 < p < 1 have been explored. Furthermore, we have shown that, the quasi-normed space under which condition is a quasi Banach space. Also, we have shown that the space if it is quasi-compact in quasi normed space then it is quasi Banach space and the converse is not true. Finally, a sufficient condition of the existence for a quasi-compact operator from Lp -> Lp has been presented and analyzed.

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20

Abed, Salwa Salman, and Hanan Sabah Lazam. "A-contraction mappings of integral type in n-normed spaces." Al-Mustansiriyah Journal of Science 31, no. 4 (December 20, 2020): 87. http://dx.doi.org/10.23851/mjs.v31i4.889.

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In this article, A-contraction type mappings in integral case are defined on a complete n-normed spaces and the existence of some fixed point theorems are proved in the complete n-normed spaces and given some results on Picard operator.

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21

Kramar, Edvard. "On reducibility of some operator semigroups and algebras on locally convex spaces." International Journal of Mathematics and Mathematical Sciences 2005, no. 24 (2005): 3951–61. http://dx.doi.org/10.1155/ijmms.2005.3951.

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22

Hoffmann, Philipp H. W., and Michael Mackey. "(m, p)-isometric and (m, ∞)-isometric operator tuples on normed spaces." Asian-European Journal of Mathematics 08, no. 02 (June 2015): 1550022. http://dx.doi.org/10.1142/s1793557115500229.

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We generalize the notion of m-isometric operator tuples on Hilbert spaces in a natural way to operator tuples on normed spaces. This is done by defining a tuple analogue of (m, p)-isometric operators, so-called (m, p)-isometric operator tuples. We then extend this definition further by introducing (m, ∞)-isometric operator tuples and study properties of and relations between these objects.

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23

Cross, R. W. "Some continuity properties of linear transformations in normed spaces." Glasgow Mathematical Journal 30, no. 2 (May 1988): 243–47. http://dx.doi.org/10.1017/s0017089500007291.

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Let X and Y be normed spaces and let L(X, Y) denote the set of linear transformations (henceforth called “operators”) T with domain a linear subspace D(T) of X and range R(T) contained in Y. The restriction of T to a subspace E is denoted by T/E; by the usual convention T|E = T|E∩ D(T). For a given linear subspace E the family of infinite dimensional ssubspaces of E is denoted by (E). An operator Tis said to have a certain property ℙ ubiquitously if every E ∈ (X) contains an F ∈(E) for which T|F has property ℙ For example, T is ubiquitously continuous if each E ∈(X) contains an F∈ (E) for which T|F is continuous. In the present note we shall characterize ubiquitous continuity, isomorphy, precompactness and smallness. A subspace of X is called a principal subspace if it is closed and of finite codimension in X. The restriction of an operator to a principal subspace will be called a principal restriction. The symbol T will always denote an arbitrary operator in L(X, Y).

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24

Mursaleen, M., Sunil K. Sharma, S. A. Mohiuddine, and A. Kılıçman. "New Difference Sequence Spaces Defined by Musielak-Orlicz Function." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/691632.

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25

Nurnugroho, Burhanudin Arif, Supama Supama, and A. Zulijanto. "Operator Linear-2 Terbatas pada Ruang Bernorma-2 Non-Archimedean." Jurnal Fourier 8, no. 2 (October 31, 2019): 43–50. http://dx.doi.org/10.14421/fourier.2019.82.43-50.

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Di dalam paper ini dikonstruksikan operator linear-2 terbatas dari X2 ke Y , dengan X ruang bernorma-2 non-Archimedean dan ruang bernorma non-Archimedean. Di dalam paper ini ditunjukan bahwa himpunan semua operator linear-2 terbatas dari X2 to Y , ditulis B(X2, Y) merupakan ruang bernorma non-Archimedean. Selanjutnya, ditunjukan bahwa B(X2, Y), apabila Y ruang Banach non-Archimedean. [In this paper we construct bounded 2-linear operators from X2 to Y, where X is non-Archimedean 2-normed spaces and is a non-Archimedean-normed space. We prove that the set of all bounded 2-linear operators from X2 to Y , denoted by B(X2, Y) is a non-Archimedean normed spaces. Furthermore, we show that B(X2, Y) is a non-Archimedean Banach normed space, whenever Y is a non-Archimedean Banach space.]

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26

Heikkilä, Seppo. "On extremal solutions of operator equations in ordered normed spaces." Applicable Analysis 44, no. 1-2 (April 1992): 77–97. http://dx.doi.org/10.1080/00036819208840069.

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27

JAMISON, JAMES, and RAENA KING. "UNBOUNDED HERMITIAN OPERATORS ON KOLASKI SPACES." Glasgow Mathematical Journal 56, no. 3 (August 30, 2013): 507–17. http://dx.doi.org/10.1017/s0017089513000426.

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AbstractWe investigate strongly continuous one-parameter (C0) groups of isometries acting on certain spaces of analytical functions which were introduced by Kolaski (C. J. Kolaski, Isometries of some smooth normed spaces of analytic functions, Complex Var. Theory Appl. 10(2–3) (1988), 115–122). We characterize the generators of these groups of isometries and also the spectrum of the generators. We provide an example on the Bloch space of an unbounded hermitian operator with non-compact resolvent.

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28

Kissin, Edward, and Victor S. Shulman. "CLASSES OF OPERATOR-SMOOTH FUNCTIONS. III. STABLE FUNCTIONS AND FUGLEDE IDEALS." Proceedings of the Edinburgh Mathematical Society 48, no. 1 (February 2005): 175–97. http://dx.doi.org/10.1017/s001309150300018x.

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AbstractThis paper continue to study the interrelation and hierarchy of the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded and operator stable. It also examines various properties of symmetrically normed ideals, introduces new classes of ideals: regular and Fuglede, and investigates them.AMS 2000 Mathematics subject classification: Primary 47A56; 47L20

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KARN, ANIL KUMAR, and DEBA PRASAD SINHA. "AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES." Glasgow Mathematical Journal 56, no. 2 (August 13, 2013): 427–37. http://dx.doi.org/10.1017/s0017089513000360.

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AbstractLet 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

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ALBANESE, ANGELA A., JOSÉ BONET, and WERNER J. RICKER. "THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓp+ AND Lp−." Glasgow Mathematical Journal 59, no. 2 (June 13, 2016): 273–87. http://dx.doi.org/10.1017/s001708951600015x.

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AbstractThe classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

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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 (March 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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32

Henson, C. Ward, Yves Raynaud, and Andrew Rizzo. "On Axiomatizability of Non-Commutative Lp-Spaces." Canadian Mathematical Bulletin 50, no. 4 (December 1, 2007): 519–34. http://dx.doi.org/10.4153/cmb-2007-051-7.

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AbstractIt is shown that Schatten p-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative Lp-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative Lp spaces. As a consequence, the class of non-commutative Lp-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative Lp-spaces. For p = 1 this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

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33

Kalton, Nigel, and Fyodor Sukochev. "Rearrangement-Invariant Functionals with Applications to Traces on Symmetrically Normed Ideals." Canadian Mathematical Bulletin 51, no. 1 (March 1, 2008): 67–80. http://dx.doi.org/10.4153/cmb-2008-009-3.

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AbstractWe present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional ϕ fails the condition that if (Hardy-Littlewood-Polya submajorization) and 0 ≤ x, y, then 0 ≤ ϕ(x) ≤ ϕ(y). We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola.

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Raj, Kuldip, Kavita Saini, and Anu Choudhary. "Orlicz lacunary sequence spaces of 𝑙-fractional difference operators." Journal of Applied Analysis 26, no. 2 (December 1, 2020): 173–83. http://dx.doi.org/10.1515/jaa-2020-2018.

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AbstractRecently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. Finally, some characterizations of the newly formed sequence spaces are given.

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35

Ludkowski, Sergey V. "Structure of Normed Simple Annihilator Algebras." Mathematics 7, no. 4 (April 11, 2019): 347. http://dx.doi.org/10.3390/math7040347.

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This article is devoted to normed simple annihilator algebras. Their structure is investigated in the paper. Maximal families of orthogonal irreducible idempotents of normed simple annihilator algebras are scrutinized. Division subalgebras of annihilator algebras are studied. Realizations of these algebras by operator algebras in Banach spaces are described. For this purpose, quasi finite dimensional operators are investigated.

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Fang, Jin-Xuan, and Gui-An Song. "-contractor and the solutions for nonlinear operator equations in fuzzy normed spaces." Fuzzy Sets and Systems 121, no. 2 (July 2001): 267–73. http://dx.doi.org/10.1016/s0165-0114(00)00016-6.

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37

Delgado, J. M., and C. Piñeiro. "On Banach ideals satisfying $c_0(\mathcal{A}(X,Y))=\mathcal{A}(X,c_0(Y))$." MATHEMATICA SCANDINAVICA 103, no. 1 (September 1, 2008): 130. http://dx.doi.org/10.7146/math.scand.a-15073.

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We characterize Banach ideals $[\mathcal{A},a]$ satisfying the equality $c_0(\mathcal{A}(X,Y))= \mathcal{A}(X,c_0(Y))$ for all Banach spaces $X$ and $Y$. Among other results we have proved that $\mathcal{K}$ (the normed operator ideal of all compact operators with the operator norm) is the only injective Banach ideal satisfying the equality.

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Bakery, Awad A. "The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces." Open Mathematics 18, no. 1 (July 29, 2020): 846–57. http://dx.doi.org/10.1515/math-2020-0045.

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Abstract In this article, we study some topological properties of the multiplication operator on Orlicz-Cesáro mean sequence spaces equipped with the pre-quasi norm and the pre-quasi operator ideal constructed by this sequence space and s-numbers.

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Iranmanesh, Mahdi, M. Saeedi Khojasteh, and M. K. Anwary. "On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality." European Journal of Pure and Applied Mathematics 11, no. 3 (July 31, 2018): 793–802. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3237.

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In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {μ : μ ∈ C, p(x − μy, y) = 1} is upper and lower semi continuous, where p(x, y) = sup{pz1,...,zn−2 (x, y) : z1, . . . , zn−2 ∈ X} and pz1,...,zn−2 (x, y) = kPx,z1,...,zn−2,yk−1 where Px,z1,...,zn−2,y denotes the projection parallel to y from X to the subspace generated by {x, z1, . . . , zn−2}. This can be considered as an alternative definition for numerical range in linear spaces.

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El-Shobaky, Entisarat, Sahar Mohammed Ali, and Wataru Takahashi. "On projection constant problems and the existence of metric projections in normed spaces." Abstract and Applied Analysis 6, no. 7 (2001): 401–11. http://dx.doi.org/10.1155/s1085337501000732.

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We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spacesl p,1≤p<∞andc 0. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator froml p,1≤p<∞orc 0onto anyone of their maximal proper subspaces.

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Zamani, Ali, and Mohammad Sal Moslehian. "Exact and Approximate Operator Parallelism." Canadian Mathematical Bulletin 58, no. 1 (March 1, 2015): 207–24. http://dx.doi.org/10.4153/cmb-2014-029-4.

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AbstractExtending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(ℋ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on B(ℋ). We also characterize the parallel elements of a C*-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert C*-module.

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Wahab, O. T., and K. Rauf. "On Faster Implicit Hybrid Kirk-Multistep Schemes for Contractive-Type Operators." International Journal of Analysis 2016 (September 19, 2016): 1–10. http://dx.doi.org/10.1155/2016/3791506.

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The purpose of this paper is to prove strong convergence and T-stability results of some modified hybrid Kirk-Multistep iterations for contractive-type operator in normed linear spaces. Our results show through analytical and numerical approach that the modified hybrid schemes are better in terms of convergence rate than other hybrid Kirk-Multistep iterative schemes in the literature.

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Afkhami, Taba, and Hossein Dehghan. "Operator inequalities related to p-angular distances." Filomat 33, no. 7 (2019): 2107–11. http://dx.doi.org/10.2298/fil1907107a.

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For any nonzero elements x,y in a normed space X, the angular and skew-angular distance is respectively defined by ?[x,y] = ||x/||x|| - y/||y|||| and ?[x,y] = ||x/||y|| - y/||x||||. Also inequality ? ? ? characterizes inner product spaces. Operator version of ? p has been studied by Pecaric, Rajic, and Saito, Tominaga, and Zou et al. In this paper, we study the operator version of p-angular distance ?p by using Douglas? lemma. We also prove that the operator version of inequality ? p ? ?p holds for normal and double commute operators. Some examples are presented to show essentiality of these conditions.

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Bonales, Fernando Garibay, and Rigoberto Vera Mendoza. "A formula to calculate the spectral radius of a compact linear operator." International Journal of Mathematics and Mathematical Sciences 20, no. 3 (1997): 585–88. http://dx.doi.org/10.1155/s0161171297000793.

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There is a formula (Gelfand's formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operatorTdefined on a complete topological vector space, locally convex. We also show an easy way to find a non-trivialT-invariant closed subspace in terms of Minkowski functional.

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Baran, Mirosław, and Paweł Ozorka. "On Vladimir Markov type inequality in Lp norms on the interval [-1; 1]." Science, Technology and Innovation 7, no. 4 (December 31, 2019): 9–12. http://dx.doi.org/10.5604/01.3001.0013.7225.

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We prove inequality ||P(k)||Lp(-1;1)≤Bp||Tn(k)||Lp(-1;1)n^(2/p) ||P||Lp(-1;1); where Bp are constants independent of n = deg P with 1 ≤ p ≤ 2, which is sharp in the case k ≥ 3. A method presented in this note is based on a factorization of linear operator of k-th derivative throughout normed spaces of polynomial equipped with a Wiener type norm.

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Slezák, B. "ON THE PRIMITIVES OF IFFERENTIAL ONE-MAPS." Studia Scientiarum Mathematicarum Hungarica 36, no. 3-4 (December 1, 2000): 317–30. http://dx.doi.org/10.1556/sscmath.36.2000.3-4.4.

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If a i .erential one-map between normed spaces is continuous on a star-like open set an is i .erentiable possibly except of one point then it has a primitive if an only if its erivative as a bilinear form is symmetric at every point where it exists.A generalization of the theorem on the existence of the potential operator is also obtained.The proof is based on the idea of the Goursat Lemma.

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Fishel, B. "Generalized translations associated with an unbounded self-adjoint operator." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 3 (May 1986): 519–28. http://dx.doi.org/10.1017/s030500410006446x.

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Delsarte [2], Povzner [9], Levitan [8], Leblanc [7], Dunford and Schwartz [3] (p. 1626) and Hutson and Pym [5] have discussed generalized translation operators (GTO) ‘associating with a differential operator’. The latter authors have also considered the topic in an abstract setting-the GTO ‘associates’ with a compact operator in a normed space. GTO are to have properties generalizing those of the translation operators defined by members of a group on a vector space E of functions defined on the group:(πεE, s and t are group elements). In the case of a locally compact topological group the integration spaces E = L1,L2,L∞, for a Haar measure of the group, are of especial interest.

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Sain, Debmalya. "On the Norm Attainment Set of a Bounded Linear Operator and Semi-Inner-Products in Normed Spaces." Indian Journal of Pure and Applied Mathematics 51, no. 1 (March 2020): 179–86. http://dx.doi.org/10.1007/s13226-020-0393-9.

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Raghad I. Sabri. "Fuzzy Open Mapping and Fuzzy Closed Graph Theorems in Fuzzy Length Space." Al-Qadisiyah Journal Of Pure Science 25, no. 4 (September 26, 2020): 32–39. http://dx.doi.org/10.29350/qjps.2020.25.4.1189.

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The theory of fuzzy set includes many aspects that regard important and significant in different fields of science and engineering in addition to there applications. Fuzzy metric and fuzzy normed spaces are essential structures in the fuzzy set theory. The concept of fuzzy length space has been given analogously and the properties of this space are studied few years ago. In this work, the definition of a fuzzy open linear operator is presented for the first time and the fuzzy Barise theorem is established to prove the fuzzy open mapping theorem in a fuzzy length space. Finally, the definition of a fuzzy closed linear operator on fuzzy length space is introduced to prove the fuzzy closed graph theorem.

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Cho, YeolJe, Heng-you Lan, and Nan-jing Huang. "A System of Nonlinear Operator Equations for a Mixed Family of Fuzzy and Crisp Operators in Probabilistic Normed Spaces." Journal of Inequalities and Applications 2010, no. 1 (2010): 152978. http://dx.doi.org/10.1155/2010/152978.

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