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Journal articles on the topic 'Bounded linear operator'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Kudryashov, Yu L. "Dilatations of Linear Operators." Contemporary Mathematics. Fundamental Directions 66, no. 2 (December 15, 2020): 209–20. http://dx.doi.org/10.22363/2413-3639-2020-66-2-209-220.

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The article is devoted to building various dilatations of linear operators. The explicit construction of a unitary dilation of a compression operator is considered. Then the J -unitary dilatation of a bounded operator is constructed by means of the operator knot concept of a bounded linear operator. Using the Pavlov method, we construct the self-adjoint dilatation of a bounded dissipative operator. We consider spectral and translational representations of the self-adjoint dilatation of a densely defined dissipative operator with nonempty set of regular points. Using the concept of an operator knot for a bounded operator and the Cayley transform, we introduce an operator knot for a linear operator. By means of this concept, we construct the J -self-adjoint dilatation of a densely defined operator with a regular point. We obtain conditions of isomorphism of extraneous dilations and their minimality.

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2

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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3

Al-Muttalibi, Rana, and Radhi M.A Ali. "Certain types of linear operators on probabilistic Hilbert space." Global Journal of Mathematical Analysis 3, no. 2 (May 24, 2015): 81. http://dx.doi.org/10.14419/gjma.v3i2.4664.

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<p>The purpose of this paper is to introduce some definitions, properties and basic results that show the relation between F-bounded of linear operator in probabilistic Hilbert space and bounded operator in norm. In the paper, we prove that the adjoint operator in probabilistic Hilbert space is bounded. The notion of the continuous operators in probabilistic Hilbert space and some basic results are given. In addition, we note that every operator in probabilistic real Hilbert space is a self-adjoint Operator.</p>

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4

Mehd, Sadiq A., Salim Dawood Mohsen, and Mohammed J. Fari. "Properties (RB) and (gRB) for Bounded Linear Operators." Journal of Physics: Conference Series 2322, no. 1 (August 1, 2022): 012053. http://dx.doi.org/10.1088/1742-6596/2322/1/012053.

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Abstract In this article, we consider pseudo invertible operators for study of the relationship between the space of relatively regular operators and some generalizations of Weyl and Browder theorems. By using the analysis and representation of pseudo invertible operators, some new properties in connection with Browder’s type theorems, were presented for bounded linear operators T ∈ B(X). These properties, which we refer to as property (RB), imply that All poles of the resolvent of T of finite rank in the typical spectrum are precisely those places of the spectrum for which a reasonably regular operator with its pseudo inverse operator is surjective. (γI – T)† ∈ SU(X), In the usual spectrum, the set of all poles of the resolvent is exactly those points of the spectrum for which we call property (gRB). γI – T is a B-relatively regular operator with its pseudo inverse operator is surjective (γI – T)† ∈ SU(X). In addition, several sufficient and necessary conditions for which properties (RB) and (gRB) hold are given.

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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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Mohsen, Salim Dawood, and Hanan Khalid Mousa. "Another Results Related of Fuzzy Soft Quasi Normal Operator in Fuzzy Soft Hilbert Space." Journal of Physics: Conference Series 2322, no. 1 (August 1, 2022): 012050. http://dx.doi.org/10.1088/1742-6596/2322/1/012050.

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Abstract The goal of this paper, is to introduce another classes of the fuzzy soft bounded linear operator in the fuzzy soft Hilbert space which is a fuzzy soft quasi normal operator, as well as, give some properties about this concept with investigating the relationship among this types of the fuzzy soft bounded linear operator on fuzzy soft Hilbert space with other kinds of fuzzy soft bounded linear operators.

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7

Djolović, Ivana, Katarina Petković, and Eberhard Malkowsky. "Matrix mappings and general bounded linear operators on the space bv." Mathematica Slovaca 68, no. 2 (April 25, 2018): 405–14. http://dx.doi.org/10.1515/ms-2017-0111.

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Abstract If X and Y are FK spaces, then every infinite matrix A ∈ (X, Y) defines a bounded linear operator LA ∈ B(X, Y) where LA(x) = Ax for each x ∈ X. But the converse is not always true. Indeed, if L is a general bounded linear operator from X to Y, that is, L ∈ B(X, Y), we are interested in the representation of such an operator using some infinite matrices. In this paper we establish the representations of the general bounded linear operators from the space bv into the spaces ℓ∞, c and c0. We also prove some estimates for their Hausdorff measures of noncompactness. In this way we show the difference between general bounded linear operators between some sequence spaces and the matrix operators associated with matrix transformations.

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8

ASSADI, AMANOLLAH, MOHAMAD ALI FARZANEH, and HAJI MOHAMMAD MOHAMMADINEJAD. "ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES." Bulletin of the Australian Mathematical Society 99, no. 2 (January 4, 2019): 274–83. http://dx.doi.org/10.1017/s0004972718001363.

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We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

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9

Liu, Xiaoji, Miao Zhang, and Yaoming Yu. "Note on the Invariance Properties of Operator Products Involving Generalized Inverses." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/213458.

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We investigate further the invariance properties of the bounded linear operator productAC1 B1 Dand its range with respect to the choice of the generalized inversesXandYof bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses.

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10

Frank, Michael. "Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules." Journal of K-theory 2, no. 3 (March 4, 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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11

Brattka, Vasco. "Effective representations of the space of linear bounded operators." Applied General Topology 4, no. 1 (April 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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Hejazian, Shirin, Madjid Mirzavaziri, and Omid Zabeti. "Bounded operators on topological vector spaces and their spectral radii." Filomat 26, no. 6 (2012): 1283–90. http://dx.doi.org/10.2298/fil1206283h.

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In this paper, we consider three classes of bounded linear operators on a topological vector space with respect to three different topologies which are introduced by Troitsky. We obtain some properties for the spectral radii of a linear operator on a topological vector space. We find some sufficient conditions for the completeness of these classes of operators. Finally, as a special application, we deduce some sufficient conditions for invertibility of a bounded linear operator.

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13

Yang, Dachun, and Dongyong Yang. "Boundedness of linear operators via atoms on Hardy spaces with non-doubling measures." gmj 18, no. 2 (June 2011): 377–97. http://dx.doi.org/10.1515/gmj.2011.0018.

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Abstract Let μ be a non-negative Radon measure on which satisfies only the polynomial growth condition. Let 𝒴 be a Banach space and H 1(μ) be the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H 1(μ) to 𝒴 if and only if T maps all (p, γ)-atomic blocks into uniformly bounded elements of 𝒴; moreover, the authors prove that for a sublinear operator T bounded from L 1(μ) to L 1, ∞(μ), if T maps all (p, γ)-atomic blocks with p ∈ (1, ∞) and γ ∈ ℕ into uniformly bounded elements of L 1(μ), then T extends to a bounded sublinear operator from H 1(μ) to L 1(μ). For the localized atomic Hardy space h 1(μ), the corresponding results are also presented. Finally, these results are applied to Calderón–Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón–Zygmund operators or fractional integral operators with Lipschitz functions to simplify the existing proofs in the related papers.

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14

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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15

Topuzu, Elena, and Paul Topuzu. "Remarks on bounded solutions of linear systems." Bulletin of the Australian Mathematical Society 53, no. 3 (June 1996): 459–67. http://dx.doi.org/10.1017/s0004972700017226.

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In the case of continuous time systems with bounded operators (coefficients) the following result, of Perron type is well known: “The linear differential system ẋ = Ax + f(t) has, for every function f continuous and bounded on ℝ, a unique bounded solution on ℝ, if and only if the spectrum of the operator A has no points on the imaginary axis”.

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16

Bakery, Awad A. "Operator Ideal of Cesaro Type Sequence Spaces Involving Lacunary Sequence." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/419560.

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The aim of this paper is to give the sufficient conditions on the sequence spaceCesθ,pdefined in Lim (1977) such that the class of all bounded linear operators between any arbitrary Banach spaces withnth approximation numbers of the bounded linear operators inCesθ,pform an operator ideal.

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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

Bahreini, Manijeh, Elizabeth Bator, and Ioana Ghenciu. "Complemented Subspaces of Linear Bounded Operators." Canadian Mathematical Bulletin 55, no. 3 (September 1, 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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19

Johnson, P. Sam, and G. Ramu. "Class of bounded operators associated with an atomic system." Tamkang Journal of Mathematics 46, no. 1 (March 22, 2014): 85–90. http://dx.doi.org/10.5556/j.tkjm.46.2015.1601.

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$K$-frames, more general than the ordinary frames, have been introduced by Laura G{\u{a}}vru{\c{t}}a in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Using the frame operator, we find a class of bounded linear operators in which a given Bessel sequence is an atomic system for every member in the class.

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20

Fedorov, V. E., A. D. Godova, and B. T. Kien. "Integro-differential equations with bounded operators in Banach spaces." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 106, no. 2 (June 30, 2022): 93–107. http://dx.doi.org/10.31489/2022m2/93-107.

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The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.

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21

Barnes, Bruce A. "A note concerning the ideal of nuclear operators." Glasgow Mathematical Journal 38, no. 2 (May 1996): 233–36. http://dx.doi.org/10.1017/s0017089500031487.

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22

He, Z., and M. W. Wong. "Wavelet multipliers and signals." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 40, no. 4 (April 1999): 437–46. http://dx.doi.org/10.1017/s0334270000010523.

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AbstractThe Schatten-von Neumann property of a pseudo-differential operator is established by showing that the pseudo-differential operator is a multiplier defined by means of an admissible wavelet associated to a unitary representation of the additive group Rn on the C*-algebra of all bounded linear operators from L2(Rn) into L2(Rn). A bounded linear operator on L2(R) arising in the Landau, Pollak and Slepian model in signal analysis is shown to be a wavelet multiplier studied in this paper.

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23

Mezrag, Lahcène, and Abdelmoumene Tiaiba. "On the sublinear operators factoring throughLq." International Journal of Mathematics and Mathematical Sciences 2004, no. 50 (2004): 2695–704. http://dx.doi.org/10.1155/s0161171204303145.

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Let0<p≤q≤+∞. LetTbe a bounded sublinear operator from a Banach spaceXinto anLp(Ω,μ)and let∇Tbe the set of all linear operators≤T. In the present paper, we will show the following. LetCbe a positive constant. For alluin∇T,Cpq(u)≤C(i.e.,uadmits a factorization of the formX→u˜Lq(Ω,μ)→MguLq(Ω,μ), whereu˜is a bounded linear operator with‖u˜‖≤C,Mguis the bounded operator of multiplication byguwhich is inBLr+(Ω,μ)(1/p=1/q+1/r),u=Mgu∘u˜andCpq(u)is the constant ofq-convexity ofu) if and only ifTadmits the same factorization; This is under the supposition that{gu}u∈∇Tis latticially bounded. Without this condition this equivalence is not true in general.

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24

Ge, Zhaoqiang. "Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces." Mathematics 10, no. 17 (August 30, 2022): 3118. http://dx.doi.org/10.3390/math10173118.

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A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the MILS solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems.

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Khatskevich, V. A., M. I. Ostrovskii, and V. S. Shulman. "Analogues of the Liouville theorem for linear fractional relations in Banach spaces." Bulletin of the Australian Mathematical Society 73, no. 1 (February 2006): 89–105. http://dx.doi.org/10.1017/s000497270003865x.

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Consider a bounded linear operator T between Banach spaces ℬ, ℬ′ which can be decomposed into direct sums ℬ = ℬ1 ⌖ ℬ2, ℬ′ = ℬ1′ ⌖ ℬ2′. Such linear operator can be represented by a 2 × 2 operator matrix of the form where Tij ∈ ℒ(ℬj, ℒi′) i, j = 1, 2. (By ℒ(ℬj, ℒi′) we denote the space of bounded linear operators acting from ℬj to ℬi′ (i, j = 1, 2).) The map GT from L (B1, B2) into the set of closed affine subspaces of ℒ(ℬ1′ ℬ2′), defined by is called a linear fractional relation associated with T.Such relations can be considered as a generalisation of linear fractional transformations which were studied by many authors and found many applications. Many traditional and recently discovered areas of application of linear fractional transformations would benefit from a better understanding of the behaviour of linear fractional relations. The present paper is devoted to analogues of the Liouville theorem “a bounded entire function is constant” for linear fractional relations.

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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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Fomin, Vasiliy I. "About a complex operator resolvent." Russian Universities Reports. Mathematics, no. 138 (2022): 183–97. http://dx.doi.org/10.20310/2686-9667-2022-27-138-183-197.

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A normed algebra of bounded linear complex operators acting in a complex normed space consisting of elements of the Cartesian square of a real Banach space is constructed. In this algebra, it is singled out a set of operators for each of which the real and imaginary parts commute with each other. It is proved that in this set, any operator for which the sum of squares of its real and imaginary parts is a continuously invertible operator, is invertible itself; a formula for the inverse operator is found. For an operator from the indicated set, the form of its regular points is investigated: conditions under which a complex number is a regular point of the given operator are found; a formula for the resolvent of a complex operator is obtained. The set of unbounded linear complex operators acting in the above complex normed space is considered. In this set, a subset of those operators for each of which the domains of the real and imaginary parts coincide is distinguished. For an operator from the specified subset, conditions on a complex number under which this number belongs to the resolvent set of the given operator are found; a formula for the resolvent of the operator is obtained. The concept of a semi-bounded complex operator as an operator in which one component is a bounded and the other is an unbounded operator is introduced. It is noted that the first and second resolvent identities for complex operators can be proved similarly to the case of real operators.

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Amara, Zouheir, Mourad Oudghiri, and Khalid Souilah. "On maps preserving skew symmetric operators." Filomat 36, no. 1 (2022): 243–54. http://dx.doi.org/10.2298/fil2201243a.

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Given a conjugation C on a separable complex Hilbert space H, a bounded linear operator T on H is said to be C-skew symmetric if CTC = -T*. This paper describes the maps, on the algebra of all bounded linear operators acting on H, that preserve the difference of C-skew symmetric operators for every conjugation C on H.

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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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Przyłuski, K. Maciej. "On a discrete-time version of a problem of A. J. Pritchard and J. Zabczyk." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 101, no. 1-2 (1985): 159–61. http://dx.doi.org/10.1017/s0308210500026251.

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SynopsisIt is shown that every weakly l1-stable linear and bounded operator (which represents a linear discrete-time system) on a Hilbert space is power stable. It solves (at least partially) a discrete-time version of a problem posed by A. J. Pritchard and J. Zabczyk for strongly continuous semigroups of bounded linear operators.

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WARK, H. M. "A NON-SEPARABLE REFLEXIVE BANACH SPACE ON WHICH THERE ARE FEW OPERATORS." Journal of the London Mathematical Society 64, no. 3 (December 2001): 675–89. http://dx.doi.org/10.1112/s0024610701002393.

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It is shown that there exists a non-separable reflexive Banach space on which every bounded linear operator is the sum of a scalar multiple of the identity operator and an operator of separable range. There is a strong sense that such a Banach space has as few operators as its linear and topological properties allow.

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An, Il, and Jaeseong Heo. "Weyl type theorems for selfadjoint operators on Krein spaces." Filomat 32, no. 17 (2018): 6001–16. http://dx.doi.org/10.2298/fil1817001a.

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In this paper, we introduce a notion of the J-kernel of a bounded linear operator on a Krein space and study the J-Fredholm theory for Krein space operators. Using J-Fredholm theory, we discuss when (a-)J-Weyl?s theorem or (a-)J-Browder?s theorem holds for bounded linear operators on a Krein space instead of a Hilbert space.

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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 (February 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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Minculete, Nicuşor. "About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators." Symmetry 13, no. 2 (February 11, 2021): 305. http://dx.doi.org/10.3390/sym13020305.

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The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

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35

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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36

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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37

Liu, Ming, and Xia Zhang. "L0-Linear Modulus of a Random Linear Operator." Abstract and Applied Analysis 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/183197.

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RANJBAR, HASSAN, and ASADOLLAH NIKNAM. "FURTHER INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS." Journal of Inequalities and Special Functions 12, no. 4 (December 31, 2021): 25–32. http://dx.doi.org/10.54379/jiasf-2021-4-3.

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By use of some non-negative Hermitian forms defined for n-tuple of bounded linear operators on the Hilbert space (H, h·, ·i) we establish new numerical radius and operator norm inequalities for sum of products of operators

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39

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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40

Kluvánek, Igor. "Scalar operators and integration." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 45, no. 3 (December 1988): 401–20. http://dx.doi.org/10.1017/s1446788700031116.

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AbstractThe notion of a scalar operator on a Banach space, in the sense of N. Dunford, is widened so as to cover those operators which can be approximated in the operator norm by linear combinations of disjoint values of an additive and multiplicative operator valued set function, P, on an algebra of sets in a space Ω such that P(Ω) = I, subject to some conditions guaranteeing that this definition is unambiguous. An operator T turns out to be scalar in this sense, if and only if, there exists a (not necessarily bounded) Boolean algebra of bounded projections such that the Banach algebra of operators it generates is semisimple and contains T.

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Tajmouati, Abdelaziz, Hamid Boua, and Abdeslam El Bakkali. "Descent spectrum equality." Asian-European Journal of Mathematics 12, no. 02 (April 2019): 1950029. http://dx.doi.org/10.1142/s1793557119500293.

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A bounded operator [Formula: see text] in a Banach space [Formula: see text] is said to satisfy the descent spectrum equality, if the descent spectrum of [Formula: see text] as an operator on [Formula: see text] coincides with the descent spectrum of [Formula: see text] as an element of the algebra [Formula: see text] of all bounded linear operators on [Formula: see text]. In this paper, we give some conditions under which the equality [Formula: see text] holds for a single operator [Formula: see text].

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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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43

HOWLETT, PHIL. "THE BEST WEIGHTED GRADIENT APPROXIMATION TO AN OBSERVED FUNCTION." Journal of the Australian Mathematical Society 98, no. 1 (October 14, 2014): 54–68. http://dx.doi.org/10.1017/s1446788713000621.

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AbstractWe find the potential function whose gradient best approximates an observed square integrable function on a bounded open set subject to prescribed weight factors. With an appropriate choice of topology, we show that the gradient operator is a bounded linear operator and that the desired potential function is obtained by solving a second-order, self-adjoint, linear, elliptic partial differential equation. The main result makes a precise analogy with a standard procedure for the best approximate solution of a system of linear algebraic equations. The use of bounded operators means that the definitive equation is expressed in terms of well-defined functions and that the error in a numerical solution can be calculated by direct substitution into this equation. The proposed method is illustrated with a hypothetical example.

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Yang, Yixuan, Yuchao Tang, and Chuanxi Zhu. "Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces." Mathematics 7, no. 2 (February 1, 2019): 131. http://dx.doi.org/10.3390/math7020131.

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The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.

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45

Solikhin, Solikhin, Susilo Hariyanto, Y. D. Sumanto, and Abdul Aziz. "NORMA OPERATOR PADA RUANG FUNGSI TERINTEGRAL DUNFORD." Journal of Fundamental Mathematics and Applications (JFMA) 2, no. 2 (November 30, 2019): 93. http://dx.doi.org/10.14710/jfma.v2i2.42.

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We are discussed operator norms on spce of Dunford integral function. We show that for a function which Dunford integral, operator from dual space into space of Lebesgue integral is a bounded linear operator. Furthermore, sets of all bounded linear operator is a linear space and it is a normed space by norm certain. Finally, the distance function generated by the norm is metrix space.

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Eskandari, Rasoul, Farzollah Mirzapour, and Ali Morassaei. "More on -Normal Operators in Hilbert Spaces." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/204031.

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We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebraℬ() of all bounded linear operators , where is Hilbert space.

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47

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 (November 30, 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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48

Journal, Baghdad Science. "Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators." Baghdad Science Journal 7, no. 1 (March 7, 2010): 191–99. http://dx.doi.org/10.21123/bsj.7.1.191-199.

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Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

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49

Crane, Daniel K., and Mark S. Gockenbach. "The Singular Value Expansion for Arbitrary Bounded Linear Operators." Mathematics 8, no. 8 (August 12, 2020): 1346. http://dx.doi.org/10.3390/math8081346.

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The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

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50

McPolin, P. T. N., and A. W. Wickstead. "The order boundedness of band preserving operators on uniformly complete vector lattices." Mathematical Proceedings of the Cambridge Philosophical Society 97, no. 3 (May 1985): 481–87. http://dx.doi.org/10.1017/s0305004100063052.

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1. Introduction. A linear operator T on a vector lattice is band preserving if x⊥ y implies Tx ⊥ y. Much is known about the order bounded band preserving operators on an Archimedean vector lattice. The collection of all of these forms an Abelian algebra under composition and a vector lattice for the operator order (see [7], [8] and [13] amongst others). Very little appears to be known about band preserving operators which are not order bounded apart from some isolated examples ([11], [13], [1] and [17]) and some non-existence results ([11] and [1]).

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