scholarly journals Isomorphisms from the Space of Multilinear Operators

2019 ◽  
Vol 27 (2) ◽  
pp. 101-106
Author(s):  
Kazuhisa Nakasho
Keyword(s):  
Linear Operators ◽  
Multilinear Operators ◽  
Normed Spaces ◽  
Bilinear Operators

Summary In this article, using the Mizar system [5], [2], the isomorphisms from the space of multilinear operators are discussed. In the first chapter, two isomorphisms are formalized. The former isomorphism shows the correspondence between the space of multilinear operators and the space of bilinear operators. The latter shows the correspondence between the space of multilinear operators and the space of the composition of linear operators. In the last chapter, the above isomorphisms are extended to isometric mappings between the normed spaces. We referred to [6], [11], [9], [3], [10] in this formalization.

scholarly journals Weighted Estimates for Bilinear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hua Zhu ◽  
Heping Liu
Keyword(s):  
Schrödinger Operator ◽  
Multilinear Operators ◽  
Schrodinger Operator ◽  
Weighted Estimates ◽  
Bilinear Operators ◽  
By Products

We study the boundedness of weighted multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We also investigate weighted estimates for bilinear operators related to Schrödinger operator.

Linear Operators on Normed Spaces

2020 ◽  
pp. 115-152
Author(s):  
James K. Peterson
Keyword(s):  
Linear Operators ◽  
Normed Spaces

scholarly journals Bidual Spaces and Reflexivity of Real Normed Spaces

2014 ◽  
Vol 22 (4) ◽  
pp. 303-311
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Uniform Boundedness ◽  
Linear Operators ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Boundedness Theorem ◽  
Natural Mapping ◽  
Real Normed Space ◽  
Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

scholarly journals Pietsch’s variants of s-numbers for multilinear operators

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
D. L. Fernandez ◽  
M. Mastyło ◽  
E. B. Silva
Keyword(s):  
Linear Operator ◽  
Linear Operators ◽  
Multilinear Operators ◽  
Dual Operator ◽  
Range Space ◽  
Linear Forms ◽  
Fundamental Properties

AbstractWe study variants of s-numbers in the context of multilinear operators. The notion of an $$s^{(k)}$$ s ( k ) -scale of k-linear operators is defined. In particular, we shall deal with multilinear variants of the $$s^{(k)}$$ s ( k ) -scales of the approximation, Gelfand, Hilbert, Kolmogorov and Weyl numbers. We investigate whether the fundamental properties of important s-numbers of linear operators are inherited to the multilinear case. We prove relationships among some $$s^{(k)}$$ s ( k ) -numbers of k-linear operators with their corresponding classical Pietsch’s s-numbers of a generalized Banach dual operator, from the Banach dual of the range space to the space of k-linear forms, on the product of the domain spaces of a given k-linear operator.

scholarly journals Quantities related to upper and lower semi-Fredholm type linear relations

2002 ◽  
Vol 66 (2) ◽  
pp. 275-289 ◽  
Author(s):  
Teresa Alvarez ◽  
Ronald Cross ◽  
Diane Wilcox
Keyword(s):  
Perturbation Theory ◽  
General Setting ◽  
Linear Operators ◽  
Normed Spaces ◽  
Fredholm Type ◽  
Linear Relations ◽  
Lower Semi ◽  
Strictly Singular

Certain norm related functions of linear operators are considered in the very general setting of linear relations in normed spaces. These are shown to be closely related to the theory of strictly singular, strictly cosingular, F+ and F− linear relations. Applications to perturbation theory follow.

scholarly journals Arens regularity andweakly compact operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4993-5002
Author(s):  
Janko Bracic
Keyword(s):  
Tensor Product ◽  
Weak Compactness ◽  
Linear Operators ◽  
Bilinear Operator ◽  
Tensor Operator ◽  
Projective Tensor Product ◽  
Bilinear Operators ◽  
Projective Tensor ◽  
Arens Regularity ◽  
Topological Centers

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.

scholarly journals Multilinear Operator and Its Basic Properties

2019 ◽  
Vol 27 (1) ◽  
pp. 35-45
Author(s):  
Kazuhisa Nakasho
Keyword(s):  
Algebraic Structure ◽  
Multilinear Operator ◽  
Multilinear Operators ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Basic Properties ◽  
Normed Linear Spaces ◽  
Real Linear

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.

scholarly journals An Extension of Hypercyclicity forN-Linear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Juan Bès ◽  
J. Alberto Conejero
Keyword(s):  
Difference Equations ◽  
Entire Functions ◽  
Linear Operators ◽  
Fréchet Space ◽  
Infinite Dimensional ◽  
Bilinear Operators ◽  
Hypercyclic Vectors ◽  
Countable Product ◽  
Dense Orbits ◽  
Residual Sets

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit forN-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclicN-linear operators, for eachN≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines supportN-linear operators with residual sets of hypercyclic vectors, forN=2.

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