This book presents modern methods in functional analysis and operator theory along with their applications in recent research. The book also deals with the solvability of infinite systems of linear equations in various sequence spaces. It uses the classical sequence spaces, generalized Cesaro and difference operators to obtain calculations and simplifications of complicated spaces involving these operators. In order to make it self-contained, comprehensive and of interest to a larger mathematical community, the authors have presented necessary concepts with results for advanced research topics. This book is intended for graduate and postgraduate students, teachers and researchers as a basis for further research, advanced lectures and seminars.
Author(s): Bruno de Malafosse, Eberhard Malkowsky, Vladimir Rakočević
Edition: 1
Publisher: Springer Nature
Year: 2021
Language: English
Pages: 366
City: Singapore
Tags: Sequence Spaces, Matrix Transformations
Preface
Contents
About the Authors
Acronyms
1 Matrix Transformations and Measures of Noncompactness
1.1 Linear Metric and Paranormed Spaces
1.2 FK and BK Spaces
1.3 Matrix Transformations into the Classical Sequence Spaces
1.4 Multipliers and Dual Spaces
1.5 Matrix Transformations Between the Classical Sequence Spaces
1.6 Crone's Theorem
1.7 Remarks on Measures of Noncompactness
1.8 The Axioms of Measures of Noncompactness
1.9 The Kuratowski and Hausdorff Measures of Noncompactness
1.10 Measures of Noncompactness of Operators
References
2 Matrix Domains
2.1 General Results
2.2 Bases of Matrix Domains of Triangles
2.3 The Multiplier Space M(XΣ,Y)
2.4 The α-, β- and γ-duals of XΣ
2.5 The α- and β-duals of XΔ(m)
2.6 The β-duals of Matrix Domains of Triangles in FK Spaces
References
3 Operators Between Matrix Domains
3.1 Matrix Transformations on W(u,v;X)
3.2 Matrix Transformations on XT
3.3 Compact Matrix Operators
3.4 The Class mathcalK(c)
3.5 Compact Operators on the Space bv+
References
4 Computations in Sequence Spaces and Applications to Statistical Convergence
4.1 On Strong τ-Summability
4.2 Sum and Product of Spaces of the Form sξ, sξ0, or sξ(c)
4.3 Properties of the Sequence C(τ)τ
4.4 Some Properties of the Sets sτ(Δ), sτ0(Δ) and sτ(c)(Δ)
4.5 The Spaces wτ(λ), wτ°(λ) and wτ(λ)
4.6 Matrix Transformations From wτ(λ)+wν(µ) into sγ
4.7 On the Sets cτ(λ,µ), cτ°(λ,µ) and cτ(λ,µ)
4.8 Sets of Sequences of the Form [ A1,A2]
4.9 Extension of the Previous Results
4.10 Sets of Sequences that are Strongly τ-Bounded With Index p
4.11 Computations in Wτ and Wτ0 and Applications to Statistical Convergence
4.12 Calculations in New Sequence Spaces
4.13 Application to A-Statistical Convergence
4.14 Tauberian Theorems for Weighted Means Operators
4.15 The Operator C(λ)
References
5 Sequence Spaces Inclusion Equations
5.1 Introduction
5.2 The (SSIE) FsubsetEa+Fx with einF and FsubsetM(F,F)
5.3 The (SSIE) FsubsetEa+Fx with E,F,Fin{c0,c,s1,ellp,w0,winfty}
5.4 Some (SSIE) and (SSE) with Operators
5.5 The (SSIE) FsubsetEa+Fx for e-.25ex-.25ex-.25ex-.25exF
5.6 Some Applications
References
6 Sequence Space Equations
6.1 Introduction
6.2 The (SSE) Ea+Fx=Fb with einF
6.3 Some Applications
6.4 The (SSE) with Operators
6.5 Some (SSE's) with the Operators Δ and Σ
6.6 The Multiplier M((Ea)Δ,F) and the (SSIE) Fbsubset(Ea)Δ+Fx
6.7 The (SSE) (Ea)Δ+sx(c)=sb(c)
6.8 More Applications
References
7 Solvability of Infinite Linear Systems
7.1 Banach Algebras of Infinite Matrices
7.2 Solvability of the Equation Ax=b
7.3 Spectra of Operators Represented by Infinite Matrices
7.4 Matrix Transformations in χ(Δm)
7.5 The Equation Ax=b, Where A Is a Tridiagonal Matrix
7.6 Infinite Linear Systems with Infinitely Many Solutions
7.7 The Hill and Mathieu Equations
References
Appendix Inequalities
A.1 Inequalities
A.2 Functional Analysis
References
Index
