Operators Between Sequence Spaces and Applications

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. Preface 5 Contents 8 About the Authors 11 Acronyms 13 1 Matrix Transformations and Measures of Noncompactness 19 1.1 Linear Metric and Paranormed Spaces 20 1.2 FK and BK Spaces 28 1.3 Matrix Transformations into the Classical Sequence Spaces 33 1.4 Multipliers and Dual Spaces 37 1.5 Matrix Transformations Between the Classical Sequence Spaces 41 1.6 Crone's Theorem 45 1.7 Remarks on Measures of Noncompactness 52 1.8 The Axioms of Measures of Noncompactness 53 1.9 The Kuratowski and Hausdorff Measures of Noncompactness 55 1.10 Measures of Noncompactness of Operators 60 References 62 2 Matrix Domains 64 2.1 General Results 64 2.2 Bases of Matrix Domains of Triangles 72 2.3 The Multiplier Space M(XΣ,Y) 83 2.4 The α-, β- and γ-duals of XΣ 85 2.5 The α- and β-duals of XΔ(m) 95 2.6 The β-duals of Matrix Domains of Triangles in FK Spaces 107 References 118 3 Operators Between Matrix Domains 122 3.1 Matrix Transformations on W(u,v;X) 122 3.2 Matrix Transformations on XT 129 3.3 Compact Matrix Operators 145 3.4 The Class mathcalK(c) 156 3.5 Compact Operators on the Space bv+ 162 References 174 4 Computations in Sequence Spaces and Applications to Statistical Convergence 176 4.1 On Strong τ-Summability 177 4.2 Sum and Product of Spaces of the Form sξ, sξ0, or sξ(c) 179 4.3 Properties of the Sequence C(τ)τ 183 4.4 Some Properties of the Sets sτ(Δ), sτ0(Δ) and sτ(c)(Δ) 189 4.5 The Spaces wτ(λ), wτ°(λ) and wτ(λ) 191 4.6 Matrix Transformations From wτ(λ)+wν(μ) into sγ 196 4.7 On the Sets cτ(λ,μ), cτ°(λ,μ) and cτ(λ,μ) 197 4.8 Sets of Sequences of the Form [ A1,A2] 200 4.9 Extension of the Previous Results 206 4.10 Sets of Sequences that are Strongly τ-Bounded With Index p 209 4.11 Computations in Wτ and Wτ0 and Applications to Statistical Convergence 215 4.12 Calculations in New Sequence Spaces 221 4.13 Application to A-Statistical Convergence 224 4.14 Tauberian Theorems for Weighted Means Operators 231 4.15 The Operator C(λ) 239 References 243 5 Sequence Spaces Inclusion Equations 245 5.1 Introduction 246 5.2 The (SSIE) FsubsetEa+Fx with einF and FsubsetM(F,F) 252 5.3 The (SSIE) FsubsetEa+Fx with E,F,Fin{c0,c,s1,ellp,w0,winfty} 254 5.4 Some (SSIE) and (SSE) with Operators 261 5.5 The (SSIE) FsubsetEa+Fx for e-.25ex-.25ex-.25ex-.25exF 266 5.6 Some Applications 269 References 279 6 Sequence Space Equations 280 6.1 Introduction 281 6.2 The (SSE) Ea+Fx=Fb with einF 286 6.3 Some Applications 291 6.4 The (SSE) with Operators 299 6.5 Some (SSE's) with the Operators Δ and Σ 304 6.6 The Multiplier M((Ea)Δ,F) and the (SSIE) Fbsubset(Ea)Δ+Fx 312 6.7 The (SSE) (Ea)Δ+sx(c)=sb(c) 315 6.8 More Applications 322 References 328 7 Solvability of Infinite Linear Systems 329 7.1 Banach Algebras of Infinite Matrices 329 7.2 Solvability of the Equation Ax=b 334 7.3 Spectra of Operators Represented by Infinite Matrices 340 7.4 Matrix Transformations in χ(Δm) 344 7.5 The Equation Ax=b, Where A Is a Tridiagonal Matrix 352 7.6 Infinite Linear Systems with Infinitely Many Solutions 357 7.7 The Hill and Mathieu Equations 363 References 370 Appendix Inequalities 373 A.1 Inequalities 373 A.2 Functional Analysis 374 References 375 Index 376