Matrix Transformations in Sequence Space Theory

Matrix transformations in sequence space theory form a cornerstone of functional analysis, investigating how infinite matrices act as operators on spaces of sequences. These spaces, often Banach sequence spaces such as ℓp, c₀ and specialised Orlicz or BK-spaces, provide frameworks for convergence, stability and approximation. Central topics include the boundedness and compactness of operators induced by classical matrices (for example, Cesàro, Hilbert, Euler and Copson matrices), spectral analysis of difference and weighted-mean operators, and the identification of dual spaces and operator ideals. Recent extensions to variable-exponent and weighted spaces have uncovered novel inclusion relations and geometric features. Practical applications range from signal processing and numerical analysis to summability theory, where explicit operator norms and spectral bounds inform algorithm design. The synergy between operator theory and sequence-space geometry underpins advances in approximation methods and the stability of iterative schemes.

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Matrix Transformations in Sequence Space Theory publication trend

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Technical terms

Sequence space: A vector space whose elements are infinite sequences, equipped with a norm or topology that defines convergence and continuity.

Matrix transformation: A rule assigning to each input sequence a new sequence via multiplication by an infinite matrix, representing a linear operator.

Banach space: A complete normed vector space in which every Cauchy sequence converges, ensuring robustness of limit processes under operators.

Bounded linear operator: A linear map between normed spaces with finite operator norm, preserving convergence and controlling output magnitude.

Spectrum: The set of scalar values for which a linear operator fails to have a bounded inverse, generalising eigenvalues to infinite-dimensional contexts.

Operator ideal: A class of operators closed under composition and limit processes, often defined by approximation numbers or s-number conditions.

References

1. Small operator ideals formed by s numbers on generalized Cesáro and Orlicz sequence spaces. Journal of Inequalities and Applications (2018).
2. A study on Copson operator and its associated sequence space. Journal of Inequalities and Applications (2020).
3. Norm of Hilbert operator on sequence spaces. Journal of Inequalities and Applications (2020).
4. On Generalized (p, q)‐Euler Matrix and Associated Sequence Spaces. Journal of Function Spaces (2021).
5. A unifying approach to the difference operators and their applications. Boletim da Sociedade Paranaense de Matemática (2013).
6. The spectrum and some subdivisions of the spectrum of discrete generalized Cesàro operators on ℓp (1. Journal of Inequalities and Applications (2017).