Generalization of the space derived by absolute Euler summability and matrix operators

Area of Science:

• Mathematical Analysis
• Summability Theory
• Sequence Spaces

Background:

• Maddox defined and studied the sequence space $$.
• This space is significant in the field of summability theory.

Purpose of the Study:

• To generalize the existing sequence space $$.
• To define and investigate a new sequence space $$ derived from the absolute summability of the Euler mean.

Main Methods:

• The study employs methods from functional analysis and summability theory.
• It involves the characterization of dual spaces and the construction of a Schauder basis.

Main Results:

• The newly defined space $$ is shown to be a paranormed space.
• Linear isomorphism is established between the new space and $$.
• The alpha-, beta-, and gamma-duals of the space are determined, and a Schauder basis is constructed.

Conclusions:

• The paper successfully extends the concept of the sequence space $$.
• The established properties and basis provide a foundation for further research in summability theory and related areas.