Abstract
We consider the generalised Hahn sequence space \(h_{d}\), where d is an unbounded monotone increasing sequence of positive real numbers, and characterise several classes of bounded linear operators or matrix transformations from \(h_{d}\) into the spaces of all bounded, convergent and null sequences, and into the space of all abolutely convergent series and \(h_{d}\), and also from spaces of all absolutely convergent series, all null, convergent and bounded sequences into \(h_{d}\). Furthermore, we establish identities or estimates for the norms of the corresponding bounded linear operators. We also derive identities for the Hausdorff measure of noncompactness for the operators in the above classes with the exception of the final space being the space of all bounded sequences and charcaterise the classes of all corresponding compact operators. Finally, we apply our results to present a Fredholm operator from \(h_{d}\) into itself given by a tridiagonal matrix.
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Communicated by Gerald Teschl.
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Malkowsky, E., Rakočević, V. & Tuǧ, O. Compact operators on the Hahn space. Monatsh Math 196, 519–551 (2021). https://doi.org/10.1007/s00605-021-01588-8
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DOI: https://doi.org/10.1007/s00605-021-01588-8
Keywords
- The Hahn sequence space
- Bounded linear operators
- Hausdorff measure of noncompactness
- Compact operators
- Fredholm operators