Abstract
It was recently shown in Ter Horst et al. (Bull Lond Math Soc 51:1005–1014, 2019) that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we derive new cases where EAE and SC coincide and provide a new example for which EAE and SC do not coincide and where the Banach spaces are not essentially incomparable.
| Original language | English |
|---|---|
| Article number | 40 |
| Pages (from-to) | 1-23 |
| Number of pages | 23 |
| Journal | Integral Equations and Operator Theory |
| Volume | 92 |
| Issue number | 5 |
| Early online date | 26 Aug 2020 |
| DOIs | |
| Publication status | Published - Oct 2020 |
Funding
This work is based on research supported in part by the National Research Foundation of South Africa (NRF) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF and CoE-MaSS do not accept any liability in this regard.
| Funders | Funder number |
|---|---|
| DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences | |
| National Research Foundation | 118513, 127364 |
Keywords
- Equivalence after extension
- Fredholm operators
- Generalized invertible operators
- Relatively regular operators
- Schur coupling