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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces
AU - Horst, Sanne ter
AU - Laustsen, Niels
PY - 2024/7/15
Y1 - 2024/7/15
N2 - Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC.Motivated by this result, we investigate when EAE and SC coincide for Fredholm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer k and every pair of Banach spaces (X,Y), either no pair of Fredholm operators of index k acting on X and Y, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, whether EAE and SC coincide for Fredholm operators of index k depends only on the geometry of the underlying Banach spaces X and Y, not on the properties of the operators themselves.We quantify this finding by introducing two numerical indices which capture the coincidence of EAE and SC and provide a number of examples illustrating the possible values of these indices. Notably, this includes an example showing that the above-mentioned result of Ter Horst et al, which is based on a pair of essentially incomparable Banach spaces, does not extend to projectively incomparable Banach spaces.
AB - Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC.Motivated by this result, we investigate when EAE and SC coincide for Fredholm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer k and every pair of Banach spaces (X,Y), either no pair of Fredholm operators of index k acting on X and Y, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, whether EAE and SC coincide for Fredholm operators of index k depends only on the geometry of the underlying Banach spaces X and Y, not on the properties of the operators themselves.We quantify this finding by introducing two numerical indices which capture the coincidence of EAE and SC and provide a number of examples illustrating the possible values of these indices. Notably, this includes an example showing that the above-mentioned result of Ter Horst et al, which is based on a pair of essentially incomparable Banach spaces, does not extend to projectively incomparable Banach spaces.
KW - Equivalence after extension
KW - Schur coupling
KW - Fredholm operator
KW - incomparable Banach spaces
KW - Gowers-Maurey spaces
U2 - 10.1016/j.jfa.2024.110463
DO - 10.1016/j.jfa.2024.110463
M3 - Journal article
VL - 287
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 2
M1 - 110463
ER -