TY - JOUR
T1 - Equivalence after extension and Schur coupling coincide for inessential operators
AU - ter Horst, S.
AU - Messerschmidt, M.
AU - Ran, A.C.M.
AU - Roelands, M.
AU - Wortel, M.
PY - 2018/10
Y1 - 2018/10
N2 - In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE ⇒ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators.
AB - In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE ⇒ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators.
KW - Compact operators
KW - Equivalence after extension
KW - Fredholm operators
KW - Inessential operators
KW - Schur coupling
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U2 - 10.1016/j.indag.2018.07.001
DO - 10.1016/j.indag.2018.07.001
M3 - Article
AN - SCOPUS:85050653784
VL - 29
SP - 1350
EP - 1361
JO - Indagationes Mathematicae
JF - Indagationes Mathematicae
SN - 0019-3577
IS - 5
ER -