Equivalence After Extension and Schur Coupling for Relatively Regular Operators
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Integral Equations and Operator Theory
Equivalence After Extension and Schur Coupling for Relatively Regular Operators S. ter Horst , M. Messerschmidt
and A. C. M. Ran
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 noncoincidence 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. Mathematics Subject Classification. Primary 47A62; Secondary 47A53, 47A08. Keywords. Equivalence after extension, Schur coupling, Relatively regular operators, Generalized invertible operators, Fredholm operators.
1. Introduction Equivalence After Extension (EAE) and Schur Coupling (SC) are two relations on (bounded linear) Banach space operators that originated in the study of integral operators [5], along with the relation Matricial Coupling (MC), and which have since found many other applications, cf., [10,11,13,14,21,24,27] for a few recent references. The applications often rely on the fact that EAE, MC and SC coincide, in the specific context of the application, and it is important that one can easily and explicitly move between the three operator ´ Tsekanovskii relations. This led to the question, posed by H. Bart and V.E. in [8], whether these three operator relations might coincide at the level of general Banach space operators. By then it was known that EAE and MC This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 118513 and 127364). 0123456789().: V,-vol
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coincide [5,6] and in [8] it was proved that EAE and MC are implied by SC. All these implications are obtained by explicit constructions, cf., [25, Section 2]. Various attempts to prove that EAE (or MC) implies SC followed and several positive results in special cases were obtained [9,25,27,29]. However, in the recent paper [28] an explicit counterexample showing that EAE need not imply SC was obtained from the characterization of EAE and SC on Banach spaces that are essentially incomparable. This reiterated the observation from [26] that the Banach space geometries of the underlying spaces play an important role. The counterexample of [28] involves Fredholm operators, which motivated us into a further investigation of EAE and SC for this class of operators, and more generally for the class of relatively regular operators, without the essential incomparability as
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