Regular linear relations on Banach spaces
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00092-9 ORIGINAL PAPER
Regular linear relations on Banach spaces Teresa Alvarez1 · Adrian Sandovici2 Received: 17 May 2020 / Accepted: 8 September 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract For a Banach space, the notion of a regular linear relation is introduced and studied. Also, the regular resolvent set for a closed linear relation is introduced and investi‑ gated. Certain characterizations of regular resolvents are obtained in terms of the gap metric between corresponding null spaces, and in terms of generalized resol‑ vents of the linear relation itself, respectively. Keywords Banach space · Regular linear relation · Regular resolvent · Minimum modulus · Gap metric · Generalized resolvent Mathematics Subject Classification 47A06 · 47A05 · 47B48
1 Introduction The notion of the regular operator, with a different name, was first introduced by Kato [17] in his treatment of perturbation theory. Later, this class of operators was studied by several authors as, for instance, Mbekhta [23, 25–27], Mbekhta and Ouahab [29], Schmoeger [33], among others. Originally the regular spectrum was defined for operators in Hilbert spaces by Apostol [6] with the name of Apostol spectrum. Later the results of Apostol were generalized by Mbekhta [22, 23], Mbekhta and Ouahab [29] for operators defined on Banach spaces. The spectral mapping theorem for the regular spectrum has been proved by Mbekhta [23] for Hilbert space operators and by Schmoeger [34] for Banach space Communicated by Raul Curto. * Adrian Sandovici [email protected]; [email protected] Teresa Alvarez [email protected] 1
Department of Mathematics, University of Oviedo, 33007 Oviedo, Asturias, Spain
2
Department of Mathematics, “Gheorghe Asachi” Technical University, B‑dul Carol I nr. 11, 700506 Iaşi, Romania
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T. Alvarez and A. Sandovici
operators. For a closed linear relation in a Banach space, the concept of regularity was first introduced and studied in [3]. In the cited paper it is shown that many of the results for operators above mentioned due to Mbekhta and other authors remain valid in the context of linear relations. The study of regular operators (resp. linear relations) has received a lot of attention in many articles since it is of substantial help in the analysis of some interesting classes of operators (resp. linear relations) as, for example, the classes of Kato, generalized Kato and quasiFredholm operators (resp. linear relations). The notion of generalized Kato operator on a Banach space was first intro‑ duced by Mbekhta [22] as an extension of the notion of Kato operator defined in [17]. The generalized Kato operators have been studied in many papers by Mbekhta [22, 24, 25], Aiena [1], Bouamama [11], Benharrat and Messirdi [7], Jiang and Zhong [15, 16], Müler [30, 31], among others. In 2017, Benharrat et al. [8] introduced the concept of generalized Kato linear relation and they showed that many of the results of [1, 24] concern
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