On essentially semi regular linear relations (original) (raw)
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Acta Mathematica Sinica, English Series, 2012
Multivalued semi-Fredholm type linear operators with complemented ranges and null spaces are introduced. Conditions are obtained under which the classes given are stable under compact, strictly singular and strictly cosingular additive perturbations. We adher to the notation and terminology of the book [3]: X and Y are normed spaces, B X the closed unit ball of X, X the dual space of X and P(X) denotes the class of all closed finite-codimensional subspaces of X. If M is a subspace of X, then M ⊥ := {x ∈ X : x (x) = 0, x ∈ M }.
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Certain norm related functions of linear operators are considered in the very general setting of linear relations in normed spaces. These are shown to be closely related to the theory of strictly singular, strictly cosingular, F+ and F-linear relations. Applications to perturbation theory follow. Serial-fee code: 0004-9727/02 SA2.00+0.00.
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R. Croisot in , 1953, stated a very interesting problem of classification of all types of the regularity of semigroups defined by equations of the form a = a m xa n , with m, n ≥ 0 , m + n ≥ 2 . He proved that any of these equations determines either the ordinary regularity, left, right or complete regularity (see also the book by A. H. Clifford and G. B. Preston [3], Section 4.1). A similar problem, concerning all types of the regularity of semigroups and their elements defined by equations of the form a = a p xa q ya r , with p, q, r ≥ 0 , was treated by S. Lajos and G. Szász in [7], 1975. The purpose of this paper is to generalize the results by Croisot, Lajos and Szász considering all types of the regularity of semigroups and their elements determined by more general equations called linear. We determine all types of the regularity of elements defined by linear equations, and prove that there are exactly 14 types of the regularity of semigroups defined by such equations. We also give implication diagrams for linear equations and regularity conditions.
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We establish in this paper a Kato-type decomposition of quasi-Fredholm relations on Banach spaces. This generalizes the corresponding result of Labrousse for Hilbert space relations. The result is then applied to study and give some properties of the class of B-Fredholm linear relations.