Projections in the convex hull of two isometries of absolutely continuous function spaces
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Projections in the convex hull of two isometries of absolutely continuous function spaces Maliheh Hosseini1 Received: 10 August 2020 / Accepted: 26 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we provide a complete description of projections in the convex hull of two surjective linear isometries (carrying a weighted composition operator form) on absolutely continuous function space AC(X , E), where X is a compact subset of R with at least two points and E is a strictly convex normed space. Among the consequences of the main result, it is shown that generalized bi-circular projections are the only projections on AC(X ) expressed as the convex combination of two surjective linear isometries, and an affirmative answer is given to the question in Botelho and Jamison (Canad Math Bull 53:398–403, 2010) for such spaces. Keywords Convex combination of isometries · Absolutely continuous function · Generalized bi-circular projection · Surjective linear isometry Mathematics Subject Classification Primary 47B38 · Secondary 46J10 · 47B33
1 Introduction Let X be a normed space. A projection P on X (a linear map on X with P 2 = P) is said to be a generalize bi-circular projection if there is a unimodular scalar λ different from 1 such that P + λ(I − P) is an isometry, where I is the identity operator on X . It easily follows that such an isometry must be in Iso(X ), the group of all surjective linear isometries on X . Let us recall that the study of this class of projections was initiated in [4] for many finite dimensional Banach spaces, and since then it has been a matter of interest. It is known that the average of the identity and an isometric reflection (an isometry T on X with T 2 = I ) is a generalized bi-circular projection. This raises the question whether any projection which is average of two isometries is a generalized bi-circular projection (see the question at the end of [2]). In this relation,
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Maliheh Hosseini [email protected] Faculty of Mathematics, K. N. Toosi University of Technology, 16315-1618 Tehran, Iran
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the first result in the context of function spaces is due to Botelho [1] who proved that any projection in the convex hull of two surjective linear isometries on C(X ) is a generalized bi-circular projection, where C(X ) is the Banach space of all complexvalued continuous functions on a connected compact Hausdorff space X . Then a vector-valued version of this problem was considered in [2], and Botelho and Jamison classified projections given as a convex combination of two distinct isometries on the Banach space of all continuous functions with values in a strictly convex Banach space. Moreover, the question was answered by Botelho, Jamison and Jiménez-Vargas for Lipschitz function spaces Lip(X ) and lip(X α ), where 0 < α < 1 and X is a compact 1-connected metric space with diameter at most 2. More precisely, they showed that generalized bi-circular projections are the only projections on these spaces which are in the convex hull of
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