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Duality in the space of regulated functions and the play operator. (English) Zbl 1055.46023
Let \(G(a,b;X)\) be the space of all regulated functions defined on the interval \([a,b]\subset \mathbb R\) with values in the Hilbert space \(X\). A regulated function is a function having two-sided limits at each point in \([a,b]\). With respect to the uniform norm \(\| f\| _{[a,b]},\; G(a,b;X)\) is a Banach space that contains \(C(a,b;X)\), the space of continuous functions. For a regulated function \(f:[a,b]\to X\) and a partition \(d:a=t_ 0<...<t_ m=b\) of the interval \([a,b]\), the essential variation of \(f\) on \([a,b]\) is defined by \(\overline{\mathcal V}_ d(f) = \sum_{i=1}^ m | f(t_ i-)-f(t_{i-1}+)| + \sum_{i=0}^ m | f(t_ i+)-f(t_ i-)| ,\) and the total essential variation by \(\overline{Var}_{[a,b]}=\sup_ d \overline{\mathcal V}_ d(f).\) The space of all functions with finite total essential variation, denoted by \(\overline{BV}(a,b;X)\), contains the space \(BV(a,b;X)\) of all functions with bounded variation.
In Theorem 2.6, the authors give a representation for the dual of the Banach space \(G(a,b;X)\) in terms of the Young integral and of some functions in \(BV(a,b;X)\). Using this representation, they establish some relationships between weak and wbo-convergence (wbo comes from “weak bounded oscillation”). The play operator, considered by P. Krejči and Ph. Laurençot [J. Conv. Anal. 9, 159–183 (2002; Zbl 1001.35118)] is used in an essential manner throughout the paper. This is a natural extension of the classical play operator from the mathematical theory of hysteresis.

MSC:
46E40 Spaces of vector- and operator-valued functions
26A45 Functions of bounded variation, generalizations
26A39 Denjoy and Perron integrals, other special integrals
26E20 Calculus of functions taking values in infinite-dimensional spaces
34C55 Hysteresis for ordinary differential equations
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