On functions of bounded semivariation. (English) Zbl 1387.26021

This paper is a survey on functions of bounded semivariation, a generalization of functions of bounded variation in the context of Banach spaces.
The author presents a summary of properties of functions of bounded semivariation, an extensive account of examples and the connection of this class of functions with the so called Kurzweil integral.
For two Banach spaces \(X\), \(Y\), \(L(X,Y)\) stands for the Banach space of bounded linear operators from \(X\) to \(Y\). For a closed interval \([a,b]\) in the real line one denotes by \(\mathcal{D}[a,b]\) the set of partitions \(D\) of the form \[ D=\{\alpha_{0},\alpha_{1},\dotsc,\alpha_{\nu(D)}\},\quad a=\alpha_{0}<\alpha_{1}<\dotsb <\alpha_{\nu(D)}=b. \] Then, given a function \(F: [a,b]\to L(X,Y)\) and \(D\in \mathcal{D}[a,b]\) let \[ V(F,D,[a,b])=\sup\left\{\left\|\sum^{\nu(D)}_{j=1}[F(\alpha_{j})-F(\alpha_{j-1})](x_{j})\right\|_{Y}: x_{j}\in X,\, \|x_{j}\|_{X}\leq 1\right\}. \] The semivariation of \(F\) is \[ \mathrm{SV}_{a}^{b} (F)=\sup \{V(F,D,[a,b]):D\in \mathcal{D}[a,b])\} \] and \(F\) has bounded semivariation when \(\mathrm{SV}_{a}^{b}(F)<\infty\) and one writes \(F\in \mathrm{SV}([a,b],L(X,Y))\).
Among the general properties of the class \(\mathrm{SV}([a,b],L(X,Y))\) it is worth mentioning the fact that it is a Banach space with respect to the norm given by \[ \|F\|_{\mathrm{SV}}=\|F(a)\|_{L(X,Y)}+\mathrm{SV}_{a}^{b}(F). \]
One has the inclusion \[ \mathrm{BV}([a,b],L(X,Y))\subseteq \mathrm{SV}([a,b],L(X,Y)), \] where \(\mathrm{BV}\) denotes the class of functions of bounded variation, i.e., functions \(F\) with \(\text{var}^{b}_{a}(F)<\infty\), where \[ \text{var}^{b}_{a}(F)=\sup\left\{\sum^{\nu(D)}_{j=1}\|F(\alpha_{j})-F(\alpha_{j-1})\|_{L(X,Y)}:D\in\mathcal{D}[a,b]\right\}. \]
An interesting question is to know when equality holds in the above inclusion. The answer is
\(\bullet\) Every function in \(\mathrm{SV}([a,b],L(X,Y))\) is of bounded variation on \([a,b]\) if and only if the dimension of the space \(Y\) is finite.
It is well known that a function of bounded variation is regulated, that is the one-sided limits exist at every point of the domain; moreover such a function is continuous except at a countable set of points. For the case of bounded semivariation, one has the following result:
\(\bullet\) Let \(F\in\mathrm{SV}([a,b],L(X,Y))\). If \(Y\) is a weakly sequentially complete Banach space, then \(F\) is continuous on \([a,b]\) except at a countable set.
Functions of bounded semivariation are regulated in some weak sense. For operator-valued functions a more general notion of regulated function can be defined:
Given \(F: [a,b]\to L(X,Y)\), we say that \(F\) is simply regulated on \([a,b]\) if, for each \(x\in X\), the function \(t\to F(t)(x)\in Y\), \(t\in [a,b]\) is regulated.
Then, one can prove:
\(\bullet\) The space \(X\) does not contain an isomorphic copy of \(c_{0}\) if and only if every function \(F: [a,b] \to L(X)\) of bounded semivariation is simply regulated.
The concept of semivariation is applied to derive some convergence results for Stieltjes type integrals and to prove a new characterization of semivariation by means of the Kurzweil-Stieltjes integral.
Given a gauge on \([a,b]\), that is a positive function \(\delta: [a,b]\to\mathbb{R}^{+}\), a partition \(P=(\tau_{j},[\alpha_{j-1},\alpha_{j}])\) with \(\tau_{j}\in [\alpha_{j-1},\alpha_{j}]\) is \(\delta\)-fine if \[ [\alpha_{j-1},\alpha_{j}]\subset (\tau_{j}-\delta(\tau_{j}),\tau_{j}+\delta(\tau_{j})),\quad j=1,\dotsc,\nu(P). \]
A function \(U : [a, b] \times [a, b] \to X\) is Kurzweil integrable if there exists \(I \in X\) such that for every \(\varepsilon > 0\), there is a gauge \(\delta\) on \([a,b]\) such that \[ \left\|\sum^{\nu(P)}_{j=1}[U(\tau_{j},\alpha_{j})-U(\tau_{j},\alpha_{j-1})]-I\right\|_{X}<\varepsilon \text{ for all \(\delta\)-fine partitions of \([a,b]\)}. \]
Choosing \(U(\tau,t)=F(\tau)g(t)\) one gets the Kurzweil-Stieltjes integral \[ \int^{b}_{a}F\,\mathrm{d}[g]. \]
The connection between semivariation and Kurzweil-Stieltjes integrals leads to some typical results of Helly type, i.e., convergence results for the integral.
Moreover Kurzweil-Stieltjes integrals let us obtain a new characterization of semivariation. Introducing the class \(S_{L}([a,b],X)\) of all finite step functions \(g: [a,b]\to X\) which are left-continuous on \([a,b]\) and such that \(g(a)=0\), one can prove:
\(\bullet\) If \(F\in\mathrm{SV} ([a,b],L(X))\), then \[ \mathrm{SV}^{b}_{a}(F)=\sup\left\{\left\|F(b)g(b)-\int^{b}_{a} F\,\mathrm{d}[g]\right\|_{X}:g\in S_{L}([a,b],X),\,\|g\|_{\infty}\leq 1\right\}. \]


26A45 Functions of bounded variation, generalizations
46B99 Normed linear spaces and Banach spaces; Banach lattices
46G10 Vector-valued measures and integration
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