The Experts below are selected from a list of 78 Experts worldwide ranked by ideXlab platform
Candido Pineiro - One of the best experts on this subject based on the ideXlab platform.
-
The Bartle–Dunford–Schwartz and the Dinculeanu–Singer theorems revisited
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Abstract Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S ∈ L ( C ( Ω ) , Y ) admits an integral representation with respect to a Y ⁎ ⁎ -valued measure. By the Dinculeanu–Singer theorem, each operator U ∈ L ( C ( Ω , X ) , Y ) admits an integral representation with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. We establish an integral representation for any operator S ∈ L ( C ( Ω ) , L ( X , Y ) ) with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.
-
the bartle dunford schwartz and the Dinculeanu singer theorems revisited
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Abstract Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S ∈ L ( C ( Ω ) , Y ) admits an integral representation with respect to a Y ⁎ ⁎ -valued measure. By the Dinculeanu–Singer theorem, each operator U ∈ L ( C ( Ω , X ) , Y ) admits an integral representation with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. We establish an integral representation for any operator S ∈ L ( C ( Ω ) , L ( X , Y ) ) with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.
-
operator valued operators that are associated to vector valued operators
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:This paper is motivated by a long-standing conjecture of Dinculeanu from 1967. Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. Dinculeanu conjectured that there exist operators S∈L(C(Ω),L(X,Y))S∈L(C(Ω),L(X,Y)) which are not associated to any U∈L(C(Ω,X),Y)U∈L(C(Ω,X),Y). We study this existence problem systematically on three possible levels of generality: the classical case C(Ω,X)C(Ω,X) of continuous vector-valued functions, p -continuous vector-valued functions, and tensor products. On each level, we establish necessary and sufficient conditions for an L(X,Y)L(X,Y)-valued operator to be associated to a Y-valued operator. Among others, we see that examples, proving Dinculeanu's conjecture, come out on the all three levels of generality.
-
The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited
arXiv: Functional Analysis, 2016Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorff space. Denote by $\mathcal{C}_{p}(\Omega,X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq \infty$. For operators $S\in\mathcal{L}(\mathcal{C}(\Omega),\mathcal{L}(X,Y))$ and $U\in\mathcal{L}(\mathcal{C}_{p}(\Omega,X),Y)$, we establish integral representation theorems with respect to a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borel subsets of $\Omega$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$.
Francesco Russo - One of the best experts on this subject based on the ideXlab platform.
-
Generalized covariation for Banach space valued processes, Itô formula and applications
2015Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If X and Y take respectively values in Banach spaces B1 and B2 and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 (denoted by (B1⊗̂πB2) ∗), we define the so-called χ-covariation of X and Y. If X = Y, the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗̂πB1) ∗ then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = C([−τ, 0]) for some τ> 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([−τ, 0])-valued process X: = X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type h = H(XT (·)), H: C([−T, 0]) − → R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u: [0, T]×C([−T, 0]) − → R solving an infinite dimensional partial differential equation
-
generalized covariation for banach space valued processes ito formula and applications
Osaka Journal of Mathematics, 2014Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Ito formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Metivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Ito formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
-
Generalized covariation for Banach space valued processes, Itô formula and applications
2013Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
-
Generalized covariation for Banach valued processes and Itô formula
2010Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper concerns the notion of quadratic variation and covariation for Banach valued processes and related Ito formula. If X and Y take respectively values in Banach spaces B1 and B2 (denoted by (B1ˆ �B2) � ) andis a suitable subspace of the dual of the projective tensor product of B1 and B2 we define the so-called �-covariation of X and Y. If X = Y the �-covariation is called �-quadratic variation. The notion of �-quadratic variation is a natural generalization of the one introduced by Metivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if � is the whole space (B1ˆ �B1) � then the �-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the �-covariation of various processes for several examples ofwith a particular attention to the case B1 = B2 = C(( �,0)) for some � > 0 and X and Y being window processes. If X is a real process, we call window process associated with X the C(( �,0))-valued process X := X(·) defined by Xt(y) = Xt+y, where y 2 ( �,0). (2010 Math Subject Classification: ) 60G05, 60G07, 60G22, 60䠰5, 60䠹9.
-
Generalized covariation for Banach space valued processes, It\^o formula and applications
arXiv: Probability, 2010Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related It\^o formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by M\'etivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The It\^o formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
Fernando Munoz - One of the best experts on this subject based on the ideXlab platform.
-
The Bartle–Dunford–Schwartz and the Dinculeanu–Singer theorems revisited
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Abstract Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S ∈ L ( C ( Ω ) , Y ) admits an integral representation with respect to a Y ⁎ ⁎ -valued measure. By the Dinculeanu–Singer theorem, each operator U ∈ L ( C ( Ω , X ) , Y ) admits an integral representation with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. We establish an integral representation for any operator S ∈ L ( C ( Ω ) , L ( X , Y ) ) with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.
-
the bartle dunford schwartz and the Dinculeanu singer theorems revisited
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Abstract Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S ∈ L ( C ( Ω ) , Y ) admits an integral representation with respect to a Y ⁎ ⁎ -valued measure. By the Dinculeanu–Singer theorem, each operator U ∈ L ( C ( Ω , X ) , Y ) admits an integral representation with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. We establish an integral representation for any operator S ∈ L ( C ( Ω ) , L ( X , Y ) ) with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.
-
operator valued operators that are associated to vector valued operators
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:This paper is motivated by a long-standing conjecture of Dinculeanu from 1967. Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. Dinculeanu conjectured that there exist operators S∈L(C(Ω),L(X,Y))S∈L(C(Ω),L(X,Y)) which are not associated to any U∈L(C(Ω,X),Y)U∈L(C(Ω,X),Y). We study this existence problem systematically on three possible levels of generality: the classical case C(Ω,X)C(Ω,X) of continuous vector-valued functions, p -continuous vector-valued functions, and tensor products. On each level, we establish necessary and sufficient conditions for an L(X,Y)L(X,Y)-valued operator to be associated to a Y-valued operator. Among others, we see that examples, proving Dinculeanu's conjecture, come out on the all three levels of generality.
-
The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited
arXiv: Functional Analysis, 2016Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorff space. Denote by $\mathcal{C}_{p}(\Omega,X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq \infty$. For operators $S\in\mathcal{L}(\mathcal{C}(\Omega),\mathcal{L}(X,Y))$ and $U\in\mathcal{L}(\mathcal{C}_{p}(\Omega,X),Y)$, we establish integral representation theorems with respect to a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borel subsets of $\Omega$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$.
Oana Mocioalca - One of the best experts on this subject based on the ideXlab platform.
-
LOCALLY INTEGRABLE PROCESSES WITH RESPECT TO LOCALLY ADDITIVE SUMMABLE PROCESSES
2015Co-Authors: Oana MocioalcaAbstract:Abstract. In [MD] we defined and studied a class of summable processes, called additive summable processes, that is larger than the class previously studied by Dinculeanu and Brooks [D–B]. We also defined a stochastic integral with respect to an additive summable process and proved several properties of the integral. In this article we consider examples of processes that are integrable with respect to an additive summable process or locally integrable with respect to a locally additive summable processes. In particular, we show that if X is a locally additive summable process, then X − is integrable with respect to X. This is essential, for example, in proving an Ito formula for locally additive summable processes. 1
-
Additive summable processes and their stochastic integral
Rendiconti del Circolo Matematico di Palermo, 2006Co-Authors: Nicolae Dinculeanu, Oana MocioalcaAbstract:We define and study a class of summable processes, called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D-B]. We relax the definition of a summable processes X :Ω×ℝ_+→ E ⊂ L ( F, G ) by asking for the associated measure I _X to have just an additive extension to the predictable σ -algebra ℘, such that each of the measures ( I _X)_ z , for z ∈( L _G ^p )*, being σ -additive, rather than having a σ -additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the integral. After that we show that this class of summable processes contains all processes X :Ω×ℝ_+→ E ⊂ L ( F, G ) with integrable semivariation if c _0 ∋ G .
-
Additive summable processes and their stochastic integral
Rendiconti del Circolo Matematico di Palermo, 2006Co-Authors: Nicolae Dinculeanu, Oana MocioalcaAbstract:We define and study a class of summable processes, called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D-B].
Eve Oja - One of the best experts on this subject based on the ideXlab platform.
-
The Bartle–Dunford–Schwartz and the Dinculeanu–Singer theorems revisited
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Abstract Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S ∈ L ( C ( Ω ) , Y ) admits an integral representation with respect to a Y ⁎ ⁎ -valued measure. By the Dinculeanu–Singer theorem, each operator U ∈ L ( C ( Ω , X ) , Y ) admits an integral representation with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. We establish an integral representation for any operator S ∈ L ( C ( Ω ) , L ( X , Y ) ) with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.
-
the bartle dunford schwartz and the Dinculeanu singer theorems revisited
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Abstract Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S ∈ L ( C ( Ω ) , Y ) admits an integral representation with respect to a Y ⁎ ⁎ -valued measure. By the Dinculeanu–Singer theorem, each operator U ∈ L ( C ( Ω , X ) , Y ) admits an integral representation with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. We establish an integral representation for any operator S ∈ L ( C ( Ω ) , L ( X , Y ) ) with respect to an L ( X , Y ⁎ ⁎ ) -valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.
-
operator valued operators that are associated to vector valued operators
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:This paper is motivated by a long-standing conjecture of Dinculeanu from 1967. Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. Dinculeanu conjectured that there exist operators S∈L(C(Ω),L(X,Y))S∈L(C(Ω),L(X,Y)) which are not associated to any U∈L(C(Ω,X),Y)U∈L(C(Ω,X),Y). We study this existence problem systematically on three possible levels of generality: the classical case C(Ω,X)C(Ω,X) of continuous vector-valued functions, p -continuous vector-valued functions, and tensor products. On each level, we establish necessary and sufficient conditions for an L(X,Y)L(X,Y)-valued operator to be associated to a Y-valued operator. Among others, we see that examples, proving Dinculeanu's conjecture, come out on the all three levels of generality.
-
The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited
arXiv: Functional Analysis, 2016Co-Authors: Fernando Munoz, Eve Oja, Candido PineiroAbstract:Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorff space. Denote by $\mathcal{C}_{p}(\Omega,X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq \infty$. For operators $S\in\mathcal{L}(\mathcal{C}(\Omega),\mathcal{L}(X,Y))$ and $U\in\mathcal{L}(\mathcal{C}_{p}(\Omega,X),Y)$, we establish integral representation theorems with respect to a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borel subsets of $\Omega$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$.