In this paper we deepen the study of a sequence of positive linear operators acting on $L^1([0,1]^N)$, $N \geq 1$, that have been introduced in \cite{AltomareCappellettiLeonessa} and that generalize the multidimensional Kantorovich operators (see \cite{zhou}). We show that particular iterates of these operators converge on $\mathscr{C}([0,1]^N)$ to a Markov semigroup and on $L^p([0,1]^N)$, $1 \leq p <+\infty$, to a positive $C_0$-semigroup (that is an extension of the previous one). The generators of these $C_0$-semigroups are determined in a core of their domains, where they coincide with an elliptic second-order differential operator whose principal part degenerates on the vertices of the hypercube $[0,1]^N$.
Iterates of multidimensional Kantorovich-type operators and their associated positive C_0-semigroups
ALTOMARE, Francesco;CAPPELLETTI MONTANO, MIRELLA;
2011-01-01
Abstract
In this paper we deepen the study of a sequence of positive linear operators acting on $L^1([0,1]^N)$, $N \geq 1$, that have been introduced in \cite{AltomareCappellettiLeonessa} and that generalize the multidimensional Kantorovich operators (see \cite{zhou}). We show that particular iterates of these operators converge on $\mathscr{C}([0,1]^N)$ to a Markov semigroup and on $L^p([0,1]^N)$, $1 \leq p <+\infty$, to a positive $C_0$-semigroup (that is an extension of the previous one). The generators of these $C_0$-semigroups are determined in a core of their domains, where they coincide with an elliptic second-order differential operator whose principal part degenerates on the vertices of the hypercube $[0,1]^N$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
