{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there
exists a vector lattice~$F$ and a positive, disjointness preserving
orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \,
T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}:
Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to
the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?
In the present work, using the concept of
A
-statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued
B
-continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.
Upon prior investigation on statistical convergence of fuzzy sequences, we
study the notion of pointwise ??-statistical convergence of fuzzy mappings
of order ?. Also, we establish the concept of strongly ??-summable sequences
of fuzzy mappings and investigate some inclusion relations. Further, we get
an analogue of Korovkin-type approximation theorem for fuzzy positive linear
operators with respect to ??-statistical convergence. Lastly, we apply fuzzy
Bernstein operator to construct an example in support of our result.
It is well known that for an associative ring R, if ab has g-Drazin inverse
then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula
is so-called Cline?s formula for g-Drazin inverse, which plays an elementary
role in matrix and operator theory. In this paper, we generalize Cline?s
formula to the wider case. In particular, as applications, we obtain new
common spectral properties of bounded linear operators.
CHARACTERIZATION OF k-DISJOINTNESS PRESERVING NON-LINEAR OPERATORS BETWEEN BANACH LATTICES
