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.
This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.
Three main themes run through this chapter: compact linear operators, measures of non-compactness, and Fredholm and semi-Fredholm maps. Connections are established between these themes so as to derive important results later in the book.