SUMMABLE FAMILIES AND SUMMABLE OPERATORS IN RIESZ SPACES

In this paper, we study the concept of a summable family in a Riesz space and introduce a new class of operators that preserve summability, called the class of summable operators. For summable families, we generalize the classical result on numerical series and sequences in normed spaces by showing that a summable net in a Riesz space order converges to zero. We also extend the formula for the sum of a finite family of pairwise disjoint positive elements to arbitrary (countable or uncountable) families in laterally complete Riesz spaces, and we formulate versions of Fubini's theorem for double-indexed series and the block summation theorem in this setting. For summable operators, we examine their relationships with other wellknown classes of operators, establish conditions under which an operator is summable, and provide criteria ensuring that the image of the sum of a summable family coincides with the sum of the individual images. Finally, we discuss several results concerning the extension of summable operators.