RANDOM BOUNDED OPERATORS AND THEIR EXTENSION

In this paper, the random bounded operators from a Banach space $ X $ into a Banach space $ Y $ and the problem of extending random operators are investigated. It is shown that unlike the deterministic bounded operators, the random version of the principle of uniform boundedness and the Banach-Steinhaus theorem do not hold for random bounded operators. In addition, a random operator can be extended to apply to all random inputs if and only if it is bounded. As a consequence, we conclude that the Ito stochastic integral cannot be extended in a natural fashion to all random functions with square-integrable sample paths