An interesting feature of the giant planets of our solar system is the existence of regions around these objects where no irregular satellites are observed. Surveys have shown that, around Jupiter, such a region extends from the outermost regular satellite Callisto, to the vicinity of Themisto, the innermost irregular satellite. To understand the reason for the existence of such a satellite-void region, we have studied the dynamical evolution of Jovian irregulars by numerically integrating the orbits of several hundred test particles, distributed in a region between 30 and 80 Jupiter-radii, for different values of their semimajor axes, orbital eccentricities, and inclinations. As expected, our simulations indicate that objects in or close to the influence zones of the Galilean satellites become unstable because of interactions with Ganymede and Callisto. However, these perturbations cannot account for the lack of irregular satellites in the entire region between Callisto and Themisto. It is suggested that at distances between 60 and 80 Jupiter-radii, Ganymede and Callisto may have long-term perturbative effects, which may require the integrations to be extended to times much longer than 10 Myr. The interactions of irregular satellites with protosatellites of Jupiter at the time of the formation of Jovian regulars may also be a destabilizing mechanism in this region. We present the results of our numerical simulations and discuss their applicability to similar satellite void-regions around other giant planets.