In this paper the following sequence space is presented. Let [t] be a positive double sequence and define the sequence space ?''(t) = {complex sequences x : xk,l = O(tk,l)}. The set of geometrically dominated double sequences is defined as G'' = U r,s?(0,1) G(r, s) where G(r, s) = {complex sequences x : x k,l = O(rk sl)} for each r, s in the interval (0, 1). Using this definition, four dimensional matrix characterizations of l?,?, c'', and c0'' into G'' and into ?''(t) are presented. In addition to these definitions and characterizations it should be noted that this ensure a rate of converges of at least as fast as [t]. Other natural implications will also be presented.