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For linear random dynamical systems in a separable Banach space $X$, wederived a series of Krein-Rutman type Theorems with respect to co-invariantcone family with rank-$k$, which present a (quasi)-equivalence relation betweenthe measurably co-invariant cone family and the measurably dominated splittingof $X$. Moreover, such (quasi)-equivalence relation turns out to be anequivalence relation whenever (i) $k=1$; or (ii) in the frame of theMultiplicative Ergodic Theorem with certain Lyapunov exponent being greaterthan the negative infinity.  For the second case, we thoroughly investigated the relations between theLyapunov exponents, the co-invariant cone family and the measurably dominatedsplitting for linear random dynamical systems in $X$.

On random linear dynamical systems in a Banach space. I. Multiplicative Ergodic Theorem and Krein-Rutmann type Theorems