Strong Monotonicity of Spectral Radius of Positive Operators

The classic result of Perron and Frobenius states that if $A$ and $B$ are matrices with nonnegative elements, such that $A \leq B$, $A$ is irreducible, and $ρ(A) = ρ(B)$ then $A = B$. We extend this result to a large class of band irreducible positive operators on a large class of Banach lattices and provide examples to show that the conditions we put on operators and Banach lattices cannot be weakened.

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