Inequalities \`a la Fr\"olicher and cohomological decompositions

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality `a la Fr"olicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equality in such an inequality `a la Fr"olicher characterizes the validity of the so-called cohomological property of satisfying the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, generalized-complex manifolds.