...le, then it follows by [15, Theorem 2.4, pp. 212] that there exists a constant β > 0 such that for arbitrary y ∈ Y , we have∫ ∞ 0 ‖B∗e−tA∗y‖2Udt ≥ β‖y‖2Y . We have the following result due to Kesavan =-=[3]-=-. Theorem 4.1. Let U and Z be the Hilbert spaces. Let A : D(A) ⊂ Z → Z be the infinitesimal generator of a C0-semigroup. Let B ∈ L(U,Z). Then the following are equivalent. (i) −A is the exponentially ...