Uniform injectivity of linear parabolic operators (Q1594298)

Let \(X\) be a complex Banach space and let \(L_p\) be the space \(L_p(\mathbb{R}_+, X)\) (of all \(X\)-valued, Bochner measurable and \(p\)-summable functions on \(\mathbb{R}_+\)). Every strongly continuous family \(U= \{{\mathcal U}(t, s); 0\leq s\leq t<\infty\}\) of evolution operators acting on \(X\) can be associated to a linear operator \({\mathcal L}_{{\mathcal U}}: D({\mathcal L}_{{\mathcal U}})\subset L_p\to L_p\), whose domain \(D({\mathcal L}_{{\mathcal U}})\) consists of those functions \(x\in L_p\) for which there exists a function \(f\in L_p\) such that \(x(t)= -\int^t_0{\mathcal U}(t, \tau) f(\tau) d\tau\), \(t\geq 0\). One puts \({\mathcal L}_{{\mathcal U}} x=f\). An (abstract) parabolic operator is an operator of the form \({\mathcal L}_{{\mathcal U}}= -d/dt+ A(t): D({\mathcal L}_{{\mathcal U}})\subset L_p\to L_p\) with \({\mathcal U}\) a family of evolution operators for the differential equation \(dx(t)/dt= A(t) x(t)\), \(t\in\mathbb{R}_+\), where \(A(t): D(A(t))\subset X\to X\), \(t\geq 0\), is a family of closed linear operators generating a well-posed Cauchy problem. The operator \({\mathcal L}_{{\mathcal U}}\) (which is always injective) is said to be uniformly injective if \(\inf_{\|x\|= 1,x\in D}\|{\mathcal L}_{{\mathcal U}}x\|> 0\). In this paper, the author states conditions under which the operator \({\mathcal L}_{{\mathcal U}}\) is uniformly injective, using the property of exponential dichotomy of the family \({\mathcal U}\).