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\(B\)-convexity, the analytic Radon-Nikodým property, and individual stability of \(C_0\)-semigroups. (English) Zbl 0943.47029

Let \(X\) be a Banach space, \(\{T(t):t\geq 0\}\) a strongly continuous semigroup of linear operators on \(X\), and denote by \(A\) the infinitesimal generator of the semigroup. Also, denote by \(\omega_0\) the (logarithmic) growth bound of \(T(t)\). The author gives conditions under which orbits of the semigroup are stable (i.e., tend to zero in norm or at least weakly). Thus, assume that \(x_0\) is a vector in \(X\) such that the function \((\lambda-A)^{-1}x_0\) extends holomorphically for \(\text{Re} \lambda>0\). The author deduces then that \(T(t)(\lambda-A)^{-\beta}x_0\) tends to zero weakly if \(\beta>1\) and \(\lambda>\omega_0\). The convergence is in norm if, in addition, \(X\) has the analytic Radon-Nikodým property. A similar result holds if \(X\) has Fourier type \(>1\), i.e. a vector valued version of the Hausdorff-Young inequality holds for some \(p\in(1,2]\).

MSC:

47D06 One-parameter semigroups and linear evolution equations
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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