Spaces of infinitely differentiable vectors of a nonnegative self-adjoint operator (Q794252)

Let G be a differentiable increasing function on \([0,\infty)\) with \(G(\lambda)\geq 1\), \(G(\lambda)\geq c\lambda G(\alpha_ 0\lambda)\) for some \(c,\alpha_ 0>0\). Define \(m_ n=\sup_{\lambda \geq 1}\lambda^ n/G(\lambda).\) For A a positive operator in the Hilbert space H, denote \(H_{\alpha}={\mathcal D}(G(\alpha A)),\) with \(\| f\|_{H_{\alpha}}=\| G(\alpha A)f\|,\) and \(C_{\alpha}\{m_ n\}\) the Banach space of the \(C^{\infty}\)-vectors of A, with \(\| f\|_{C_{\alpha}}=\sup_{n}\| \quad A^ nf\| /m_ n\alpha^ n<\infty.\) Theorem: \(\lim pr_{\alpha \to 0}C_{\alpha}\{m_ n\}=\lim pr_{\alpha \to \infty}H_{\alpha}\) and \(\lim ind_{\alpha \to \infty}C_{\alpha}\{m_ n\}=\lim ind_{\alpha \to 0}H_{\alpha}.\) In particular the Gevrey classes of order \(\beta\) and Beurling (Roumieu) type for A are projective (inductive) limits of \(H_{\alpha}\) for \(G(\lambda)=\exp(\lambda^{1/\beta}).\)