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Spaces of infinitely differentiable vectors of a nonnegative self-adjoint operator. (English. Russian original) Zbl 0541.47021

Ukr. Math. J. 35, 531-543 (1983); translation from Ukr. Mat. Zh. 35, No. 5, 617-621 (1983).
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 \(G(\lambda)=\exp(\lambda^{1/\beta}).\)
Reviewer: N.Angelescu

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46M40 Inductive and projective limits in functional analysis
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References:

[1] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag (1972).
[2] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Linear Spaces [in Russian], Fizmatgiz, Moscow (1959). · Zbl 0127.06102
[3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis. Self-Adjointness, Academic Press, New York (1975). · Zbl 0308.47002
[4] G. E. Shilov, Mathematical Analysis, Pergamon (1965). · Zbl 0137.26203
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