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On quasi-analytic vectors for some classes of operators. (English) Zbl 0564.47001

Let \(E\) be a Banach space and \(T\) be an operator defined in \(E\). An element \(x\) in \(\cap_{n\geq 1}{\mathfrak D}(T^n)\), (\({\mathfrak D}(T)\) denotes the domain of an operator \(T\)) is called quasi-analytic if \(\sum^{\infty}_{n=1}1/\| T^nx\|^{1/n}=\infty.\) The class of analytic vectors of E. Nelson [Ann. Math. (2) 70, 572–615 (1959; Zbl 0091.10704)] is contained in this class.
The aim of this note is to give some results concerning quasi-analytic vectors for some classes of operators including symmetric operators. The following class of operators is considered: an operator \(T\) is of class (N) if for each \(x\in\mathfrak D(T)\cap \mathfrak D(T^n)\) we have \(\| Tx\|^2\leq \| T^2x\|\, \| x\|.\) A characterization of the quasi-analytic vectors is given by
Theorem 2.1. An element \(x\in \cap_{n\geq 1}{\mathfrak D}(T^ n)\) is a quasi-analytic vector for an operator of the class (N) if and only if \(\sum^{\infty}_{n=1}\) \(1/\| T^{2n}x\|^{1/2n}=\infty.\)
If \(x\) is a quasi-analytic vector for the operator \(T\) of class (N) in the Hilbert space \(E\) and if \(A\) and \(A^*\) are two operators in \(E\) which are adjoint to each other and if the following holds:
(i)\(\, Ax\in \cap_{n\geq 1}{\mathfrak D}(T^n),\quad\) (ii) \(\, T^nAx=AT^nx,\quad n=1,2,3,\ldots\quad\) (iii) \(\, T^ 2\text{ is symmetric },\)
then \(Ax\) is a quasi-analytic vector for \(T\).
An element \(x\) in a Banach space is called quasi-analytic vector for the family of commuting operators \(T_1,\ldots,T_n\) if for each \(k\) there exists a constant \(L_k\) such that \(\| T_1^{\alpha_1}\cdots T_n^{\alpha_n}x\| \leq C^{k+1}L^k_k\), \(\alpha_1+\cdots+\alpha_n=k\), \(k=0,1,\ldots\) and \(\int^{\infty}_{0}\log (\sum^{\infty}_{k=0}(\frac{t}{L_k})^ k)\frac{dt}{1+t^2}=\infty.\)
If \(\mathfrak g\) is the Lie algebra of a Lie group \(G\), \(\partial \pi\) be the corresponding representation of the universal enveloping algebra of \(\mathfrak g\) on the space \(H^{\infty}(\pi)\) of \(C^{\infty}\) vectors for \(\pi\) and \(X_1,\ldots,X_d\) a basis for \(\mathfrak g\) and set \(\Delta =X^2_1+\cdots+X^2_d,\) \(A=\partial \pi (1-\Delta)^-\) \((^-\) denotes the operator closure). It is known that \(A\) is a selfadjoint and positive operator.
Conjecture: the set of all quasi-analytic vectors for \(\pi\) is precisely the set of quasi-analytic vectors for the operator \(B=A^{1/2}.\)
A weaker condition for two operators to generate contraction semigroups: Theorem 3.3. Let \(A\) and \(B\) two closed dissipative operators with common dense domain and also commuting. If the set \(\{A^kB^jx\), \(x\) is a quasi-analytic vector for \(A\) and \(B\}\) is total in \(X\) then \(A\) and \(B\) generate contraction semigroups.
In the last part analytic and quasi-analytic elements in Banach algebras are defined.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B44 Linear accretive operators, dissipative operators, etc.
46H05 General theory of topological algebras
17B35 Universal enveloping (super)algebras

Citations:

Zbl 0091.10704
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