Spectral distributions and Gelfand's theorem (Q1884562)

In [Rec. Math. Moscow, n. Ser. 9, 49--50 (1941; Zbl 0024.32301; JFM 67.0407.02)], \textit{I. M. Gelfand} obtained the following result: If a bounded operator \(T\) satisfies \(\text{Sp}(T)= \{1\}\) and \(\sup_{n\in\mathbb{Z}}\,\| T^n\|< +\infty\), then \(T= I\). The version for operator groups says that if \(A\) is the generator of a uniformly bounded group of operators \(e^{itA}\) and \(\text{Sp}(A)= \{0\}\), then \(A= 0\). The authors give here a short elegant proof of the following generalization: Let \(A\) be the generator of a \(k\)-times integrated group \(G(t)\) satisfying \(\| G(t)\|\leq C|t|^k\) for some positive constant \(C\). If \(\text{Sp}(A)= \{0\}\), then \(A= 0\). The case \(k=1\) is Gelfand's theorem. The proof is based on the theory of spectral distributions and Stone's theorem for \(k\)-times integrated temperate groups [see \textit{M. Balabane}, \textit{H. Emamirad} and \textit{M. Jazar}, Acta Appl. Math. 31, No. 3, 275--295 (1993; Zbl 0802.47013)]. The paper under review also contains the following resolution of the identity result in the case of a discrete spectrum: Let \(A\) be the generator of a bounded group \(G(t)\) and suppose that \(\text{Sp}(A)= \{\lambda_n: n\in\mathbb{Z}\}\). Then there exist projections \(P_n\), \(n\in\mathbb{Z}\), such that \(\sum_n P_n= I\) on \(D(A)\) and \(\sum_n\lambda_n P_n= A\) on \(D(A^2)\).