Inequalities of Taikov type for selfadjoint operators in Hilbert space (Q2879964)

The Taikov inequality estimates the \(L_{\infty}\)-norm of an intermediate derivative by the \(L_2\)-norms of a function and its higher derivatives (cf.\ \textit{L. V. Taĭkov} [Proc. Steklov Math. Inst. 78 (1965), 43--48 (1967); translation from Tr. Mat. Inst. Steklova 78, 43--47 (1965; Zbl 0151.06802)]; [Math. Notes 50, No.~4, 1062--1067 (1991); translation from Mat. Zametki 50, No.~4, 114--122 (1991; Zbl 0774.26010)]). In the present paper, this inequality is extended to arbitrary powers of selfadjoint operators acting in Hilbert space (see also [\textit{V. F. Babenko} and \textit{R. O. Bilichenko}, Ukr. Mat. Zh. 61, No.~10, 1299--1305 (2009); translation in Ukr. Math. J. 61, No.~10, 1533--1540 (2009; Zbl 1224.26068)]). More specifically, let \(A\) be linear non-bounded selfadjoint operator in a Hilbert space \(H\), let \(D(A)\) denote its domain of definition and let \(f\) be a linear continuous functional in \(H\). In Theorem~1, an additional generalization of the Taikov inequality on arbitrary powers of \(A\), in which the \(L_{\infty}\)-norm of the intermediate derivative is replaced by \(| (A^kx, f)| \), is obtained. The obtained additional inequality, which estimates \(| (A^kx, f)| \) by \(\| x\| \) and \(\| A^rx\| \) for \(x\in D(A^r)\), \(r, k \in {\mathbb N}\), \(k<r\), is precise as one may find some \(x\) for which the corresponding equality holds. Theorem~2 states that some multiplicative analogue of the inequality in Theorem~1 holds. Once more, the above inequality cannot be improved. Finally, some important particular cases of operators are considered and some remarks are given. The obtained theorems can be applied to different problems of mathematical analysis.