On some properties of imaginary powers of positive operators (Q2760715)

The aim of the article is to study properties of imaginary powers of a positive operator \(A\) in complex Banach and Hilbert spaces and expose conditions ensuring that the imaginary powers \(A^{is}\) (\(s\in\mathbb R\)) of this operator are bounded. The boundedness of the operators \(A^{is}\) is used when studying the following questions: the boundedness of Riesz projections corresponding to a given unbounded component of the spectrum of \(A\); the property of a given operator to be exponentially dichotomous; solvability of the Cauchy problem for the equation \(u_t + Au = f\) in a Banach space; and others.NEWLINENEWLINENEWLINEThe main result of the article reads as follows: NEWLINENEWLINENEWLINE(i) Let \(A\:H_0\to H_0 \) be a given positive operator such that the set \(S_0 = \{z: |\text{arg } z|\geq \pi - \theta_0\}\) is included in \(\rho(A)\) (\(0 < \theta_0 < \pi\)) and the estimate \(\|(A - zI)^{-1}\|_{L(H_0,H_0)}\leq c/(1 + |z|)\) holds for every \(z\in S_0\), where \(c\) is a constant independent of \(z\). Assume that \((H_1,H_{-1})_{1/2,2} = H_0\) (\(H_k = D(A^k)\)) or, in other words, \(B_2^0 = H_0\) (\(B_q^s = (H_m,H_k)_{\theta,q}\) denotes the interpolation pair with \(1\leq q\leq\infty\), \(k<s<m\), \(\theta=(m-s)/(m+k)\)), then \(A^{is} \in L(H_0,H_0)\) for every \(s\in\mathbb R\), the group \(s\to A^{is}\) is strongly continuous, and the estimate NEWLINE\[NEWLINE \|A^{is}\|_{L(H_0,H_0)} \leq c\frac{e^{(\pi - \theta_0)s}}{1 + |s|} NEWLINE\]NEWLINE holds. NEWLINENEWLINENEWLINE(ii) Suppose that \(A\) is a positive operator and \(\|A^{is}\|_{L(H_0,H_0)} \leq c\) for every \(s\) in some neighborhood of zero, where \(c <\infty\) is a constant. Then \(B_2^0 = H_0\).NEWLINENEWLINENEWLINEIn the case of a Banach space, a similar result is also proved by the author.