Applications of the purely imaginary powers of operators in Hilbert spaces

In C. R. Acad. Sci. Paris299 (1984), 173–176, we discussed purely imaginary powers A^iy(−∞ < y < + ∞) of linear operators A in Hilbert spaces. Here we utilize the results to consider the various problems: generation of cosine families in Hilbert spaces, coincidence of the definition domains of the fractional powers of operators, differentiability of the functions of the form A(·)⁰ (0 < θ < 1) where A(·) is an operator valued function defined on an interval 0, T], and so forth.