On uniqueness of fractional powers of multi-valued linear operators and the incomplete Cauchy problem

Abstract

We obtain uniqueness of additive families {A t } t>0 of fractional powers of a multi-valued sectorial linear operator A ≡ A 1 in a Banach space, satisfying a certain kind of continuity with respect to the exponent and a spectral property, from uniqueness of the solutions of the second-order incomplete Cauchy problem associated with A. We show the close relationship between the multiplicativity and the uniqueness of fractional powers.

Estimates for Fractional Powers and Logarithm of Operators With Hilbert-Schmidt Resolvents and Perturbation Results

“…For the classical results, see [8,9]. The recent investigations can be found in [1,2,4,5], [10]- [15] and the references given therein.…”

On a class of abstract degenerate fractional differential equations of parabolic type

“…A is continuously embedded in E. For more details about fractional powers of multivalued linear operators, the interpolation spaces and their mutual relations, we refer the reader to [10] (see, especially, the equations [10, (1.2)-(1.5)]), [12,Section 1.4], [25] and [27].…”

Abstract degenerate fractional differential inclusions in Banach spaces

“…forms a Banach space which is continuously embedded in E. We refer the reader to [8], [21] and [23] for further information concerning fractional powers of multivalued linear operators.…”