Chebyshev polynomials for higher order differential equations and fractional powers

Abstract

In this paper we show the characterization of the fractional powers of a class of positive operators by Chebyshev polynomials of the second kind. We consider the following higher order abstract Cauchy problems

$$\begin{aligned} \dfrac{d^nu}{dt^n} +Au = 0,\quad t>0, \end{aligned}$$

(0.1)

with initial conditions given by

$$\begin{aligned} \dfrac{d^iu}{dt^i}(0)=u_i\in X^{\frac{n-(i+1)}{n}},\quad i\in \{0,1,\ldots , n-1\},\quad n\geqslant 1, \end{aligned}$$

where X be a separable Hilbert space and $A:D(A)\subset X\rightarrow X$ is an unbounded linear, closed, densely defined, self-adjoint and positive definite operator, and its fractional counterpart. Here, $X^\alpha$ ($0\leqslant \alpha \leqslant 1$) denotes the domain of the fractional powers $A^\alpha$ endowed with graphic norm.