Chebyshev polynomials for higher order differential equations and fractional powers

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 0.1dnudtn+Au=0,t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{d^nu}{dt^n} +Au = 0,\quad t>0, \end{aligned}$$\end{document}with initial conditions given by diudti(0)=ui∈Xn-(i+1)n,i∈{0,1,…,n-1},n⩾1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{d^iu}{dt^i}(0)=u_i\in X^{\frac{n-(i+1)}{n}},\quad i\in \{0,1,\ldots , n-1\},\quad n\geqslant 1, \end{aligned}$$\end{document}where X be a separable Hilbert space and A:D(A)⊂X→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:D(A)\subset X\rightarrow X$$\end{document} is an unbounded linear, closed, densely defined, self-adjoint and positive definite operator, and its fractional counterpart. Here, Xα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^\alpha $$\end{document} (0⩽α⩽1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqslant \alpha \leqslant 1$$\end{document}) denotes the domain of the fractional powers Aα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^\alpha $$\end{document} endowed with graphic norm.