Differential Equations in Hilbert Space with Dissipative Symbols

In this article equations of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$T\left( { - i\frac{d}{{dt}}} \right)u\left( t \right) = T_0 - iT_1 u'\left( t \right) + ... + \left( { - i} \right)^n T_n u^{\left( n \right)} \left( t \right) = 0t \in \left( {0,\infty } \right)$$ \end{document} are studied; here u(t) is a function with values in the Hilbert space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{H}$$ \end{document} and the coefficients Tj, j = 1,...,n are linear operators, possibly unbounded, in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{H}$$ \end{document}. The operator symbol T(λ) is assumed to be dissipative, that is, to satisfy the condition: Im(T(λ)x,x) ≥ 0 for λ ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document} and x ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{D}$$ \end{document}(T). When the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{H}$$ \end{document} is finite-dimensional, theorems of factorization for the symbol T(λ) and theorems on the unique solvability for the truncated Cauchy problem on the half-axis t ∈ [0,∞) are proved. In the infinite-dimensional space we can obtain identities for solutions of the equations considered. From these identities it is possible to deduce a priori estimates for the solutions.