Differential Equations in Hilbert Space with Dissipative Symbols

Abstract

In this article equations of the form

$$T\left( { - i\frac{d}{{dt}}} \right)u\left( t \right) = T_0 - iT_1 u'\left( t \right) + ... + \left( { - i} \right)^n T_n u^{\left( n \right)} \left( t \right) = 0t \in \left( {0,\infty } \right)$$

are studied; here u(t) is a function with values in the Hilbert space $\mathfrak{H}$ and the coefficients T _j , j = 1,...,n are linear operators, possibly unbounded, in $\mathfrak{H}$. The operator symbol T(λ) is assumed to be dissipative, that is, to satisfy the condition: Im(T(λ)x,x) ≥ 0 for λ ∈ $\mathbb{R}$ and x ∈ $\mathcal{D}$(T). When the space $\mathfrak{H}$ 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.