Analyticity of Semigroups Related to a Class of Block Operator Matrices

Abstract

We derive various properties, e.g., analyticity of the associated semigroup and existence of a Riesz basis consisting of eigenfunctions, of the operator matrix \( \mathcal{A} = \left[ {\begin{array}{*{20}c} 0 & I { - A_0 } & { - D} \end{array} } \right] \). Here the entries A 0 and D are unbounded operators. Such operator matrices are associated with second-order problems of the form \( \ddot z\left( t \right) + A_0 z\left( t \right) + D\dot z\left( t \right) = 0 \) which are used as models for small motions of some hydrodynamical systems.