STRUCTURE FOR NONNEGATIVE SQUARE ROOTS OF UNBOUNDED NONNEGATIVE SELF-ADJOINT OPERATORS

It is well known that, for an unbounded nonnegative selfadjoint operat or A on a Hilbert space, there is a unique nonnegative square root A(1 /2), which is frequently associated with the structural damping in man y practical vibration systems. In this paper we develop a general theo ry for the structure of A(1/2), which includes the expression of A(1/2 ) and a program to find the domain of A(1/2) explicitly from the domai n of A. The relationship between A(1/2) and related differential opera tors is determined for the selfadjoint differential operator A. Finall y, the theoretical results given in this paper are applied to fourth-o rder ''beam'' operators and n-dimensional ''wave'' operators with suff icient complexity for applications to elastic vibration systems.