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It is well known that, for an unbounded nonnegative selfadjoint operator      A A       on a Hilbert space, there is a unique nonnegative square root      A 1 / 2 {A^{1/2}}      , which is frequently associated with the structural damping in many practical vibration systems. In this paper we develop a general theory for the structure of      A 1 / 2 {A^{1/2}}      , which includes the expression of      A 1 / 2 {A^{1/2}}       and a program to find the domain of      A 1 / 2 {A^{1/2}}       explicitly from the domain of      A A      . The relationship between      A 1 / 2 {A^{1/2}}       and related differential operators is determined for the selfadjoint differential operator      A A      . Finally, the theoretical results given in this paper are applied to fourth-order “beam” operators and      n n      -dimensional “wave” operators with sufficient complexity for applications to elastic vibration systems.

Structure for nonnegative square roots of unbounded nonnegative selfadjoint operators