On the positive square root of the fourth derivative operator

It is only for a special subset of the natural boundary conditions for the operator \[ Aw = \frac {{{d^4}w}}{{d{x^4}}}\] that its positive square root is the negative second derivative operator. In this paper we develop a procedure for parametric description of all natural boundary conditions, we show which ones admit ${A^{1/2}}$ in the form just noted, and we show that in the other cases \[ - {D^2}w \equiv - \frac {{{d^2}w}}{{d{x^2}}} = \left [ {I + P} \right ]{A^{1/2}}w\] where $P$ is a bounded, but in general not compact, operator on the Hilbert space ${L^2}\left [ {0, \pi } \right ]$. Possible applications to the theory of the partial differential equation \[ \rho \frac {{{\partial ^2}w}}{{\partial {t^2}}} - 2\gamma \frac {{{\partial ^3}w}}{{\partial t\partial {x^2}}} + EI\frac {{{\partial ^4}w}}{{\partial {x^4}}} = 0\] are indicated.