  The universal Teichm\"uller space is an infinitely dimensional generalization
of the classical Teichm\"uller space of Riemann surfaces. It carries a natural
Hilbert structure, on which one can define a natural Riemannian metric, the
Weil-Petersson metric. In this paper we investigate the Weil-Petersson
Riemannian curvature operator $\tilde{Q}$ of the universal Teichm\"uller space
with the Hilbert structure, and prove the following:
  (i) $\tilde{Q}$ is non-positive definite.
  (ii) $\tilde{Q}$ is a bounded operator.
  (iii) $\tilde{Q}$ is not compact; the set of the spectra of $\tilde{Q}$ is
not discrete.
  As an application, we show that neither the Quaternionic hyperbolic space nor
the Cayley plane can be totally geodesically immersed in the universal
Teichm\"uller space endowed with the Weil-Petersson metric.
