  Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative
self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian
upper bound. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with the
operator $L$ and we define its distance from the subspace $L^{\infty}(X)$ under
the $ {\rm BMO}_L(X)$ norm as follows: $$
  {\rm dist} (f, L^{\infty}):= \inf_{g\in L^{\infty}} \|f -g\|_{{\rm
BMO}_L(X)}. $$ In this paper we prove that ${\rm dist} (f, L^{\infty})$ is
equivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberg
inequality for the space ${\rm BMO}_L(X)$: $$
  \sup_B { \mu\big(\{ x\in B: |f(x)-e^{-{r_B^2}L}f(x)|>\lambda\}\big) \over
\mu(B)} \leq e^{-\lambda/\varepsilon}\ \ \ \ {\rm for\ large\ } \lambda. $$
This extends the well-known result of Garnett and Jones \cite{GJ1} for the
classical ${\rm BMO}$ space (introduced by John and Nirenberg). As an
application, we show that a ${\rm BMO}_L(X)$ function with compact support can
be decomposed as the summation of an $L^\infty$-function and the integral of
the heat kernel (associated with $L$) against a finite Carleson measure on
$X\times[0,\infty)$. The key new technique is a geometric construction
involving the semigroup $e^{-tL}$. We also resort to several fundamental tools
including the stopping time argument and the random dyadic lattice.
