  Let $(E,\mathcal F,\mu)$ be a probability space, and let $P$ be a Markov
operator on $L^2(\mu)$ with $1$ a simple eigenvalue such that $\mu P=\mu$ (i.e.
$\mu$ is an invariant probability measure of $P$). Then $\hat P:=\ff 1 2
(P+P^*)$ has a spectral gap, i.e. $1$ is isolated in the spectrum of $\hat P$,
if and only if $$\|P\|_\tau:=\lim_{R\to\infty} \sup_{\mu(f^2)\le
1}\mu\big(f(Pf-R)^+\big)<1.$$ This strengthens a conjecture of Simon and
H$\phi$egh-Krohn on the spectral gap for hyperbounded operators solved recently
by L. Miclo in \cite{M}. Consequently, for a symmetric, conservative,
irreducible Dirichlet form on $L^2(\mu)$, a Poincar\'e/log-Sobolev type
inequality holds if and only if so does the corresponding defective inequality.
Extensions to sub-Markov operators and non-conservative Dirichlet forms are
also presented.
