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1. CJM 1997 (vol 49 pp. 3)
Sweeping out properties of operator sequences Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach
Spaces of real valued functions on a complete non-atomic
probability space. Let $(T_1,\ldots,T_{K})$ be
$L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a
function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if
$\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where
$Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned
by $(T_1f,\ldots,T_kf)$. Call a function $h$ a
$(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$,
$\|h\|_1<\varepsilon$, and if
$\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of
measure greater than $1-\varepsilon$. The following is the main
technical result, which is obtained by elementary estimates. There
is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a
$\delta$-spanning function, and if the joint distribution
of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge
M)\vee(-M)\bigr)/M$
is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$.
Furthermore, if $T_k$s are the averages of operators induced by
the iterates of a measure preserving ergodic transformation, then a
similar result is true without requiring that the joint distribution
is normal. This gives the following theorem on a sequence $(T_i)$ of
these averages. Assume that for each $K\geq1$ there is a subsequence
$(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning
function $f_K$ for this subsequence. Then for each $\varepsilon>0$
there is a function $h$,
$0\leq h\leq1$,
$\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e..
Another application of the main result gives a refinement of a part
of Bourgain's ``Entropy Theorem'', resulting in a
different, self contained proof of that theorem.
Keywords:Strong and $\delta$-sweeping out, Gaussian distributions, Bourgain's entropy theorem. Categories:28D99, 60F99 |