Abstract view
Sweeping out properties of operator sequences


Published:19970201
Printed: Feb 1997
Mustafa A. Akcoglu
Dzung M. Ha
Roger L. Jones
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Abstract
Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach
Spaces of real valued functions on a complete nonatomic
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
$\T_kfQ_{k1}T_kf\_2\geq\delta$ 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.