Operator and System Theory Seminar
SPEAKER: Michael Lin, BGU
TITLE:   On convergence of power series of contractions
DATE:    Monday, December 19, 2011
PLACE:   Room -101, Mathematics Building, BGU
TIME:	 14:10
Let $T$ be a power-bounded operator in a (real or complex) Banach space $X$.
We study the convergence properties of $\sum_{n=0}^\infty \beta_n T^n$
when $\{\beta_n\}$ is a Kaluza sequence
($\beta_0=1, \quad \beta_n>0$ with $\beta_{n+1}/\beta_n$ increasing to one)
with divergent sum and $\beta_n \to 0$.
This study is motivated by the one-sided ergodic Hilbert transform,
whose convergence is obtained when $\beta_n =1/2n$ for $n>0$.
Theorem 1.
Let $T$ be a power-bounded operator on $X$ and $f \in X$.
Then $ \sum_{k=0}^\infty \beta_k T^kf $ converges in norm if and only if
it converges weakly.
Theorem 2.
If $T^{**}$ is mean ergodic (in $X^{**}$), then
$\sum_{k=0}^\infty \beta_k T^kf$ converges  if (and only if)
$\liminf_{n\to \infty} \Big\| \sum_{k=0}^n \beta_k T^k f \Big\| < \infty$.
Theorem 3.
Let $T$ be a power-bounded operator on $X$.
If $ \sum_{k=0}^\infty \beta_k T^kf $ converges, then
$\big\| \frac1n \sum_{k=0}^{n-1} T^k f\big\| = o(1/\sum_{k=0}^n \beta_k)$.
Theorem 4.
Let $T$ be a mean ergodic positive  contraction on $L_1$
and $f \in L_1$.  Assume $\beta_n=O(1/n)$, and the coefficients of
$1/\sum_{n=0}^\infty \beta_n z^n = \sum_{n=0}^\infty \alpha_n z^n$ satisfy
$\{|\alpha_n|\}$ eventually non-increasing.
Then norm convergence of $\sum_{k=0}^\infty \beta_k T^kf$ implies
almost everywhere convergence.
Theorem 4 as stated is false for the rate $\beta_n =O(1/n^{1-\gamma})$ with
$\gamma \in (0,1)$.
Examples of $\{\beta_n\}$ Kaluza with $\beta_n \to 0$ and $\{|\alpha_n|\}$
non-increasing are given by Hausdorff moment sequences
($\beta_n= \int_0^1 t^n d\nu(t)$ for some probability $\nu$ on $[0,1]$
with no atom at 1).
(Joint work with Guy Cohen and Christophe Cuny)
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