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Modulated and subsequential ergodic theorems in Hilbert and Banach spaces. (English) Zbl 1036.47002

The main results of the present paper are the following.

Theorem 2.2. Let ${\left\{{\alpha }_{n,k}\right\}}_{n>0,k\in ℤ}$ be a matrix of complex numbers, such that ${\sum }_{k=-\infty }^{\infty }|{\alpha }_{n,k}|<\infty$ for every $n$. Then the following are equivalent.

1. The matrix satisfies the following two conditions:

(a) for every ${\Gamma }=\left\{\lambda :|\lambda |=1\right\}$, ${lim}_{n\to \infty }{\sum }_{k=-\infty }^{\infty }{\alpha }_{n,k}{\lambda }^{k}$ exists;

(b) ${sup}_{n\ge 1}{sup}_{|\lambda |=1}|{\sum }_{k=-\infty }^{\infty }{\alpha }_{n,k}{\lambda }^{k}|<\infty$.

2. For every unitary operator $U$ on a Hilbert space $H$ and every $x\in H$ the sequence $\left\{{\sum }_{k=-\infty }^{\infty }{\alpha }_{n,k}{U}^{k}x\right\}$ converges in norm.

3. For every unitary operator $U$ on a Hilbert space $H$ and every $x\in H$ the sequence $\left\{{\sum }_{k=-\infty }^{\infty }{\alpha }_{n,k}{U}^{k}x\right\}$ converges weakly.

Taking ${\alpha }_{n,k}={a}_{k}/n$ for $n>0$ and $0\le k\le n-1$ and 0 otherwise, the following corollary follows.

Corollary 2.3. Let ${\left\{{a}_{k}\right\}}_{k\ge 0}$ be a sequence of complex numbers. Then the following are equivalent.

1.The sequence satisfies the following two conditions:

(a) for every $\lambda \in {\Gamma }$ there exists $c\left(\lambda \right)$ such that $\frac{1}{n}{\sum }_{k=0}^{n-1}{a}_{k}{\overline{\lambda }}^{k}\to c\left(\lambda \right)$ $\left(n\to \infty \right)$;

(b) ${sup}_{n\ge 1}{sup}_{|\lambda |=1}|\frac{1}{n}{\sum }_{k=0}^{n-1}{a}_{k}{\lambda }^{k}|<\infty$.

2. For every unitary operator $U$ on a Hilbert space $H$ and every $x\in H$ the sequence $\left\{{\sum }_{k=-\infty }^{\infty }{a}_{k}{U}^{k}x\right\}$ converges in norm.

3. For every contraction $T$ on a Hilbert space $H$ and every $x\in H$ the sequence $\left\{{\sum }_{k=-\infty }^{\infty }{a}_{k}{T}^{k}x\right\}$ converges in norm.

4. For every unitary operator $U$ on a Hilbert space $H$ and every $x\in H$ the sequence $\left\{{\sum }_{k=-\infty }^{\infty }{a}_{k}{U}^{k}x\right\}$ converges weakly.

When 1. holds, $c\left(\lambda \right)\ne 0$ for at most countably many $\lambda \in {\Gamma }$, and the limit in 3. has the form

$\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=0}^{n}{a}_{k}{T}^{k}x=\sum _{\lambda \in {\Gamma }}c\left(\overline{\lambda }\right)E\left(\lambda ,T\right)x,$

where $E\left(\lambda ,T\right)$ is the orthogonal projection on the eigenspace of $T$ corresponding to the eigenvalue $\lambda$, and the series converges strongly. In this situation, condition

(c) ${sup}_{n}\frac{1}{n}{\sum }_{k=0}^{n-1}|{a}_{k}|<\infty$

is stronger than the condition (b). The authors give several examples of sequences such as a sequence (Ex. 2.4) satisfying (c) but not (a), (Ex. 2.5) satisfying (a) and (b) but not (c), (Ex. 2.6) satisfying (a) but not (b).

Theorem 3.1. Let $\left\{{\alpha }_{n,k}\right\}$ be a matrix of non-negative numbers, auch that ${\sum }_{k=0}^{\infty }{\alpha }_{n,k}=1$ for every $n$. If ${\sum }_{k=0}^{\infty }{\alpha }_{n,k}{\overline{\lambda }}^{k}\to 0$ for every $\lambda \ne 1$ with $|\lambda |=1$, then for every almost periodic operator $T$ on a Banach space $X$ and every contraction $T$ on a Hilbert space, ${lim}_{n\to \infty }{\sum }_{k=0}^{\infty }{\alpha }_{n,k}{T}^{k}x=E\left(T\right)x$.

From this theorem, the next proposition follows.

Proposition 3.2. Let $\left\{{k}_{j}\right\}$ be a strictly increasing sequence of positive integers satisfying ${lim}_{n\to \infty }{n}^{-1}{\sum }_{j=1}^{n}{\lambda }^{{k}_{j}}=0$ for every $\lambda \ne 1$ with $|\lambda |=1$.

1. For every almost periodic operator $T$ on a Banach space $X$ and every contraction $T$ on a Hilbert space,

$\underset{n\to \infty }{lim}\frac{1}{n}\sum _{j=1}^{n}{T}^{{k}_{j}}x=E\left(T\right)x·\phantom{\rule{2.em}{0ex}}\left(3·2\right)$

2. If $\left\{{k}_{j}\right\}$ has positive lower density, then (3.2) holds for every weakly almost periodic operator $T$ on a Banach space $X$.

3. If $\left\{{k}_{j}\right\}$ is thin, then there exists a weakly almost periodic operator $T$ for which (3.2) does not hold.

The authors then give a negative answer to the following question (Proposition 3.3): If $\left\{{\mu }_{n}\right\}$ satisfies ${\mu }_{n}f\to M\left(f\right)$ pointwise for every almost periodic function $f$, does then ${\mu }_{n}f\to M\left(f\right)$ hold pointwise for every weakly almost periodic functions? In the last section, the results in §2 are generalized to unitary representations of $\sigma$-compact locally compact berian groups (Proposition 4.1 and Theorem 4.2).

##### MSC:
 47A35 Ergodic theory of linear operators 37A30 Ergodic theorems, spectral theory, Markov operators