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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 {α n,k } n>0,k be a matrix of complex numbers, such that k=- |α n,k |< for every n. Then the following are equivalent.

1. The matrix satisfies the following two conditions:

(a) for every Γ={λ:|λ|=1}, lim n k=- α n,k λ k exists;

(b) sup n1 sup |λ|=1 | k=- α n,k λ k |<.

2. For every unitary operator U on a Hilbert space H and every xH the sequence { k=- α n,k U k x} converges in norm.

3. For every unitary operator U on a Hilbert space H and every xH the sequence { k=- α n,k U k x} converges weakly.

Taking α n,k =a k /n for n>0 and 0kn-1 and 0 otherwise, the following corollary follows.

Corollary 2.3. Let {a k } k0 be a sequence of complex numbers. Then the following are equivalent.

1.The sequence satisfies the following two conditions:

(a) for every λΓ there exists c(λ) such that 1 n k=0 n-1 a k λ ¯ k c(λ) (n);

(b) sup n1 sup |λ|=1 |1 n k=0 n-1 a k λ k |<.

2. For every unitary operator U on a Hilbert space H and every xH the sequence { k=- a k U k x} converges in norm.

3. For every contraction T on a Hilbert space H and every xH the sequence { k=- a k T k x} converges in norm.

4. For every unitary operator U on a Hilbert space H and every xH the sequence { k=- a k U k x} converges weakly.

When 1. holds, c(λ)0 for at most countably many λΓ, and the limit in 3. has the form

lim n 1 n k=0 n a k T k x= λΓ c(λ ¯)E(λ,T)x,

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

(c) sup n 1 n k=0 n-1 |a k |<

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 {α n,k } be a matrix of non-negative numbers, auch that k=0 α n,k =1 for every n. If k=0 α n,k λ ¯ k 0 for every λ1 with |λ|=1, then for every almost periodic operator T on a Banach space X and every contraction T on a Hilbert space, lim n k=0 α n,k T k x=E(T)x.

From this theorem, the next proposition follows.

Proposition 3.2. Let {k j } be a strictly increasing sequence of positive integers satisfying lim n n -1 j=1 n λ k j =0 for every λ1 with |λ|=1.

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

lim n 1 n j=1 n T k j x=E(T)x·(3·2)

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

3. If {k j } 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 {μ n } satisfies μ n fM(f) pointwise for every almost periodic function f, does then μ n fM(f) hold pointwise for every weakly almost periodic functions? In the last section, the results in §2 are generalized to unitary representations of σ-compact locally compact berian groups (Proposition 4.1 and Theorem 4.2).

MSC:
47A35Ergodic theory of linear operators
37A30Ergodic theorems, spectral theory, Markov operators