We obtain a uniform ergodic theorem for the sequence \$ rac1{s(n)} sum_{k=0}^n(arDelta s)(n-k),T^k\$, where \$arDelta\$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums, \$T\$ is a bounded linear operator on a Banach space and \$s\$ is a divergent nondecreasing sequence of strictly positive real numbers, such that \$lim_{n ightarrow+infty} s(n+1)/s(n)=1\$ and \$arDelta^qsinell_1\$ for some positive integer \$q\$. Indeed, we prove that if \$T^n/s(n)\$ converges to zero in the uniform operator topology, then the sequence of averages above converges in the same topology if and only if 1 is either in the resolvent set of \$T\$, or a simple pole of the resolvent function of \$T\$.

### On the uniform ergodic theorem for some Norlund means

#### Abstract

We obtain a uniform ergodic theorem for the sequence \$ rac1{s(n)} sum_{k=0}^n(arDelta s)(n-k),T^k\$, where \$arDelta\$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums, \$T\$ is a bounded linear operator on a Banach space and \$s\$ is a divergent nondecreasing sequence of strictly positive real numbers, such that \$lim_{n ightarrow+infty} s(n+1)/s(n)=1\$ and \$arDelta^qsinell_1\$ for some positive integer \$q\$. Indeed, we prove that if \$T^n/s(n)\$ converges to zero in the uniform operator topology, then the sequence of averages above converges in the same topology if and only if 1 is either in the resolvent set of \$T\$, or a simple pole of the resolvent function of \$T\$.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11567/928164`
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