Opuscula Math. 41, no. 3 (2021), 437-446

Opuscula Mathematica

Quadratic inequalities for functionals in l

Gerd Herzog
Peer Chr. Kunstmann

Abstract. For a class of operators \(T\) on \(l^{\infty}\) and \(T\)-invariant functionals \(\varphi\) we prove inequalities between \(\varphi(x)\), \(\varphi(x^2)\) and the upper density of the sets \[P_r:=\{n \in \mathbb{N}_0: \varphi((T^{n}x)\cdot x) \gt r\}.\] Applications are given to Banach limits and integrals.

Keywords: Banach algebras of bounded functions, operator-invariant functionals, Banach limits.

Mathematics Subject Classification: 47B37, 47B48, 47B60.

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  • Gerd Herzog (corresponding author)
  • Karlsruhe Institute for Technology, Institute for Analysis, D-76128 Karlsruhe, Germany
  • Peer Chr. Kunstmann
  • Karlsruhe Institute for Technology, Institute for Analysis, D-76128 Karlsruhe, Germany
  • Communicated by Alexander Gomilko.
  • Received: 2020-11-20.
  • Accepted: 2021-01-23.
  • Published online: 2021-04-19.
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Cite this article as:
Gerd Herzog, Peer Chr. Kunstmann, Quadratic inequalities for functionals in l, Opuscula Math. 41, no. 3 (2021), 437-446, https://doi.org/10.7494/OpMath.2021.41.3.437

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