Order-to-topology continuous operators

Abstract

An operator T from vector lattice E into topological vector space \((F,\tau )\) is said to be order-to-topology continuous whenever \(x_\alpha \xrightarrow {o}0\) implies \(Tx_\alpha \xrightarrow {\tau }0\) for each \((x_\alpha )_\alpha \subset E\). The collection of all order-to-topology continuous operators will be denoted by \(L_{o\tau }(E,F)\). In this paper, we will study some properties of this new class of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and b-weakly compact operators. Under some sufficient and necessary conditions we show that the adjoint of order-to-norm continuous operators is also order-to-norm continuous and vice verse.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments.

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Correspondence to Kazem Haghnejad Azar.

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Jalili, S.A., Azar, K.H. & Moghimi, M.B.F. Order-to-topology continuous operators. Positivity (2021). https://doi.org/10.1007/s11117-021-00817-6

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Keywords

  • Vector lattice
  • Order-to-topology continuous operator
  • B-weakly compact operator

Mathematics Subject Classification

  • 47B65
  • 46B40
  • 46B42