Gelfand-Philips and L-limited properties of order P on Banach spaces

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Abstract

In this paper we show the stability of Gelfand-Phillips property of order $p$ under taking injective tensor product, compact operators, and Bochner integrable functions. We introduce the concept of $ L $-limited sets of order $p,$ and obtain some characterizations of limited $p$-convergent operators. Also we define the notion of $L$-limited of order $p$ and characterize this property in terms of weak compact operators.\ Furthermore, we give a new dual characterization of the class of weak$^{\ast}$ $p$-convergent operators through $ L $-limited sets of order $ p. $\ Moreover, some characterizations of the Gelfand-Phillips property of order $p$ in terms of limited $ p $-convergent operators are given.\ In addition by applying our results on the limited $ p $-convergent operators, we obtain some characterizations of the Dunford-Pettis$ ^{\ast} $ property of order $p.$