# Compact-like operators in lattice-normed spaces

2017
Aydın, Abdullah
Let \$(X,p,E)\$ and \$(Y,m,F)\$ be two lattice-normed spaces. A linear operator \$T:Xto Y\$ is said to be \$p\$-compact if, for any \$p\$-bounded net \$x_alpha\$ in X, the net \$Tx_alpha\$ has a \$p\$-convergent subnet in Y. That is, if \$x_alpha\$ is a net in X such that there is a \$ein E_+\$ satisfying \$p(x_alpha) ≤ e\$ for all \$alpha\$, then there exists a subnet \$x_{alpha_beta}\$ and \$y_in Y\$ such that \$m(Tx_{alpha_beta} −y) xrightarrow{o}0\$ in \$F\$. A linear operator \$T:Xto Y\$ is called \$p\$-continuous if \$p(x_alpha) xrightarrow{o}0\$ in \$E\$ implies \$m(Tx_alpha) xrightarrow{o}\$ in \$F\$, where \$x_alpha\$ is a net in \$X\$. \$p\$-compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, \$AM\$-compact operators, etc. Also, \$p\$-continuous operators generalize many classes of operators such as order continuous, norm con- tinuous, Dunford-Pettis, etc. Similar to \$M\$-weakly and \$L\$-weakly compact operators, we deﬁne \$p\$-\$M\$-weakly and \$p\$-\$L\$-weakly compact operators and study some of their properties. We also study up-continuous and up-compact operators between lattice- normed vector lattices. We give some results about acting mixed-normed spaces on lattice normed spaces.

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Citation Formats
A. Aydın, “Compact-like operators in lattice-normed spaces,” Ph.D. - Doctoral Program, Middle East Technical University, 2017. 