Compact-like operators in lattice-normed spaces

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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.