Let T be a bounded, linear operator on a complex, separable, infinite dimensional Hilbert space H. We assume that T is an essentially isometric (resp. normal) operator, that is, IH-T*T (resp. TT* -T*T) is COMPACT. For the COMPACTness of S from the commutant of T, some necessary and sufficient conditions are found on S. Some related problems are also discussed.

Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous if $x_\alpha\xrightarrow{Fo}0$ implies $T(x_\alpha)\xrightarrow{\Vert.\Vert}0$ for every $(x_{\alpha})_{\alpha \in A}\subseteq E$. This paper aims to investigate the properties of this new class of OPERATORS and explore their relationships with existing classifications of OPERATORS. We introduce a new class of OPERATORS called $\tilde{o}$rder weakly COMPACT OPERATORS. A continuous operator $T: E \rightarrow X $ is considered $\tilde{o}$rder weakly COMPACT if $ T(A) $ in $X$ is a relatively weakly COMPACT set for every $Fo$-bounded $A\subseteq E$. In this manuscript, we examine various properties of this class of OPERATORS and explore their connections with $\tilde{o}$rder-norm continuous OPERATORS.

A continuous operator $T$ between two Banach lattices $E$ and     $F$ is called almost order-weakly COMPACT, whenever for each almost order bounded subset $A$ of $E$,  $T(A)$ is a relatively weakly COMPACT subset of $F$. We show that the positive operator $T$ from  $E$ into a  Dedekind complete Banach lattice $F$  is almost order-weakly COMPACT iff  $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of OPERATORS and its relationships with the others known classes of OPERATORS.

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