turkmath.org

The concept of Kaplansky$-$Hilbert module was introduced by I. Kaplansky (in I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), no. 4, 839--858.) under the name $AW^\ast$-module. Cyclically compact sets and operators in lattice-normed spaces were introduced by A. G. Kusraev, (see A.G. Kusraev, Dominated Operators, Kluwer Academic Publishers, Dordrecht, 2000.) In particular, a general form of cyclically compact operators in Kaplansky$-$Hilbert modules, which is similar to the Schmidt representation of compact operators on Hilbert spaces, as well as a variant of the Fredholm alternative for cyclically compact operators, were also given. In this talk, we will present some characterizations of cyclically compact operators and Schatten-type classes of operators on Kaplansky$-$Hilbert modules.