–Hilbert module orAW∗-module arose naturally in Kaplansky’s study ofAW∗algebras of type I [2]. I. Kaplansky proved some deep and <...> legant results for such structures, and therefore they have many properties of Hilbert spaces. In [7] A. G. Kusraev established functional representations of Kaplansky–Hilbert modules and AW∗-algebras of type I by <...> rators in lattice-normed spaces were introduced by A. G. Kusraev in [5] and [6], respectively. In [8] (see also [9]) a general form of cyclically compact operators in Kaplansky–Hilbert modules, which, like the Schmidt representation of compact operators in Hilbert spaces, as <...> Banach–Kantorovich spaces over a ring of measurable functions were investigated in [1, 3, 4]. In this paper, we introduce and study the concepts of the trace class operators and global eigenvalue and multiplicity of a global eigenvalue, and give a v <...> t in X is called relatively cyclically compact if it is contained in a cyclically compact set. An operator T ∈ BΛ(X,Y ) is called cyclically <...> ote that θx,y ∈K (X,Y ). The techniques employed in [1] yield the following theorem: U = S 3. The Trace Class In this section, we study the trace class operators on Kaplansky–Hilbert modules and investigate the dualities of the trace class. u is a cyclically compact )k∈N, G¨ ull¨ on¨ u U. Trace class and Lidski˘ı trace formula on Kaplansky–Hilbert modules 31 From now onward, it will be assumed that (ek)k∈N, (fk)k∈N, and (µ <...> es (xi)i∈I in X and (yi)i∈I in Y such that xi yi i∈I is o-summable and T = bo-i∈I θxi,yi . In particular, if (xi)i∈I and (yi)i∈I are projection orthonormal <...> X. A scalar λ ∈ Λ is said to be an eigenvalue if there exists nonzero x ∈ X such that Tx = λx. A nonzero eigenvalue λ is called a global eigenvalue if for ev <...> , l,m,n ∈ N with n = m. So, (τλ(n))n∈N is a partition of [λ] where τλ(n) := supl∈N {τλ,l(n)}. Now, we define the multiplicity of global eigenvalues of cyclically compact operators on of T. The following conditions are <...>
Владикавказский_математический_журнал_№2_2014.pdf
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