Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces (Q2896203)

The main goal of the paper is to finish the program of \textit{J. Ernest} [``Charting the operator terrain'', Mem. Am. Math. Soc. 171 (1976; Zbl 0331.47001)] on exploring the realm of unitary equivalence classes of closed densely defined operators by making no assumptions neither on the dimension of Hilbert spaces nor on the boundedness of operators. Another aim is related to various results on decompositions of operators.NEWLINENEWLINEAn ideal of \(N\)-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their reduced parts. New decomposition theorems for \(N\)-tuples of closed densely defined linear operators acting on a common arbitrary Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class \(\mathcal{CDD}_N\) of all unitary equivalence classes of such \(N\)-tuples are established and certain ideals in \(\mathcal{CDD}_N\) are distinguished. Prime decomposition in \(\mathcal{CDD}_N\) is proposed and its uniqueness in a special sense is established. The issue of classification of ideals in \(\mathcal{CDD}_N\) up to isomorphism is discussed. A model for \(\mathcal{CDD}_N\) is described and its concrete realization is presented. A new partial order of \(N\)-tuples of operators is introduced and its fundamental properties are established.