Strongly irreducible decomposition and similarity classification of operators (Q2382954)

An infinite-dimensional analogue of the Jordan theorem for matrices of finite rank is shown in this paper as the main result: if two bounded linear operators \(A\) and \(B\) on an infinite-dimensional Hilbert space have the decompositions into finite diagonal sums of strongly irreducible operators \(A_i\) and \(B_j\) (with finite multiplicities and not similar to each other, respectively) such that the \(K_0\)-groups of commutant algebras of \(A_i\) and \(B_j\) are the group \(\mathbb Z\) of all integers, and the quotients of the commutant algebras by their radicals for \(A, B\), and their finite diagonal sums \(A^{(n)}\) and \(B^{(n)}\), restricted to the ranges of minimal idempotents of the commutant algebras, are assumed to be commutative, then \(A\) is similar to \(B\) if and only if the \(K_0\)-group of the commutant algebra of the diagonal sum \(A\oplus B\) is isomorphic to \(\mathbb Z^k\) and its positive cone is \(\mathbb N^{k}\) for the semigroup \(\mathbb N\) of all natural numbers, and the quotient of the commutant algebra by any maximal ideal is isomorphic to a matrix algebra.