K₀-groups and strongly irreducible decompositions of operator tuples

An operator tuple T= (T₁, , T₍) is called strongly irreducible (SI), if the joint commutant of T does not any nontrivial idempotent operator. In this paper, we study the uniqueness of finitely strong irreducible decomposition of operator tuples up to similarity by K-theory of operator algebra, and give the algebraically similarity invariants of the Cowen-Douglas tuple with index 1 by using K₀-group of the commutant of operator tuples. As an application, we calculate K₀-groups of some multiplier algebras, and describe the similarity of backwards multishifts on Drury-Arveson space by means of inflation theory.