K-Group Of The Commutants Of One Kind Of Operators

In recent years, K-theory has played an important role in study-ing the operator algebras, Chunlan Jiang, Junsheng Fang and Yang Cao have found the connection between the uniqueness of the strongly irreducible decomposition of the type of I operator and the K0-group of the commutants operators in 1990s. One begins to concern the cal-culation of K0-groups.Let H be a complex, separable, infinite dimensional Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T in L(H) is said to be strongly irreducible, if A’(T) has no non-trivial idempotent, equivalently, T does not exist non-trivial invariant subspace for M and H, Such that M+N=H and M∩N={0}, T is a strong irreducible operator.In the paper, through the use of strongly irreducible opera-tor, commutant algebra, induction sequence, K0 group, CFJ the-orem and other related knowledge we discuss the following oper-ator class:F= {T∈L(H)σ(T) is connected } and A’(T) {R(T) |R is the holomorphic function in or(T)}. For T ∈F, we prove that T is a stongly irreducible operator and V(A’(T))(?)N, Ko(A’(T))～Z And obtained if T ∈L(H) under certain conditions are met, then for all positive integers n, T(n) for finite strongly irreducible decompo-sition operator, and its strong irreducible decomposition uniquely in a similar sense . where N= {0, 1, 2, 3,...}, Z is integer group.