TAILIEUCHUNG - Báo cáo toán học: "Singly generated algebras containing compact operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đại số có đơn lẻ được tạo ra có chứa các nhà khai thác nhỏ gọn. | Copyright by INCREST 1979 J. OPERATOR THEORY 2 1979 211-214 SINGLY GENERATED ALGEBRAS CONTAINING COMPACT OPERATORS J. A. ERDOS Let T be an invertible operator on a complex Hilbert space H and let jtf T be the strongly closed algebra generated by T and the identity operator I. The problem of finding conditions which imply that has been considered in 1 2 3 4 In 3 Feintuch asks if 71-1 6 Jsf T whenever jrf T contains a compact injective operator. A partial answer is given below. Standard terminology and notation will be used see for example 6 . For any operator X the lattice of all invariant subspaces of X is denoted by Lat X. For any positive integer n H n denotes the direct sum of n copies of H and XM denotes the operator on HM which is the direct sum of n copies of X. For any set y of operators on H Y s e y . A well known result 6 Theorem states that if sđ is any algebra of operators on H containing I then the strong closure of VÍ is B Lat Lat BM for all n . We say that an invariant subspace of an operator X is full if XfN N and say that X is full if every invariant subspace of X is full. In 5 full subspaces are called regular but this term is avoided since regular operator has a different meaning. A subspace of the form N 0 M where N and M are invariant under X and M A is called a semi-invariant subspace of X cf. 7J . An obvious necessary condition for T-1 to be in s T is that Lat T Lat T-1. It is easy to see that a paraphrase of this condition is that T be a full operator. It is not known whether this condition is sufficient. From the result quoted above T ỵ e T if and only if Lat T Lat 7 1 for all n or in other words if and only if is full for all n. In Theorem 1 we give a condition on j F which implies that T is full. The subsequent results consist of finding cases when this condition is satisfied by j2 T n for every n. Theorem 1. Let T be an invertible operator and suppose that the compression of the strongly closed algebra generated by T and I to any .

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