TAILIEUCHUNG - Báo cáo toán học: "Averages of projections "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trung bình của dự. | J. OPERATOR THEORY 13 1985 219-225 Copyright by INCREST 1985 AVERAGES OF PROJECTIONS c K. FONG and G. J. MURPHY INTRODUCTION Let H be an infinite-dimensional Hilbert space and H the algebra of all bounded linear operators on H. It is well known that the closed convex hull of the projections in is the set of all positive contractions . positive operators of norm less than or equal to one . However there appears to be no characterization of the convex hull of the projections known. This latter set cannot be closed for as we shall see below it cannot contain any infinite-rank compact operator. This was proved originally in 2 . In this paper we give a number of conditions either necessary or sufficient for an operator A to be a convex combination or even an average of projections. Our main result is that if A is a positive noncompact contraction whose essential spectrum ƠC A contains a positive number 2 1 then A is an average of projections. Interest in this circle of ideas was initiated by Fillmore in 2 where he showed that every operator is a linear combination of projections for an easy proof of this result see 4 and that every positive invertible operator is a linear combination of projections with the coefficients positive. We considerably extend this result giving also an estimate on the size of the coefficients. TERMINOLOGY All operators will be assumed to act on Hilbert spaces and all Hilbert spaces will be assumed to be separable. We say an operator A is a rational convex combination of projections if A .yPy . 2 p with 2j non-negative rational numbers and Py projections and 2X . 2 1. We say A is an average of projections if A A . p n where A . p are projections. The following two easily-proved lemmas will be very useful. Lemma 1. If Ay . A are averages of projections so is Ay . A . 220 c. K. FONG and G. J. MURPHY Proof. This follows immediately from the observation that if an operator is an average of n projections then it is an average also of nk .

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