tailieunhanh - Báo cáo toán học: "A note on positive operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một lưu ý các nhà khai thác tích cực. | J. OPERATOR THEORY 5 1981 73 76 Copyright by ĨNCREST 1981 A NOTE ON POSITIVE OPERATORS CHE-KAO FONG and SZE-KAI TSUI In the present note we give a generalization of the elementary fact that a complex number z is a nonnegative real number if z Re z . Let T be a bounded linear operator on a Hilbert space. We denote T T j2 by Re T and the positive square root of T T by T . The generalization is the following. Theorem 1. If T SỈ Re T then T is positive. The above theorem gives a characterization of positive operators. In what follows we shall prove a stronger result from which Theorem 1 can be derived immediately. Theorem 2. Let T VP where T V p are bounded linear operators on a Hilbert space with p 0 and. V being power bounded . K Ị X k for a fixed k and n 1 2. . If p Re T then T p. Note that Theorem 1 follows from Theorem 2 by considering the polar decomposition of T. An immediate consequence of Theorem 2 is Corollary 3. If V is a power bounded operator and Re F I then V I. We remark that the hypothesis RelV I in the above Corollary cannot be replaced by a weaker condition such as jRe Jz I. For example let Vbe a 2x2 0 2 matrix of the form I I then Re F Z and V2 0. Now we proceed to prove Theorem 2. Firstly we show the following lemma. Lemma 4. Suppose that p V are operators on a Hilbert space đi and p is positive. If p Re FP then p -c VPV . Furthermore if p sj Re FP and p VPV then VP p. Proof. For each vector X in we have Px x - Re LP x x Re VPx x SỈ 1 . . VPx x Px X 1 2 PL X P x 2 74 CHE-KAO FONG and SZE-KAI TSUI by applying Schwarz s inequality to the positive semi-definite form x y -H- Px ỳ x F F in obtaining the last inequality in 1 . Hence Px x VPV X x for all X in jf that is p íC VPV . In addition to P Re VP if p VPV is assumed then 1 yields Px x Re FPx x VPx x VPx x for all X in yf and hence p VP. . Proof of Theorem 2. Since VPV - p 0 by Lemma 4 it follows that V VPV P V 0 that is F2P JZ 2 VPV . Repeating the process n times we have F ip jz i VnP y y.