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Báo cáo toán học: "Stability of operator inequalities "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bất bình đẳng ổn định của nhà điều hành. | J. OPERATOR THEORY 10 1983 133 139 Copyright by INCREST 1983 STABILITY OF OPERATOR INEQUALITIES FUMIO KUBO 1. INTRODUCTION Let Ao Ji .An be a finite sequence of bounded linear operators on a Hilbert space It often occurs that the mutual relationship among these operators is governed by a finite or infinite number of inequalities F . A A A 0 1 2 where X 0 means that X is a positive semidefinite operator on . One might be interested in when the system of inequalities is stable under Cesáro averaging. More precisely let Bk be the Cesáro average of Bk A A T Ăk .k 1 L 0 1 . n . Does the system of inequalities imply that 0 i 1 2 . 2V In this paper these stability properties are discussed. The first example concerns the convex sequences of operators. Together with this the second section discusses the stability of log-convexity of operator sequences. The following two sections discuss the stability of the relations governed by truncated Hankel and Toe-plitz operator matrices. The last section gives an application and a continuous analogue of the stability. In concluding this introduction the author would like to express his heartly thanks to Prof. T. Ando for many valuable suggestions and discussions. 2. CONVEX SEQUENCES An operator sequence is convex if k-1 Jk I 2 - A 0 k 1 2 . n - 1 . 134 FVM1O KUBO After some routine calculations it follows that Fk - l fc 4- 2 - k 2 3 . n 1 where pk - ƠV1 4- 5s 1 2 - Bk k - 1 2 . n 1 and hence the Cesáro average Ổ J is also convex. A stronger condition on an operator sequence Ak is given by Vi 4k0 k -1 2 . Ak Ak 1 J and called log-convexity. Remark. From this definition it is implied that all the terms of the sequence Ak are positive semidefinite. If the terms Ak commute with each other the stability of the log-convexity under Cesáro average is already known in essence through the spectral theory cf. 4 and 5 . But their proof does not work for this general case. Note that if the operators A B and c are positive semidefinite and A has .