# A note on Furuta type operator equation

Volume 18, Issue 1, pp 94--97 Publication Date: January 05, 2018       Article History
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### Authors

Xiaolin Zeng - School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, P. R. China Jian Shi - College of Mathematics and Information Science, Hebei University, Baoding, 071002, P. R. China

### Abstract

In this paper, we will show the existence of positive semidefinite solution of Furuta type operator equation $\displaystyle\sum_{j=0}^{n-1}A^{j}XA^{n-j-1}=Y,$ where $Y$ can be expressed by a comprehensive form.

### Keywords

• Furuta type operator equation
• generalized Furuta inequality
• positive definite operator and positive semidefinite operator

•  47A62
•  47A63

### References

• [1] T. Furuta, An extension of order preserving operator inequality, Math. Inequal. Appl., 13 (2010), 49–56

• [2] T. Furuta, Positive semidefinite solutions of the operator equation $\sum^n_{ j=1} A_{n-j}XA_{j-1} = B$, Linear Algebra Appl., 432 (2010), 949–955

• [3] J. Shi, An application of grand Furuta inequality to a type of operator equation, Global Journal of Mathematical Analysis, 2 (2014), 281–285