JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (8): 39-44.doi: 10.6040/j.issn.1671-9352.0.2020.421

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The left-star order for idempotent operators

HAO Hong-yan, LI Yuan*   

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xian 710119, Shaanxi, China
  • Published:2021-08-09

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Abstract: It is investigated that the characterizations of all idempotent operators on Hilbert space H with respect to the left-star order, which is defined by A*A=A*B and R(A)⊆R(B). Let A and B be two idempotent operators, the equivalent condition of A*≤B and the representation of operator matrix form is given. Meanwhile, the existence and representation of A∨B(supremum)and A∧B(infimum)of the star order when A*≤B is discussed.

Key words: idempotent operator, left-star order, operator matrix

CLC Number: 

  • O153.1
[1] DJIKIC M S, FONGI G, MAESTRIPIERI A. The minus order and range additivity[J]. Linear Algebra Appl, 2017, 531:234-256.
[2] DARZIN M P. Natural structures on semigroups with involution[J]. Bull Amer Math Soc, 1978, 84:139-141.
[3] LI Yuan, NIU Jiajia, XU Xiaoming. The minus order for idempotent operators[J/OL]. [2019-12-20]. Filomat. https://arxiv.org/abs/1912.09718.
[4] DENG Chunyuan, WANG Shunqin. On some characterizations of the partial orderings for bounded operators[J]. Math Ineq Appl, 2012(3):619-630.
[5] DENG Chunyuan, YU Anqi. Some relations of projection and star order in Hilbert space[J]. Linear Algebra Appl, 2015, 474:158-168.
[6] HARTWIG R E, DARZIN M P. Lattice properties of the star order for complex matrices[J]. J Math Anal Appl, 1982, 86:539-578.
[7] ANTEZANA J, CANO C, MOSCONI I, et al. A note on the star order in Hilbert spaces[J]. Linear and Multilinear Algebra, 2010, 58:1037-1051.
[8] XU Xiaoming, DU Hongke, LI Yuan, et al. The supremum of linear operators for the *-order[J]. Linear Algebra Appl, 2010, 433:2198-2207.
[9] LI Yuan, GAO Shuhui, XU Xiaoming. The star order for idempotent operators[J/OL]. [2021-02-14]. Linear and Multilinear Algebra. https://doi.org/10.1080/03081087.2021.1887069.
[10] DOLINAR G, MAROVT J. Star partial order on B(H)[J]. Linear Algebra Appl, 2011, 434:319-326.
[11] MITRA S K. The minus partial order and the shorted matrix[J]. Linear Algebra Appl, 1986, 83:1-27.
[12] C(-overL)RULIS J. One-sided star partial orders for bounded linear operators[J]. Operators and Matrices, 2015(9):891-905.
[13] BAKSALARY J K, BAKSALARY O M, LIU Xiaoji. Further properties of the star, left-star, rightstar, and minus partial orderings[J]. Linear Algebra Appl, 2003, 375:83-94.
[1] YANG Yuan, ZHANG Jian-hua. A local characterization of centralizers on B(H) [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 1-4.
[2] XIE Tao, ZUO Ke-zheng. [J]. J4, 2013, 48(4): 95-103.
[3] GENG Wan-hai, CHEN Yi-ming, LIU Yu-feng, WANG Xiao-juan. The approximation of definite integration by using Haar wavelet and operator matrix [J]. J4, 2012, 47(4): 84-88.
[4] DU Gui-chun1,2, SHAO Chun-fang1. Closedness of ranges of upper triangular operator matrix [J]. J4, 2012, 47(4): 42-46.
[5] DUAN Ying-tao. The reverse order law for ｛1,3,4｝-inverse of the product of two operators [J]. J4, 2012, 47(4): 53-56.
[6] XU Jun-lian1，2. Solutions to the operator equation AXB*-BX*A*=C [J]. J4, 2012, 47(4): 47-52.
[7] XU Junlian. Eigen-value functions of some functions of orthogonal projections [J]. J4, 2011, 46(6): 70-74.
[8] FANG Li1, BAI Wei-zu2. Maps preserving the idempotency of products or triple Jordan products of idempotent operators [J]. J4, 2010, 45(12): 98-105.
[9] LI Yuan . Weyl's theorem for 2×2 upper triangular operator matrices [J]. J4, 2007, 42(6): 69-73 .
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