On submajorization inequalities for matrices of measurable operators
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Advances in Operator Theory https://doi.org/10.1007/s43036-020-00101-6
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ORIGINAL PAPER
On submajorization inequalities for matrices of measurable operators Sawlet Junis1
•
Azhar Oshanova2
Received: 11 May 2020 / Accepted: 12 August 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract
x z and Let M be a semifinite von Neumann algebra. We proved that if z y x z are positive matrices with entries in M, then z is logarithmically subz y 1 1 majorized by x2 y2 . Using this, we proved some related submajorization inequalities. Keywords Matrix of operators Submajorization Semi-finite von Neumann algebra
Mathematics Subject Classification 46L52 47L05
1 Introduction Let Mm be the set of all m m complex matrices. Bourin proved that if A X and are positive block-matrix with entries in Mm , then X B k Y
sj ðXÞ
j¼1
k Y
1
1
sj ðA2 B2 Þ;
k ¼ 1; 2; . . .; m;
j¼1
Communicated by Lyudmila Turowska. & Sawlet Junis [email protected] Azhar Oshanova [email protected] 1
School of Mathematics and Statistics, Yili Normal University, Yining 835000, China
2
Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Nur-Sultan 010008, Kazakhstan
A X
X B
ð1Þ
S. Junis and A. Oshanova
where sj ðZÞ ðj ¼ 1; 2; . . .; mÞ is singular value of Z 2 Mm (see [9, Theorem 4.1]). Using (1), Lin [10] gave a new proof of the main inequality in [1], it is that if Aj ; Bj ; j ¼ 1; 2; . . .; n are positive operators in Mm such that Aj Bj ¼ Bj Aj ; j ¼ 1; 2; . . .; n, then for any unitarily invariant norm k k, !2 ! ! X n n n n X X X 1 1 2 2 ð2Þ Aj Bj Aj Bj A B j k : j¼1 j¼1 j¼1 j¼1 By Ky Fan dominance principle (see [2, Theorem IV.2.2]), this is equivalent to 0 ! !2 1 ! !! k n k n k n n X X X X X X X 1 1 2 2 si Aj Bj si @ Aj Bj A si Aj Bj ; ð3Þ i¼1
j¼1
i¼1
j¼1
i¼1
j¼1
j¼1
x z
z y
for all 1 k m. In [8], Han obtained the following extension of (3): if x z and are positive matrices with entries in a semifinite von Neumann z y algebra, then Z t Z t 1 1 ð4Þ ls ðzÞ ds ls x2 y2 ds; t [ 0: 0
0
From this, one can extend (2) for the norm on noncommutative Lp -spaces (also see [7]). x z x z In this paper, we extend (4) as following: if 0 and 0; z y z y 1 1 then Dt ðzÞ Dt ðx2 y2 Þ holds for all t [ 0, where Dt ðaÞ is the determinant function associated with a 2 M. We use this inequality to obtain a generalization of (3) for measurable operators. Also, we use properties of positive 2-by-2 matrix with entries in M to prove a Schwarz type inequality and an inequality for absolute values of operator sums.
2 Preliminaries Let L0 ð0; aÞ ð0\a 1Þ the space of all l-measurable real-valued functions f on ð0; aÞ. We define the decreasing rearrangement function f : ð0; aÞ7!ð0; aÞ for f 2 L0 ð0; aÞ by f ðtÞ ¼ inffs [ 0 : lðfx 2 ð0; aÞ : jf ðxÞj [ sgÞ tg; t 0: Rt Rt Let f ; g 2 L0 ð0; aÞ. If 0 f ðsÞds 0 g ðsÞds for all t
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