Ando, Tsuyoshi

English

Ando Tsuyoshi

University of Szeged

Acta Sci. Math., 45 (1983), 19—22 An inequality between symmetric function means of positive operators T. A N D O Dedicated to Professor B. Szokefalvi-Nagy on his seventieth birthday 1. There are various methods of averaging of an «-tuple A=(Ai, ..., A„) of bounded positive (semi-definite) operators on a Hilbert space. The most basic are the arithmetic mean ( A + ••• +An)/n and the harmonic mean n(A^1+...+A~1)~1 (provided all Al are invertible). ANDERSON and TRAPP [2] called (A~1 + ...+A~1)~1 the parallel sum of the «-tuple A, and denoted it by A^. ... :A„, or in short JJ: Av ¡=i ' Further they gave a variational description for parallel sum; (1) (x, ( / 7 : A^ = i n f { J <x;, A,x,) x = J xf}, where (x, y) denotes the inner product of the vectors x and y. Formula (1) was then used to define the parallel sum for a general «-tuple of positive operators. For an «-tuple of positive numbers, a= (a 1 ; ...,a„), MARCUS and LOPES [5] defined symmetric function means (or Marcus—Lopes means) Ekn{a) by (2) = fU*L> k = l,2,...,n ek-l,nW where ek„(a) is the normalized £>th elementary symmetric function of a = =(<*!, ...,a„), that is, k • 2 n«ij e0,n& = 1 and ek,n(S) = I S ' . - - - ' ^ » ^ (3 Using an equivalent version of definition (2), ANDERSON, MORLEY, and TRAPP [3] introduced two kinds of symmetric function means for an «-tuple A = ( A 1 , ..., A„) Received June 24, 1982. 2* 20 T. Ando of positive operators; "»l.nC?) = ( . Z ^ ; ) / " (arithmetic mean), s«,n(^) = n [ i l - (harmonic mean), and s |{(dhr4 Ml' = 2' '» n ¡=1 where denotes the («—l)-tuple ..., Ai_i, Ai+1, ...,A„). By definition both 6t>„(^4) and sk n(A) are invariant under permutations of Ax, ..., A„, and the maps A~<ZKn(A). and A>-—s,k n(A) are positively homogeneous and monotone with respect to coordinatewise ordering. If all At are invertible, then = *n-k+i,n(A), k = l, ..., n, where A*1 = (A{~1, ..., A'1). For any «-tuple A Si,„(i) = ©i,„(i) and ©„,„ 6*) = sB> „(,?). Besides the easily proved inequalities not much is known about the order relation among <BJn(A) and s k>n(A), j , k =2 , ... ...,n — 1. If all Ai are scalars, that is, A=a, then both <5M(a) and skn(a) coincide with the Marcus—Lopes mean Ek n{a). Therefore it follows via spectral theory that if A is a commuting «-tuple then The equality <5ktn(A)=skn(A), k—2,...,«— 1 is not valid in general for a non-commuting «-tuple. 'ANDERSON, MORLEY, and TRAPP [3] asked if the inequalities S/t.nOi) = <Sk+1>„(A), k = 2, ..., n —2 (or equivalently s*.n(^) = St+i,nU)> k = 2, . . . , n - 2 ) and Sk,„(A) ^ sk>n(A), k = 2, ..., n —1 are valid for every «-tuple A. They mentioned, without proof, that in case n=3 the inequality <32i304)£s2j3(/i) could be derived via electrical network consideration. Symmetric function means 21 The purpose of the present paper is to give a mathematical proof to the in-equality S 2 > 3 ( i )Ss 2 j 3 ( i ) . 2. Our proof is based on a solution of an extremal problem, due to F L A N D E R S [4]. Lemma. Given two set of vectors Xi, ..., xm and yx, ..., y„, define a functional i¡/(A) for an invertible positive operator A by m n t(A) = 2 <*., Axt)+ 2 (yj» A-'yj)-¡=i • j=i Then inf \l/(A)=2\\[(xi,yj)]\\1, where ||[(x;, y})]||! is the trace norm of the mXn matrix [<x;,jy>]. See [1] and [4] for a proof. T h e o r e m . For any triple A = (A1, A2, A3) of positive operators (3) S 2 , 3 ( i ) S H 3 ( A ) . P roo f . All Ai can be assumed invertible. Since S2j3(-4)=S2 3 ( Y | - 1 ) - 1 , and operator inequality (3) is equivalent to < x , ® 2 , 3 ( A ) x y i * . ( y , < 5 2 , 3 ( A - i ) y y * ^ \ ( x , y ) \ for all x, y, which is equivalent, in view of the arithmetic-geometric means inequality, to (4) < x , S 2 , 3 a ) x > + < 3 ; , 8 2 , 3 ( i - i ) 7 > s 2 K x , j > | for all x, Since = j {Ai: (A2 + A3)+A2: (Aa + A,) + A3: (Ax+A2)}, formula (1) gives <x, <52>3(A)x) = (5) 1 3 XV xt' x3 £ ¡ = 1 where Xj=Xj_3 for 7 = 4 , 5 , and similarly (y, &*i3(A-*)y) = (6) 1 3 = inf T 2 {(y+yi, AfHy+ydy + i y i ^ A ^ y ^ + iy^Ar^y^)}, y\*yi»y3 ^ /=3 22 T. Ando: Symmetric function means where y } = y j - z for / = 4 , 5. Then Lemma yields, for fixed x, x1, x2 , x3, y, ylt y2, and ya, (x+xi, ^¡(x+xi)>4-<x1.+1, AiXi+1)+(xi+2, AiX^ + iy+yi, Ar1(y+y,))+ (7) + ( Y I + I , ^ R ^ + i ) s 2115,-11!, i = 1, 2 , 3, where (x+x . - . j + .y,) <x+Xi , j i + i ) <x+x i 5j; i+2> = (xi+1 ,y + yt) <Xi+1,yi + 1> <XI + 1,J>I + 2> .<*.+2, Ji> < x i + 2 , >' i+1> (xi+2,yi+2) Now according to (5), (6) and (7), the inequality (4) will follow from (8) 2 ||S,||i s 2 | < * , y)\. i=1 To see (8), consider a 3X3 Hermitian matrix 2 - 1 - l " - 1 - 1 2 - 1 2 - 1 Since T has —3, 0 and 3 as its eigenvalues, | | r | | „ , the operator norm of T, is 3 equal to 3. Easy computation shows 2 ir(SiT)=6(x, y). Then ¡=i • l i i s . - i u ^ l i i i s . - i k - i m u ^ l ¡ = 1 J ¡ = 1 -> 2 t r ( 5 ; r ) >=i — 2|<x, This completes the proof. The method in the above proof can be used to prove <»2,nG4)=sn-i,n0?) for every «-tuple A. But the inequality <Sk n(A)^sk n(A) stands still open. Added in proof. In the revised version of [3] a different proof is presented. References [1] T. W. ANDERSON and I. OLKIN, An extremal problem for positive definite matrices, Linear and Multilinear Algebra, 6 (1978), 257—262. [2] W. N. ANDERSON and G. E. TRAPP, Shorted operators. It, SIAM J. Appl. Math., 28 (1975), 160—175. [3] W. N . ANDERSON, T . D . MORELY, a n d G . E . TRAPP, S y m m e t r i c f u n c t i o n m e a n s o f p o s i t i v e operators, Linear Alg. Appl., to appear. [4] H. FLANDERS, An extremal problem in the space of positive definite matrices, Linear and Multilinear Algebra, 3 (1975), 33—39. [5] M. MARCUS and L. LOPES, Inequalities for symmetric functions and Hermitian matrices, Canad. J. Math., 9 (1957), 305—312. DIVISION OF APPLIED MATHEMATICS RESEARCH INSTITUTE O F APPLIED ELECTRICITY HOKKAIDO UNIVERSITY SAPPORO 060, JAPAN 

An inequality between symmetric function means of positive operators

http://acta.bibl.u-szeged.hu/14867/1/math_045_019-022.pdf

An inequality between symmetric function means of positive operators

Abstract

Similar works

Full text

Available Versions

University of Szeged