Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS
Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem
Introduction PP: The set of all positive definite operators on a Hilbert space. Operator mean Let σσ: PP 22 PP. If σσ satisfies the following conditions, then σσ is called an operator mean. 1. σσ AA, BB σσ CC, DD if AA CC and BB DD, 2. XX σσ AA, BB XX σσ(xx AAAA, XX BBBB) for all bounded linear operator XX, 3. σσ is upper semi-continuous on PP 22, 4. σσ II, II = II. F. Kubo and T. Ando, Math. A., 246 (1980), 205-224.
Introduction M: The set of all operator monotone functions on 0,. Representing function of an operator mean Let σσ be an operator mean. Then ff MM such that ff 11 = 11 and for all AA, BB PP. σσ AA, BB = AA 11 22ff AA 11 22BBAA 11 22 AA 11 22 ff xx = σσ(1, xx) F. Kubo and T. Ando, Math. A., 246 (1980), 205-224.
Examples of operator means Weighted geometric mean. For λλ [0,1] and AA, BB PP, a AA λλ BB = AA 1 2 AA 1 2BBAA 1 2 λλ AA 1 2. Representing function is gg λλ xx = xx λλ. Weighted power mean. For λλ 0,1, tt [ 1,1] and AA, BB PP, a PP tt λλ; AA, BB = AA 1 1 2 (1 λλ)i + λλ AA 1 2BBAA 1 2 tt tt 1 AA 2. Representing function is pp tt,λλ xx = 1 λλ + λλxx tt 1 tt.
Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem
The generalized Karcher equation (GKE) L = {gg M: gg 1 = 0 aaaaaa gg 1 = 1}, Δ = { ww 1,, ww 0,1 : ww ii = 1} Definition. Let gg LL. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, the generalized Karcher equation (GKE) is defined by Theorem P1. ww ii gg(xx 11 22AA ii XX 11 22) = 00. 1 The GKE has a unique solution XX PP. M. Palfia, Adv. Math., 289 (2016), 951 1007
Properties of the solution of GKE Theorem P2. Let gg LL. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, the solution of GKE XX = σσ gg (ωω; AA) satisfies the properties. (1) σσ gg ωω; XXAAXX = XXσσ gg ωω; AA XX for XX PP, (2) σσ gg ωω; AA σσ gg ωω; BB if AA ii BB ii, where BB = (BB 11,, BB ) PP, (3) σσ gg ωω; AA σσ ff ωω; AA if gg(xx) ff(xx) for all xx > 00. Especially, σσ gg 11 ww, ww; AA, BB is a ww-weighted operator mean. M. Palfia, Adv. Math., 289 (2016), 951 1007
Examples of a solutions of GKE 1. Let AA = (AA 1,, AA ) PP and ωω = (ww 1,, ww ) Δ. a The Karcher mean XX = Λ(ωω; AA) (gg xx = log xx L). ww ii log(xx 1 2AA ii XX 1 2) = 0 Relative operator entropy SS AA BB = AA 1 2 log(aa 1 2BBAA 1 2) AA 1 2 a a Λ 1 ww, ww; AA 1, AA 2 = AA 1 ww AA 2 Rep. ft. xx ww ww xxww ww=0 = log xx
Examples of a solutions of GKE 2. Let AA = AA 1,, AA PP, ωω = (ww 1,, ww ) Δ and tt 1,1. a The power mean P t (ωω; AA) (gg xx = xxtt 1 1 ww ii XX 1 2AA ii XX 1 2 tt II tt tt L). = 0 a a PP tt 1 ww, ww; AA 1, AA 2 Rep. ft. (1 ww) + wwxx tt 1 tt ww Tsallis Relative operator entropy TT tt AA BB = AA 1 2 1 AA (1 ww) + wwxxtt 2BBAA 1 2 tt II 1 tt tt ww=0 AA 1 2 = xxtt 1 tt
Operator means via GKE. Let AA = (AA 1,, AA ) PP, ωω = (ww 1,, ww ) Δ and gg L. a The solution of GKE XX = σσ gg (ωω; AA). Prop. 1 ww ii gg(xx 1 2AA ii XX 1 2) = 0 Generalized relative operator entropy A a σσ gg 1 ww, ww; AA 1, AA 2 Rep. ft. ff ww (xx) Prop. 2 GG AA BB = AA 1 2gg(AA 1 2BBAA 1 2)AA 1 2 ww ff ww xx ww=00 = gg(xx)
Representing function Let AA, BB PP, ww (0,1), aa > 0 and gg L. GKE 11 ww gg XX 11 22AAXX 11 22 + wwww XX 11 22BBXX 11 22 = 00 Solution (Operator mean) XX = σσ gg (11 ww, ww; AA, BB) GKE 11 ww gg 11 xx + wwww aa Solution (representing function) xx = 00 xx = ff ww aa = σσ gg 11 ww, ww; 11, aa
Representing function L = {gg M: gg 1 = 0 aaaaaa gg 1 = 1} Proposition 1. Let gg LL. For aa > 00, ww (00, 11), and let xx = ff ww (aa) be a solution of 11 ww gg 11 xx + wwww aa xx = 00. Then the inverse function of ff ww is ff ww 11 xx = xxgg 11 11 ww ww gg xx 11. cf. M. Palfia, ArXiv 1208.5603
Proof GKE 11 ww gg 11 xx + wwww aa Solution (representing function) xx = 00 xx = ff ww aa = σσ gg 11 ww, ww; 11, aa 1 ww gg 1 xx + wwww aa xx = 0 gg aa xx = 1 ww ww gg 1 xx ff ww 1 xx = aa = xxgg 1 1 ww ww gg 1 xx
Generalized relative operator entropy Proposition 2. Let gg LL. For aa > 00, ww 00, 11, let xx = ff ww (aa) be a solution of 11 ww gg 11 xx + wwww aa xx = 00. Then ww ff ww(xx) ww=00 = gg xx. J.I. Fujii and Kamei, Math. Japon. 34 (1989), 541 547. Paalfia and Petz, Linear Algebra Appl. 463 (2014), 134 153.
Proof 1 ww gg 1 ff ww (aa) + wwww aa ff ww (aa) = 0 ww 1 ww gg 1 ff ww aa + wwww aa ff ww aa = 0 Let ww = 0, then by gg 1 = 0, gg 1 = 1 and ff 0 xx = 1, ff ww(xx) ww=0 = gg xx.
Operator means via GKE. Let AA = (AA 1,, AA ) PP, ωω = (ww 1,, ww ) Δ and gg L. a The solution of GKE XX = σσ gg (ωω; AA). Prop. 1 ww ii gg(xx 1 2AA ii XX 1 2) = 0 Generalized relative operator entropy A a σσ gg 1 ww, ww; AA 1, AA 2 Rep. ft. ff ww (xx) Prop. 2 GG AA BB = AA 1 2gg(AA 1 2BBAA 1 2)AA 1 2 ww ff ww xx ww=00 = gg(xx)
Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem
Ando-Hiai inequality Theorem AH. Let AA, BB PP and ww (00, 11). Then AA ww BB II AA pp ww BB pp II holds for all pp 11. Ando and Hiai, Linear Algebra Appl. 197/198 (1994), 113 131.
Extension of Ando-Hiai inequality 1 Theorem PLY. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, a (1) ΛΛ ωω; AA II ΛΛ ωω; AA pp II, (Karcher mean) (2) for tt (00, 11], PP tt ωω; AA II PP tt/pp ωω; AA pp II a hold for all pp 11. (Power mean) Y., Oper. Matrices, 6 (2012), 577 588. M. Palfia and Y. Lim, J. Funct. Anal. 262 (2012), 1498 1514. Y. Lim and Y., Linear Algebra Appl. 438 (2013), 1293 1304.
Extension of Ando-Hiai inequality 2 Theorem W. Let σσ be a operator mean with a representing function ff MM. TFAE (1) ff xx pp ff xx pp for all xx > 00 and pp 11. (2) For AA, BB PP, σσ(aa, BB) II σσ(aa pp, BB pp ) II for all pp 11. All statements hold for, respectively. S. Wada, Linear Algebra Appl. 457 (2014), 276 286.
Motivation Theorem AH. AA ww BB II AA pp ww BB pp II Rep. Function Karcher equation Theorem W. TFAE (1) ff xx pp ff xx pp (2) σσ(aa, BB) II σσ(aa pp, BB pp ) II Theorem PLY. For tt 00, 11, (1) ΛΛ ωω; AA II ΛΛ ωω; AA pp II, (2) PP tt ωω; AA II PP tt/pp ωω; AA pp II GKE and Rep. Function (Generalized relative operator entropy)
PLY type Ando-Hiai inequality L = {gg M: gg 1 = 0 aaaaaa gg 1 = 1} Theorem 1. Let gg LL. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, σσ gg ωω; AA II σσ ggpp ωω; AA pp II for all pp 11, where gg pp xx = pppp xx 11/pp LL. Moreover, if ff is a representing function of σσ gg, then ff pp xx = ff xx 11/pp pp is a representing function of σσ ggpp.
A key lemma Lemma. Let gg LL. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, for all pp 11. ww ii gg(aa ii ) 00 σσ gg ωω; AA II 1 mm ww ii gg XX 1 2AA ii XX 1 2 = 0 XX = σσ gg (ωω; AA) mm ww ii gg AA ii = 0 σσ gg ωω; AA = II
Proof ww ii gg(aa ii ) 0 XX II; wwii 2 gg AA ii + 1 gg XX = 0 2 (gg 1 = 0) Then σσ gg ωω 2, 1 ; AA, XX = II. 2 AA = (AA 1,, AA ) ωω = (ww 1,, ww ) Let XX 0 = II, XX kk = σσ gg ωω 2, 1 2 ; AA, XX kk 1.
Proof XX 0 = II, XX kk = σσ gg ωω 2, 1 2 ; AA, XX kk 1. Then II = σσ gg ωω 2, 1 2 ; AA, XX σσ gg ωω 2, 1 2 ; AA, II σσ gg ωω 2, 1 2 ; AA, XX 1 XX kk > 0 XX 11 XX 22 Hence XX = lim kk XX kk. XX II II XX 1
Proof It satisfies XX = σσ gg ωω 2, 1 2 ; AA, XX ωω II = σσ gg 2, 1 2 ; XX 12 12 AAXX, II wwii 2 gg XX 12 12 AA ii XX + 1 gg II = 0 2 ww ii gg XX 12 AA ii XX 12 = 0 (gg 1 = 0) Hence XX = σσ gg ωω; AA II.
PLY type Ando-Hiai inequality Theorem 1. Let gg LL. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, σσ gg ωω; AA II σσ ggpp ωω; AA pp II for all pp 11, where gg pp xx = pppp xx 11/pp LL. Moreover, if ff is a representing function of σσ gg, then ff pp xx = ff xx 11/pp pp is a representing function of σσ ggpp.
Proof of Theorem 1 In the case 1 pp 2. Hansen s inequality gg pp xx : = pppp xx 1/pp By lemma, Let XX = σσ gg ωω; AA II. Then XX 1 II and 0 = pp ww ii gg XX 1 2AA ii XX 1 2 L = pp ww ii gg XX 1 2AA ii XX 1 2 pp ww ii gg XX 1 2AA pp ii XX 1 2 = ww ii gg pp XX 1 2AA pp ii XX 1 2 Hence σσ ggpp ωω; AA pp XX = σσ gg ωω; AA II holds for all 1 pp 2. ww ii gg pp XX 1 2AA pp ii XX 1 2 0 σσ ggpp ωω; XX 1 2AA pp XX 1 2 II pp pp 1 pp
Proof of Theorem 1 Let ff pp be a representing function of σσ ggpp (1 ww, ww; AA, BB). Then ff pp 1 xx = xxgg pp 1 1 ww ww gg pp xx 1 gg pp xx : = pppp xx 11/pp gg pp 11 xx = gg xx/pp pp = xxgg 1 1 pp 1 ww ww = xx 1/pp gg 1 1 ww ww pppp xx 1/pp gg xx 1/pp pp pp = ff 1 xx1/pp pp Hence ff pp xx = ff xx 1/pp pp.
Wada s type Ando-Hiai inequality Theorem 2. Let gg LL. Then TFAE. A (1) ff ww xx pp ff ww xx pp for all pp 11, xx > 00 and ww 00, 11, a (2) gg xx pp pppp(xx) for all pp 11 and xx > 00, a (3) σσ gg ωω; AA II σσ gg ωω; AA pp II for all pp 11, AA = AA 11,, AA PP aaaaaa ωω = (ww 11,, ww ) ΔΔ, a where σσ gg ωω; AA and ff λλ (xx) mean a solution of GKE and representing function of σσ gg, respectively. All statements hold for, respectively.
Proof of Theorem 2 Proof of (1) (2). By Proposition 1, ff ww 1 xx = xxxx ff ww xx pp ff ww xx pp ff ww 1 xx pp ff ww 1 xx pp. 1 ww ww gg xx 1 = xxxx λλgg xx 1 PPPPPP λλ: = 11 ww ww > 00 gg 1 λλ gg xx 1 pp 1 λλ gg 1 λλλλ xx pp 1 λλ λλ +0 pppp(xx 1 ) gg xx pp Hence gg xx pp pppp(xx) holds for all xx > 0.
Proof of Theorem 2 Proof of (2) (3). Put XX = σσ gg ωω; AA II. Then XX 1 II. The case 1 pp 2. 0 = pp ww ii gg(xx 1 2AA ii XX 1 2) = pp ww ii gg XX 1 2 AA ii XX 1 2 Hansen s inequality (2) By lemma, gg xx pp pppp(xx) pp ww ii gg XX 1 2AA pp ii XX 1 2 ww ii gg(xx 1 2AA pp ii XX 1 2) Hence σσ gg ωω; AA pp XX = σσ gg ωω; AA II holds for all 1 pp 2. ww ii gg XX 1 2AA pp ii XX 1 2 0 σσ gg ωω; XX 1 2AA pp XX 1 2 II. pp pp 1 pp
Proof of Theorem 2 Proof of (3) (1). By (3), for AA, BB PP and ww 0,1, σσ gg 1 ww, ww; AA, BB II σσ gg 1 ww, ww; AA pp, BB pp II. By Theorem W, ff ww xx pp ff ww xx pp.
Results Theorem AH. AA ww BB II AA pp ww BB pp II Rep. Function Karcher equation Theorem W. TFAE (1) ff xx pp ff xx pp (2) σσ(aa, BB) II σσ(aa pp, BB pp ) II Theorem PLY. (1) ΛΛ ωω; AA II ΛΛ ωω; AA pp II, (2) PP tt ωω; AA II PP tt/pp ωω; AA pp II Theorem 2. Theorem 1.
Weighted power mean case tt (00, 11] ff ww xx = 1 ww + wwxx tt 1 tt, gg xx = xxtt 1 tt Theorem PLY. PP tt ωω; AA II PP tt/pp ωω; AA pp II Theorem 2. TFAE. a (1) ff ww xx pp ff ww (xx pp ) for all pp 11. (2) pppp xx gg(xx pp ) for all pp 11. (3) PP tt ωω; AA II PP tt ωω; AA pp II for all pp 11.
Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem
Norm inequality Theorem HP. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, ΛΛ ωω; AA eexxxx ww ii log AA ii holds for all unitarily invariant norm. 1 If {A 1,, AA } is commuting, then Λ ωω; AA = exp ww ii log AA ii Hiai and Petz, Linear Algebra Appl. 436 (2012), 2117 2136.
Norm inequality Theorem LY-BLY-DDF. For AA = AA 11,, AA PP and ωω = (ww 11,, ww ) ΔΔ, tt PP tt ωω; AA pp ww ii AA ii 1 holds for all Schatten pp-norm for pp 11. 11 tt pp If {A 1,, AA } is commuting, then PP tt ωω; AA = ww ii AA ii tt 1 tt Lim and Y., Linear Algebra Appl. 438 (2013), 1293 1304. Bhatia, Lim and Y., Linear Algebra Appl. 501 (2016), 112 122. Dinh, Dumitru and Franco, Linear Algebra Appl. 532 (2017) 140-145.
Problem Problem. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, σσ gg (ωω; AA) gg 11 ww ii gg AA ii 1 holds for any unitarily invariant norm?
A key lemma Lemma. Let gg LL. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, for all pp 11. gg 11 ww ii gg AA ii 1 II σσ gg ωω; AA II L = {gg M: gg 1 = 0 aaaaaa gg 1 = 1}, Δ = { ww 1,, ww 0,1 : ww ii = 1} gg 1 ww ii gg AA ii = II σσ gg ωω; AA = II
Problem Let gg L and σσ gg be a solution of GKE. Problem. For AA = (AA 11,, AA ) PP and ωω = (ww 11,, ww ) ΔΔ, σσ gg (ωω; AA) gg 11 ww ii gg AA ii 1 holds for any unitarily invariant norm? Proposition 4. Let AA, BB PP and ww (00, 11). If AAAA = BBBB, then TFAE. 1. σσ gg 11 ww, ww; AA, BB is a power mean. 2. σσ gg 11 ww, ww; AA, BB = gg 11 11 ww gg(aa) + wwww(bb).
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