MORE COMMUTATOR INEQUALITIES FOR HILBERT SPACE OPERATORS
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1 terat. J. Fuctioal alyi Operator Theory ad pplicatio 04 Puhpa Publihig Houe llahabad dia vailable olie at Volue Nuber 04 Page MORE COMMUTTOR NEQULTES FOR HLERT SPCE OPERTORS Wai udeh Departet of aic Sciece Petra Uiverity a Jorda e-ail: waudeh@uop.edu.o btract We preet geeral igular value iequalitie for th order udeh geeralized coutator fro the recet reult for coutator due to hatia-kittaeh Kittaeh Hirzallah-Kittaeh Hirzallah ad Wag-Due are pecial cae. Several applicatio are give.. troductio Let (H deote the pace of bouded liear operator o a coplex eparable Hilbert pace H ad let K(H deote the two-ided ideal of copact operator i (H. operator of the for i called a coutator ad a operator of the for i called a geeralized coutator. Variou igular value iequalitie for the coutator or the geeralized coutator are obtaied by differet author. thi paper the author ue the th order udeh geeralized coutator to geeralize the coutator ad coider aalogou igular value iequalitie. Received: March 04; Revied: pril 8 04; ccepted: May Matheatic Subect Claificatio: Keyword ad phrae: coutator copact operator iequality poitive operator igular value.
2 Wai udeh Kittaeh ha proved i [7] that if ( H uch that are copact the ( ax( ( (. for = Moreover he geeralized reult of hatia-kittaeh [] Kittaeh [6] ad Wag-Due [8]. oe of thee geeralizatio he proved that if ( H uch that ad are poitive ad i copact the ( ax ( ( (. for =... f i additio i poitive Kittaeh ha proved i [5] that for =.... ( ( (.3 Hirzallah i [4] ha proved a geeralizatio to the iequality (.: Let ( H uch that ad are copact. The for =.... ( ( ( (.4 t ha bee how by Zha i [9] that if K( H are poitive the for =.... ( ( (.5 Kittaeh i [5] geeralized the iequality (.5 for geeralized coutator: f ( H uch that ad are copact ad poitive the for =.... ( ( (.6
3 More Coutator equalitie for Hilbert Space Operator 3 Hirzallah i [4] geeralized the iequality (.6: Let be -by- atrice uch that ad are poitive eidefiite. The ( ( for =. Moreover Hirzallah ha proved i [4]: (.7 Let be -by- atrice with polar decopoitio = U = V. The U V ( ( for =. particular U U ( ( for =. (.8 (.9 Our ai i thi paper i to prove iequalitie for igular value of thorder udeh geeralized coutator which will geeralize the iequalitie (. to (.9.. Mai Reult We will preet the aor theore which i a iequality for igular value of the th order udeh geeralized coutator. To prove thi iequality we eed the followig lea which i a iediate coequece of the i-ax priciple (ee e.g. [ p.75] or [4 p.7]. Lea.. Let ( H uch that i copact. The for =.... ( ( (.
4 Wai udeh 4 Our aor theore i a geeralizatio of the iequality (.4. Theore.. Let ( H uch that are copact. The ( ( ( N M (. for = where M = ad. N = Proof. Sice ( 0 = (.3 the (
5 More Coutator equalitie for Hilbert Space Operator 5 = (.4 ( ( ( ( ( for. = the iequality (.4 replacig by t t t t for 0 > t repectively we get
6 6 Wai udeh ( for ( t t ( t t t ( (.5 = ad all t > 0. Sice i t > 0 ( t t ( t t t = ( M N (.6 where M = ad N =. t follow fro the iequalitie (.5 (.6 that ( ( M N ( for =.... ~ Reark. Whe replacig = = = = = = = = = = = = 0 i the iequality (. we get the iequality (.4. Let K( H ad let α be coplex uber. The operator α will be copact if H i r-dieioal Hilbert pace or i r-by-r atrix. So our ext reult will be for r-by-r atrice (or operator o r-dieioal Hilbert pace H. To prove our ext theore we eed the followig two lea. Lea.3. Let be r-by-r poitive eidefiite atrix ad let ( α = for = r. The for = r. ( α = α (.7
7 More Coutator equalitie for Hilbert Space Operator 7 Lea.4. Let K( H. The for =.... ( ( (.8 a applicatio of Theore. we will preet the followig theore which i a geeralizatio of the iequalitie (.5 (.6 ad (.7. Theore.5. Let be r-by-r atrice uch that are poitive eidefiite The M = ad N =. ( ( M N ( (.9 for = r. particular ( ( (.0 for = r. Proof. t i well kow that ( T = ( T for =... Thi iplie that ( = ( for = r. y direct coputatio we ee that
8 8 Wai udeh = ( γ ( γ ( γ ( γ γ( (. where γ i coplex uber. Now apply the iequality (.8 we get ( ( ( γ ( γ ( γ ( γ γ( (. for = r. t follow fro Theore. that ( ( M N (( γ ( γ ( γ ( γ γ( for = r (.3 where M = ad N =. we get Lettig γ = γ = ( for = r ( ( M N (( γ ( γ ( γ ( γ ( Sice. (.4 Lea.3 that are poitive eidefiite it follow fro
9 More Coutator equalitie for Hilbert Space Operator 9 (( γ ( γ ( γ ( γ = ( (.5 for = r. Now fro the iequalitie (.4 ad (.5 we get ( ( M N ( for = r. Note that the iequality (.9 i a geeralizatio of the iequality (.7. To ee thi replace = = = = = = = = = 0 we get ( = = ( for = r. a applicatio of Theore. we will preet the followig theore which i a geeralizatio of the iequality (.8. Theore.6. Let be r-by-r atrice with polar decopoitio = U = U = U. The for = r. ( U U U ( (.6 Proof. pecial cae fro Theore. aue = = = = 0 = = = = i i = ad i = i for i = we get
10 0 Wai udeh ( ( L ( (.7 where L = for =... Uig the polar decopoitio of ad applyig Theore.5 we get ( = ( U U U = ( U U U U ( ( U U U U U U U U ( U U ( ( ( U U U U ( (.8 for = r.
11 More Coutator equalitie for Hilbert Space Operator Referece [] R. hatia Matrix alyi GTM69 Spriger-Verlag New ork 997. [] R. hatia ad F. Kittaeh Coutator pichig ad pectral variatio Oper. Matrice ( [3]. C. Gohberg ad M. G. Krei troductio to the Theory of Liear No-elfadoit Operator er. Math. Soc. Providece R 969. [4] O. Hirzallah Coutator iequalitie for Hilbert pace operator Liear lgebra ppl. 43 ( [5] F. Kittaeh equalitie for coutator of poitive operator J. Fuc. al. 50 ( [6] F. Kittaeh Nor iequalitie for coutator of elf-adoit operator tegral Equatio Operator Theory 6 ( [7] F. Kittaeh Sigular value iequalitie for coutator of Hilbert pace operator Liear lgebra ppl. 430 ( [8].-Q. Wag ad H.-K. Due Nor for coutator of elf-adoit operator J. Math. al. ppl. 34 ( [9]. Zha Sigular value of differece of poitive eidefiite atrice SM J. Matrix al. ppl. (3 (
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