Singular Value Inequalities for Compact Normal Operators

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dvance in Linear lgebra & Matrix Theory, 3, 3, 34-38 Publihed Online December 3 (http://www.cirp.org/ournal/alamt) http://dx.doi.org/.436/alamt.3.347 Singular Value Inequalitie for Compact Normal Operator Waim udeh Department of Baic Science, Petra Univerity, mman, Jordan Email: waudeh@uop.

1 dvance in Linear lgebra & Matrix Theory, 3, 3, Publihed Online December 3 ( Singular Value Inequalitie for Compact Normal Operator Waim udeh Department of Baic Science, Petra Univerity, mman, Jordan waudeh@uop.edu.o Received September 5, 3; revied October 8, 3; accepted November 7, 3 Copyright 3 Waim udeh. Thi i an open acce article ditributed under the Creative Common ttribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. BSTRCT We give ingular value inequality to compact normal operator, which tate that if i compact normal operator on a complex eparable Hilbert pace, where i i the carteian decompoition of, then for,, Moreover, we give inequality which aert that if i compact normal operator, then i for,, Several inequalitie will be proved. Keyword: Compact Operator; Inequality; Normal Operator; Self-doint Operator; Singular Value. Introduction Let BH denote the pace of all bounded linear op- erator on a complex eparable Hilbert pace H, and let K H denote the two-ided ideal of compact operator in B H. For T K H, the ingular value of T, denoted by,, T T are the eigenvalue of the poitive operator T TT T T and repeated according to multiplicity. Note that T T T for,, It follow Weyl monotonicity principle (ee, e.g., [, p. 63] or [, p. 6]) that if ST, KH are poitive and S T, then S T for,, The ingular value T of ST and are the ame, and they conit S of thoe of S together with thoe of T. Here, we ue the direct um notation ST for the blockdiagonal S operator defined on H H. T The Jordan decompoition for elf-adoint operator aert that every elf-adoint operator can be expreed a the difference of two poitive operator. In fact, if B H i elf-adoint, then, where a and are the poitive operator given by and, ee []. Let be any operator, we can write in the form i where and are i elf-adoint operator, thi i called the Carteian decompoition of the operator. If i normal, then. udeh and Kittaneh have proved in [3] that if B BC, KH uch that, then B C B C (.) for,, lo, udeh and Kittaneh have proved in [3] that if B KH uch that i elf-adoint, B, and B, then B B (.) for,, In addition to thi, udeh and Kittaneh have proved in B K H be elf-adoint operator, then [3] that if for,, B B B (.3) Zhan ha proved in [4] that if B KH are poi-

2 W. UDEH 35 tive, then B B (.4) for,, Moreover, it ha proved in [3] that (.3) i a generalization of (.4). Hirzallah and Kittaneh have proved in [5] that if B K H, then B B (.5) In thi paper, we will give ingular value inequalitie for normal operator: Let be normal operator in K H. (.6) for,, We will give ingular value inequality to the normal operator i, where i normal: Let be normal operator in K H. for,, i (.7). Main Reult We will begin by preenting the following theorem for complex number Theorem.. Let x a ib be complex number. lo, a b x a b (.) a b x a b (.) Proof. The right hand ide of the inequalitie i well known. To prove the left hand ide, ab ab a abb a a b b a b a b x ab ab a abb a b x. Moreover, Now, we will preent operator verion of Theorem., inequality (.). Theorem.. Let be normal operator in K H, where i be the Carteian decompoition of. for,, Proof. Let i be the Carteian decompoition of the normal operator, which implie that. Now,, it follow that. In fact for,, By uing Weyl monotonicity principle [] and the inequality, we get the right hand ide of the theorem. To prove the left hand ide of the inequality, we will ue the inequality which i well known for commuting elf-adoint operator and it aert that Thi implie that (.3) (.4) But it i known that, it follow Weyl monotonicity principle [] and the inequality (.4) that (.5) for,, Inequality (.5) i equivalent to aying that for,, Remark. (i) Equality hold in the right hand ide of Theorem. if either or. (ii) Equality hold in the left hand ide of theorem. if. We will preent operator verion of Theorem., inequality (.). Remark. Let, where i normal operator. i normal operator with i i the Carteian decompoition of.

3 36 W. UDEH It follow that, and and i. i. Now, by direct calculation and applying Theorem. we get (.6) for,, Remark 3. We note that the right hand ide of the inequality (.6) i the ame a the inequality (.6), but the left hand ide of the inequalitie (.6) and (.6) ay that the ingular value of the addition or ubtraction of the Carteian decompoition for the normal operator divided by i le than or equal to the ingular value of the normal operator itelf. an application of the Theorem., we will determine upper and lower bound for ingular value of the normal operator i, where i normal. Theorem.3. Let KH be normal operator, where i i the Carteian decompoition of. i for,, Proof. Note that T i i normal operator, o we can write the Carteian decompoition of T a T T it, where T i, and i T, i where the carteian decompoition of i given by i. By making comparion of T and T we ee eaily that T T. It follow that T T i. Moreover, T i i i i i i. 4 Similarly, T. Now, apply Theorem. to get T (.7) for,, Thi i equivalent to aying that i for,, We will give imple and new proof to the inequality (.). Theorem.4. Let B KH uch that i elf-adoint, B,and B, then B B for,, Proof. Since i elf-adoint operator, we can write B B in the form, apply the in- equality (.4) we get B B B B which i equivalent to aying that B B

4 W. UDEH 37 for,, udeh and Kittaneh eparate Jordan of elf-adoint operator in the inequality (.3). Here we will give a horter proof. Theorem.5. Let B KH be elf-adoint operator. B B B for,, Proof. Since and B are elf-adoint operator, we can write in the form, and imilarly we will write B in the form B B B. pply the inequality (.4) we get B B B B B B B B B B for,, We will preent the following two theorem a an application to the inequality (.5). Theorem.6. Let KH be elf-adoint operator. (.8) for,, Proof. It wa proved in Theorem. that if i normal operator with Carteian decompoition i then for,, from thi, it follow that for,, The following theorem i the econd application of the inequality (.5). Theorem.7. Let KH be elf-adoint operator. for,, Moreover, (.9) for,, Proof. It i well known that the inequality (.5) we get (.), o uing for,, Similarly,, o uing the inequality (.5) we get for,, Bhatia and Kittaneh have proved in [6] that if B K H, then B B B B BB for,, For related Cauchy-Schwarz type inequalitie, we refer to [] and reference therein. Here, we will preent imilar new inequality. Theorem.8. Let B KH be operator. B B B B BB (.) for,, Proof. Suppoe Thi implie that B B B, and BB On the other hand, we have YY BB B B and Y BB. B BB, and YY.. Since and YY are poitive operator, then BB BB YY i poitive B B BB operator. Now by applying the inequality (.), we get B B B B BB for,, REFERENCES [] R. Bhatia, Matrix nalyi, GTM69, Springer-Verlag, New York, [] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonelfadoint Operator, merican Mathematical Society, Providence, 969. [3] W. udeh and F. Kittaneh, Singular Value Inequalitie for Compact Operator, Linear lgebra pplication, Vol. 437,, pp [4]. Zhan, Singular Value of Difference of Poitive Semidefinite Matrice, SIM Journal on Matrix nalyi and pplication, Vol., No. 3,, pp [5] O. Hirzallah and F. Kittaneh, Inequalitie for Sum and

5 38 W. UDEH Direct Sum of Hilbert Space Operator, Linear lgebra pplication, Vol. 44, 7, pp [6] R. Bhatia and F. Kittaneh, The Matrix rithmetic-geo- metric Mean Inequality Reviited, Linear lgebra pplication, Vol. 48, 8, pp

SINGULAR VALUE INEQUALITIES FOR COMPACT OPERATORS

SINGULAR VALUE INEQUALITIES FOR OMPAT OPERATORS WASIM AUDEH AND FUAD KITTANEH Abstract. A singular value inequality due to hatia and Kittaneh says that if A and are compact operators on a complex separable