SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES

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1 italian jounal of pue and applied mathematics n (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics Yamouk Univesity Ibed Jodan Abstact. We give some shap inequalities involving powes of the numeical adii fo the off-diagonal pats of opeato matices. These inequalities, which ae based on some classical convexity inequalities fo the nonnegative eal numbes, genealize ealie numeical adius inequalities. Keywods: numeical adius, opeato matix, off-diagonal pat, Catesian decomposition, Jensen s inequality, mixed Schwaz inequality. 000 Mathematics Subject Classification: 47A1, 47A30, 47A63, 47B Intoduction Let B(H) denote the C - algeba of all bounded linea opeatos on a complex Hilbet space H with inne poduct,. Fo A B(H), let ω(a) and A denote the numeical adius and the usual opeato nom of A, espectively. It is well known that ω( ) defines a nom on B(H), which is equivalent to the usual opeato nom. In fact, fo evey A B(H), (1.1) 1 A ω(a) A. The inequalities in (1.1) ae shap. The fist inequality becomes an equality if A = 0. The second inequality becomes an equality if A is nomal. Fo basic popeties of the numeical adius, we efe to [4] and [6]. The inequalities in (1.1) have been impoved consideably by Kittaneh. It has been shown in [10] and [11], espectively, that if A B(H), then (1.) ω(a) 1 A + A 1 ( ) A + A 1,

2 434 w. bani-domi whee A = (A A) 1 is the absolute value of A, and (1.3) 1 4 A A + AA ω (A) 1 A A + AA. The inequalities in (1.), which efine the second inequality in (1.1), have been utilized in [10] to deive an estimate fo the numeical adius of the Fobenius companion matix. Such an estimate can be employed to give new bounds fo the zeos of polynomials (see, e.g., [9],[10], and efeences theein). If A = B + ic is the Catesian decomposition of A, then B and C ae selfadjoint, and A A + AA = (B + C ). Thus, the inequalities in (1.3) can be witten as 1 (1.4) B + C ω (A) B + C. The pupose of this pape is to establish a geneal inequalities involving powes of the numeical adii fo the off-diagonal pats of opeato matices that ae based on the classical convexity inequalities fo nonnegative eal numbes and some opeato inequalities. Othe ecent numeical adius inequalities have been obtained by Dagomi [3], El-Haddad [5], and Yamazaki [1]. The inequalities in [3] ae elated to the Euclidean adius of two Hilbet space opeatos, the inequalities in [5] involving powes of the numeical adii fo Hilbet space opeatos, and those in [1] involve the Aluthge tansfom.. Main esults To pove ou genealized numeical adius inequalities fo the off-diagonal pats of opeato matices, we need seveal well known lemmas. The fist lemma is a simple consequence of the classical Jensen s inequality concening the convexity o the concavity of cetain powe functions. It is a special case of Schlömilch s inequality fo weighted means of nonnegative eal numbes (see, e.g., [7, p. 6]). Lemma.1 Fo a, b 0, 0<α<1, and 0, let M (a, b, α)=(αa +(1 α)b ) 1 let M 0 (a, b, α) = a α b 1 α. Then and M (a, b, α) M s (a, b, α) fo s. The second lemma is anothe application of Jensen s inequality (see. e.g., [7, p. 8]). Lemma. Fo a, b 0, and > 0, let N (a, b) = (a + b ) 1. Then N s (a, b) N (a, b) fo s > 0. The thid lemma follows fom the spectal theoem fo positive opeatos and Jensen s inequality (see, e.g., [8]).

3 some geneal numeical adius inequalities Lemma.3 Let A B(H) be positive, and let x H be any unit vecto. Then (a) Ax, x A x, x fo 1. (b) A x, x Ax, x fo 0 < 1. The fouth lemma is an immediate consequence of the spectal theoem fo self-adjoint opeatos. Fo genealizations of this lemma, we efe to [8]. Lemma.4 Let A B(H) be self-adjoint, and let x H be any vecto. Then Ax, x A x, x. The fifth lemma is a genealized fo the mixed Schwaz inequality which has been poved by Kittaneh [8]. Lemma.5 Let T be an opeato in B(H) and let f and g be nonnegative functions on [0, ) which ae continuous and satisfying the elation f(t)g(t) = t fo all t [0, ). Then T x, y f ( T x g ( T x fo all x, y H. The sixth lemma contains two pats. Pat (a) is well known and can be found in [, p. 10]. Pat (b) is also known and can be found in [1]. Lemma.6 Let X, Y B(H). Then (a) ω ([ X 0 0 Y ]) = max {ω(x), ω(y )}. ([ ]) X Y (b) ω = max {ω(x + Y ), ω(x Y )}. Y X In paticula, ω ([ 0 Y Y 0 ]) = ω(y ). Ou fist esult is a genealization of the fist inequality in (1.). [ ] Theoem.7 Let S = be a opeato matix in B(H C 0 1 H ), and let f and g be nonnegative functions on [0, ) which ae continuous and satisfying the elation f(t)g(t) = t fo all t [0, ), and 1. Then (.1) ω (S) 1 max{ f ( C + g ( B, f ( B + g ( C }.

4 436 w. bani-domi Poof. Fo evey unit vecto X = ( x1 Lemma.1, and Lemma.3(a) we have x SX, X f ( S X g ( S X ) (H 1 H ), by using Lemma.5, = f ( S X, X 1 g ( S X, X 1 ([ ]) 1 ([ C 0 = f B X, X g ]) B 0 C X, X [ f = ] 1 ( C [ g 0 f X, X ( B ] 1 ( B 0 g ( C X, X 1 ( [ f ] [ ( C g 0 f X, X + ( B ] ) ( B 0 g ( C X, X ( ( [ 1 f ] [ ( C g 0 f X, X + ( B ] ( B 0 g ( C X, X ( ( [ 1 f ] [ ( C g 0 f X, X + ( B ] ( B 0 g ( C X, X ( [ 1 f ( C + g ( B ] ) 1 0 f ( B + g ( C X, X. Thus, SX, X 1 and so [ f ( C + g ( B 0 f ( B + g ( C ω (S) = sup { SX, X : X (H 1 H ), X = 1} 1 { [ ] } λ 0 sup X, X : X (H 0 µ 1 H ), X = 1 = 1 max{ λ, µ }, ] X, X )) 1 )) 1, whee and λ = f ( C + g ( B µ = f ( B + g ( C, as equied. Inequality (.1) includes seveal numeical adius inequalities fo opeato matices. Samples of inequalities ae demonstated in the following emaks.

5 some geneal numeical adius inequalities Remak.8 Fo f(t) = t α and g(t) = t 1 α, α (0, 1), in inequality (.1), we get the following inequality ω (S) 1 max{ C α + B (1 α), B α + C (1 α) }. Remak.9 If B = C in the above Remak, and by using Lemma.6(b), then ω (B) = ω (S) 1 B α + B (1 α), and this inequality is given in Theoem 1 in [5]. Now, the second esult is a genealization of the second inequality in (1.3). [ ] Theoem.10 Let S = be a opeato matix in B(H C 0 1 H ), and let f and g be nonnegative functions on [0, ) which ae continuous and satisfying the elation f(t)g(t) = t fo all t [0, ), and 1 and 0 < k < 1. Then ω (S) max{ kf k ( C +(1 k)g 1 k ( B, kf k ( B +(1 k)g 1 k ( C }. Poof. Fo evey unit vecto X = ( x1 Lemma.3(b), Lemma.1, and Lemma.3(a), we have x ) (H 1 H ), by using Lemma.5, SX, X f ( S X, X g ( S X, X [ f = ] [ ( C g 0 f X, X ( B ] ( B 0 g ( C X, X [ ] k f k ( C X, X g 1 k ( B X, X 0 f k ( B 0 g 1 k ( C ( [ ] f k ( C k X, X 0 f k ( B +(1 k) g 1 k ( B X, X 0 g 1 k ( C ( [ ] f k ( C k X, X 0 f k ( B +(1 k) g 1 k ( B X, X 1 0 g 1 k ( C 1. 1 k

6 438 w. bani-domi Thus, SX, X k = and so [ f k ( C 0 f k ( B ] + (1 k) g 1 k ( B 0 g 1 k ( C kf k ( C + (1 k)g 1 k ( B 0 kf k ( B + (1 k)g 1 k ( C ω (S) = sup { SX, X : X (H 1 H ), X = 1} { [ ] } β 0 sup X, X : X (H 0 γ 1 H ), X = 1 = max{ β, γ }, X, X X, X, whee and as equied. β = kf k ( C + (1 k)g 1 k ( B γ = kf k ( B + (1 k)g 1 k ( C, Now, Theoem.10 includes seveal numeical adius inequalities fo opeato matices, and so we give some inequalities in the following emaks. Remak.11 If f(t) = t k and g(t) = t 1 k, k (0, 1), in Theoem.10, then we get the following inequality ω (S) max{ k C +(1 k) B, k B +(1 k) C }. Remak.1 If B = C in the above Remak, and by using Lemma (.6)b, then we have ω (B) = ω (S) k B +(1 k) B, and this inequality can be found in Theoem in [5]. Remak.13 If we take = 1 and k = 1 in the last Remak, we find which is the second inequality in (1.3). ω (B) 1 B + B, Ou next esults ae genealizations of the second inequality in (1.4).

7 some geneal numeical adius inequalities [ ] Theoem.14 Let R = be a opeato matix in B(H C 0 1 H ), with the Catesian decomposition R = S + it and 1. Then ω (R) 1 max{ C + B + C B, B + C + B C }. Poof. Fo evey unit vecto X= ( ) x1 x RX, X = (S + it )X, X = SX, X + i T X, X = SX, X + T X, X (H 1 H ), and fo 1, we have SX, X + T X, X (by Lemma.) S X, X + T X, X (by Lemma.4) S X, X + T X, X (by Lemma (.3)a) = ( S + T )X, X. Thus, and so, RX, X ( S + T )X, X ω (R) = sup { RX, X : X (H 1 H ), X = 1} sup { ( S + T )X, X : X (H 1 H ), X = 1} = 1 max{ C + B + C B, B + C + B C }, as equied. Remak.15 Let B = C and = in Theoem (.14). Then we get ([ ]) ω = ω B 0 (B) (by Lemma (.6)b) 1 4 max{ B+B + B B, B+B + B B } = 1 4 B+B + B B = 1 B B+BB, which is the second inequality in (1.3).

8 440 w. bani-domi Theoem.16 Let R = [ C 0 ] be a opeato matix in B(H 1 H ), with the Catesian decomposition R = S + it and. Then ω (R) 1 max{ C + B + C B, B + C + B C }. Poof. Fo evey unit vecto X = ( x1 1 RX, X = 1 (S + it )X, X x = 1 SX, X + i T X, X ) (H 1 H ), we have = SX, X + T X, X SX, X + T X, X (by Lemma.1) 1 ( S X, X + T X, X ) 1 (by Lemma.4) 1 ( S X, X + T X, X ) 1 (by Lemma (.3)a) = 1 ( S + T ) X, X 1 = 1 1 [ η 0 0 θ ] X, X 1. Thus, and so, RX, X 1 [ η 0 0 θ ] X, X. ω (R) = sup { RX, X : X (H 1 H ), X = 1} { [ ] } η 0 1 sup X, X : X (H 0 θ 1 H ), X = 1 = 1 max{ η, θ }, whee and as equied. η = C + B + C B θ = B + C + B C,

9 some geneal numeical adius inequalities Remak.17 Let B = C and = in Theoem (.16). Then we get ([ ω B 0 ]) = ω (B) (by Lemma (.6)b) 1 4 max{ B+B + B B, B + B + B B } = 1 4 B+B + B B = 1 B B+BB, which is the second inequality in (1.3). Acknowledgment. This wok was suppoted by the deanship of scientific eseach and gaduated studies at Yamouk Univesity. Refeences [1] Bani-Domi, W., Kittaneh, F., Nom equalities and inequalities fo opeato matices, Linea Algeba Appl., 49 (008), [] Bhatia, R., Matix Analysis, Spinge, New Yok (1997). [3] Dagomi, S.S., Some inequalities fo the Euclidean opeatoe adius of two opeatos in Hilbet spaces, Linea Algeba Appl., 419 (006), [4] Gustafson, K.E., Rao, D.K.M., Numeical Range, Spinge-Velag, New Yok, [5] El-Haddad, M., Kittaneh, F., Numeical adius inequalities fo Hilbet space opeatos. II, Studia Math., 18 (007), [6] Halmos, P.R., A Hilbet Space Poblem Book, nd ed., Spinge-Velag, New Yok, 198. [7] Hady, G.H., Littlewood, J.E., P ólya, G., Inequalities, nd ed., Cambidge Univesity Pess, Cambidge, [8] Kittaneh, F., Notes on some inequalities fo Hilbet space opeatos, Res. Inst. Math. Sci., 4 (1988), [9] Kittaneh, F., Bounds fo the zeos of polynomials fom matix inequalities, Ach. Math. (Basel), 81 (003),

10 44 w. bani-domi [10] Kittaneh, F., A numeical adius inequality and an estimate fo the numeical adius of the Fobenius companion matix, Studia Math., 158 (003), [11] Kittaneh, F., Numeical adius inequalities fo Hilbet space opeatos, Studia Math., 168 (005), [1] Yamazaki, T., On uppe and lowe bounds of the numeical adius and an equality condition, Studia Math., 178 (007), Accepted:

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

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