MORE NUMERICAL RADIUS INEQUALITIES FOR OPERATOR MATRICES. Petra University Amman, JORDAN

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1 International Journal of Pure and Applied Mathematics Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v118i3.0 PAijpam.eu MORE NUMERICAL RADIUS INEQUALITIES FOR OPERATOR MATRICES Wasim Audeh 1, Manal Al-Labadi 1, Department of Mathematics Petra University Amman, JORDAN Abstract: In this work, we will prove several numerical radius inequalities from them we get recently proved numerical radius inequalities as special cases, and we will present numerical radius inequalities which are sharper than recently proved numerical radius inequalities. AMS Subject Classification: 47B47, 15A4, 47B15, 47A63 Key Words: inequality, numeriacl radius, operator, norm 1. Fundamental Principles Let H be a complex Hilbert space with inner product.,.. Let B(H) indicate the space of all bounded linear operators on H. For T B(H), let w(t) and T denote the numerical radius and the usual operator norm, respectively where w(t) = sup Tx,x and X = sup Tx. X =1 X =1 Received: November 7, 017 Revised: February 6, 018 Published: April 15, 018 c 018 Academic Publications, Ltd. url: Correspondence author

2 738 W. Audeh, M. Al-Labadi It is well-known that w(.) defines a norm on B(H), which is equivalent to the usual operator norm. For T B(H), we have T w(t) T (1.1) where w(t) = T if T = 0, and w(t) = T if T is normal. In the case where w(t) = T, the author in [7] has proved that ReT = ImT = T (1.) An improvement of the second part in inequality (1.1) has been proved in [6]: Let X B(H).Then which implies that if X = 0, then w(x) X + X 1/ w(x) = X The equality (1.4) follows from inequalities (1.1) and (1.3). For X B(H), the power inequality states that (1.3) (1.4) w(x n ) (w(x)) n (1.5) for n = 1,,... see []. For X B(H) where X = U X, the Aluthge transform of X is given by X = X 1/ U X 1/. Here U is a partial isometry and X = (X X) 1/.It can be shown that w( X) X 1/ (1.6) Inequality (1.6) was the first step to improve the inequality (1.3) which was proved in [8].( see, e.g., [8]).

3 MORE NUMERICAL RADIUS INEQUALITIES The numerical radius norm is weak unitary invariance, where for T H and for every unitary U B(H). w(u TU) = w(t) (1.7) The numerical radius for operator matrices has the following well known properties for X, Y B(H). Equality (1.8) can be found in [1], equalities (1.9), (1.10), and (1.11) where proved in [3]. ([ ]) X 0 w = max(w(x),w(y )) (1.8) 0 Y ([ 0 X w Y 0 ([ ]) X Y w Y X ]) ([ 0 Y = w X 0 ]) (1.9) = max(w(x +Y),w(X Y)) (1.10) and from equality (1.10) we get, ([ ]) 0 Y w = w(y) (1.11) Y 0 For useful numerical radius compression inequalities, together with an improvement and applications, we refer to [5]. In [4] the authors proved that if X,Y B(H), then n max(w((xy) n ).w((yx) n )) w([ 0 X Y 0 for n = 1,,... ]) X + Y (1.1). Introduction In this paper we prove new forms of numerical radius equalities and inequalities for,3 3,4 4, and n n operator matrices. 0 A 0 At the begining of this work we proved that if T = B 0 C be D 0 operator matrix. Then

4 740 W. Audeh, M. Al-Labadi w(t) 1 B B +AA 0 B C +AD 0 A A+D D +BB +CC 0 C B +DA 0 C C +DD (.1) As special cases when we replace B = D = 0, we get ([ ]) w A C A A+CC (.) From inequality (.) we will prove that if A and C are normal operators in H, then w ([ A 0 0 C ]) = 1 A A+CC (.3) In other special case when C = D = 0, then ([ ]) w 0 A 1 B 0 A A+BB (.4) ([ ]) A C We proved that if T =. Then 0 B w(t) Then 1 ( AA +CC + C C +B B +max( AA, BB )) (.5) We generalize equation (1.10) by proving that if T = X 0 Y 0 Y 0 Y 0 X. Then w(t) = max(w(x +Y),w(X Y),w(Y)) (.6) and again we generalize equation (1.10) by proveing that if X 0 0 Y T = 0 0 Y 0 0 Y 0 0. Y 0 0 X w(t) = max(w(x +Y),w(X Y),w(Y)) (.7)

5 MORE NUMERICAL RADIUS INEQUALITIES Attheendofourresearchwegivelowerboundforn noff-diagonal operator T T 0 matrix T =. 0.., where we prove the following two cases: 0 0 T n (i) If n is even, then w(t) n max w(t it n i+1 ) n,w(t n i+1 T i ) n (.8) i=1,...,n (ii) If n is odd, then w(t) n max i=1,...,n,i n+1 w(t i T n i+1 ) n,w(t n i+1 T i ) n,w(tn+1) (.9) 3. Main Results We will present the following lemma from which we get bound for 3 3 operator matrices. 0 A 0 Lemma 3.1. Let T = B 0 C be 3 3 operator matrix. Then 0 D 0 w(t) 1 Proof. 0 A 0 B 0 C 0 D 0 B B +AA 0 B C +AD 0 A A+D D +BB +CC 0 C B +DA 0 C C +DD X,X 0 A 0 B 0 C 0 D 0 X,X 0 A 0 B 0 C 0 D 0 X,X

6 74 W. Audeh, M. Al-Labadi = B B 0 B 1/ C 0 A A+D D 0 X,X C B 0 C C AA 0 AD 0 BB +CC 0 DA 0 DD 1/ X,X B B 0 B C 0 A A+D D 0 C B 0 C C X,X 1/ AA 0 AD 0 BB +CC 0 DA 0 DD X,X 1/ = 1 B B 0 B C C B 0 C C AA 0 AD DA 0 DD + + B B 0 B C + C B 0 C C AA 0 AD + DA 0 DD 0 A A+D D 0 0 BB +CC 0 X,X X,X 0 A A+D D 0 0 BB +CC 0 1/ 1/ X,X. X,X B B 0 B C + 0 A A+D D 0 + = 1 C B 0 C C AA 0 AD X,X + 0 BB +CC 0 DA 0 DD B B +AA 0 B C +AD = 1 0 A A+D D +BB +CC 0 X,X C B +DA 0 C C +DD Now take the supremum for all unit vectors and note that the operator +

7 MORE NUMERICAL RADIUS INEQUALITIES B B +AA 0 B C +AD 0 A A+D D +BB +CC 0 is normal operator C B +DA 0 C C +DD to get w(t) 1 B B +AA 0 B C +AD 0 A A+D D +BB +CC 0 C B +DA 0 C C +DD Remark 1. Let B = D = 0 in the inequality (.1) we get ([ ]) A 0 w 1 AA C 0 A A+CC C C = 1 A A+CC From inequality (.) and using some well known facts we can conclude the following equality. Remark. Let A,C be normal operators in H. Then w ([ A 0 0 C ]) = 1 A A+CC Proof. From inequality (.), max(w (A),w (C)) 1 A A+CC (3.1) On the other side, = 1 A ( A+CC, A + C ), by CC = C 1 max( A, C ), by C = C = max(w (A),w (C)), because A,B are normal operators. This means 1 A A+CC max(w (A),w (C)) (3.) From inequalities (3.1) and (3.) we reach our aim. Remark 3. Replace C = D = 0 in inequality (.1) to get the inequality

8 744 W. Audeh, M. Al-Labadi w ([ 0 A B 0 ]) 1 B B +AA A A+BB 0 = 1 A A+BB The following theorem is numerical radius inequality for 3 3 operator matrix. E A F Theorem 3.. Let T = B G C. Then H D I 1 Proof. T = w(t) w w(t) max(w(e),w(g),w(i))+ B B +AA 0 B C +AD 0 A A+D D +BB +CC 0 C B +DA 0 C C +DD E G I E G I + + F + H +w 0 A 0 B 0 C 0 D 0 0 A 0 B 0 C 0 D 0 + +w = max(w(e),w(g),w(i)) + 1 by using inequalites (1.1) and (.1) and equalitiy (1.8). (3.3) 0 0 F. This implies H F H 0 0 BB +CC + A A+D D + F + H We will prove the following theorem for numerical radius inequality for upper triangular operator matrices. ([ ]) A C Theorem 3.3. Let T =. Then 0 B w(t) 1 ( AA +CC + C C +B B +max( AA, BB ))

9 MORE NUMERICAL RADIUS INEQUALITIES [ ] Proof. A C [ ] [ ] A C A C X,X X,X X,X 0 B 0 B 0 B [ A A A ] 1/ [ C AA C A C C +B X,X +CC CB ] 1/ X,X B BC BB [ A A A ] 1/ [ C AA C A C C +B X,X +CC CB ] 1/ B BC BB X,X ( [ A 1 A A ] [ C AA C A C C +B X,X + +CC CB ] ) B BC BB X,X ([ A = 1 A A ] [ C CC CB C A C + ] [ ]) AA 0 C BC BB + 0 BB X,X [ A but the operator A A ] [ C CC CB C A C + ] [ ] AA 0 C BC BB + 0 BB is self-adjoint, this implies( that [ A w (T) 1 A A ] [ C CC CB C A C + ] [ ] ) AA 0 C BC BB + 0 BB ( [ ] A 1 A A [ ] C + CC CB [ ] ) C A C C + AA 0 BC BB 0 BB ( [ ] AA = 1 +CC [ ] 0 + C C +B [ ] ) B AA BB = 1 ( AA +CC + C C +B B +max( AA, BB ) as required. Remark 4. Replacing B = C = 0 in inequality (.5) we get the right side of inequality (1.1): 1 w(a) ( AA + AA ) = AA = A = A Now we will present numerical radius equalities for 3 3, 4 4, operator matrices. X 0 Y Theorem 3.4. Let T = 0 Y 0. Then Y 0 X w(t) = max(w(x +Y),w(X Y),w(Y))

10 746 W. Audeh, M. Al-Labadi I 0 I Proof. Let U = 0 I 0. Then U is unitary, and I 0 I X +Y 0 0 UTU = 0 Y 0, it is well known that numerical radius 0 0 X Y is weakly unitarily invariant, this means that w(utu ) = w(t). but w(utu ) = max(w(x+y),w(x Y),w(Y)). This implies that w(t) = max(w(x +Y),w(X Y),w(Y)). Remark 5. Let X = 0 in equality (.6) we get w 0 0 Y 0 Y 0 Y 0 0 = w(y) (3.4) Theorem 3.5. Let T = X 0 0 Y 0 0 Y 0 0 Y 0 0 Y 0 0 X. Then w(t) = max(w(x +Y),w(X Y),w(Y)) I 0 0 I Proof. LetU = 1 0 I I 0.ThenU isunitary, andu TU = I 0 0 I X +Y 0 0 Y 0 0 Y 0 0. Using the fact that numerical radius is weakly X Y unitarily invariant gives our result. Now we will prove the following Theorem which gives lower bound for offdiagonal operator matrices.

11 MORE NUMERICAL RADIUS INEQUALITIES T T 0 Theorem 3.6. Let T = Then T n (1) If n is even, then () If n is odd, then w(t) n w(t) n max w(t it n i+1 ) n,w(t n i+1 T i ) n i=1,...,n max i=1,...,n,i n+1 w(t i T n i+1 ) n,w(t n i+1 T i ) n,w(tn+1) T T 0 Proof. Let T =. 0.., where n is even integer. Then 0 0 T n T 1 T n 0 0 T T n T = Tn T n Tn +1 Tn, T n 1 T 0 0 T n T 1 which implies that T n = (T 1 T n) n 0 0 (T T n 1 ) n ( 0 0 n Tn Tn +1) 0 ( ) n 0 Tn +1Tn (T n 1 T ) n 0 0 (T nt 1 ) n

12 748 W. Audeh, M. Al-Labadi for n = 1,,..., and so max w(t it n i+1 ) n,w(t n i+1 T i ) n = w(t n ) w n (T) which is inequality i=1,...,n (.8) ( by inequality ()) T T 0 Let T =. 0.. where n is odd integer. Then 0 0 T n T = T 1 T n 0 T T n Tn+1 Tn+1, T n 1 T 0 T n T 1 which implies that (T 1 T n ) n 0 (T T n 1 ) n ( ) T n n = T n+1t n (T n 1 T ) n 0 (T n T 1 ) n for n = 1,,..., which implies that max w(t i T n i+1 ) n,w(t n i+1 T i ) n,w(tn+1) = w(t n ) w n (T) i = 1,...,n, i n+1 Acknowledgement The author is grateful to the University of Petra for its Support.

13 MORE NUMERICAL RADIUS INEQUALITIES References [1] Bhatia, R.: Matrix Analysis. Springer, New York (1997) [] Halmos, P.R.: A Hilbert space problem book, nd ed. Springer, New York (198) [3] Hirzallah, O., Kittaneh, F., Shebrawi, K.: Numerical radius inequalities for commutators of Hilbert space operators. Numer. Funct. Anal. Optim. 3, (011) [4] Hirzallah, O., Kittaneh, F., Shebrawi, K.: Numerical radius inequalities for certain operator matrices. Integr.Equ. Oper. Theory. 71, (011) [5] Hou, J.C., Du, H.K.: Norm inequalities of positive operator matrices., (1995 ) [6] Kittaneh, F.: A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 158, (003). [7] Levon, G.: On operators with large self-commutators. Oper. Matrices 4, (010) [8] Yamazaki, T.: On upper and lower bounds of the numerical radius and an equality condition. Studia Math. 178, (007)

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