International Mathematical Forum, Vol. 6, 011, no. 48, 379-385 Generalized Numerical Radius Inequalities for Operator Matrices Wathiq Bani-Domi Department of Mathematics Yarmouk University, Irbed, Jordan Wathiq @yahoo.com Abstract We prove a generalized numerical radius inequality for operator matrices, which improves and generalized an earlier inequality due to Bani- Domi and Kittaneh. Mathematics Subject Classification: 47A05, 47A10, 47A1, 47A30, 47A63, 15A60 Keywords: Numerical radius, usual operator norm, spectral radius, inequality, operator matrix, mixed Schwarz inequality 1 Introduction. Let B(H denote the C algebra of all bounded linear operators on a complex Hilbert space H with inner product <.,.>.ForA B(H, let ω(a = sup{ Ax, x : x H, x =1}, A = sup { Ax : x H, x =1}, where x = x, x, r(a = sup { λ : λ σ (A}, where σ (A is the spectrum of A, A =(A A 1 denote the numerical radius of A, the usual operator norm of A, the spectral radius of A, and the absolute value of A. It is well known that ω (. is a norm on B (H, which is equivalent to the usual operator norm.. In fact, for every A B (H, 1 A ω (A A (1 Several numerical radius inequalities improving the inequalities in (1 have been recently given in [3], [13], [15], and [16]. ( Let H 1,H,..., H n be complex Hilbert spaces. Then every operator A B H i can be represented as an n n operator matrix of the form A =
380 W. Bani-Domi [A ij ] with A ij is a bounded linear operator from H j into H i. Note that for every [ ] T vector x =[x 1 x... x n ] T H i,ax= A 1j x j A j x j... A nj X j. In this paper, we are interested in estimating a general numerical radii of operator matrices. Such estimates have been very useful in obtaining bounds for the zeros of polynomials (see [, 14]. Since the numerical radius norm is weakly unitarily invariant, in the sense that ω (U AU =ω (A for all A B (H and for all unitary operators U B (H, it follows that this norm satisfies( a pinching inequality. More precisely, if A =[A ij ] is an operator matrix in B H i, then A 11 A ω ω (A, that is,... A nn max {ω (A ii :i =1,,..., n} ω (A. ( Related pinching type inequalities for the general class of weakly unitarily invariant norms have been recently given in [1]. A very useful numerical radius inequalitty for operator matrices, ( due to Hou and Du [8], asserts that if A = [A ij ] is an operator matrix in B H i, and if à =[ A ij ] is the n n matrix whose (i, j entry is A ij, then ω (A ω (Ã. (3 It should be mentioned here that similar inequalities hold for the usual operator norm and the spectral radius (see [8]. Recently, Bani-Domi and Kittaneh [] improves the ( inequality in (3, which asserts that if A =[A ij ] is an operator matrix in B H i, then ω (A ω ([a ij ], (4 where, { ( 1 A a ij = ii + A ii 1 if i = j A ij if i j. In the next section of this paper, we prove several generalized numerical radii inequalities for operator matrices.
Generalized numerical radius inequalities 381 Main results. In order to state our results, we shall need the following well-known lemmas. The first lemma is a generalized mixed Schwarz inequality (see, e.g., [5, pp. 75-76], or [10]. Lemma 1 Let A B(H. Then for all x, y H, Ax, y A x, x A y, y. The second lemma contains a special case of a more general norm inequality that is equivalent to some Löwner-Heinz type inequalities (see, e.g., [4] or [11]. Lemma Let A, B B(H be positive operators. Then A 1 B 1 AB 1. The third lemma is a generalized form of the mixed Schwarz inequality which has been proved by Kittaneh [10]. Lemma 3 Let T be an operator in B(H and let f and g be nonnegative functions on [0, which are continuous and satisfying the relation f(tg(t = t for all t [0,. Then Tx,y f ( T x g ( T y for all x, y H. The fourth lemma is very useful in computing the numerical radius for matrices (see [7]. Lemma 4 If A =[a ij ] M n (C, then ω (A ω ([ a ij ] = 1 r ([ a ij + a ji ], where r (B is the spectral radius of B. The fifth lemma contains a recent norm inequality for sums of positive operators that is sharper than the triangle inequality (see [1]. Lemma 5 Let A, B B(H ( be positive operators. Then A + B 1 A + B + ( A B +4 A 1 B 1. The sixth lemma which is based on the Geršgorin theorem for location of eigenvalues (see [6], enables us to obtain a resonable estimate for the spectral radius. Lemma 6 If B =[b ij ] ( M n (C, then r (B min max 1 i n b ij, max 1 j n b ij. Our first result is a generalization of the inequality in (4.
38 W. Bani-Domi Theorem 7 Let A =[A ij ] be an operator matrix in B H i, and let f and g be nonnegative functions on [0, which are continuous and satisfying the relation f(tg(t =t for all t [0,. Then ω (A ω ([a ij ], (5 where, { 1 a ij = ( f ( A ii +g ( A ii if i = j f ( A ij g ( A. ij if i j Proof. Let x =[x 1 x... x n ] T H i, with x =1. Then we have Ax, x = = A ij x j,x i i, A ij x j,x i i, A ii x i,x i + A ij x j,x i f ( A ii x i,x i 1 g ( A ii x i,x i 1 + f ( A ij x j ( g A ij xi ( by lemma 3 1 [ f ( A ii +g ( A ii ] x i,x i + f ( A ij ( g A ij xi x j ( by the arithmetic-geometric mean inequality 1 f ( A ii +g ( A ii xi + f ( A ij ( g A ij xi x j = [a ij ] x, x, where x =[ x 1 x... x n ] T. Now the result follows by taking the supremum over all unit vectors in H i. Inequality (5 includes several numerical radius inequalities for operator matrices. Samples of inequalities are demonstrated in what follows. For f (t =t α and g (t =t 1 α,α (0, 1, in inequality (5, we get the following inequality that generalizes (4.
Generalized numerical radius inequalities 383 Corollary 8 Let A =[A ij ] be an operator matrix in B H i, and α (0, 1. Then ω (A ω ([a ij ], where, a ij = ( 1 Aii α + A ii (1 α if i = j A ij α A (1 α. ij if i j Remark. It follows as a special case of the previous corollary that if α = 1 and by Lemmas and 5 we get the inequality (4. Now using Lemma 4, one can formulate the inequality (5 in terms of the spectral radius as follows. Theorem 9 Let A =[A ij ] be an operator matrix in B H i, and let f and g be nonnegative functions on [0, which are continuous and satisfying the relation f(tg(t =t for all t [0,. Then ω (A r ([b ij ], (6 where, { 1 b ij = ( f ( A ii +g ( A ii if i = j 1 ( f ( A ij ( g A ij + f ( Aji ( g A ji if i j. When n =, we have the following sharp generalized numerical radius estimate for operator matrices. [ ] A11 A Corollary 10 Let A = 1 be an operator matrix in B A 1 A and let f and( g be as in Lemma 3. Then ω (A 1 a + c + (H 1 H, (a c +4b, (7 where, a = 1 f ( A 11 +g ( A 11, b = 1( f ( A 1 g ( A 1 + f ( A 1 g ( A 1, and c = 1 f ( A +g ( A. For n>, it is difficult to compute the right-hand side of the inequality (7. So, by using Lemma 6, we have the following theorem, which gives an estimate for the right-hand side of the inequality (6. Theorem 11 Let A =[A ij ] be an operator matrix in B H i, and let f and g be as in Lemma 3. Then
384 W. Bani-Domi ω (A max 1 i n 0 @ 1 flf ( A ii +g fia ii fi fl nx + 1 flf fiaij fi fl flg fia ijfi fl + fl f fiaji fi fl flg fia jifi fl 1 A. (8 Now the inequality (8 generalizes a related inequality for the numerical radii of matrices due to Johnson [8]. Based on Theorem 11, when n =we obtain the following simple estimate for the numerical radius of an operator matrix. [ ] A11 A Corollary 1 Let A = 1 be an operator matrix in B A 1 A and let f and g be as in Lemma 3. Then (H 1 H, ω (A 1 max flf ( A 11 +g fia 11 fi fl, flf ( A +g fia fi fl + 1 ( f ( A 1 fl g fia 1 fi fl + f ( A1 fl g fia 1 fi fl. References 1. W. Bani-Domi and F. Kittaneh, Norm equalitiies and inequalities for operator matrices, Linear Algebra Appl., 49(008, 57-67.. W.Bani-Domi and F. Kittaneh, Numerical radius inequalities for operator matrices, Linear and Multilinear algebra, 57(009, 41-47. 3. M. El-Haddad and F. Kittaneh, Numerical radius inequalities for Hilbert space operators.ii, Studia Math. 18(007, 133-140. 4. T. Furuta, Norm inequalities equivalent to Löwner-Heinz theorem, Rev. Math. 1(1989, 135-137. 5. P. R. Halmos, A Hilbert Space Problem Book, nd ed., Springer-Verlag, New York, 198. 6. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. 7. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge, 1991. 8. J. C. Hou and H. K. Du, Norm inequalities of positive operator matrices, Integral Equations Operator Theory (1995, 81-94. 9. C. R. Johnson, Geršgorin sets and the field of values, J. Math. Anal. Appl. 45(1974, 416-419. 10. F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst.Math. Sci. 4(1988, 83-93. 11. F. Kittaneh, Norm inequalities for certain operator sums, J. Funct. Anal. 143(1997, 337-348.
Generalized numerical radius inequalities 385 1. Kittaneh, Norm inequalities for sums of positive operator, J. Operator Theory 48(00, 95-103. 13. F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158(003, 11-17. 14. F. Kittaneh, Bounds for the zeros of polynomials from matrix inequalities, Arch. Math. (Basel 81(003,. 601-608. 15. F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168(005, 73-80. 16. T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178(007, 83-89. Received: April, 011