INEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES

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1 INEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. In this paper various inequalities between the operator norm its numerical radius are provided. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Goldstein-Ryff- Clarke, Dragomir-Sándor the author. 1. Introduction Let (H;, ) be a complex Hilbert space. The numerical range of an operator T is the subset of the complex numbers C given by 10, p. 1]: (1.1) W (T ) = { T x, x, x H, x = 1}. It is well known (see for instance 10]) that: (i) The numerical range of an operator is convex (the Toeplitz-Hausdorff theorem); (ii) The spectrum of an operator is contained in the closure of its numerical range; (iii) T is self-adjoint if only if W is real. The numerical radius w (T ) of an operator T on H is defined by 10, p. 8]: (1.) w (T ) = sup { λ, λ W (T )} = sup { T x, x, x = 1}. It is well known that w ( ) is a norm on the Banach algebra B (H) the following inequality holds true (1.3) w (T ) T w (T ), for any T B (H). In the previous work 6], the following reverse inequalities have been proved: (1.4) (0 ) T w (T ) 1 r λ, provided that λ C\ {0}, r > 0 (1.5) T λi r. If, in addition λ > r (1.5) holds true, then ( ) 1 (1.6) 1 r λ w (T ) ( 1), T Date: 9 August, Mathematics Subject Classification. 47A1. Key words phrases. Numerical range, Numerical radius, Bounded linear operators, Hilbert spaces. 1

2 S.S. DRAGOMIR which provides a refinement of the general inequality (1.7) 1 w (T ) T, in the case when r λ satisfy the assumption r/ λ 3/. With the same assumption on λ r, i.e. λ > r, we also have the inequality (1.8) (0 ) T w r (T ) w (T ), λ + λ r provided (1.5) holds true. In the same paper, on assuming that (T ϕi) (φi T ) is accreative 10, p. 5] (or sufficintly, self-adjoint nonnegative in the operator order of B (H)), where ϕ, φ C, φ ϕ, ϕ, we have proved the following inequality as well: (1.9) (0 ) T w (T ) 1 4 φ ϕ φ + ϕ. If we assume more, i.e., Re (φ ϕ) > 0 (which implies φ ϕ), then for T as above, we also have: (1.10) Re (φ ϕ) φ + ϕ (1.11) (0 ) T w (T ) w (T ) T ( 1) φ + ϕ ] Re (φ ϕ) w (T ). The main aim of this paper is to establish other inequalities between the operator norm its numerical radius. We employ, amongst others, the Buzano inequality as well as some results for vectors in inner product spaces due to Goldstein-Ryff- Clarke 9], Dragomir-Sándor 7] Dragomir 5].. The Results The following result may be stated as well: Theorem 1. Let (H;, ) be a Hilbert space T : H H a bounded linear operator on H. Then (.1) w (T ) 1 w ( T ) + T ]. The constant 1 is best possible in (.1). Proof. We need the following refinement of Schwarz s inequality obtained by the author in 1985, Theorem ] (see also 7] 5]): (.) a b a, b a, e e, b + a, e e, b a, b, provided a, b, e are vectors in H e = 1. Observing that a, b a, e e, b a, e e, b a, b, hence by the first inequality in (.) we deduce 1 (.3) ( a b + a, b ) a, e e, b. This inequality was obtained in a different way earlier by M.L. Buzano in 1].

3 INEQUALITIES FOR THE NORM & NUMERICAL RADIUS 3 Now, choose in (.3), e = x, x = 1, a = T x b = T x to get 1 ( (.4) T x T x + T x, x ) T x, x for any x H, x = 1. Taking the supremum in (.4) over x H, x = 1, we deduce the desired inequality (.1). Now, if we assume that (.1) holds with a constant C > 0, i.e., (.5) w (T ) C w ( T ) + T ] for any T B (H), then if we choose T a normal operator use the fact that for normal operators we have w (T ) = T w ( T ) = T = T, then by (.5) we deduce that C 1 which proves the sharpnes of the constant. Remark 1. From the above result (.1) we obviously have { 1 (.6) w (T ) w ( T ) + T ]} 1/ { 1 ( T )} 1/ + T T (.7) w (T ) { 1 w ( T ) + T ]} 1/ { 1 (w (T ) + T )} 1/ T, that provide refinements for the first inequality in (1.3). The following result may be stated. Theorem. Let T : H H be a bounded linear operator on the Hilbert space H λ C\ {0}. If T λ, then (.8) T r + λ r T r 1 λ r w (T ) + r λ r T λi, where r 1. Proof. We use the following inequality for vectors in inner product spaces due to Goldstein, Ryff Clarke 9]: (.9) a r + b r a r b r Re a, b a b r a r a b if r 1, b r a b if r < 1, provided r R a, b H with a b. Now, let x H with x = 1. From the hypothesis of the theorem, we have that T x λ x applying (.9) for the choices a = λx, x = 1, b = T x, we get (.10) T x r + λ r T x r 1 λ r T x, x µ r λ r T x λx for any x H, x = 1 r 1. Taking the supremum in (.10) over x H, x = 1, we deduce the desired inequality (.8). The following result may be stated as well:

4 4 S.S. DRAGOMIR Theorem 3. Let T : H H be a bounded linear operator on the Hilbert space (H,, ). Then for any α 0, 1] t R one has the inequality: (.11) T (1 α) + α ] w (T ) + α T ti + (1 α) T iti. Proof. We use the following inequality obtained by the author in 5]: α tb a + (1 α) itb a ] b to get: (.1) a b (1 α) Im a, b + α Re a, b ] ( 0) a b (1 α) Im a, b + α Re a, b ] + α tb a + (1 α) itb a ] b (1 α) + α ] a, b + α tb a + (1 α) itb a ] b for any a, b H, α 0, 1] t R. Choosing in (.1) a = T x, b = x, x H, x = 1, we get (.13) T x (1 α) + α ] T x, x + α tx T x + (1 α) itx T x. Finally, taking the supremum over x H, x = 1 in (.13), we deduce the desired result. The following particular cases may be of interest. Corollary 1. For any T a bounded linear operator on H, one has: inf T ti (.14) (0 ) T w t R (T ) inf T iti t R (.15) T 1 w (T ) + 1 inf T ti + T iti ]. t R Remark. The inequality (.14) can in fact be improved taking into account that for any a, b H, b 0, (see for instance 3]) the bound actually implies that inf a λ C λb = a b a, b b (.16) a b a, b b a λb for any a, b H λ C. Now if in (.16) we choose a = T x, b = x, x H, x = 1, then we obtain (.17) T x T x, x T x λx

5 INEQUALITIES FOR THE NORM & NUMERICAL RADIUS 5 for any λ C, which, by taking the supremum over x H, x = 1, implies that (.18) (0 ) T w (T ) inf λ C T λi. Remark 3. If we take a = x, b = T x in (.16), then we obtain (.19) T x T x, x + T x x µt x for any x H, x = 1 µ C. Now, if we take the supremum over x H, x = 1 in (.19), then we get (.0) (0 ) T w (T ) T inf µ C I µt. From a different view point we may state: Theorem 4. Let T : H H be a bounded linear operator on H. If p, then: (.1) T p + λ p 1 ( T + λi p + T λi p ), for any λ C. Proof. We use the following inequality obtained by Dragomir Sándor in 7]: (.) a + b p + a b p ( a p + b p ) for any a, b H p. Now, if we choose a = T x, b = λx, then we get (.3) T x + λx p + T x λx p ( T x p + λ p ) for any x H, x = 1. Taking the supremum in (.3) over x H, x = 1, we get the desired result (.1). Remark 4. For p =, we have the simpler result: (.4) T + λ 1 ( T + λi + T λi ) for any λ C. This can easily be obtained from the parallelogram identity as well. Theorem 5. Let T : H H be a bounded linear operator on H. Then: (.5) T + λi T + λ w (T ) + λ (.6) T + λi T T + λ I + λ w (T ), for any λ C. Proof. Observe that, for λ C, T x + λx = T x + λ + Re λ (.7) T x, x T x + λ + λ T x, x. Taking the supremum over x H, x = 1, we deduce the desired inequality (.5). Now, since for x = 1, we have ) T x + λ x = T x, T x + λ Ix, x = T T + λ I x, x, then by (.7) we obtain: (.8) T x + λx ) T T + λ I x, x + λ T x, x.

6 6 S.S. DRAGOMIR Taking the supremum in (.8) noticing that T T + λ I is a self-adjoint operator, we get the desired inequality (.6). Theorem 6. Let T : H H be a bounded linear operator on H α, β C. Then (.9) αt + βt α T T + β T T ( + αβ w T ) (.30) α T T + β T T αt βt + αβ w ( T ). Proof. Observe that for x H, x = 1, we have: αt x + βt x = α T x + β T x + Re α β T x, T x ] α T x, T x + β T x, T x + αβ T x, x = α T T + β T T ) x, x + αβ T x, x. Taking the supremum over x H, x = 1, we get αt + βt w ( α T T + β T T ) + αβ w ( T ) ) since α T T + β T T is self-adjoint, hence w ( α T T + β T T = α T T + β T T the inequality (.9) is obtained. Similarly, αt x βt x = α T T + β T T ) x, x Re α β T x, x ] α T T + β T T ) x, x αβ T x, x giving that (.31) αt x βt x + αβ T x, x α T T + β T T ) x, x for any x H, x = 1. Taking the supremum over x H, x = 1 in (.31) we deduce (.30). References 1] M.L. BUZANO, Generalizzatione della disiguaglianza di Cauchy-Schwaz (Italian), Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971/73), (1974). ] S.S. DRAGOMIR, Some refinements of Schwarz inequality, Simposional de Matematică şi Aplicaţii, Polytechnical Institute Timişoara, Romania, 1- Nov., 1985, ZBL 0594: ] S.S. DRAGOMIR, Some Grüss type inequalities in inner product spaces, J. Ineq. Pure & Appl. Math., 4() (003), Art. 4. ONLINE 4] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner product spaces (I), J. Ineq. Pure & Appl. Math., 6(3) (005), Art. 59. ONLINE 5] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner product spaces (II), RGMIA Res. Rep. Coll., 8(Supp) (005), Art. 5. ONLINE 6] S.S. DRAGOMIR, Reverse inequalities for the numerical radius of linear operators in Hilbert spaces, RGMIA Res. Rep. Coll. 8(005), Article 9, ONLINE ].

7 INEQUALITIES FOR THE NORM & NUMERICAL RADIUS 7 7] S.S. DRAGOMIR J. SÁNDOR, Some inequalities in prehilbertian spaces, Studia Univ. Babeş-Bolyai - Mathematica, 3(1) (1987), ] C.F. DUNKL K.S. WILLIAMS, A simple norm inequality, Amer. Math. Monthly, 71(1) (1964), ] A. GOLDSTEIN, J.V. RYFF L.E. CLARKE, Problem 5473, Amer. Math. Monthly, 75(3) (1968), ] K.E. GUSTAFSON D.K.M. RAO, Numerical Range, Springer-Verlag, New York, Inc., ] P.R. HALMOS, Introduction to Hilbert Space the Theory of Spectral Multiplicity, Chelsea Pub. Comp, New York, N.Y., ] G.H. HILE, Entire solutions of linear elliptic equations with Laplacian principal part, Pacific J. Math., 6(1) (1976), School of Computer Science Mathematics, Victoria University of Technology, PO Box 1448, Melbourne City, Victoria 8001, Australia. address: sever.dragomir@vu.edu.au URL: