- Email: [email protected]
A generalization of the numerical radius
Linear Algebra and its Applications 569 (2019) 323–334
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
A generalization of the numerical radius Amer Abu-Omar a , Fuad Kittaneh b,∗ a
College of Sciences and Humanities, Fahad Bin Sultan University, Tabuk, Saudi Arabia b Department of Mathematics, The University of Jordan, Amman, Jordan
a r t i c l e
i n f o
Article history: Received 15 November 2018 Accepted 24 January 2019 Available online 29 January 2019 Submitted by P. Semrl MSC: primary 47A12, 47A30, 47A36 secondary 47B10, 47B15
a b s t r a c t We define a norm on the space of bounded linear operators on a Hilbert space, which generalizes the numerical radius norm. We investigate basic properties of this norm and prove inequalities involving it. A concrete example of this norm is also given. © 2019 Elsevier Inc. All rights reserved.
Keywords: Numerical radius Spectral radius Usual operator norm Hilbert–Schmidt norm Inequality
1. Introduction Let B (H) denote the C ∗ -algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B (H), let r (A), w (A), and A denote the spectral radius, the numerical radius, and the usual operator norm of A, respectively. Recall that the numerical radius is defined as
* Corresponding author. E-mail addresses: [email protected] (A. Abu-Omar), [email protected] (F. Kittaneh). https://doi.org/10.1016/j.laa.2019.01.019 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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w (A) = sup |Ax, x| , x=1
where · and ·, · are the inner product and its corresponding norm on H. It is wellknown that w (A) defines a norm on B (H), which is equivalent to the usual operator norm ·. In fact, the following inequalities hold for every A ∈ B (H): max{r (A) ,
1 A} ≤ w (A) ≤ A . 2
(1.1)
It is well-known that if A is normal, then r(A) = w(A) = A, and if A2 = 0, then w(A) = 12 A. Hence, the inequalities in (1.1) are sharp. For proofs and more facts about the numerical radius, we refer the reader to [4] and [5]. In this paper, a norm N (·) on B (H) is an algebra norm if N (AB) ≤ N (A) N (B) for every A, B ∈ B(H), self-adjoint if N (A∗ ) = N (A) for every A ∈ B (H), and weakly unitarily invariant if N (U ∗ AU ) = N (A) for every A ∈ B (H) and every unitary U ∈ B (H). Obviously, w (·) is self-adjoint and weakly 0 1 unitarily invariant, but, it is not an algebra norm. Consider the example A = 0 0 0 0 . and B = 1 0 In Section 2, we define a norm on B (H), which generalizes the numerical radius and prove some basic properties. In Section 3, many inequalities involving this norm are given. These inequalities generalize known numerical radius inequalities. In Section 4, we give a concrete example of such a norm and prove some results concerning it. 2. Definition and basic properties In this section, we define our new norm, which generalizes the numerical radius. Basic properties of this norm are also investigated. Definition 1. Let N (·) be a norm on B (H). The function wN (·) : B (H) → R+ is defined as wN (A) = sup N Re eiθ A θ∈R
for every A ∈ B(H).
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325
Theorem 1. wN (·) is a norm on B(H). Proof. Let A ∈ B(H). Since N (·) is a norm on B(H), we have N Re eiθ A ≥ 0 for every θ ∈ R. Hence wN (A) = sup N Re eiθ A ≥ 0. Now, assume that wN (A) = 0. θ∈R Then N Re eiθ A = 0 for every θ ∈ R. Taking θ = 0 and θ = −π/2, we have N (Re A) = N (Im A) = 0. Since N (·) is a norm, it follows that Re A = Im A = 0, and so A = Re A + i Im A = 0. Also, if λ ∈ C and if φ ∈ R is such that λ = |λ| eiφ , then wN (λA) = sup N Re λeiθ A θ∈R
= sup N |λ| Re ei(θ+φ) A θ∈R
= |λ| sup N Re ei(θ+φ) A θ∈R
= |λ| wN (A) . The triangle inequality for wN (·) follows from the triangle inequality for the norm N (·) as follows: wN (A + B) = sup N Re eiθ (A + B) θ∈R
= sup N Re eiθ A + Re eiθ B θ∈R
≤ sup N Re eiθ A + N Re eiθ B θ∈R
≤ sup N Re eiθ A + sup N Re eiθ B θ∈R
θ∈R
= wN (A) + wN (B).
2
Remark 1. It is well-known (see, e.g., [9]) that w (A) = sup Re eiθ A . θ∈R
In view of the previous relation, it is now obvious that the norm wN (·) generalizes the numerical radius norm w (·). Theorem 2. Let A ∈ B(H). Then wN (A) ≥
1 N (A) . 2
(2.1)
Moreover, if N (·) is self-adjoint, then wN (A) ≤ N (A) .
(2.2)
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Proof. Since wN (A) = sup N Re eiθ A ≥ N Re eiθ A θ∈R
for every θ ∈ R, we have (by taking θ = 0 and θ = −π/2) that wN (A) ≥ N (Re A) and wN (A) ≥ N (Im A) . Hence, 2wN (A) ≥ N (Re A) + N (Im A) . By the triangle inequality for the norm N (·), we get 2wN (A) ≥ N (Re A + i Im A) = N (A) as required. When N (·) is self adjoint, we have 1 N Re eiθ A = N eiθ A + e−iθ A∗ 2 1 iθ 1 −iθ ∗ ≤ N e A + N e A 2 2 = N (A) for every θ ∈ R. Hence, wN (A) = sup N Re eiθ A ≤ N (A) .
2
θ∈R
Remark 2. The inequality (2.2) is sharp. For instance, if A is self-adjoint, then wN (A) = sup N Re eiθ A θ∈R
= sup N A Re eiθ θ∈R
= N (A) sup Re eiθ θ∈R
= N (A) . In the following theorem, a refinement of the inequality (2.1) is given.
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Theorem 3. Let A ∈ B(H) and let m1 = max N (Re A) , 12 N (A) , d1 = |N (Re A) −
1 1 1 2 N (A)|, m2 = max N (Im A) , 2 N (A) , and d2 = N (Im A) − 2 N (A) . Then 1 1 1 N (A) + (d1 + d2 ) + |m1 − m2 | ≤ wN (A) . 2 4 2 Proof. Recall that wN (A) ≥ N (Re A), wN (A) ≥ N (Im A), and wN (A) ≥ Hence,
(2.3) 1 2N
(A).
wN (A) ≥ max {m1 , m2 } 1 1 (m1 + m2 ) + |m1 − m2 | 2 2 1 1 1 1 = (N (Re A) + N (Im A)) + N (A) + (d1 + d2 ) + |m1 − m2 | 4 4 4 2 1 1 1 1 ≥ N (Re A + i Im A) + N (A) + (d1 + d2 ) + |m1 − m2 | 4 4 4 2 1 1 1 = N (A) + (d1 + d2 ) + |m1 − m2 | . 2 2 4 2
=
Corollary 1. Let A ∈ B(H). The following conditions are equivalent: (a) wN (A) = 12 N (A). (b) N Re eiθ A = 12 N (A) for all θ ∈ R. Remark 3. (a) When N (·) is the usual operator norm, the inequality (2.3) becomes 1 1 1 A + (d1 + d2 ) + |m1 − m2 | ≤ w (A) , 2 4 2
where m1 = max Re A , 12 A , d1 = Re A − 12 A , m2 = max{ Im A, 1 1 (4.5) in [6]. 2 A}, and d2 = | Im A − 2 A|. This inequality refines the inequality iθ 1 1 (b) It follows from the Corollary 1 that w (A) = 2 A if and only if Re e A = 2 A for all θ ∈ R. In [9], it has been shown using a different technique that w (A) = 12 A if and only if Re eiθ A + Im eiθ A = A for all θ ∈ R. Theorem 4. The following properties hold: (a) The norm wN (·) is self-adjoint. (b) If the norm N (·) is weakly unitarily invariant, then so is wN (·).
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Proof. (a) We have wN (A∗ ) = sup N Re eiθ A∗ θ∈R
= sup N Re e−iθ A θ∈R
= wN (A) . (b) Assume that N (·) is weakly unitarily invariant and let U ∈ B(H) be unitary. Then wN (U ∗ AU ) = sup N Re eiθ U ∗ AU θ∈R
= sup N U ∗ Re eiθ A U . θ∈R
By the assumption, N U ∗ Re eiθ A U = N Re eiθ A . Hence, wN (U ∗ AU ) = sup N Re eiθ A θ∈R
= wN (A) .
2
3. Inequalities involving wN (·) The idea of this section is to establish inequalities for the norm wN (·) using inequalities on the norm N (·). This idea enables us to derive new numerical radius inequalities using known operator norm inequalities. Theorem 5. Let A, B, X ∈ B (H). If N (·) is an algebra norm, then wN (AXB + B ∗ XA∗ ) ≤ (N (A) N (B) + N (A∗ ) N (B ∗ )) wN (X) .
(3.1)
In particular, wN (AXA∗ ) ≤ N (A) N (A∗ ) wN (X) .
(3.2)
Proof. First, notice that N Re eiθ (AXB + B ∗ XA∗ ) = N A Re eiθ X B + B ∗ Re eiθ X A∗ . Now, since N (·) is an algebra norm, we have N Re eiθ (AXB + B ∗ XA∗ ) ≤ (N (A) N (B) + N (A∗ ) N (B ∗ )) N Re eiθ X . Hence,
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wN (AXB + B ∗ XA∗ ) = sup N Re eiθ (AXB + B ∗ XA∗ ) θ∈R
≤ sup (N (A) N (B) + N (A∗ ) N (B ∗ )) N Re eiθ X θ∈R
= (N (A) N (B) + N (A∗ ) N (B ∗ )) wN (X) . This proves the inequality (3.1). The inequality (3.2) follows from the inequality (3.1) by letting B = A∗ . 2 The following lemma contains the classical norm inequalities of Heinz and Kato. For generalizations of these inequalities to the wider class of unitarily invariant norms, we refer the reader to [2] and [7]. Lemma 1. Let A, B, X ∈ B (H) such that A and B are positive and let 0 < ν < 1. The following inequalities hold: ν
1−ν
Aν XB ν ≤ AXB X , 1/2 2 A XB 1/2 ≤ Aν XB 1−ν + A1−ν XB ν ≤ AX + XB ,
(3.4)
ν A XB 1−ν − A1−ν XB ν ≤ |2ν − 1| AX − XB .
(3.5)
(3.3)
and
Theorem 6. Let A, X ∈ B (H) such that A is positive and let 0 < ν < 1. The following inequalities hold: w (Aν XAν ) ≤ wν (AXA)w1−ν (X) , 2w A1/2 XA1/2 ≤ w Aν XA1−ν + A1−ν XAν ≤ w (AX + XA) ,
(3.6) (3.7)
and w Aν XA1−ν − A1−ν XAν ≤ |2ν − 1| w (AX − XA) .
(3.8)
Proof. Let θ ∈ R be arbitrary. Since Re eiθ Aν XAν = Aν Re eiθ X Aν , we have by the inequality (3.3) iθ ν Re e A XAν ≤ A Re eiθ X Aν Re eiθ X 1−ν . Hence, w (Aν XAν ) = sup Re eiθ Aν XAν θ∈R
ν 1−ν ≤ sup A Re eiθ X A Re eiθ X θ∈R
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ν 1−ν = sup Re eiθ AXA Re eiθ X θ∈R
ν 1−ν ≤ sup Re eiθ AXA sup Re eiθ X θ∈R
θ∈R
ν
= w (AXA)w
1−ν
(X) .
This proves the inequality (3.6). To prove the inequalities (3.7), recall that Re eiθ A1/2 XA1/2 = A1/2 Re eiθ X A1/2 . Hence, by the inequalities (3.4), 2 Re eiθ A1/2 XA1/2 ≤ Aν Re eiθ X A1−ν + A1−ν Re eiθ X Aν ≤ A Re eiθ X + Re eiθ X A . Equivalently, 2 Re eiθ A1/2 XA1/2 ≤ Re eiθ Aν XA1−ν + A1−ν XAν ≤ Re eiθ (AX + XA) . (Notice that Aν Re(eiθ X)A1−ν + A1−ν Re(eiθ X)Aν = Re(eiθ (Aν XA1−ν + iθ iθ iθ 1−ν ν XA )) and A Re e X + Re e X A = Re e (AX + XA) ). The inequalities A (3.7) now follow from the previous inequalities by taking the supremum over all θ ∈ R. The inequality (3.8) can be similarly proved. 2 In [1], several numerical radius inequalities for matrices have been obtained based on Schur’s multipliers technique. We remark that these inequalities can be derived in the general setting of Hilbert space operators in a simple way using our analysis presented in this section. 4. A concrete example In this section, we study the norm wN (·) when N (·) is the Hilbert–Schmidt norm. Let H be a separable complex Hilbert space. Recall that an operator A ∈ B (H) is ∞ said to belong to the trace class C1 if |A| ei , ei is finite and to the Hilbert–Schmidt class C2 if
∞
2
|Aei , ej | =
i,j=1
∞
i=1 2
Aei is finite for some (hence, for any) orthonormal
i=1
∞ ∞ basis {ei }i=1 for H. For A ∈ C1 , let trA = Aei , ei be the trace of A. Also, for i=1 1/2
∞ 2 A ∈ C2 , let A2 = Aei be the Hilbert–Schmidt norm of A. Note that for i=1
A ∈ C2 , A2 = trA∗ A. The classes C1 and C2 are examples of the Schatten p-classes Cp (1 ≤ p ≤ ∞). For more details about these classes, we refer the reader to [3] and [8]. 2
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When N (·) is the Hilbert–Schmidt norm ·2 , the norm wN (·) is denoted by w2 (·). iθ That is, w2 (A) = sup Re e A 2 . θ∈R
In the following theorem, we derive a formula for w2 (A) in terms of A2 and trA2 . Theorem 7. Let A ∈ C2 . Then w2 (A) =
1 1 2 A2 + |trA2 |. 2 2
Proof. Since iθ 2 Re e A = tr Re eiθ A 2 2 1 = tr A∗ A + AA∗ + 2 Re e2iθ A2 4 1 2 = 2 A2 + 2tr Re e2iθ A2 4 1 2 = 2 A2 + 2 Re e2iθ trA2 , 4 we have w2 (A) = sup Re eiθ A 2 θ∈R
= sup θ∈R
=
1 1 2 A2 + Re (e2iθ trA2 ) 2 2
1 1 2 A2 + |trA2 |, 2 2
as required. 2 In the following theorem, we prove that the norms w2 (·) and ·2 are equivalent. Theorem 8. Let A ∈ C2 . Then 1 √ A2 ≤ w2 (A) ≤ A2 . 2
(4.1)
Proof. The first inequality is obvious. The second inequality follows from Theorem 2 The second inequality also follows from and recalling that the norm · 2 is2 self-adjoint. 2 Theorem 7 and the fact that trA ≤ A2 . 2 In the rest of the paper, the following two facts are needed. Let A ∈ C2 and let λ1 , λ2 , . . . be the eigenvalues of A, repeated according to their multiplicities. The well∞ 2 2 known Schur’s inequality (see, e.g., [3, p. 107]) states that A2 ≥ |λi | , with the i=1
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equality occurs if and only if A is normal. The other fact is Lidiskii’s theorem (see, e.g., ∞ [3, p. 101]), which states that trA = λi . i=1
The inequalities (4.1) are sharp. In the following corollary, we state necessary and sufficient conditions for the equality cases in the inequalities (4.1). Corollary 2. Let A ∈ C2 be nonzero. Then (a) w2 (A) = √12 A2 if and only if trA2 = 0. (b) w2 (A) = A2 if and only if A is normal and the squares of its nonzero eigenvalues have the same argument. Proof. (a) This is a direct consequence of Theorem 7. 2 2 2 (b) Assume that w2 (A) = A2 . By Theorem 7, we have A2 = trA . Hence, A2 = ∞ ∞ trA2 = λ2 ≤ |λi |2 ≤ A2 . This implies that i 2 i=1
i=1
∞ i=1
2
2
|λi | = A2
(4.2)
and ∞ ∞ λ2i = λ2i . i=1
(4.3)
i=1
The normality of A follows from the equality (4.2). The equality (4.3) implies that all the squares of the nonzero eigenvalues of A have the same argument. Conversely, assume that A is normal and the squares of its nonzero eigenvalues 2 have the same argument. Let θ = arg λ2i for i = 1, 2, . . .. Then λ2i = |λi | eiθ for i = 1, 2, . . ., and hence ∞ trA2 = λ2i i=1
∞
=
i=1
2
2
|λi | = A2 (by the normality of A).
Thus, by Theorem 7, we have w2 (A) = A2 .
2
Theorem 9. Let A ∈ C2 . Then
k k 1 1 2 |λi | + λ2i ≤ w2 (A) 2 i=1 2 i=1
for k = 1, 2, . . .. In particular,
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r (A) ≤ w2 (A) . Proof. By Lidiskii’s theorem and Schur’s inequality, we have 2 A2
∞ ∞ 2 2 2 + trA ≥ |λi | + λi i=1
=
k
i=1
2
|λi | +
i=1
∞ i=k+1
∞ 2 |λi | + λ2i . i=1
Hence, by the triangle inequality, k ∞ ∞ 2 2 2 2 2 A2 + trA ≥ |λi | + λi + λi i=k+1 i=1 i=1 k ∞ ∞ 2 2 2 ≥ |λi | + λi − λi i=1 i=1 i=k+1 k k 2 = |λi | + λ2i . i=1
i=1
Thus, w2 (A) ≥
k k 1 1 2 |λi | + λ2i . 2 2 i=1 2 i=1
Remark 4. For every A ∈ C2 and every θ ∈ R, we have Re eiθ A ≤ Re eiθ A 2 . Hence, w (A) ≤ w2 (A), which is better than the inequality r (A) ≤ w2 (A). However, 0 1 the example A = shows that the inequality A ≤ w2 (A) need not be true in 0 0 general. Finally, it is tempting to study the norm wN (·) in the case of other unitarily invariant norms N (·), including the Schatten p-norms (p = 2). References [1] G. Aghamollaei, A. Sheikh Hosseini, Some numerical radius inequalities with positive definite functions, Bull. Iranian Math. Soc. 41 (2015) 889–900. [2] R. Bhatia, C. Davis, A Cauchy–Schwarz inequality for operators with applications, Linear Algebra Appl. 223/224 (1995) 119–129. [3] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Nonselfadjoint Linear Operators, Transl. Math. Monogr., vol. 18, Amer Math. Soc., Providence, RI, 1969. [4] K.E. Gustafson, D.K.M. Rao, Numerical Range, Springer, New York, 1997. [5] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982. [6] O. Hirzallah, F. Kittaneh, K. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices, Integral Equations Operator Theory 71 (2011) 129–147.
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[7] F. Kittaneh, Norm inequalities for fractional powers of positive operators, Lett. Math. Phys. 27 (1993) 279–285. [8] B. Simon, Trace Ideals and Their Applications, 2nd ed., Amer. Math. Soc., Providence, RI, 2005. [9] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178 (2007) 83–89.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
A generalization of the numerical radius Amer Abu-Omar a , Fuad Kittaneh b,∗ a
College of Sciences and Humanities, Fahad Bin Sultan University, Tabuk, Saudi Arabia b Department of Mathematics, The University of Jordan, Amman, Jordan
a r t i c l e
i n f o
Article history: Received 15 November 2018 Accepted 24 January 2019 Available online 29 January 2019 Submitted by P. Semrl MSC: primary 47A12, 47A30, 47A36 secondary 47B10, 47B15
a b s t r a c t We define a norm on the space of bounded linear operators on a Hilbert space, which generalizes the numerical radius norm. We investigate basic properties of this norm and prove inequalities involving it. A concrete example of this norm is also given. © 2019 Elsevier Inc. All rights reserved.
Keywords: Numerical radius Spectral radius Usual operator norm Hilbert–Schmidt norm Inequality
1. Introduction Let B (H) denote the C ∗ -algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B (H), let r (A), w (A), and A denote the spectral radius, the numerical radius, and the usual operator norm of A, respectively. Recall that the numerical radius is defined as
* Corresponding author. E-mail addresses: [email protected] (A. Abu-Omar), [email protected] (F. Kittaneh). https://doi.org/10.1016/j.laa.2019.01.019 0024-3795/© 2019 Elsevier Inc. All rights reserved.
324
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w (A) = sup |Ax, x| , x=1
where · and ·, · are the inner product and its corresponding norm on H. It is wellknown that w (A) defines a norm on B (H), which is equivalent to the usual operator norm ·. In fact, the following inequalities hold for every A ∈ B (H): max{r (A) ,
1 A} ≤ w (A) ≤ A . 2
(1.1)
It is well-known that if A is normal, then r(A) = w(A) = A, and if A2 = 0, then w(A) = 12 A. Hence, the inequalities in (1.1) are sharp. For proofs and more facts about the numerical radius, we refer the reader to [4] and [5]. In this paper, a norm N (·) on B (H) is an algebra norm if N (AB) ≤ N (A) N (B) for every A, B ∈ B(H), self-adjoint if N (A∗ ) = N (A) for every A ∈ B (H), and weakly unitarily invariant if N (U ∗ AU ) = N (A) for every A ∈ B (H) and every unitary U ∈ B (H). Obviously, w (·) is self-adjoint and weakly 0 1 unitarily invariant, but, it is not an algebra norm. Consider the example A = 0 0 0 0 . and B = 1 0 In Section 2, we define a norm on B (H), which generalizes the numerical radius and prove some basic properties. In Section 3, many inequalities involving this norm are given. These inequalities generalize known numerical radius inequalities. In Section 4, we give a concrete example of such a norm and prove some results concerning it. 2. Definition and basic properties In this section, we define our new norm, which generalizes the numerical radius. Basic properties of this norm are also investigated. Definition 1. Let N (·) be a norm on B (H). The function wN (·) : B (H) → R+ is defined as wN (A) = sup N Re eiθ A θ∈R
for every A ∈ B(H).
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325
Theorem 1. wN (·) is a norm on B(H). Proof. Let A ∈ B(H). Since N (·) is a norm on B(H), we have N Re eiθ A ≥ 0 for every θ ∈ R. Hence wN (A) = sup N Re eiθ A ≥ 0. Now, assume that wN (A) = 0. θ∈R Then N Re eiθ A = 0 for every θ ∈ R. Taking θ = 0 and θ = −π/2, we have N (Re A) = N (Im A) = 0. Since N (·) is a norm, it follows that Re A = Im A = 0, and so A = Re A + i Im A = 0. Also, if λ ∈ C and if φ ∈ R is such that λ = |λ| eiφ , then wN (λA) = sup N Re λeiθ A θ∈R
= sup N |λ| Re ei(θ+φ) A θ∈R
= |λ| sup N Re ei(θ+φ) A θ∈R
= |λ| wN (A) . The triangle inequality for wN (·) follows from the triangle inequality for the norm N (·) as follows: wN (A + B) = sup N Re eiθ (A + B) θ∈R
= sup N Re eiθ A + Re eiθ B θ∈R
≤ sup N Re eiθ A + N Re eiθ B θ∈R
≤ sup N Re eiθ A + sup N Re eiθ B θ∈R
θ∈R
= wN (A) + wN (B).
2
Remark 1. It is well-known (see, e.g., [9]) that w (A) = sup Re eiθ A . θ∈R
In view of the previous relation, it is now obvious that the norm wN (·) generalizes the numerical radius norm w (·). Theorem 2. Let A ∈ B(H). Then wN (A) ≥
1 N (A) . 2
(2.1)
Moreover, if N (·) is self-adjoint, then wN (A) ≤ N (A) .
(2.2)
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Proof. Since wN (A) = sup N Re eiθ A ≥ N Re eiθ A θ∈R
for every θ ∈ R, we have (by taking θ = 0 and θ = −π/2) that wN (A) ≥ N (Re A) and wN (A) ≥ N (Im A) . Hence, 2wN (A) ≥ N (Re A) + N (Im A) . By the triangle inequality for the norm N (·), we get 2wN (A) ≥ N (Re A + i Im A) = N (A) as required. When N (·) is self adjoint, we have 1 N Re eiθ A = N eiθ A + e−iθ A∗ 2 1 iθ 1 −iθ ∗ ≤ N e A + N e A 2 2 = N (A) for every θ ∈ R. Hence, wN (A) = sup N Re eiθ A ≤ N (A) .
2
θ∈R
Remark 2. The inequality (2.2) is sharp. For instance, if A is self-adjoint, then wN (A) = sup N Re eiθ A θ∈R
= sup N A Re eiθ θ∈R
= N (A) sup Re eiθ θ∈R
= N (A) . In the following theorem, a refinement of the inequality (2.1) is given.
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327
Theorem 3. Let A ∈ B(H) and let m1 = max N (Re A) , 12 N (A) , d1 = |N (Re A) −
1 1 1 2 N (A)|, m2 = max N (Im A) , 2 N (A) , and d2 = N (Im A) − 2 N (A) . Then 1 1 1 N (A) + (d1 + d2 ) + |m1 − m2 | ≤ wN (A) . 2 4 2 Proof. Recall that wN (A) ≥ N (Re A), wN (A) ≥ N (Im A), and wN (A) ≥ Hence,
(2.3) 1 2N
(A).
wN (A) ≥ max {m1 , m2 } 1 1 (m1 + m2 ) + |m1 − m2 | 2 2 1 1 1 1 = (N (Re A) + N (Im A)) + N (A) + (d1 + d2 ) + |m1 − m2 | 4 4 4 2 1 1 1 1 ≥ N (Re A + i Im A) + N (A) + (d1 + d2 ) + |m1 − m2 | 4 4 4 2 1 1 1 = N (A) + (d1 + d2 ) + |m1 − m2 | . 2 2 4 2
=
Corollary 1. Let A ∈ B(H). The following conditions are equivalent: (a) wN (A) = 12 N (A). (b) N Re eiθ A = 12 N (A) for all θ ∈ R. Remark 3. (a) When N (·) is the usual operator norm, the inequality (2.3) becomes 1 1 1 A + (d1 + d2 ) + |m1 − m2 | ≤ w (A) , 2 4 2
where m1 = max Re A , 12 A , d1 = Re A − 12 A , m2 = max{ Im A, 1 1 (4.5) in [6]. 2 A}, and d2 = | Im A − 2 A|. This inequality refines the inequality iθ 1 1 (b) It follows from the Corollary 1 that w (A) = 2 A if and only if Re e A = 2 A for all θ ∈ R. In [9], it has been shown using a different technique that w (A) = 12 A if and only if Re eiθ A + Im eiθ A = A for all θ ∈ R. Theorem 4. The following properties hold: (a) The norm wN (·) is self-adjoint. (b) If the norm N (·) is weakly unitarily invariant, then so is wN (·).
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Proof. (a) We have wN (A∗ ) = sup N Re eiθ A∗ θ∈R
= sup N Re e−iθ A θ∈R
= wN (A) . (b) Assume that N (·) is weakly unitarily invariant and let U ∈ B(H) be unitary. Then wN (U ∗ AU ) = sup N Re eiθ U ∗ AU θ∈R
= sup N U ∗ Re eiθ A U . θ∈R
By the assumption, N U ∗ Re eiθ A U = N Re eiθ A . Hence, wN (U ∗ AU ) = sup N Re eiθ A θ∈R
= wN (A) .
2
3. Inequalities involving wN (·) The idea of this section is to establish inequalities for the norm wN (·) using inequalities on the norm N (·). This idea enables us to derive new numerical radius inequalities using known operator norm inequalities. Theorem 5. Let A, B, X ∈ B (H). If N (·) is an algebra norm, then wN (AXB + B ∗ XA∗ ) ≤ (N (A) N (B) + N (A∗ ) N (B ∗ )) wN (X) .
(3.1)
In particular, wN (AXA∗ ) ≤ N (A) N (A∗ ) wN (X) .
(3.2)
Proof. First, notice that N Re eiθ (AXB + B ∗ XA∗ ) = N A Re eiθ X B + B ∗ Re eiθ X A∗ . Now, since N (·) is an algebra norm, we have N Re eiθ (AXB + B ∗ XA∗ ) ≤ (N (A) N (B) + N (A∗ ) N (B ∗ )) N Re eiθ X . Hence,
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wN (AXB + B ∗ XA∗ ) = sup N Re eiθ (AXB + B ∗ XA∗ ) θ∈R
≤ sup (N (A) N (B) + N (A∗ ) N (B ∗ )) N Re eiθ X θ∈R
= (N (A) N (B) + N (A∗ ) N (B ∗ )) wN (X) . This proves the inequality (3.1). The inequality (3.2) follows from the inequality (3.1) by letting B = A∗ . 2 The following lemma contains the classical norm inequalities of Heinz and Kato. For generalizations of these inequalities to the wider class of unitarily invariant norms, we refer the reader to [2] and [7]. Lemma 1. Let A, B, X ∈ B (H) such that A and B are positive and let 0 < ν < 1. The following inequalities hold: ν
1−ν
Aν XB ν ≤ AXB X , 1/2 2 A XB 1/2 ≤ Aν XB 1−ν + A1−ν XB ν ≤ AX + XB ,
(3.4)
ν A XB 1−ν − A1−ν XB ν ≤ |2ν − 1| AX − XB .
(3.5)
(3.3)
and
Theorem 6. Let A, X ∈ B (H) such that A is positive and let 0 < ν < 1. The following inequalities hold: w (Aν XAν ) ≤ wν (AXA)w1−ν (X) , 2w A1/2 XA1/2 ≤ w Aν XA1−ν + A1−ν XAν ≤ w (AX + XA) ,
(3.6) (3.7)
and w Aν XA1−ν − A1−ν XAν ≤ |2ν − 1| w (AX − XA) .
(3.8)
Proof. Let θ ∈ R be arbitrary. Since Re eiθ Aν XAν = Aν Re eiθ X Aν , we have by the inequality (3.3) iθ ν Re e A XAν ≤ A Re eiθ X Aν Re eiθ X 1−ν . Hence, w (Aν XAν ) = sup Re eiθ Aν XAν θ∈R
ν 1−ν ≤ sup A Re eiθ X A Re eiθ X θ∈R
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ν 1−ν = sup Re eiθ AXA Re eiθ X θ∈R
ν 1−ν ≤ sup Re eiθ AXA sup Re eiθ X θ∈R
θ∈R
ν
= w (AXA)w
1−ν
(X) .
This proves the inequality (3.6). To prove the inequalities (3.7), recall that Re eiθ A1/2 XA1/2 = A1/2 Re eiθ X A1/2 . Hence, by the inequalities (3.4), 2 Re eiθ A1/2 XA1/2 ≤ Aν Re eiθ X A1−ν + A1−ν Re eiθ X Aν ≤ A Re eiθ X + Re eiθ X A . Equivalently, 2 Re eiθ A1/2 XA1/2 ≤ Re eiθ Aν XA1−ν + A1−ν XAν ≤ Re eiθ (AX + XA) . (Notice that Aν Re(eiθ X)A1−ν + A1−ν Re(eiθ X)Aν = Re(eiθ (Aν XA1−ν + iθ iθ iθ 1−ν ν XA )) and A Re e X + Re e X A = Re e (AX + XA) ). The inequalities A (3.7) now follow from the previous inequalities by taking the supremum over all θ ∈ R. The inequality (3.8) can be similarly proved. 2 In [1], several numerical radius inequalities for matrices have been obtained based on Schur’s multipliers technique. We remark that these inequalities can be derived in the general setting of Hilbert space operators in a simple way using our analysis presented in this section. 4. A concrete example In this section, we study the norm wN (·) when N (·) is the Hilbert–Schmidt norm. Let H be a separable complex Hilbert space. Recall that an operator A ∈ B (H) is ∞ said to belong to the trace class C1 if |A| ei , ei is finite and to the Hilbert–Schmidt class C2 if
∞
2
|Aei , ej | =
i,j=1
∞
i=1 2
Aei is finite for some (hence, for any) orthonormal
i=1
∞ ∞ basis {ei }i=1 for H. For A ∈ C1 , let trA = Aei , ei be the trace of A. Also, for i=1 1/2
∞ 2 A ∈ C2 , let A2 = Aei be the Hilbert–Schmidt norm of A. Note that for i=1
A ∈ C2 , A2 = trA∗ A. The classes C1 and C2 are examples of the Schatten p-classes Cp (1 ≤ p ≤ ∞). For more details about these classes, we refer the reader to [3] and [8]. 2
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When N (·) is the Hilbert–Schmidt norm ·2 , the norm wN (·) is denoted by w2 (·). iθ That is, w2 (A) = sup Re e A 2 . θ∈R
In the following theorem, we derive a formula for w2 (A) in terms of A2 and trA2 . Theorem 7. Let A ∈ C2 . Then w2 (A) =
1 1 2 A2 + |trA2 |. 2 2
Proof. Since iθ 2 Re e A = tr Re eiθ A 2 2 1 = tr A∗ A + AA∗ + 2 Re e2iθ A2 4 1 2 = 2 A2 + 2tr Re e2iθ A2 4 1 2 = 2 A2 + 2 Re e2iθ trA2 , 4 we have w2 (A) = sup Re eiθ A 2 θ∈R
= sup θ∈R
=
1 1 2 A2 + Re (e2iθ trA2 ) 2 2
1 1 2 A2 + |trA2 |, 2 2
as required. 2 In the following theorem, we prove that the norms w2 (·) and ·2 are equivalent. Theorem 8. Let A ∈ C2 . Then 1 √ A2 ≤ w2 (A) ≤ A2 . 2
(4.1)
Proof. The first inequality is obvious. The second inequality follows from Theorem 2 The second inequality also follows from and recalling that the norm · 2 is2 self-adjoint. 2 Theorem 7 and the fact that trA ≤ A2 . 2 In the rest of the paper, the following two facts are needed. Let A ∈ C2 and let λ1 , λ2 , . . . be the eigenvalues of A, repeated according to their multiplicities. The well∞ 2 2 known Schur’s inequality (see, e.g., [3, p. 107]) states that A2 ≥ |λi | , with the i=1
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equality occurs if and only if A is normal. The other fact is Lidiskii’s theorem (see, e.g., ∞ [3, p. 101]), which states that trA = λi . i=1
The inequalities (4.1) are sharp. In the following corollary, we state necessary and sufficient conditions for the equality cases in the inequalities (4.1). Corollary 2. Let A ∈ C2 be nonzero. Then (a) w2 (A) = √12 A2 if and only if trA2 = 0. (b) w2 (A) = A2 if and only if A is normal and the squares of its nonzero eigenvalues have the same argument. Proof. (a) This is a direct consequence of Theorem 7. 2 2 2 (b) Assume that w2 (A) = A2 . By Theorem 7, we have A2 = trA . Hence, A2 = ∞ ∞ trA2 = λ2 ≤ |λi |2 ≤ A2 . This implies that i 2 i=1
i=1
∞ i=1
2
2
|λi | = A2
(4.2)
and ∞ ∞ λ2i = λ2i . i=1
(4.3)
i=1
The normality of A follows from the equality (4.2). The equality (4.3) implies that all the squares of the nonzero eigenvalues of A have the same argument. Conversely, assume that A is normal and the squares of its nonzero eigenvalues 2 have the same argument. Let θ = arg λ2i for i = 1, 2, . . .. Then λ2i = |λi | eiθ for i = 1, 2, . . ., and hence ∞ trA2 = λ2i i=1
∞
=
i=1
2
2
|λi | = A2 (by the normality of A).
Thus, by Theorem 7, we have w2 (A) = A2 .
2
Theorem 9. Let A ∈ C2 . Then
k k 1 1 2 |λi | + λ2i ≤ w2 (A) 2 i=1 2 i=1
for k = 1, 2, . . .. In particular,
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r (A) ≤ w2 (A) . Proof. By Lidiskii’s theorem and Schur’s inequality, we have 2 A2
∞ ∞ 2 2 2 + trA ≥ |λi | + λi i=1
=
k
i=1
2
|λi | +
i=1
∞ i=k+1
∞ 2 |λi | + λ2i . i=1
Hence, by the triangle inequality, k ∞ ∞ 2 2 2 2 2 A2 + trA ≥ |λi | + λi + λi i=k+1 i=1 i=1 k ∞ ∞ 2 2 2 ≥ |λi | + λi − λi i=1 i=1 i=k+1 k k 2 = |λi | + λ2i . i=1
i=1
Thus, w2 (A) ≥
k k 1 1 2 |λi | + λ2i . 2 2 i=1 2 i=1
Remark 4. For every A ∈ C2 and every θ ∈ R, we have Re eiθ A ≤ Re eiθ A 2 . Hence, w (A) ≤ w2 (A), which is better than the inequality r (A) ≤ w2 (A). However, 0 1 the example A = shows that the inequality A ≤ w2 (A) need not be true in 0 0 general. Finally, it is tempting to study the norm wN (·) in the case of other unitarily invariant norms N (·), including the Schatten p-norms (p = 2). References [1] G. Aghamollaei, A. Sheikh Hosseini, Some numerical radius inequalities with positive definite functions, Bull. Iranian Math. Soc. 41 (2015) 889–900. [2] R. Bhatia, C. Davis, A Cauchy–Schwarz inequality for operators with applications, Linear Algebra Appl. 223/224 (1995) 119–129. [3] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Nonselfadjoint Linear Operators, Transl. Math. Monogr., vol. 18, Amer Math. Soc., Providence, RI, 1969. [4] K.E. Gustafson, D.K.M. Rao, Numerical Range, Springer, New York, 1997. [5] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982. [6] O. Hirzallah, F. Kittaneh, K. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices, Integral Equations Operator Theory 71 (2011) 129–147.
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[7] F. Kittaneh, Norm inequalities for fractional powers of positive operators, Lett. Math. Phys. 27 (1993) 279–285. [8] B. Simon, Trace Ideals and Their Applications, 2nd ed., Amer. Math. Soc., Providence, RI, 2005. [9] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178 (2007) 83–89.