Furtherance of Numerical radius inequalities of Hilbert space operators
aa r X i v : . [ m a t h . F A ] F e b FURTHERANCE OF NUMERICAL RADIUS INEQUALITIES OFHILBERT SPACE OPERATORS
PINTU BHUNIA AND KALLOL PAUL
Abstract. If A, B are bounded linear operators on a complex Hilbert space,then w ( A ) ≤ (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) ,w ( AB ± BA ) ≤ √ k B k r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 , where w ( . ) , k . k , c ( . ) and r ( . ) are the numerical radius, the operator norm, theCrawford number and the spectral radius respectively, and ℜ ( A ), ℑ ( A ) are thereal part, the imaginary part of A respectively. The inequalities obtained heregeneralize and improve on the existing well known inequalities. Introduction
Let H be a complex Hilbert space with inner product h ., . i and let B ( H ) be thecollection of all bounded linear operators on H . As usual the norm induced by theinner product h ., . i is denoted by k . k . For A ∈ B ( H ), let k A k be the operator normof A, i.e., k A k = sup k x k =1 k Ax k . For A ∈ B ( H ), A ∗ denotes the adjoint of A and | A | , | A ∗ | respectively denote the positive part of A, A ∗ , i.e., | A | = ( A ∗ A ) , | A ∗ | =( AA ∗ ) . Let S H denote the unit sphere of the Hilbert space H . The numericalrange of A , denoted by W ( A ) , is defined as W ( A ) := (cid:8) h Ax, x i : x ∈ S H (cid:9) . Considering the continuous mapping x Ax, x i from S H to the scalar field C , it is easy to see that W ( A ) is a compact subset of C if H is finite dimensional. Thefamous Toeplitz-Hausdorff theorem states that the numerical range is a convexset. The numerical radius and the Crawford number of A , denoted as w ( A ) and c ( A ), respectively, are defined as w ( A ) := sup x ∈ S H |h Ax, x i| and c ( A ) := inf x ∈ S H |h Ax, x i| . Mathematics Subject Classification.
Key words and phrases.
Numerical radius, Spectral radius, Operator norm, Bounded linearoperator, Inequality.First author would like to thank UGC, Govt. of India for the financial support in the formof SRF.
The numerical radius is a norm on B ( H ) satisfying the following inequality12 k A k ≤ w ( A ) ≤ k A k . (1.1)Clearly, (1.1) implies that the numerical radius norm is equivalent to the operatornorm. The inequality (1.1) is sharp, w ( A ) = k A k if AA ∗ = A ∗ A and w ( A ) = k A k if A = 0 . For further readings on the numerical range and the numerical radiusof bounded linear operators, we refer to the book [12]. The spectral radius of A ,denoted as r ( A ) , is defined as r ( A ) := sup λ ∈ σ ( A ) | λ | , where σ ( A ) is the spectrum of A . Since σ ( A ) ⊆ W ( A ), r ( A ) ≤ w ( A ). Also, r ( A ) = w ( A ) if A ∗ A = AA ∗ . Kittaneh [16, Th. 1] and [17, Th. 1] improved onthe inequality (1.1), to prove that14 k A ∗ A + AA ∗ k ≤ w ( A ) ≤ k A ∗ A + AA ∗ k (1.2)and w ( A ) ≤ (cid:16) k A k + p k A k (cid:17) , (1.3)respectively. Bhunia and Paul [10, Cor. 2.5] improved on the right hand inequal-ities of both (1.1) and (1.2) to prove that w ( A ) ≤ min ≤ α ≤ (cid:13)(cid:13) α | A | + (1 − α ) | A ∗ | (cid:13)(cid:13) . (1.4)In [9, Th. 2.1], Bhunia and Paul also improved on the left hand inequalities ofboth (1.1) and (1.2) to prove that14 k A ∗ A + AA ∗ k ≤ (cid:0) k A + A ∗ k + k A − A ∗ k (cid:1) ≤ (cid:0) k A + A ∗ k + k A − A ∗ k (cid:1) + 18 c (cid:0) A + A ∗ (cid:1) + 18 c (cid:0) A − A ∗ (cid:1) ≤ w ( A ) . Fong and Holbrook [11] obtained the remarkable numerical radius inequality that w ( AB + BA ) ≤ √ k B k w ( A ) . (1.5)Hirzallah and Kittaneh [14] improved on the inequality (1.5) in the followingform: w ( AB ± BA ) ≤ √ k B k r w ( A ) − | kℜ ( A ) k − kℑ ( A ) k | . (1.6)Over the years many mathematicians have developed various inequalities improv-ing (1.1), we refer to [1, 3, 4, 5, 6, 7, 8] and references therein.In this paper, we obtain an improvement and generalization of the inequality(1.3). Some inequalities for the numerical radius of the commutators of boundedlinear operators are also obtained, which improve on (1.5). UMERICAL RADIUS INEQUALITIES OF HILBERT SPACE OPERATORS 3 Improvement of inequality (1.3)
Our improvement of the inequality (1.3), is stated as the following theorem:
Theorem 2.1.
Let A ∈ B ( H ) . Then, w ( A ) ≤ (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) . Remark 2.2. If A ∈ B ( H ), then r ( | A || A ∗ | ) ≤ w ( | A || A ∗ | ) ≤ k ( | A || A ∗ | ) k = k A k . Hence, Theorem 2.1 improves (1.3). To show proper improvement weconsider A = (cid:18) (cid:19) . Then | A | = (cid:18) (cid:19) and | A ∗ | = (cid:18) (cid:19) . It is easyto see that r ( | A || A ∗ | ) = 9 < k ( | A || A ∗ | ) k = k A k = p
59 + 10 √ ≈ . . In order to prove Theorem 2.1 we need the following sequence of lemmas. Firstlemma can be found in [18].
Lemma 2.3. ([18, Cor. 2])
Let
A, B ∈ B ( H ) be positive operators. Then k A + B k ≤ max {k A k , k B k} + (cid:13)(cid:13) A / B / (cid:13)(cid:13) . The second lemma which contains a mixed schwarz inequality, can be found in[13, pp. 75-76].
Lemma 2.4. ([13, pp. 75-76])
Let A ∈ B ( H ) . Then |h Ax, x i| ≤ h| A | x, x i / h| A ∗ | x, x i / , ∀ x ∈ H . The third lemma is as follows.
Lemma 2.5.
Let
A, B ∈ B ( H ) be positive operators. Then r ( AB ) = (cid:13)(cid:13) A / B / (cid:13)(cid:13) . Proof.
By commutativity property of the spectral radius we have that r ( AB ) = r (cid:0) A / A / B / B / (cid:1) = r (cid:0) A / B / B / A / (cid:1) = r (cid:16) A / B / (cid:0) A / B / (cid:1) ∗ (cid:17) = (cid:13)(cid:13)(cid:13) A / B / (cid:0) A / B / (cid:1) ∗ (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) A / B / (cid:13)(cid:13) . (cid:3) Now we prove Theorem 2.1.
Proof of Theorem 2.1.
Let x ∈ S H . Then by Lemma 2.4 we get, |h Ax, x i| ≤ h| A | x, x i / h| A ∗ | x, x i / ≤
12 ( h| A | x, x i + h| A ∗ | x, x i ) ≤ k | A | + | A ∗ | k≤ (cid:0) k A k + (cid:13)(cid:13) | A | / | A ∗ | / (cid:13)(cid:13)(cid:1) , by Lemma 2.3 = 12 (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) , by Lemma 2.5 . P. BHUNIA AND K. PAUL
Hence, by taking supremum over x ∈ S H we get, w ( A ) ≤ (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) , This completes the proof.As an application of Theorem 2.1, we prove the following corollary.
Corollary 2.6.
Let A ∈ B ( H ) . If r ( | A || A ∗ | ) = 0 , then w ( A ) = k A k . Proof.
It follows from (1.1) and Theorem 2.1 that k A k ≤ w ( A ) ≤ (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) . This implies that if r ( | A || A ∗ | ) = 0, then w ( A ) = k A k . (cid:3) Remark 2.7.
It should be mentioned here that the converse of Corollary 2.6does not hold if dim( H ) ≥ . As for example, we consider A = .Then we see that w ( A ) = = k A k , but r ( | A || A ∗ | ) = 0 . The following corollary is an immediate consequnece of Theorem 2.1.
Corollary 2.8.
Let A ∈ B ( H ) . If w ( A ) = (cid:16) k A k + p k A k (cid:17) , then r ( | A || A ∗ | ) = k A k . Proof.
Using Remark 2.2, it follows from Theorem 2.1 that w ( A ) ≤ (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) ≤ (cid:16) k A k + p k A k (cid:17) . This implies that if w ( A ) = (cid:16) k A k + p k A k (cid:17) , then r ( | A || A ∗ | ) = k A k . (cid:3) Remark 2.9.
It should be mentioned that the converse of Corollary 2.8 is nottrue. Considering the same example as in Remark 2.7, i.e., A = ,we see that r ( | A || A ∗ | ) = k A k = 1 , but w ( A ) = < (cid:16) k A k + p k A k (cid:17) .We give a sufficient condition for w ( A ) = (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) , when A isa complex n × n matrix. Proposition 2.10.
Let A be a complex n × n matrix. Suppose A satisfies eitherone of the following conditions. ( i ) A is unitarily similar to [ α ] ⊕ B , where B is an ( n − × ( n − matrix with k B k ≤ | α | . ( ii ) r ( | A || A ∗ | ) = 0 . Then, w ( A ) = (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) . UMERICAL RADIUS INEQUALITIES OF HILBERT SPACE OPERATORS 5
Proof.
Let ( i ) holds. Then w ( A ) = | α | and k A k = | α | . Also it is not difficult toverify that r ( | A || A ∗ | ) = | α | . Hence, (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) = | α | . Now let ( ii )holds. Then from Corollary 2.6 we get, w ( A ) = (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) = k A k .Thus, we complete the proof. (cid:3) Next we give a generalized result of Theorem 2.1. For this purpose we needthe following lemma, which is the generalization of Lemma 2.4.
Lemma 2.11. ([19, Th. 5]) . Let
A, B ∈ B ( H ) be such that | A | B = B ∗ | A | and let f, g be non-negative continuous functions on [0 , ∞ ] satisfy f ( t ) g ( t ) = t , ∀ t ≥ . Then, |h ABx, y i| ≤ r ( B ) k f ( | A | ) x kk g ( | A ∗ | ) y k , ∀ x, y ∈ H . Using Lemma 2.11 and proceeding similarly as in Theorem 2.1, we can provethe following theorem.
Theorem 2.12.
Let
A, B ∈ B ( H ) be such that | A | B = B ∗ | A | and let f , g be asin Lemma 2.11. Then w ( AB ) ≤ r ( B )2 (cid:16) max (cid:8) k f ( | A | ) k , k g ( | A ∗ | ) k (cid:9) + k | f ( | A | ) | | g ( | A ∗ | ) | k (cid:17) . Considering f ( t ) = g ( t ) = √ t in Theorem 2.12 we get the following corollary. Corollary 2.13.
Let
A, B ∈ B ( H ) be such that | A | B = B ∗ | A | . Then w ( AB ) ≤ r ( B )2 (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) ≤ (cid:16) k B k + p r ( | B || B ∗ | ) (cid:17) (cid:16) k A k + p r ( | A || A ∗ | ) (cid:17) . Remark 2.14. If A, B ∈ B ( H ) be such that | A | B = B ∗ | A | , then Alomari [2,Cor. 3.2] proved that w ( AB ) ≤ (cid:16) k B k + p k B k (cid:17) (cid:16) k A k + p k A k (cid:17) . (2.1)Clearly our inequalities in Corollary 2.13 improve on the inequality (2.1).3. Improvement of inequality (1.5)
In order to obtain an improvement of the inequality (1.5) we need the followinglemma [9] . First, we note the Cartesian decomposition of A ∈ B ( H ), i.e., A = ℜ ( A ) + i ℑ ( A ), where ℜ ( A ) = A + A ∗ and ℑ ( A ) = A − A ∗ . Lemma 3.1. ([9, Cor. 2.3])
Let A ∈ B ( H ) . Then k AA ∗ + A ∗ A k ≤ (cid:20) w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 (cid:21) . Now we prove the desired result.
Theorem 3.2.
Let
A, B, X, Y ∈ B ( H ) . Then w ( AXB ± BY A ) ≤ √ k B k max {k X k , k Y k} r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 . P. BHUNIA AND K. PAUL
Proof.
First we assume that k X k ≤ k Y k ≤
1. Let x ∈ S H . Then we have |h ( AX ± Y A ) x, x i| ≤ |h AXx, x i| + |h Y Ax, x i| = |h Xx, A ∗ x i| + |h Ax, Y ∗ x i|≤ k A ∗ x k + k Ax k , by Cauchy Schwarz inequality ≤ p k A ∗ x k + k Ax k ) , by convexity of f ( x ) = x ≤ p k AA ∗ + A ∗ A k≤ √ r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 , by Lemma 3.1 . Hence, by taking supremum over k x k = 1 we get, w ( AX ± Y A ) ≤ √ r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 . (3.1)Now we consider the general case, i.e., X, Y ∈ B ( H ) be arbitrary operators. If X = Y = 0 then Theorem 3.2 holds trivially. Let max {k X k , k Y k} 6 = 0 . Thenclearly (cid:13)(cid:13)(cid:13) X max {k X k , k Y k} (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) Y max {k X k , k Y k} (cid:13)(cid:13)(cid:13) ≤
1. So, replacing X and Y by X max {k X k , k Y k} and Y max {k X k , k Y k} , respectively, in (3.1) we get, w ( AX ± Y A ) ≤ √ {k X k , k Y k} r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 . (3.2)Now replacing X by XB and Y by BY in (3.2) we get, w ( AXB ± BY A ) ≤ √ {k XB k , k BY k} r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 , which implies that w ( AXB ± BY A ) ≤ √ k B k max {k X k , k Y k} r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 . (cid:3) On the basis of Theorem 3.2 we prove the following corollary.
Corollary 3.3.
Let
A, B ∈ B ( H ) . Then w ( AB ± BA ) ≤ √ k B k r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 . (3.3) and w ( AB ± BA ) ≤ √ k A k r w ( B ) − c ( ℜ ( B )) + c ( ℑ ( B ))2 . (3.4) Proof.
By considering X = Y = I in Theorem 3.2 we get, (3.3). Interchanging A and B in (3.3) we get, (3.4). (cid:3) Remark 3.4.
Clearly, the inequality (3.3) is stronger than the inequality (1.5).As an application of the inequality (3.3) we prove the following result.
UMERICAL RADIUS INEQUALITIES OF HILBERT SPACE OPERATORS 7
Corollary 3.5.
Let
A, B ∈ B ( H ) and let B = 0 . If w ( AB ± BA ) = 2 √ k B k w ( A ) ,then ∈ W ( ℜ ( A )) ∩ W ( ℑ ( A )) .Proof. Let w ( AB ± BA ) = 2 √ k B k w ( A ). Then it follows from (3.3) that w ( A ) = r w ( A ) − c ( ℜ ( A )) + c ( ℑ ( A ))2 . Hence, c ( ℜ ( A )) + c ( ℑ ( A )) = 0 , i.e., c ( ℜ ( A )) = c ( ℑ ( A )) = 0. Therefore, thereexist norm one sequences { x n } and { y n } in H such that |hℜ ( A ) x n , x n i| → |hℑ ( A ) y n , y n i| → n → ∞ . So, 0 ∈ W ( ℜ ( A )) ∩ W ( ℑ ( A )). (cid:3) For our next result we need the following three lemmas, the first two of whichcan be found in [1] and [15], respectively.
Lemma 3.6. ([1, Remark 2.2])
Let
A, B, X, Y ∈ B ( H ) . Then w ( AX ± BY ) ≤ k AA ∗ + Y ∗ Y k k X ∗ X + BB ∗ k . Lemma 3.7. ([15, Th. 1.1])
Let
A, B, X, Y ∈ B ( H ) . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)
A XY B (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k A k k X kk Y k k B k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . The next lemma is as follows.
Lemma 3.8.
Let
A, B ∈ B ( H ) . Then k AA ∗ + B ∗ B k ≤ µ ( A, B ) , where µ ( A, B ) = 12 (cid:20) k A k + k B k + q ( k A k − k B k ) + 4 k BA k (cid:21) . Proof. AA ∗ + B ∗ B being a self-adjoint operator, we have k AA ∗ + B ∗ B k = r ( AA ∗ + B ∗ B )= r (cid:18) AA ∗ + B ∗ B
00 0 (cid:19) = r (cid:18)(cid:18) | A ∗ | | B | (cid:19) (cid:18) | A ∗ | | B | (cid:19)(cid:19) = r (cid:18)(cid:18) | A ∗ | | B | (cid:19) (cid:18) | A ∗ | | B | (cid:19)(cid:19) , r ( XY ) = r ( Y X )= r (cid:18) | A ∗ | | A ∗ || B || B || A ∗ | | B | (cid:19) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A ∗ | | A ∗ || B || B || A ∗ | | B | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k A k k| A ∗ || B |kk| B || A ∗ |k k B k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , by Lemma 3.7 = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k A k k BA kk BA k k B k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = 12 (cid:20) k A k + k B k + q ( k A k − k B k ) + 4 k BA k (cid:21) . P. BHUNIA AND K. PAUL
Hence, k AA ∗ + B ∗ B k ≤ µ ( A, B ) . (cid:3) Remark 3.9.
Notice that µ ( A, B ) ≤ max {k A k , k B k } + k BA k . In particular, if A = B then µ ( A, A ) = k A k + k A k . Hence, we have k AA ∗ + A ∗ A k ≤ k A k + k A k .Now we are in a position to prove the following result. Theorem 3.10.
Let
A, B, X, Y ∈ B ( H ) . Then w ( AX ± BY ) ≤ p µ ( A, Y ) µ ( B, X ) . Proof.
The proof follows from Lemma 3.6 and Lemma 3.8. (cid:3)
An application of Theorem 3.10 we get the following corollary.
Corollary 3.11.
Let
A, B ∈ B ( H ) . Then w ( AB ± BA ) ≤ p ( k A k + k A k ) ( k B k + k B k ) . Remark 3.12.
Let
A, B ∈ B ( H ) with A = B = 0 . Then it follows fromCorollary 3.11 that w ( AB ± BA ) ≤ k A kk B k < √ k B k w ( A ) = √ k A kk B k . References
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UMERICAL RADIUS INEQUALITIES OF HILBERT SPACE OPERATORS 9
14. O. Hirzallah and F. Kittaneh, Numerical radius inequalities for several operators, Math.Scand. 114(1) (2014) 110-119.15. J.-C. Hou and H.-K. Du, Norm inequalities of positive operator matrices. Integr Equ OperTheory 22(1995) 281-294.16. F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168(1)(2005), 73-80.17. F. Kittaneh, Numerical radius inequality and an estimate for the numerical radius of theFrobenius companion matrix, Studia Math. 158(1) (2003), 11-17.18. F. Kittaneh, Norm inequalities for certain operator sums, J. Funct. Anal. 143 (1997), 337-348.19. F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. RIMS KyotoUniv. 24 (1988) 283-293. (Bhunia) Department of Mathematics, Jadavpur University, Kolkata 700032,West Bengal, India
Email address : [email protected] (Paul) Department of Mathematics, Jadavpur University, Kolkata 700032,West Bengal, India Email address ::