Let B ( H ) denote the C ∗ -algebra of all bounded linear operators on a Hilbert space H. In the case when dim H = n , we identify B ( H ) with the matrix algebra M n of all n × n matrices with entries in the complex field. The numerical radius of T ∈ B ( H ) is defined by

w ( T ) = sup { | 〈 T x , x 〉 | : x ∈ H , ‖ x ‖ = 1 } . (1)

It is well-known that w ( . ) defines a norm on B ( H ) , which is equivalent to the usual operator norm. Namely, for T ∈ B ( H ) , we have

‖ T ‖ 2 ≤ w ( T ) ≤ ‖ T ‖ (2)

These inequalities are sharp. The first inequality becomes an equality if T 2 = 0 , and the second inequality becomes an equality if T is normal (see [1] ).

An important inequality for w ( T ) is the power inequality stating that w ( X n ) ≤ ( w ( X ) ) n for n = 1 , 2 , ⋯ see ( [2] : p. 118).

An important property of the numerical radius norm is its weak unitary invariance, that is, for X ∈ B ( H ) ,

w ( U ∗ X U ) = w ( X ) (3)

for every unitary U ∈ B ( H ) . For further information about the properties of numerical radius inequalities we refer the reader to [2] - [7] and references therein.

Let H 1 , H 2 be Hilbert spaces, and consider the direct sum H = H 1 ⊕ H 2 . By considering this decomposition, every operator T ∈ B ( H ) has a 2 × 2 operator matrix representation T = [ T i j ] with entries T i j ∈ B ( H 1 ⊕ H 2 ) .

Hirzallah, Kittaneh and Shebrawi have proved in [8] that:

If X ∈ B ( H ) , then:

‖ X ‖ 2 + | ‖ Re X ‖ − ‖ Im X ‖ | 2 ≤ w ( X ) (4)

also, they proved that:

If X ∈ B ( H ) , then:

‖ X ‖ 2 + | ‖ Re X ‖ − ‖ X ‖ 2 | 4 + | ‖ Im X ‖ − ‖ X ‖ 2 | 4 ≤ w ( X ) (5)

Moreover, they showed that:

if X , Y ∈ B ( H ) , then:

w ( [ 0 X Y 0 ] ) ≤ w ( X + Y ) + w ( X − Y ) 2 (6)

Shebrawi and Albadawi have proved in [9] that:

If A i , B i , X i ∈ B ( H ) , ( i = 1 , 2 , ⋯ , n ) and f , g be nonnegative continuous functions on [ 0, ∞ ) satisfying the relation f ( t ) g ( t ) = t ( t ∈ [ 0, ∞ ) ) , then:

w r ( ∑ i = 1 n   A i ∗ X i B i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( [ A i ∗ g 2 ( | X i ∗ | ) A i ] r + [ B i ∗ f 2 ( | X i | ) B i ] r ) ‖ (7)

for all r ≥ 1 .

In the special case, where f ( t ) = t k and g ( t ) = t 1 − k , α ∈ ( 0,1 ) , they proved that:

w r ( ∑ i = 1 n   A i ∗ X i B i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( [ A i ∗ | X i ∗ | 2 ( 1 − k ) A i ] r + [ B i ∗ | X i | 2 k B i ] r ) ‖ (8)

In particular, they proved the following inequalities:

1)

w ( ∑ i = 1 n   A i ∗ X i B i ) ≤ 1 2 ‖ ∑ i = 1 n ( A i ∗ | X i ∗ | A i + B i ∗ | X i | B i ) ‖ (9)

2)

w r ( ∑ i = 1 n   X i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( | X i ∗ | 2 r ( 1 − κ ) + | X i | 2 r k ) ‖ (10)

3)

w r ( ∑ i = 1 n   X i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( | X i ∗ | r + | X i | r ) ‖ (11)

4)

w r ( X ) ≤ 1 2 ‖ ( | X | r + | X ∗ | r ) ‖ (12)

The main purpose of this paper is to give considerable improvements of the inequalities (7), (8), (9), (10), (11), and (12). In order to achieve our goal, we need the following three lemmas which are essential in our analysis.

The first lemma was proved in [10] .

Lemma 1 If a , b ≥ 0 and 0 ≤ v ≤ 1 , then:

a v b ( 1 − v ) + k ( a − b ) 2 ≤ v a + ( 1 − v ) b (13)

where k = min { v , 1 − v } .

If v = 1 2 , the inequality (13) becomes an equality where

a b = a + b 2 − ( a − b ) 2 2 (14)

The second lemma follows from the spectral theorem for positive operators and Jensen’s inequality (see [11] ).

Lemma 2 Let T ∈ B ( H ) , T ≥ 0 and x ∈ H such that ‖ x ‖ ≤ 1 . Then:

1) 〈 T x , x 〉 r ≤ 〈 T r x , x 〉 for r ≥ 1 .

2) 〈 T r x , x 〉 ≤ 〈 T x , x 〉 r for 0 < r ≤ 1 .

The third lemma was proved in [11]

Lemma 3 Let T ∈ B ( H ) and x , y ∈ H be any vectors. If f , g are nonnegative continuous functions on [ 0, ∞ ) which are satisfying the relation f ( t ) g ( t ) = t ( t ∈ [ 0, ∞ ) ) , then:

| 〈 T x , y 〉 | 2 ≤ 〈 | T | x , x 〉 〈 | T ∗ | y , y 〉 (15)

and more general,

| 〈 T x , y 〉 | 2 ≤ 〈 f 2 ( | T | ) x , x 〉 〈 g 2 ( | T ∗ | ) y , y 〉 (16)

The first result in this paper is numerical radius inequality which is sharper than the inequality (7).

Theorem 3.1 Let A i , B i , X i ∈ B ( H ) , i = 1 , 2 , ⋯ , n , and f , g be nonnegative continuous functions on [ 0, ∞ ) satisfying the relation f ( t ) g ( t ) = t ( t ∈ [ 0, ∞ ) ) . Then:

w r ( ∑ i = 1 n   A i ∗ X i B i ) ≤ n r − 1 2 ‖ ∑ i = 1 n [ A i ∗ g 2 ( | X i ∗ | ) A i ] r + [ B i ∗ f 2 ( | X i | ) B i ] r ‖ − α (17)

where

α = sup ‖ x ‖ = 1 n r − 1 2 ∑ i = 1 n ( 〈 [ A i ∗ g 2 ( | X i ∗ | ) A i ] r x , x 〉 1 / 2 − 〈 [ B i ∗ f 2 ( | X i | ) B i ] r x , x 〉 1 / 2 ) 2 (18)

Proof.

| 〈 ∑ i = 1 n ( A i ∗ X i B i ) x , x 〉 | r = | ∑ i = 1 n 〈 ( A i ∗ X i B i ) x , x 〉 | r ≤ ( ∑ i = 1 n | 〈 ( A i ∗ X i B i ) x , x 〉 | ) r = ( ∑ i = 1 n | 〈 X i B i x , A i x 〉 | ) r ≤ ( ∑ i = 1 n 〈 f 2 ( | X i | ) B i x , B i x 〉 1 / 2 〈 g 2 ( | X i ∗ | ) A i x , A i x 〉 1 / 2 ) r

≤ n r − 1 ∑ i = 1 n 〈 B i ∗ f 2 ( | X i | ) B i x , x 〉 r / 2 〈 A i ∗ g 2 ( | X i ∗ | ) A i x , x 〉 r / 2 ≤ n r − 1 ∑ i = 1 n 〈 ( B i ∗ f 2 ( | X i | ) B i ) r x , x 〉 1 / 2 〈 ( A i ∗ g 2 ( | X i ∗ | ) A i ) r x , x 〉 1 / 2 = n r − 1 2 [ ∑ i = 1 n 〈 ( B i ∗ f 2 ( | X i | ) B i ) r x , x 〉 + 〈 ( A i ∗ g 2 ( | X i ∗ | ) A i ) r x , x 〉 ]       − n r − 1 2 [ ∑ i = 1 n ( 〈 ( B i ∗ f 2 ( | X i | ) B i ) r x , x 〉 1 / 2 − 〈 ( A i ∗ g 2 ( | X i ∗ | ) A i ) r x , x 〉 1 / 2 ) 2 ]

Taking the supremum over all unit vectors x ∈ H , we get

w r ( ∑ i = 1 n A i ∗ X i B i ) ≤ n r − 1 2 ‖ ∑ i = 1 n [ A i ∗ g 2 ( | X i ∗ | ) A i ] r + [ B i ∗ f 2 ( | X i | ) B i ] r ‖

    − sup ‖ x ‖ = 1 n r − 1 2 ∑ i = 1 n ( 〈 [ A i ∗ g 2 ( | X i ∗ | ) A i ] r x , x 〉 1 / 2 − 〈 [ B i ∗ f 2 ( | X i | ) B i ] r x , x 〉 1 / 2 ) 2 = n r − 1 2 ‖ ∑ i = 1 n [ A i ∗ g 2 ( | X i ∗ | ) A i ] r + [ B i ∗ f 2 ( | X i | ) B i ] r ‖ − α .

Remark 1 In view of the inequalities (7) and (17), it clears that the inequality (17) is sharper than the inequality (7).

As special case of the inequality (17), let f ( t ) = t κ and g ( t ) = t 1 − κ , κ ∈ ( 0,1 ) , we will get the following inequality which is sharper than the inequality (8).

Corollary 4 Let A i , B i , X i ∈ B ( H ) , i = 1 , 2 , ⋯ , n , r ≥ 1 , and 0 < κ < 1 . Then:

w r ( ∑ i = 1 n   A i ∗ X i B i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( [ A i ∗ | X i ∗ | 2 ( 1 − κ ) A i ] r + [ B i ∗ | X i | 2 κ B i ] r ) ‖ − β (19)

where

β = sup ‖ x ‖ = 1 n r − 1 2 ∑ i = 1 n ( 〈 [ A i ∗ | X i ∗ | 2 ( 1 − κ ) A i ] r x , x 〉 1 / 2 − 〈 [ B i ∗ | X i | 2 κ B i ] r x , x 〉 1 / 2 ) 2 (20)

In particular, if r = 1 , α = 1 2 we get the following inequality which is charper than the inequality (9),

w ( ∑ i = 1 n   A i ∗ X i B i ) ≤ 1 2 ‖ ∑ i = 1 n ( A i ∗ | X i ∗ | A i + B i ∗ | X i | B i ) ‖ − γ (21)

where

γ = sup ‖ x ‖ = 1 1 2 ∑ i = 1 n ( 〈 [ A i ∗ | X i ∗ | A i ] x , x 〉 1 / 2 − 〈 [ B i ∗ | X i | B i ] x , x 〉 1 / 2 ) 2 (22)

By letting A i = B i = 0 in the inequality (19), we obtain the following inequality which is sharper than the inequality (10).

Corollary 5 Let X i ∈ B ( H ) , i = 1 , 2 , ⋯ , n , r ≥ 1 , and 0 < κ < 1 . Then:

w r ( ∑ i = 1 n   X i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( | X i ∗ | 2 r ( 1 − κ ) + | X i | 2 r k ) ‖ − η (23)

where

η = sup ‖ x ‖ = 1 n r − 1 2 ∑ i = 1 n ( 〈 | X i ∗ | 2 r ( 1 − κ ) x , x 〉 1 / 2 − 〈 | X i | 2 r κ x , x 〉 1 / 2 ) 2

Letting κ = 1 2 in the inequality (23), we obtain the following inequality which is sharper than the inequality (11).

Corollary 6 Let X i ∈ B ( H ) , i = 1 , 2 , ⋯ , n , and r ≥ 1 . Then:

w r ( ∑ i = 1 n   X i ) ≤ n r − 1 2 ‖ ∑ i = 1 n ( | X i ∗ | r + | X i | r ) ‖ − ζ (24)

where

ζ = sup ‖ x ‖ = 1 n r − 1 2 ∑ i = 1 n ( 〈 | X i ∗ | r x , x 〉 1 / 2 − 〈 | X i | r x , x 〉 1 / 2 ) 2 (25)

In the inequality (24), replacing X 2 , X 3 , ⋯ , X n by 0, we have the following inequality which is sharper than the inequality (12).

Corollary 7 Let X ∈ B ( H ) ,   r ≥ 1 . Then:

w r ( X ) ≤ 1 2 ‖ ( | X | r + | X ∗ | r ) ‖ − ξ (26)

where

ξ = sup ‖ x ‖ = 1 1 2 ( 〈 | X ∗ | r x , x 〉 1 / 2 − 〈 | X | r x , x 〉 1 / 2 ) 2

Now, we will prove the following inequality which is another version of the inequality (6).

Theorem 3.2 Let A , B ∈ B ( H ) . Then:

w ( [ 0 A B 0 ] ) ≤ ‖ A ‖ + ‖ B ‖ 2 (27)

Proof. Let U = 1 2 [ I − I I I ] , then U is unitary, and

w ( [ 0 A B 0 ] ) = w ( U ∗ [ 0 A B 0 ] U ) ((by Equation (3))

= 1 2 w ( [ − B − A − B + A B − A B + A ] ) = 1 2 w ( [ − A A − A A ] + [ − B − B B B ] ) ≤ 1 2 [ w ( [ − A A − A A ] ) + w ( [ − B − B B B ] ) ] = ‖ A ‖ + ‖ B ‖ 2

since ( [ − A A − A A ] 2 = 0 , so ( [ − A A − A A ] ) = ‖ [ − A A − A A ] ‖ 2 = ‖ A ‖ ),

and ( [ − B − B B B ] 2 = 0 , so w ( [ − B − B B B ] ) = ‖ [ − B − B B B ] ‖ 2 = ‖ B ‖ ).

Chaining the inequality (27) with the inequality (4) yields the following inequality.

Corollary 8 Let A , B ∈ B ( H ) . Then:

w ( [ 0 A B 0 ] ) ≤ w ( A ) + w ( B ) − | ‖ Re A ‖ − ‖ Im A ‖ | 2 − | ‖ Re B ‖ − ‖ Im B ‖ | 2 (28)

Proof. In Theorem 3.2, apply the inequality (4) on the right side, we get the result.

Chaining the inequality (27) with the inequality (5) yields the following inequality.

Corollary 9 Let A , B ∈ B ( H ) . Then:

w ( [ 0 A B 0 ] ) ≤ w ( A ) + w ( B ) − | ‖ Re A ‖ − ‖ A 2 ‖ | 4 − | ‖ Im A ‖ − ‖ A 2 ‖ | 4     − | ‖ Re B ‖ − ‖ B ‖ 2 | 4 − | ‖ Im B ‖ − ‖ B ‖ 2 | 4 (29)

Proof. In Theorem 3.2, apply the inequality (5) on the right side, we get the result.

The authors declare no conflicts of interest regarding the publication of this paper.

Audeh, W. (2019) Numerical Radius Inequalities for Sums and Products of Operators. Advances in Linear Algebra & Matrix Theory, 9, 35-42. https://doi.org/10.4236/alamt.2019.93003