A convex treatment of numerical radius inequalities
aa r X i v : . [ m a t h . F A ] S e p A CONVEX TREATMENT OF NUMERICAL RADIUS INEQUALITIES
ZAHRA HEYDARBEYGI, MOHAMMAD SABABHEH AND HAMID REZA MORADI
Abstract.
In this article, we prove an inner product inequality for Hilbert space operators.This inequality, then, is utilized to present a general numerical radius inequality using convexfunctions. Applications of the new results include obtaining new forms that generalize andextend some well known results in the literature, with an application to the newly definedgeneralized numerical radius.We emphasize that the approach followed in this article is different from the approaches usedin the literature to obtain the refined versions. Introduction
Let B ( H ) denote the C ∗ -algebra of all bounded linear operators on a complex Hilbert space H with inner product h· , ·i . For T ∈ B ( H ), let ω ( T ) and k T k denote the numerical radius andthe operator norm of T , respectively. Recall that ω ( T ) = sup x ∈ H k x k =1 |h T x, x i| and k T k = sup x ∈ H k x k =1 k T x k .It is clear that ω ( · ) defines a norm on B ( H ), which is equivalent to the operator norm k·k . Infact, for every T ∈ B ( H ),(1.1) 12 k T k ≤ ω ( T ) ≤ k T k . The inequalities in (1.1) are sharp. The first inequality becomes an equality if T = 0, whilethe second inequality becomes an equality if T is normal, i.e., T ∗ T = T T ∗ .In [11], Kittaneh improved the second inequality in (1.1) as follows(1.2) ω ( T ) ≤ (cid:16) k T k + (cid:13)(cid:13) T (cid:13)(cid:13) (cid:17) . Another refinement of the second inequality in (1.1) has been established in [13]. This refine-ment asserts that if T ∈ B ( H ), then(1.3) ω ( T ) ≤ (cid:13)(cid:13) | T | + | T ∗ | (cid:13)(cid:13) . Here | T | stands for the positive operator ( T ∗ T ) . Mathematics Subject Classification.
Primary 47A12, 47A30. Secondary 15A60.
Key words and phrases.
Numerical radius, operator norm, mixed Schwarz inequality.
A generalization of the inequality (1.3) has been given in [8] that if T ∈ B ( H ) and r ≥ ω r ( T ) ≤ (cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13) . Nowadays, a considerable attention is dedicated to refinements and generalizations of the aboveinequalities [6, 14, 15, 16].Our main target in this article is to present a generalized form that leads to new refine-ments and to some already known results in the literature. Our approach is based on delicatetreatments of inner product inequalities via convex functions.In Section 3, we introduce a sharp inequality that refines the inequality (1.2). Furthermore,we make a refinement of the inequality (1.4).2.
Preliminary lemmas
In this short section, we present some lemmas that we shall need in our analysis. The firstlemma is a simple consequence of the classical Jensen and Young inequalities.
Lemma 2.1.
For a, b ≥ , ≤ α ≤ , and r ≥ , a α b − α ≤ αa + (1 − α ) b ≤ ( αa r + (1 − α ) b r ) r . The second lemma follows from the spectral theorem for positive operators and Jensen’sinequality (see e.g., [9, Theorem 1.4]).
Lemma 2.2.
Let T ∈ B ( H ) be a self adjoint operator and let x ∈ H be a unit vector. If f is a convex function on an interval containing the spectrum of T , then (2.1) f ( h T x, x i ) ≤ h f ( T ) x, x i . If f is concave, then (2.1) holds in the reverse direction. The third lemma is known as the mixed Schwarz inequality (see, e.g., [10, pp. 75–76]).
Lemma 2.3.
Let T ∈ B ( H ) and let x ∈ H be a unit vector. Then, |h T x, x i| ≤ h| T | x, x i h| T ∗ | x, x i . The fourth lemma has been shown in [12, (18)], and is considered as a refined triangleinequality for positive operators.
Lemma 2.4.
Let T ∈ B ( H ) . Then, (cid:13)(cid:13) | T | + | T ∗ | (cid:13)(cid:13) ≤ (cid:13)(cid:13) T (cid:13)(cid:13) + k T k . convex treatment of numerical radius inequalities 3 The fifth lemma, which can be found in [2, Theorem 2.3], gives a norm inequality involvingconvex function of positive operators.
Lemma 2.5.
Let f be a non-negative nondecreasing convex function on [0 , ∞ ) and let A, B ∈ B ( H ) be positive operators. Then (cid:13)(cid:13)(cid:13)(cid:13) f (cid:18) A + B (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) f ( A ) + f ( B )2 (cid:13)(cid:13)(cid:13)(cid:13) . Main Results
In this section, we present our main results. However, we present these results in consecutivesubsections, where an inner product inequality for Hilbert space operators is shown via convexfunctions in the first subsection. Then applications of this generalized form are presented inthe second and third subsections.3.1.
Inner product inequalities.
Our first main result can be stated as follows.
Theorem 3.1.
Let
A, B ∈ B ( H ) and let x ∈ H be a unit vector. If f : [0 , ∞ ) → R is anincreasing convex function, then (3.1) f (cid:0) | h Ax, x i h
Bx, x i | (cid:1) ≤ f ( | h BAx, x i | ) + D(cid:16) αf (cid:16) | A | α (cid:17) + (1 − α ) f (cid:16) | B ∗ | − α (cid:17)(cid:17) x, x E , for ≤ α ≤ . Further, (3.2) f ( | h Ax, x i h
Bx, x i | ) ≤ f ( | h BAx, x i | ) + 14 (cid:10) ( f ( | A | ) + f ( | B ∗ | )) x, x (cid:11) . Proof.
In [4], the following refinement of the Cauchy-Schwarz inequality has been established: |h a, b i| ≤ |h a, e i h e, b i| + |h a, b i − h a, e i h e, b i| ≤ k a k k b k , where a , b , e are vectors in H and k e k = 1. Since |h a, e i h e, b i| + |h a, b i − h a, e i h e, b i| ≥ |h a, e i h e, b i| − |h a, b i| + |h a, e i h e, b i| = 2 |h a, e i h e, b i| − |h a, b i| , we have (see also [7])(3.3) |h a, e i h e, b i| ≤
12 ( |h a, b i| + k a k k b k ) . Putting e = x with k x k = 1, a = Ax and b = B ∗ x in the inequality (3.3), we obtain |h Ax, x i h
Bx, x i| ≤
12 ( |h BAx, x i| + k Ax k k B ∗ x k ) . (3.4) Z. Heydarbeygi, M. Sababheh & H. R. Moradi
Therefore, |h Ax, x i h
Bx, x i| ≤ (cid:18) |h BAx, x i| + k Ax k k B ∗ x k (cid:19) ≤ (cid:0) |h BAx, x i| + k Ax k k B ∗ x k (cid:1) (3.5) = 12 (cid:0) |h BAx, x i| + h Ax, Ax i h B ∗ x, B ∗ x i (cid:1) = 12 (cid:0) |h BAx, x i| + (cid:10) | A | x, x (cid:11) (cid:10) | B ∗ | x, x (cid:11)(cid:1) = 12 (cid:18) |h BAx, x i| + D(cid:16) | A | α (cid:17) α x, x E (cid:28)(cid:16) | B ∗ | − α (cid:17) − α x, x (cid:29)(cid:19) ≤ (cid:18) |h BAx, x i| + D | A | α x, x E α D | B ∗ | − α x, x E − α (cid:19) (3.6) ≤ (cid:16) |h BAx, x i| + α D | A | α x, x E + (1 − α ) D | B ∗ | − α x, x E(cid:17) , (3.7)where in (3.5) we have used the fact that the function t t is convex, in (3.6) we have usedLemma 2.2 and in (3.7) we have used Lemma 2.1.Now since f is increasing and convex, (3.7) implies f (cid:0) | h Ax, x i h
Bx, x i | (cid:1) ≤ f |h BAx, x i| + (cid:16) α D | A | α x, x E + (1 − α ) D | B ∗ | − α x, x E(cid:17) ≤ f (cid:0) |h BAx, x i| (cid:1) + f (cid:16) α D | A | α x, x E + (1 − α ) D | B ∗ | − α x, x E(cid:17) ≤ f (cid:0) |h BAx, x i| (cid:1) + αf (cid:16)D | A | α x, x E(cid:17) + (1 − α ) f (cid:16)D | B ∗ | − α x, x E(cid:17) ≤ f (cid:0) |h BAx, x i| (cid:1) + α D f (cid:16) | A | α (cid:17) x, x E + (1 − α ) D f (cid:16) | B ∗ | − α (cid:17) x, x E ≤ f ( | h BAx, x i | ) + D(cid:16) αf (cid:16) | A | α (cid:17) + (1 − α ) f (cid:16) | B ∗ | − α (cid:17)(cid:17) x, x E , where we have used the fact that f is convex and Lemma 2.2 to obtain the above inequalities.This completes the proof of (3.1). convex treatment of numerical radius inequalities 5 On the other hand, from (3.4), we infer for any unit vector x ∈ H , |h Ax, x i h
Bx, x i| ≤ |h
BAx, x i| + h Ax, Ax i / h B ∗ x, B ∗ x i / |h BAx, x i| + h| A | x, x i / h| B ∗ | x, x i / ≤ |h BAx, x i| + h | A | x,x i + h | B ∗ | x,x i . Again, since f is increasing and convex, we obtain f ( |h Ax, x i h
Bx, x i| ) ≤ f |h BAx, x i| + h | A | x,x i + h | B ∗ | x,x i ≤ f ( |h BAx, x i| ) + f (cid:18) h | A | x,x i + h | B ∗ | x,x i (cid:19) ≤ f ( |h BAx, x i| ) + f ( h | A | x,x i ) + f ( h | B ∗ | x,x i ) ≤ f ( |h BAx, x i| ) + h f ( | A | ) x,x i + h f ( | B ∗ | ) x,x i
2= 12 f ( | h BAx, x i | ) + 14 (cid:10) ( f ( | A | ) + f ( | B ∗ | )) x, x (cid:11) , which proves the inequality (3.2) and completes the proof of the theorem. (cid:3) Noting that the function f ( t ) = t r , r ≥ Corollary 3.1.
Let
A, B ∈ B ( H ) and let x ∈ H be a unit vector. Then for any r ≥ and ≤ α ≤ , (3.8) |h Ax, x i h
Bx, x i| r ≤ (cid:16) |h BAx, x i| r + D(cid:16) α | A | rα + (1 − α ) | B ∗ | r − α (cid:17) x, x E(cid:17) , and (3.9) |h Ax, x i h
Bx, x i| r ≤ |h BAx, x i| r + 14 (cid:10) ( | A | r + | B ∗ | r ) x, x (cid:11) . Applications to numerical radius inequalities.
The first application of Theorem 3.1and Corollary 3.1 is the following numerical radius inequality for the product of two operators.
Corollary 3.2.
Let
A, B ∈ B ( H ) and let f : [0 , ∞ ) → R be an increasing convex function.Then f (cid:0) ω ( B ∗ A ) (cid:1) ≤ f (cid:0) ω ( | B | | A | ) (cid:1) + 14 (cid:13)(cid:13) f ( | A | ) + f ( | B | ) (cid:13)(cid:13) . Z. Heydarbeygi, M. Sababheh & H. R. Moradi
In particular, if r ≥ , then (3.10) ω r ( B ∗ A ) ≤ ω r (cid:0) | B | | A | (cid:1) + 14 (cid:13)(cid:13) | A | r + | B | r (cid:13)(cid:13) . Proof.
Replacing A and B by | A | and | B | respectively in Theorem 3.1, then the inequality(3.2) reduces to f (cid:0)(cid:10) | A | x, x (cid:11) (cid:10) | B | x, x (cid:11)(cid:1) ≤ f (cid:0) | (cid:10) | B | | A | x, x (cid:11) | (cid:1) + 14 (cid:10)(cid:0) f ( | A | ) + f ( | B | ) (cid:1) x, x (cid:11) . (3.11)On the other hand, |h B ∗ Ax, x i| = |h Ax, Bx i| ≤ k Ax k k Bx k (by the Cauchy–Schwarz inequality)= (cid:10) | A | x, x (cid:11)(cid:10) | B | x, x (cid:11) . Since f is increasing, it follows that f (cid:0) |h B ∗ Ax, x i| (cid:1) ≤ f (cid:0)(cid:10) | A | x, x (cid:11)(cid:10) | B | x, x (cid:11)(cid:1) . This together with (3.11) imply f (cid:0) |h B ∗ Ax, x i| (cid:1) ≤ f (cid:0) | (cid:10) | B | | A | x, x (cid:11) | (cid:1) + 14 (cid:10)(cid:0) f ( | A | ) + f ( | B | ) (cid:1) x, x (cid:11) , which implies the first desired inequality upon taking the supremum over all unit vectors x ∈ H . The second inequality follows from the first by letting f ( t ) = t r ; r ≥ . (cid:3) Remark 3.1.
Notice that the inequality (3.10) is sharp. Since for r = 1 and A = B , we geton both sides of (3.10) the same quantity k A k . Remark 3.2.
Dragomir [6] (see also [5, Remark 135] ) proved that for any
A, B ∈ B ( H ) , (3.12) ω ( B ∗ A ) ≤ (cid:13)(cid:13) | A | + | B | (cid:13)(cid:13) , and (3.13) ω r ( B ∗ A ) ≤ (cid:13)(cid:13) | A | r + | B | r (cid:13)(cid:13) , r ≥ . First, we give another proof for the inequality (3.12) . We recall the following arithmetic–geometric mean inequality obtained in [3, (3.5)](3.14) k B ∗ A k ≤ (cid:13)(cid:13) ( | A | + | B | ) (cid:13)(cid:13) . convex treatment of numerical radius inequalities 7 Hence ω ( B ∗ A ) ≤ k B ∗ A k (by the second inequality in (1.1) ) ≤ k| A | + | B |k (by (3.14) ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A | + | B | (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) | A | + | B | (cid:13)(cid:13) (by Lemma 2.5) . Remark 3.3.
In this remark, we show that Corollary 3.2 provides a refinement of Dragomir’sresult. Notice, first that ω r (cid:0) | B | | A | (cid:1) ≤ (cid:13)(cid:13) | B | | A | (cid:13)(cid:13) r ≤ (cid:13)(cid:13)(cid:13)(cid:13) k A | + | B | (cid:13)(cid:13)(cid:13)(cid:13) r = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) k A | + | B | (cid:19) r (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) | A | r + | B | r (cid:13)(cid:13) . Consequently, Corollary 3.2 implies that (3.15) ω r ( B ∗ A ) ≤ ω r (cid:0) | B | | A | (cid:1) + 14 (cid:13)(cid:13) | A | r + | B | r (cid:13)(cid:13) ≤ (cid:13)(cid:13) | A | r + | B | r (cid:13)(cid:13) , explaining why Corollary 3.2 provide a refinement of the inequality (3.13) . Further, the firstinequality in Corollary 3.2 provides a generalization of (3.13) . Now Theorem 3.1 is utilized to obtain the following one-operator numerical radius inequality.
Corollary 3.3.
Let T ∈ B ( H ) and let f : [0 , ∞ ) → R be an increasing convex function. Thenfor ≤ α ≤ ,f ( ω ( T )) ≤ (cid:16) f ( ω ( | T | | T ∗ | )) + (cid:13)(cid:13)(cid:13) (1 − α ) f (cid:16) | T | − α (cid:17) + αf (cid:16) | T ∗ | α (cid:17)(cid:13)(cid:13)(cid:13)(cid:17) , and f ( ω ( T )) ≤ f ( ω ( | T | | T ∗ | )) + 14 (cid:13)(cid:13) f ( | T | ) + f ( | T ∗ | ) (cid:13)(cid:13) . In particular, if r ≥ , then (3.16) ω r ( T ) ≤ (cid:16) ω r ( | T | | T ∗ | ) + (cid:13)(cid:13)(cid:13) (1 − α ) | T | r − α + α | T ∗ | rα (cid:13)(cid:13)(cid:13)(cid:17) , Z. Heydarbeygi, M. Sababheh & H. R. Moradi and (3.17) ω r ( T ) ≤ ω r ( | T | | T ∗ | ) + 14 (cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13) . Both inequalities (3.16) and (3.17) are sharp.Proof.
Replacing A = | T ∗ | and B = | T | in the inequality (3.1), we get f (cid:0) | h| T | x, x i h| T ∗ | x, x i | (cid:1) ≤ f ( | h| T | | T ∗ | x, x i | ) + Dn (1 − α ) f (cid:16) | T | − α (cid:17) + αf (cid:16) | T ∗ | α (cid:17)o x, x E . Since f is increasing, it follows from Lemma 2.3 that f (cid:0) | h T x, x i | (cid:1) ≤ f ( | h| T | | T ∗ | x, x i | ) + Dn (1 − α ) f (cid:16) | T | − α (cid:17) + αf (cid:16) | T ∗ | α (cid:17)o x, x E . Taking the supremum over unit vectors x implies the first desired inequality. The secondinequality follows in a similar way, but using (3.2).The other two inequalities follow by by letting f ( t ) = t r ; r ≥ . Assume that T is a normal operator. For r = 1 and α = , we get on both sides of (3.16)(resp. (3.17)) the same quantity k T k (resp. k T k ), which shows the sharpness of (3.16) (resp.(3.17)). (cid:3) The following result will be needed for further investigation; yet it is of interest by itself.
Proposition 3.1.
Let T ∈ B ( H ) . Then for any r ≥ and ≤ α ≤ , (3.18) ω r ( | T | | T ∗ | ) ≤ (cid:13)(cid:13)(cid:13) (1 − α ) | T | r − α + α | T ∗ | rα (cid:13)(cid:13)(cid:13) , and (3.19) ω r ( | T | | T ∗ | ) ≤ (cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13) . convex treatment of numerical radius inequalities 9 Proof.
Let x ∈ H be a unit vector. We have |h| T | | T ∗ | x, x i| r = |h| T ∗ | x, | T | x i| r ≤ k| T | x k r k| T ∗ | x k r (3.20) = h| T | x, | T | x i r h| T ∗ | x, | T ∗ | x i r = (cid:10) | T | x, x (cid:11) r (cid:10) | T ∗ | x, x (cid:11) r ≤ (cid:10) | T | r x, x (cid:11) (cid:10) | T ∗ | r x, x (cid:11) (3.21) = (cid:28)(cid:16) | T | r − α (cid:17) − α x, x (cid:29) D(cid:16) | T ∗ | rα (cid:17) α x, x E ≤ D | T | r − α x, x E − α D | T ∗ | rα x, x E α (3.22) ≤ (1 − α ) D | T | r − α x, x E + α D | T ∗ | rα x, x E (3.23) = D(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) x, x E , where in the inequality (3.20) we have used the Cauchy–Schwarz inequality, the inequalities(3.21) and (3.22) are obtained from Lemma 2.2, and the inequality (3.23) is a consequence ofthe first inequality in Lemma 2.1.Whence,(3.24) |h| T | | T ∗ | x, x i| r ≤ D(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) x, x E , for any unit vector x ∈ H . Taking the supremum over x ∈ H with k x k = 1 in the inequality(3.24), we obtain (3.18).Similar argument implies(3.25) |h| T | | T ∗ | x, x i| r ≤ (cid:10)(cid:0) | T | r + | T ∗ | r (cid:1) x, x (cid:11) , for any unit vector x ∈ H . Taking the supremum over x ∈ H , k x k = 1 in (3.25) produces theinequality (3.19). (cid:3) Remark 3.4.
By combining inequalities (3.17) and (3.19) , we infer that (3.26) ω r ( T ) ≤ ω r ( | T | | T ∗ | ) + 14 (cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13) ≤ (cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13) . The inequalities (3.26) provide a refinement of the inequality (1.4)The following corollary shows that the inequality (3.17) provides an improvement of theinequality (1.2).
Corollary 3.4.
Let T ∈ B ( H ) . Then ω ( T ) ≤ q ω ( | T | | T ∗ | ) + (cid:13)(cid:13) | T | + | T ∗ | (cid:13)(cid:13) ≤ (cid:16)(cid:13)(cid:13) T (cid:13)(cid:13) / + k T k (cid:17) . Proof.
We have ω ( T ) ≤ q ω ( | T | | T ∗ | ) + (cid:13)(cid:13) | T | + | T ∗ | (cid:13)(cid:13) (by (3.17)) ≤ q k| T | | T ∗ |k + (cid:13)(cid:13) | T | + | T ∗ | (cid:13)(cid:13) (by the second inequality in (1.1))= 12 q k T k + (cid:13)(cid:13) | T | + | T ∗ | (cid:13)(cid:13) (since k| T | | T ∗ |k = (cid:13)(cid:13) T (cid:13)(cid:13) ) ≤ q k T k + k T k + k T k (by Lemma 2.4) ≤ q k T k k T k + k T k + k T k (since (cid:13)(cid:13) T (cid:13)(cid:13) = (cid:13)(cid:13) T (cid:13)(cid:13) (cid:13)(cid:13) T (cid:13)(cid:13) ≤ k T k (cid:13)(cid:13) T (cid:13)(cid:13) )= 12 r(cid:16) k T k / + k T k (cid:17) = 12 (cid:16)(cid:13)(cid:13) T (cid:13)(cid:13) / + k T k (cid:17) , and the proof is complete. (cid:3) The generalized numerical radius.
In this section, we present some new inequalitiesfor the generalized numerical radius ω N ( · ), based on the inner product inequalities obtainedearlier. First, we recall the following definition from [1]. Definition 3.1.
Let T ∈ B ( H ) and let N be any norm on B ( H ) . Then the generalizednumerical radius of T , induced by the norm N , is defined by ω N ( T ) = sup θ ∈ R N ( ℜ ( e iθ T )) , where ℜ ( T ) is the real part of the operator T . In the following result, we use Proposition 3.1 to obtain a new inequality for ω N ( · ) . Thisresult is stated for the algebra of all n × n matrices, denoted by M n . Notice that since thefinite rank operators are dense in the class of compact operators in B ( H ), it follows that thefollowing result is also true for any compact operator T ∈ B ( H ) . Proposition 3.2.
Let T ∈ M n and let N ( · ) be a given unitarily invariant norm on M n . Thenfor any r ≥ and ≤ α ≤ ,ω N ( | T | | T ∗ | ) ≤ N (cid:18)n (1 − α ) | T | r − α + α | T ∗ | rα o r (cid:19) , and ω N ( | T | | T ∗ | ) ≤ N (cid:26) | T | r + | T ∗ | r (cid:27) /r ! . convex treatment of numerical radius inequalities 11 Proof.
From Proposition 3.1, we have |h| T | | T ∗ | x, x i| r ≤ D(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) x, x E . Since | e iθ | = 1 , this implies (cid:12)(cid:12)(cid:10) e iθ | T | | T ∗ | x, x (cid:11)(cid:12)(cid:12) ≤ D(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) x, x E r . But since, for any operator A , | hℜ ( A ) x, x i | ≤ | h Ax, x i | , it follows that (cid:12)(cid:12)(cid:10) ℜ (cid:8) e iθ | T | | T ∗ | (cid:9) x, x (cid:11)(cid:12)(cid:12) ≤ D(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) x, x E r . By the minimax principle, it follows that, for 1 ≤ k ≤ n,s k (cid:0) ℜ (cid:8) e iθ | T | | T ∗ | (cid:9)(cid:1) ≤ s r k (cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) = s k (cid:26)(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) r (cid:27) . This latter inequality implies that for any unitarily invariant norm N ( · ) on M n , N (cid:0) ℜ (cid:8) e iθ | T | | T ∗ | (cid:9)(cid:1) ≤ N (cid:26)(cid:16) (1 − α ) | T | r − α + α | T ∗ | rα (cid:17) r (cid:27) , which implies ω N ( | T | | T ∗ | ) ≤ N (cid:18)n (1 − α ) | T | r − α + α | T ∗ | rα o r (cid:19) upon taking the supremum over θ. This proves the first desired inequality.The second inequality can be shown similarly, and hebce we leave its proof to the reader. (cid:3)
Remark 3.5.
Notice that when N is the operator norm, N (cid:26) | T | r + | T ∗ | r (cid:27) /r ! = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:26) | T | r + | T ∗ | r (cid:27) / r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13)(cid:13)(cid:13) /r . So, when N ( · ) = k · k , Proposition 3.2 implies w r ( | T | | T ∗ | ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) | T | r + | T ∗ | r (cid:13)(cid:13)(cid:13)(cid:13) ; which has been shown earlier in Proposition 3.1. References [1] A. Abu-Omar and F. Kittaneh,
A generalization of the numerical radius,
Linear Algebra Appl., (2019),323–334.[2] J. Aujla and F. Silva,
Weak majorization inequalities and convex functions , Linear Algebra Appl., (2003), 217–233.[3] R. Bhatia and F. Kittaneh, Notes on matrix arithmetic–geometric mean inequalities, Linear Algebra Appl., (2000), 203–211.[4] M. L. Buzano,
Generalizzazione della diseguaglianza di Cauchy–Schwarz . (Italian), Rend. Sem. Mat. Univ.e Politech. Torino., (1971/73), 405–409 (1974).[5] S. S. Dragomir, Inequalities for the numerical radius of linear operators in Hilbert spaces , Springer Briefs inMathematics (Springer, Cham, Switzerland, 2013), x+120.[6] S. S. Dragomir,
Power inequalities for the numerical radius of a product of two operators in Hilbert spaces ,Sarajevo J. Math., (18) (2009), 269–278.[7] S. S. Dragomir, Some refinements of Schwarz inequality , in: Simposional de Math Si Appl PolytechnicalInst Timisoara, vols. 1–2, Romania, 1985, pp. 13–16.[8] M. El-Haddad and F. Kittaneh,
Numerical radius inequalities for Hilbert space operators. II , Studia Math., (2) (2007), 133–140.[9] T. Furuta, J. Mi´ci´c, J. Peˇcari´c and Y. Seo,
Mond–Peˇcari´c method in operator inequalities , Element, Zagreb,2005.[10] P . R. Halmos,
A Hilbert space problem book , 2nd ed., Springer, New York, 1982.[11] F. Kittaneh,
A numerical radius inequality and an estimate for the numerical radius of the Frobeniuscompanion matrix , Studia Math., (1) (2003), 11–17.[12] F. Kittaneh,
Norm inequalities for sums and differences of positive operators , Linear Algebra Appl., (2004), 85–91.[13] F. Kittaneh,
Numerical radius inequalities for Hilbert space operators , Studia Math., (1) (2005), 73–80.[14] H. R. Moradi and M. Sababheh,
More accurate numerical radius inequalities (II) , Linear Multilinear Al-gebra. https://doi.org/10.1080/03081087.2019.1703886[15] M. E. Omidvar, H. R. Moradi and K. Shebrawi,
Sharpening some classical numerical radius inequalities ,Oper. Matrices., (2) (2018), 407–416.[16] M. Sababheh and H. R. Moradi, More accurate numerical radius inequalities (I) , Linear Multilinear Algebra.https://doi.org/10.1080/03081087.2019.1651815 (Z. Heydarbeygi) Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran.
E-mail address: [email protected](M. Sababheh) Department of Basic Sciences, Princess Sumaya University For Technology, Al Jubaiha, Amman 11941, Jordan.
E-mail address: [email protected](H. R. Moradi) Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran.