On Some Numerical Radius Inequalities for Hilbert Space Operators
aa r X i v : . [ m a t h . F A ] A ug ON SOME NUMERICAL RADIUS INEQUALITIES FOR HILBERT SPACEOPERATORS
MAHDI GHASVAREH , MOHSEN ERFANIAN OMIDVAR Abstract.
This article is devoted to studying some new numerical radius inequalities forHilbert space operators. Our analysis enables us to improve an earlier bound of numericalradius due to Kittaneh. It is shown, among others, that if A ∈ B ( H ), then18 (cid:16) k A + A ∗ k + k A − A ∗ k (cid:17) ≤ ω ( A ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − m (cid:18) | A | − | A ∗ | (cid:19) ! . 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 A ∈ B ( H ), let ω ( A ) and k A k denote the numerical radius andthe operator norm of A , respectively. Recall that ω ( A ) = sup k x k =1 h Ax, x i . It is well-known that ω ( · ) defines a norm on B ( H ), which is equivalent to the operator norm k·k . In fact, for every A ∈ B ( H ),(1.1) 12 k A k ≤ ω ( A ) ≤ k A k . Also, it is a basic fact that ω ( · ) defines a norm on B ( H ) which satisfies the power inequality ω ( A n ) ≤ ω n ( A )for all n = 1 , , . . . .In [2], Kittaneh gave the following estimate of the numerical radius which refines the secondinequality in (1.1): For every A ,(1.2) ω ( A ) ≤ k| A | + | A ∗ |k . The following estimate of the numerical radius has been given in [3]:(1.3) 14 (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) ≤ ω ( A ) ≤ (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) . Mathematics Subject Classification.
Primary 47A63, Secondary 46L05, 47A60.
Key words and phrases.
Numerical radius, norm inequality, convex function.
The first inequality in (1.3) also refines the first inequality in (1.1). This can be seen by usingthe fact that for any positive operator
A, B ∈ B ( H ),max ( k A k , k B k ) ≤ k A + B k . Actually, 14 k A k = 14 max (cid:0)(cid:13)(cid:13) | A | (cid:13)(cid:13) , (cid:13)(cid:13) | A ∗ | (cid:13)(cid:13)(cid:1) ≤ (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) . For other properties of the numerical radius and related inequalities, the reader may consult[5, 6, 8]. In this article, we give several refinements of numerical radius inequalities. Our resultsmainly improve the inequalities in [3].2.
Main Results
Lemma 2.1.
Let A ∈ B ( H ) . Then k A ± A ∗ k ≤ ω ( A ) . Proof.
Since A + A ∗ is normal, we have k A + A ∗ k = ω ( A + A ∗ ) ≤ ω ( A ) + ω ( A ∗ )= 2 ω ( A ) . Therefore,(2.1) 12 k A + A ∗ k ≤ ω ( A ) . Now, by replacing A by iA in (2.1), we reach the desired result. (cid:3) Theorem 2.1.
Let A ∈ B ( H ) . Then (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) ≤ (cid:0) k A + A ∗ k + k A − A ∗ k (cid:1) ≤ ω ( A ) . Proof.
For any
A, B ∈ B ( H ), we have the following parallelogramm law | A + B | + | A − B | = 2 (cid:0) | A | + | B | (cid:1) , equivalently (cid:12)(cid:12)(cid:12)(cid:12) A + B (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) A − B (cid:12)(cid:12)(cid:12)(cid:12) = | A | + | B | . . Ghasvareh, M.E. Omidvar 3 Therefore, by the triangle inequality for the usual operator norm and Lemma 2.1, we have14 (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) = 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12) A + A ∗ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) A − A ∗ (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12) A + A ∗ (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12) A − A ∗ (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 12 (cid:13)(cid:13)(cid:13)(cid:13) A + A ∗ (cid:13)(cid:13)(cid:13)(cid:13) + 12 (cid:13)(cid:13)(cid:13)(cid:13) A − A ∗ (cid:13)(cid:13)(cid:13)(cid:13) ≤ ω ( A ) . We remark here that if T ∈ B ( H ), and if f is a non-negative increasing function on [0 , ∞ ),then k f ( | T | ) k = f ( k T k ). In particular, k| T | r k = k T k r for every r >
0. This completes theproof of the theorem. (cid:3)
We present a refinement the first inequality from (1.3). For see this, we need the followinglemma, which can found in [1].
Lemma 2.2.
Let
A, B ∈ B ( H ) . Then k A + B k ≤ p k A ∗ A + B ∗ B k + 2 ω ( B ∗ A ) . By the above lemma, we can improve the first inequality in (1.3).
Theorem 2.2.
Let A ∈ B ( H ) . Then (2.2) 14 (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) ≤ r ω ( A ) + 18 ω (cid:0) ( A ∗ − A ) ( A ∗ + A ) (cid:1) ≤ ω ( A ) . Proof.
Let A = B + iC be the Cartesian decomposition of A . Then B and C are self-adjointoperators. One can easily check that(2.3) | A | + | A ∗ | B + C , and(2.4) |h Ax, x i| = h Bx, x i + h Cx, x i for any unit vector x ∈ H . Of course, the relation (2.4) implies h Bx, x i (cid:0) resp. h Cx, x i (cid:1) ≤ |h Ax, x i| . Now, by taking supremum over x ∈ H with k x k = 1, we get(2.5) k B k (cid:0) resp. k C k (cid:1) ≤ ω ( A ) . On some numerical radius inequalities for Hilbert space operators
Whence,14 (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) = 12 (cid:13)(cid:13) B + C (cid:13)(cid:13) (by (2.3)) ≤ p k B + C k + 2 ω ( C B ) (by Lemma 2.2) ≤ p k B k + k C k + 2 ω ( C B ) ≤ p ω ( A ) + 2 ω ( C B ) (by (2.5)) ≤ p ω ( A ) + 2 k C B k (by the second inequality in (1.1)) ≤ q ω ( A ) + 2 k B k k C k (by the submultiplicativity of the usual operator norm) ≤ ω ( A ) (by (2.5))i.e., 14 (cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) ≤ p ω ( A ) + 2 ω ( C B ) ≤ ω ( A ) . Since ω (cid:0) C B (cid:1) = 116 ω (cid:0) ( A ∗ − A ) ( A ∗ + A ) (cid:1) we get the desired result (2.2). (cid:3) Remark 2.1.
Notice that if A is a self-adjoint operator, then Theorem 2.1 implies k A k ≤ k A k while from Theorem 2.2 we infer that k A k ≤ √ k A k ≤ k A k . Hence, in this case, Theorem 2.2 is better than Theorem 2.1.
The next lemma can be found in [4].
Lemma 2.3. If A and B are positive operators in B ( H ) , Then k A − B k ≤ max {k A k , k B k} − min { m ( A ) , m ( B ) } , where m ( A ) = inf {h Ax, x i : x ∈ H , k x k = 1 } . Theorem 2.3.
Let A ∈ B ( H ) . Then ω ( A ) ≤ (cid:2)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13) − m (cid:0) ( | A | − | A ∗ | ) (cid:1)(cid:3) . . Ghasvareh, M.E. Omidvar 5 Proof.
We can write (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A | + | A ∗ | (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A | − | A ∗ | (cid:19) − | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ max (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A | − | A ∗ | (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! − min m (cid:18) | A | − | A ∗ | (cid:19) ! , m | A | + | A ∗ | !! ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − m (cid:18) | A | − | A ∗ | (cid:19) ! . On the other hand, since ω ( A ) ≤ k| A | + | A ∗ |k , we have ω ( A ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A | + | A ∗ | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | A | + | A ∗ | (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Consequently, ω ( A ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − m (cid:18) | A | − | A ∗ | (cid:19) ! , as desired. (cid:3) Using some ideas of [7], we prove our last result.
Theorem 2.4.
Let A ∈ B ( H ) and let f be a continuous function on the interval [0 , ∞ ) andlet g be increasing and concave on [0 , ∞ ) , such that gof is increasing and convex on [0 , ∞ ) .Then f ( ω ( A )) ≤ (cid:13)(cid:13)(cid:13)(cid:13) g − (cid:18) gof ( | A | ) + gof ( | A ∗ | )2 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k f ( | A | ) + f ( | A ∗ | ) k . Proof.
As mentioned above, gof is increasing and convex on [0 , ∞ ), therefore, from the inequal-ity (1.2), gof ( ω ( A )) ≤ gof (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) | A | + | A ∗ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) = (cid:13)(cid:13)(cid:13)(cid:13) gof (cid:18) | A | + | A ∗ | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) gof ( | A | ) + gof ( | A ∗ | )2 (cid:13)(cid:13)(cid:13)(cid:13) . On some numerical radius inequalities for Hilbert space operators
Therefore, gof ( ω ( A )) ≤ (cid:13)(cid:13)(cid:13)(cid:13) gof ( | A | ) + gof ( | A ∗ | )2 (cid:13)(cid:13)(cid:13)(cid:13) . Now, since g − is increasing and convex, we then have f ( ω ( A )) = g − ( gof ( ω ( A ))) ≤ g − (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) gof ( | A | ) + gof ( | A ∗ | )2 (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) = (cid:13)(cid:13)(cid:13)(cid:13) g − (cid:18) gof ( | A | ) + gof ( | A ∗ | )2 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) f ( | A | ) + f ( | A ∗ | )2 (cid:13)(cid:13)(cid:13)(cid:13) = 12 k f ( | A | ) + f ( | A ∗ | ) k . Thus, f ( ω ( A )) ≤ (cid:13)(cid:13)(cid:13)(cid:13) g − (cid:18) gof ( | A | ) + gof ( | A ∗ | )2 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k f ( | A | ) + f ( | A ∗ | ) k . (cid:3) Corollary 2.1.
Let A ∈ B ( H ) . Then for any r ≥ , ω r ( A ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | A | r + | A ∗ | r + | A | r + | A ∗ | r + I − r (cid:16) | A | r + | A ∗ | r + | A | r + | A ∗ | r (cid:17) + I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k| A | r + | A ∗ | r k . Proof.
Define g ( x ) = x + √ x & f ( x ) = x r , ( r ≥ , ∞ ). Thus, gof ( x ) = x r + x r . One can quickly check that f , g , and gof satisfy all the assumptions in Theorem 2.3. Since g − ( x ) = 2 x + 1 − √ x + 12 , we get the desired result. (cid:3) . Ghasvareh, M.E. Omidvar 7 References [1] S.S. Dragomir,
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