EXTENSION OF EUCLIDEAN OPERATOR RADIUS INEQUALITIES 1 1 2 MOHAMMAD SAL MOSLEHIAN , MOSTAFA SATTARI AND KHALID SHEBRAWI 5 Abstract. ToextendtheEuclideanoperatorradius,wedefinew forann-tuples 1 p 20 ofoperators(T1,...,Tn)inB(H )bywp(T1,...,Tn):=supkxk=1( ni=1|hTix,xi|p)p1 for p 1. We generalize some inequalities including Euclidean operator radius P n ≥ of two operators to those involving w . Further we obtain some lower and upper a p J bounds for w . Our mainresult states that if f andg arenonnegative continuous p 1 functions on [0, ) satisfying f(t)g(t)=t for all t [0, ), then 3 ∞ ∈ ∞ n .FA] fworprpa(llAp∗1T1B1,1,r...,1Aa∗nnTdnoBpne)ra≤to12rs(cid:13)(cid:13)(cid:13)(cid:13)iXin=1B(cid:16)(H(cid:2)B)i∗.f2(|Ti|)Bi(cid:3)rp+(cid:2)A∗ig2(|Ti∗|)Ai(cid:3)rp(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) h ≥ ≥ (cid:13) (cid:13) t a m [ 1 1. Introduction v 3 Let B(H ) be the C∗-algebra of all bounded linear operators on a Hilbert space 8 0 (H , , ). The numerical radius of A B(H ) is defined by 0 h· ·i ∈ 0 2. w(A) = sup Ax,x : x H , x = 1 . {|h i| ∈ k k } 0 5 1 It is well known that w( ) defines a norm on B(H ), which is equivalent to the usual : · v operator norm . Namely, we have i k·k X r 1 a A w(A) A . 2k k ≤ ≤ k k for each A B(H ). It is known that if A B(H ) is self-adjoint, then w(A) = A . ∈ ∈ k k An important inequality for w(A) is the power inequality stating that w(An) ≤ wn(A) for n = 1,2,.... There are many inequalities involving numerical radius; see [2, 3, 4, 10, 11, 12] and references therein. The Euclidean operator radius of an n-tuple (T ,...,T ) B(H )(n) := B(H ) 1 n ∈ × 2010 Mathematics Subject Classification. 47A12, 47B15,47A30, 47A63. Key words and phrases. Euclideanoperatorradius,numericalradius, Cartesiandecomposition, self-adjoint operator. 1 2 M.S. MOSLEHIAN,M. SATTARI,K. SHEBRAWI ... B(H ) was defined in [9] by × 1 n 2 w (T ,...,T ) := sup T x,x 2 . e 1 n i |h i| kxk=1 ! i=1 X The particular cases n = 1 and n = 2 are numerical radius and Euclidean operator radius. Some interesting properties of this radius were obtained in [9]. For example, it is established that 1 1 1 n 2 n 2 T T∗ w (T , ,T ) T T∗ . (1.1) 2√n (cid:13) i i (cid:13) ≤ e 1 ··· n ≤ (cid:13) i i (cid:13) (cid:13)Xi=1 (cid:13) (cid:13)Xi=1 (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) We also observe tha(cid:13)t if A = B(cid:13) +iC is the Cartesian(cid:13)decompo(cid:13)sition of A, then (cid:13) (cid:13) (cid:13) (cid:13) w2(B,C) = sup Bx,x 2 + Cx,x 2 = sup Ax,x 2 = w2(A). e {|h i| |h i| } |h i| kxk=1 kxk=1 By the above inequality and A∗A+AA∗ = 2(B2 +C2), we have 1 1 A∗A+AA∗ w2(A) A∗A+AA∗ . 16k k ≤ ≤ 2k k We define w for n-tuples of operators (T ,...,T ) B(H )(n) for p 1 by p 1 n ∈ ≥ 1 n p w (T ,...,T ) := sup T x,x p . p 1 n i |h i| kxk=1 ! i=1 X It follows from Minkowski’s inequality for two vectors a = (a ,a ) and b = (b ,b ), 1 2 1 2 namely, ( a +b p + a +b p)1p ( a p + a p)p1 +( b p + b p)p1 for p > 1 1 1 2 2 1 2 1 2 | | | | ≤ | | | | | | | | that w is a norm. p Moreover w ,p 1, for n-tuple of operators (T ,...,T ) B(H )(n) satisfies the p 1 n ≥ ∈ following properties: (i) w (T ,...,T ) = 0 T = ... = T = 0. p 1 n 1 n ⇔ (ii) w (λT ,...,λT ) = λ w (T ,...,T ) for all λ C. p 1 n p 1 n | | ∈ (iii)w (T +T´,...,T +T´) w (T ,...,T )+w (T´,...,T´)for(T´,...,T´) p 1 1 n n p 1 n p 1 n 1 n ≤ ∈ B(H )(n). (iv) w (X∗T X,...,X∗T X) X 2w (T ,...,T ) for X B(H ). p 1 n p 1 n ≤ k k ∈ Dragomir[1]obtainedsomeinequalitiesfortheEuclideanoperatorradiusw (B,C) = e 1 sup ( Bx,x 2 + Cx,x 2)2 oftwoboundedlinearoperatorsinaHilbertspace. kxk=1 |h i| |h i| In section 2 of this paper we extend some his results including inequalities for the Euclidean operator radius of linear operators to w (p 1). In addition, we apply p ≥ some known inequalities for getting new inequalities for w in two operators. p EXTENSION OF EUCLIDEAN OPERATOR RADIUS INEQUALITIES 3 In section 3 we prove inequalities for w for n-tuples of operators. Some of our result p in this section, generalize some inequalities in section 2. Further, we find some lower and upper bounds for w . p 2. Inequalities for w for two operators p To prove our generalized numerical radius inequalities, we need several known lemmas. The first lemma is a simple result of the classical Jensen inequality and a generalized mixed Cauchy–Schwarz inequality [7, 8, 6]. Lemma 2.1. For a,b 0, 0 α 1 and r = 0, ≥ ≤ ≤ 6 (a) aαb1−α αa+(1 α)b [αar +(1 α)br]1r for r 1, ≤ − ≤ − ≥ (b) If A B(H ),then Ax,y 2 A 2αx,x A∗ 2(1−α)y,y for all x,y H , ∈ |h i| ≤ h| | ih| | i ∈ where A = (A∗A)1. 2 | | (c) Let A B(H ), and f and g be nonnegative continuous functions on [0, ) ∈ ∞ satisfying f (t)g(t) = t for all t [0, ). Then ∈ ∞ Ax,y f ( A )x g( A∗ )y |h i| ≤ k | | kk | | k for all x,y H . ∈ Lemma 2.2 (McCarthy inequality [5]). Let A B(H ), A 0 and let x H ∈ ≥ ∈ be any unit vector. Then (a) Ax,x r Arx,x for r 1, h i ≤ h i ≥ (b) Arx,x Ax,x r for 0 < r 1. h i ≤ h i ≤ Inequalities ofthe following lemma were obtainedfor thefirst timeby Clarkson[7]. Lemma 2.3. Let X be a normed space and x,y X. Then for all p 2 with ∈ ≥ 1 + 1 = 1, p q (a) 2( x p + y p)q−1 x+y q + x y q, k k k k ≤ k k k − k (b) 2( x p + y p) x+y p + x y p 2p−1( x p + y p), k k k k ≤ k k k − k ≤ k k k k (c) x+y p + x y p 2( x q + y q)p−1. k k k − k ≤ k k k k If 1 < p 2 the converse inequalities hold. ≤ Makingthetransformationsx x+y andy x−y weobservethatinequalities (a) → 2 → 2 and (c) in Lemma 2.3 are equivalent and so are the first and the second inequalities of (b). First of all we obtain a relation between w and w for p 1. p e ≥ 4 M.S. MOSLEHIAN,M. SATTARI,K. SHEBRAWI Proposition 2.4. Let B,C B(H ). Then ∈ wp(B,C) wq(B,C) 2q1−p1wp(B,C) ≤ ≤ for p q 1. In particular ≥ ≥ wp(B,C) we(B,C) 221−p1wp(B,C) (2.1) ≤ ≤ for p 2, and ≥ 221−p1wp(B,C) we(B,C) wp(B,C) ≤ ≤ for 1 p 2. ≤ ≤ Proof. An application of Jensen’s inequality says that for a,b > 0 and p q > 0, ≥ we have (ap +bp)p1 (aq +bq)q1 . ≤ Let x H be a unit vector. Choosing a = Bx,x and b = Cx,x , we have ∈ |h i| |h i| 1 1 Bx,x p + Cx,x p p Bx,x q + Cx,x q q. |h i| |h i| ≤ |h i| |h i| (cid:16) (cid:17) (cid:16) (cid:17) Now the first inequality follows by taking the supremum over all unit vectors in H . A simple consequence of the classical Jensen’s inequality concerning the convexity or the concavity of certain power functions says that for a,b 0,0 α 1 and ≥ ≤ ≤ p q, we have ≥ (αaq +(1 α)bq)1q (αap +(1 α)bp)p1 . − ≤ − For α = 1, we get 2 (aq +bq)q1 21q−p1 (ap +bp)p1 . ≤ Again let x H be a unit vector. Choosing a = Bx,x and b = Cx,x we get ∈ |h i| |h i| 1 1 Bx,x q + Cx,x q q 21q−p1 Bx,x p + Cx,x p p. |h i| |h i| ≤ |h i| |h i| (cid:16) (cid:17) (cid:16) (cid:17) Now the second inequality follows by taking the supremum over all unit vectors in H . (cid:3) On making use of inequality (2.1) we find a lower bound for w (p 2). p ≥ Corollary 2.5. If B,C B(H ), then for p 2 ∈ ≥ wp(B,C) 2p1−2 B∗B +C∗C 21 . ≥ k k EXTENSION OF EUCLIDEAN OPERATOR RADIUS INEQUALITIES 5 Proof. According to inequalities (1.1) and (2.1) we can write 1 we(B,C) B∗B +C∗C 21 ≥ 2√2k k and wp(B,C) 2p1−21we(B,C), ≥ (cid:3) respectively. We therefore get desired inequality. The next result is concerned with some lower bounds for w . This consequence p has several inequalities as special cases. Our result will be generalized to n-tuples of operators in the next section. Proposition 2.6. Let B,C B(H ). Then for p 1 ∈ ≥ wp(B,C) 2p1−1max(w(B +C),w(B C)). (2.2) ≥ − This inequality is sharp. Proof. We use convexity of function f(t) = tp (p 1) as follows: ≥ ( Bx,x p + Cx,x p)p1 2p1−1( Bx,x + Cx,x ) |h i| |h i| ≥ |h i| |h i| 2p1−1 Bx,x Cx,x ≥ |h i±h i| = 2p1−1 (B C)x,x . |h ± i| Taking supremum over x H with x = 1 yields that ∈ k k wp(B,C) 2p1−1w(B C). ≥ ± 1 For sharpness one can obtain the same quantity 2pw(B) on both sides of the in- equality by putting B = C. (cid:3) Corollary 2.7. If A = B + iC is the Cartesian decomposition of A, then for all p 2 ≥ wp(B,C) 2p1−1max( B +C , B C ), ≥ k k k − k and w(A) 2p1−2max( (1 i)A+(1+i)A∗ , (1+i)A+(1 i)A∗ ) ≥ k − k k − k Proof. Obviously by inequality (2.2) we have the first inequality. For the second we (cid:3) use inequality (2.1). 6 M.S. MOSLEHIAN,M. SATTARI,K. SHEBRAWI Corollary 2.8. If B,C B(H ), then for p 1 ∈ ≥ wp(B,C) 2p1−1max w(B),w(C) . (2.3) ≥ { } In addition, if A = B +iC is the Cartesian decomposition of A, then for p 2 ≥ w(A) 2p1−2max( A+A∗ , A A∗ ). ≥ k k k − k Proof. By inequality (2.2) and properties of the numerical radius, we have 2wp(B,C) 2p1−1(w(B +C)+w(B C)) 2p1−1w(B +C +B C). ≥ − ≥ − So wp(B,C) 2p1−1w(B). ≥ By symmetry we conclude that wp(B,C) 2p1−1max(w(B),w(C)). ≥ (cid:3) While the second inequality follows easily from inequality (2.1). Now we apply part (b) of Lemma 2.3 to find some lower and upper bounds for w (p > 1). p Proposition 2.9. Let B,C B(H ). Then for all p 2, ∈ ≥ (i) 2p1−1wp(B +C,B C) wp(B,C) 2−p1wp(B +C,B C); − ≤ ≤ − (ii) 2p1−1 wp(B+C)+wp(B C) p1 wp(B,C) 2−p1 wp(B+C)+wp(B C) p1. − ≤ ≤ − If 1 < p 2 these inequalities hold in the opposite direction. (cid:0) (cid:1) (cid:0) (cid:1) ≤ Proof. Let x H be a unit vector. Part (b) of Lemma 2.3 implies that for any ∈ p 2 ≥ 1 21−p( a+b p + a b p) a p + b p ( a+b p + a b p). | | | − | ≤ | | | | ≤ 2 | | | − | Replacing a = Bx,x and b = Cx,x in above inequalities we obtain the desired |h i| |h i| (cid:3) inequalities. Remark 2.10. In inequality (2.3), if we take B +C and B C instead of B and − C, then for p 1 ≥ wp(B +C,B C) 2p1−1max w(B +C),w(B C) . − ≥ { − } By employing the first inequality of part (i) of Proposition 2.9, we get wp(B,C) 2p2−2max w(B +C),w(B C) ≥ { − } EXTENSION OF EUCLIDEAN OPERATOR RADIUS INEQUALITIES 7 for p 1. ≥ Taking B +C and B C instead of B and C in the second inequality of part (ii) − of Proposition 2.9, we reach wp(B +C,B C) 21−p1 (wp(B)+wp(C))p1 . − ≤ for all p 1. ≥ Now by applying the second inequality of part (i) of Proposition 2.9, we infer for p 1 that ≥ wp(B,C) 21−p2 (wp(B)+wp(C))1p . ≤ So 2p2−2max w(B +C),w(B C) wp(B,C) 21−p2 (wp(B)+wp(C))1p . { − } ≤ ≤ Moreover if B and C are self-adjoint, then 2p2−2max B +C , B C wp(B,C) 21−p2 ( B p + C p)p1 {k k k − k} ≤ ≤ k k k k for all p 1. ≥ In the following result we find another lower bound for w (p 1). p ≥ Theorem 2.11. Let B,C B(H ). Then for p 1 ∈ ≥ wp(B,C) 2p1−1w21(B2 +C2). ≥ Proof. It follows from (2.2) that 2p2−2w2(B C) w2(B,C). ± ≤ p Hence 2w2(B,C) 2p2−2 w2(B +C)+w2(B C) p ≥ − 2p2−2(cid:2)w (B +C)2 +w (B (cid:3)C)2 ≥ − 2p2−2(cid:2)w(cid:0)(B +C)2(cid:1)+(B(cid:0) C)2 =(cid:1)2(cid:3)p2−1w(B2 +C2). ≥ − (cid:2) (cid:0) (cid:1)(cid:3) It follows that wp(B,C) 2p1−1w21(B2 +C2). ≥ (cid:3) 8 M.S. MOSLEHIAN,M. SATTARI,K. SHEBRAWI Corollary 2.12. If A = B +iC is the Cartesian decomposition of A , then wp(B,C) 2p1−1 B2 +C2 12. ≥ k k And w(A) 2p1−23 A∗A+AA∗ 12. ≥ k k for any p 2. ≥ Proof. The first inequality is obvious. For the second we have A∗A+AA∗ = 2(B2+ C2). Now by using inequality (2.1) the proof is complete. (cid:3) Corollary 2.13. If B,C B(H ), then for p 2 ∈ ≥ wp(B,C) 2p2−23w21 B2 +C2 . ≥ (cid:0) (cid:1) Proof. By choosing B + C and B C instead of B and C in Theorem 2.11 and − (cid:3) employing part (i) of Proposition 2.9 we conclude that the desired inequality. The following result providing other bound for w (p > 1) may be stated as p follows: Proposition 2.14. Let B,C B(H ). Then ∈ B +C B C w (B,C) w , − . p q ≤ 2 2 (cid:18) (cid:19) for any p 2,1 < q 2 with 1 + 1 = 1. If 1 < p 2, the reverse inequality holds. ≥ ≤ p q ≤ Proof. Let x H be a unit vector. Part (a) of Lemma 2.3 implies that ∈ a p + b p 21−1q ( a+b q + a b q)q−11 . | | | | ≤ | | | − | So ( a p + b p)p1 2p(11−q) ( a+b q + a b q)p(q1−1) . | | | | ≤ | | | − | Now replacing a = Bx,x and b = Cx,x in the above inequality we conclude that h i h i ( Bx,x p + Cx,x p)p1 B +C x,x q + B−C x,x q q1 . (2.4) |h i| |h i| ≤ 2 2 (cid:18)(cid:12)(cid:28)(cid:18) (cid:19) (cid:29)(cid:12) (cid:12)(cid:28)(cid:18) (cid:19) (cid:29)(cid:12) (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) By taking supremum over x (cid:12)H with x = 1(cid:12)we d(cid:12)educe that (cid:12) ∈(cid:12) k k (cid:12) (cid:12) (cid:12) B +C B C w (B,C) w , − p q ≤ 2 2 (cid:18) (cid:19) for any p 2,1 < q 2 with 1 + 1 = 1. (cid:3) ≥ ≤ p q EXTENSION OF EUCLIDEAN OPERATOR RADIUS INEQUALITIES 9 Corollary 2.15. Inequality (2.4) implies that 1 B +C B C q w (B,C) wq +wq − . p ≤ 2 2 (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) for any 1 < q 2,p 2 with 1 + 1 = 1. Further, if Band C are self-adjoint, then ≤ ≥ p q 1 w (B,C) ( B +C q + B C q)1q . p ≤ 2 k k k − k If 1 < p 2, the converse inequalities hold. ≤ Corollary 2.16. If B,C B(H ), then ∈ B +C B C B +C B C 1 wq , − 2pwp , − . 2 2 ≤ 2 2 (cid:18) (cid:19) (cid:18) (cid:19) for all 1 < p 2 with 1 + 1 = 1. If p 2, the above inequality is valid in the ≤ p q ≥ opposite direction. Proof. By Proposition 2.14 we have B +C B C w , − w (B,C). q p 2 2 ≤ (cid:18) (cid:19) for all 1 < p 2 with 1 + 1 = 1. Proposition 2.9 follows that ≤ p q B +C B C wp(B,C) 2p1−1wp(B +C,B C) = 2p1wp , − . ≤ − 2 2 (cid:18) (cid:19) (cid:3) We therefore get the desired inequality. 3. Inequalities of w for n-tuples of operators p In this section, we are going to obtain some numerical radius inequalities for n- tuples of operators. Some generalization of inequalities in the previous section are also established. According to the definition of numerical radius, we immediately get the following double inequality for p 1 ≥ 1 n p n w (T ,...,T ) wp(T ) w(T ). p 1 n i i ≤ ≤ ! i=1 i=1 X X An application of Holder’s inequality gives the next result, which is a generalization of inequality (2.2). 10 M.S. MOSLEHIAN,M. SATTARI,K. SHEBRAWI Theorem 3.1. Let (T ,...,T ) B(H )(n) and 0 α 1, i = 1,...n, with 1 n i n ∈ ≤ ≤ α = 1. Then i i=1 P 1−1 1−1 1−1 w (T ,...,T ) w α pT α pT ... α pT p 1 n ≥ 1 1 ± 2 2 ± ± n n (cid:18) (cid:19) for any p > 1. Proof. In the Euclidean space Rn with the standard inner product, Holder’s inequal- ity 1 1 n n p n q x y x p y q i i i i | | ≤ | | | | ! ! i=1 i=1 i=1 X X X holds, where p and q are in the open interval (1, ) with 1+1 = 1 and (x ,...,x ), ∞ p q 1 n (y ,...,y ) Rn. For (y ,...,y ) = α1−p1,...,α1−p1 we have 1 n ∈ 1 n 1 n (cid:18) (cid:19) 1 1 n n p n q q α1−p1x x p α1−p1 . i i ≤ | i| i ! ! i=1 (cid:12) (cid:12) i=1 i=1 (cid:12) (cid:12) X(cid:12) (cid:12) X X(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Thus (cid:12) (cid:12) (cid:12) (cid:12) 1 n p n x p α1−p1x . | i| ≥ i i ! i=1 i=1 (cid:12) (cid:12) X X(cid:12) (cid:12) Choosing x = T x,x ,i = 1,...n, we get (cid:12) (cid:12) i i (cid:12) (cid:12) |h i| 1 n p T x,x p i |h i| ! i=1 X n 1−1 α pT x,x ≥ i i i=1 (cid:12)(cid:28) (cid:29)(cid:12) X(cid:12) (cid:12) (cid:12)1−1 (cid:12) 1−1 1−1 α(cid:12) pT x,x (cid:12)α pT x,x ... α pT x,x ≥ 1 1 ± 2 2 ± ± n n (cid:12)(cid:28) (cid:29) (cid:28) (cid:29) (cid:28) (cid:29)(cid:12) (cid:12) (cid:12) (cid:12) 1−1 1−1 1−1 (cid:12) = (cid:12) α1 pT1 ±α2 pT2 ±...±αn pTn x,x . (cid:12) (cid:12)(cid:28)(cid:18) (cid:19) (cid:29)(cid:12) (cid:12) (cid:12) Now the result(cid:12)follows by taking the supremum over all unit(cid:12) vectors in H . (cid:3) (cid:12) (cid:12) Now we give another upper bound for the powers of w . This result has several p inequalities as special cases, which considerably generalize the second inequality of (1.1).