Extension of Euclidean operator radius inequalities

0.17 MB·English

by  M. S. Moslehian

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).