ON SOME INEQUALITIES FOR THE GENERALIZED EUCLIDEAN OPERATOR RADIUS
MOHAMMAD W. ALOMARI
Abstract. There are many criterion to generalize the concept of numerical radius; one of the most re-cent interesting generalization is what so called the generalized Euclidean operator radius. Simply, it is the numerical radius of multivariable operators. In this work, several new inequalities, refinements and generalizations are established for this kind of numerical radius.
1. Introduction
Let B(H) be the Banach algebra of all bounded linear operators defined on a complex Hilbert space (H;h·,·i) with the identity operator 1H in B(H). WhenH =Cn, we identifyB(H) with the algebra
Mn×n ofn-by-ncomplex matrices. Then, M+n×n is just the cone of n-by-npositive semidefinite matrices. For a bounded linear operator T on a Hilbert space H, the numerical rangeW(T) is the image of the unit sphere ofH under the quadratic formx→ hT x, xiassociated with the operator. More precisely,
W(T) ={hT x, xi:x∈H,kxk= 1} Also, the numerical radius is defined to be
w(T) = sup{|λ|:λ∈W(T)}= sup
kxk=1
|hT x, xi|.
We recall that, the usual operator norm of an operatorT is defined to be kTk= sup{kT xk:x∈H,kxk= 1}.
It is well known thatw(·) defines an operator norm onB(H) which is equivalent to operator normk · k. Moreover, we have
1
2kTk ≤w(T)≤ kTk (1.1)
for anyT ∈B(H) and this inequality is sharp.
It is known thatw(A) is a norm onB(H), but it is not unitarily invariant. But the numerical radius norm is weakly unitarily invariant; i.e.,w(U∗T U) =w(T) for all unitaryU. Also, let us don’t miss the chance to mention the important property thatw(T) =w(T∗) andw(T∗T) =w(T T∗) for everyT ∈B(H).
Denote|T|= (T∗T)1/2 the absolute value of the operator T. Then we have w(|T|) =kTk.
It’s well known that the numerical radius is not submultiplicative, but it satisfies
w(T S)≤4w(T)w(S) for allT, S∈B(H). In particular ifT, S commute, then
w(T S)≤2w(T)w(S). Moreover, ifT, Sare normal thenw(·) is submultiplicative, i.e.,
w(T S)≤w(T)w(S)
Date: January 1, 2020.
2000Mathematics Subject Classification. 47A12, 47B15, 47A30, 47A63.
Key words and phrases. Euclidean operator radius, numerical radius, self-adjoint operator.
1
In 2009, Popsecu [21] introduced the concept of Euclidean operator radius of ann-tupleT= (T1,· · · , Tn)∈
B(H)n := B(H)× · · · ×B(H). Namely, for T1,· · · , Tn ∈ B(H). The Euclidean operator radius of
T1,· · ·, Tn is defined by
we(T1,· · ·, Tn) := sup
kxk=1
n X
i=1
|hTix, xi|2 !1/2
.
Indeed, the Euclidean operator radius was generalized in [24] as follows:
wp(T1,· · · , Tn) := sup
kxk=1
n X
i=1
|hTix, xi| p
!1/p
, p≥1.
Ifp= 1 thenw1(T1,· · · , Tn) (also, it is denoted bywR(T1,· · ·, Tn)) is called the Rhombic numerical radius
which have been studied in [5]. In an interesting case,w1(C,· · ·, C) =n·w(C).
The Crawford number is defined to be
c(T) = inf{|λ|:λ∈W(T)}= inf
kxk=1|hT x, xi|.
Consequently, we define the generalized Crawford number as:
cp(T1,· · · , Tn) := inf
kxk=1
n X
i=1
|hTix, xi|p !1/p
, p≥1.
In case p= 1, the generalized Crawford number is called the Rhombic Crawford number and is denoted by cR(T1,· · ·, Tn).
We note that in casep=∞, the generalized Euclidean operator radius is defined as:
w∞(T1,· · ·, Tn) := sup
kxk=1
n X
i=1
|hTix, xi| − inf
kxk=1
n X
i=1
|hTix, xi|
=wR(T1,· · · , Tn)−cR(T1,· · ·, Tn).
Thus, the inequality
w∞(T1,· · ·, Tn)≤wp(T1,· · ·, Tn)≤wR(T1,· · · , Tn)
(1.2)
for allp∈(1,∞). This fact follows by Jensen’s inequality applied for the function h(p) =wp(T1,· · · , Tn), which is log-convex and decreasing for allp >1.
On the other hand, by employing the Jensen’s inequality
1 n
n X
k=1
ak !p
≤ 1 n
n X
k=1
apk,
which holds for every finite positive sequence of real numbers (ak)nk=1 andp≥1; by settingak =|hTkx, xi| for all (k= 1,2,· · ·, n), we get
n X
k=1
|hTkx, xi| ≤n1−1p
n X
k=1
|hTkx, xi|p !1p
.
Taking the supremum over all unit vectorx∈H, one could get wR(T1,· · · , Tn)≤n1−
1
pw
p(T1,· · · , Tn). (1.3)
Combining the inequalities (1.2) and (1.3) we get
w∞(T1,· · ·, Tn)≤wp(T1,· · · , Tn)≤wR(T1,· · ·, Tn)≤n1−
1
pwp(T1,· · ·, Tn).
(1.4)
More generally, in the power mean inequality
1 n
n X
k=1
apk !1p
≤ 1
n n X
k=1
aqk !1q
if one choosesak =|hTkx, xi|for all (k= 1,2,· · · , n), then we have 1
n n X
k=1
|hTkx, xi| p
!1p
≤ 1
n n X
k=1
|hTkx, xi| q
!1q
.
Taking the supremum over all unit vectorx∈H, we get wp(T1,· · · , Tn)≤n
1
p−1qw
q(T1,· · ·, Tn), ∀q≥p≥1. (1.5)
Indeed, one can refine (1.3) by applying the Jensen’s inequality
1 n
n X
k=1
ak !p
≤ 1 n
n X
k=1
apk− 1 n
n X
k=1
ak− 1 n
n X
j=1
aj p
p≥2 (1.6)
which obtained from more general result for superquadratic functions [1]. Thus, by settingak=|hTkx, xi|in (1.6) we get
n X
k=1
|hTkx, xi| !p
≤np−1 n X
k=1
|hTkx, xi| p
−np−1 n X
k=1
|hTkx, xi| − 1 n
n X
j=1
|hTjx, xi| p
≤np−1 n X
k=1
|hTkx, xi|p−np−1 n X
k=1
|hTkx, xi| −1 nksupxk=1
n X
j=1
|hTjx, xi| p
.
Taking the supremum again over all unit vectorx∈H, we get
sup
kxk=1
n X
k=1
|hTkx, xi| !p
≤ sup
kxk=1
np−1
n X
k=1
|hTkx, xi|p−np−1 n X
k=1
|hTkx, xi| −1 nksupxk=1
n X
j=1
|hTjx, xi|
p
≤np−1 sup
kxk=1
n X
k=1
|hTkx, xi| p
−np−1 inf
kxk=1
n X
k=1
|hTkx, xi| − 1 nksupxk=1
n X
j=1
|hTjx, xi| p
=np−1wpp(T1,· · ·, Tn)−np−1 inf
kxk=1
n X
k=1
|hTkx, xi| − 1
nwR(T1,· · ·, Tn) p
,
which gives
wRp (T1,· · · , Tn)≤np−1wpp(T1,· · ·, Tn)−np−1 inf
kxk=1
n X
k=1
|hTkx, xi| − 1
nwR(T1,· · ·, Tn) p
.
which refine the right hand side of (1.4). Clearly, all above mentioned inequalities generalize and refine some inequalities obtained in [20]. For recent inequalities, counterparts, refinements and other related properties concerning the generalized Euclidean operator radius the reader my refer to [5], [9] ,[12],[13], [21], [23] and[24].
2. Bounds for the generalized Euclidean operator radius
Lemma 1. We have
(1) The Power-Mean inequality
aαb1−α≤αa+ (1−α)b≤(αap+ (1−α)bp)p1
(2.1)
for allα∈[0,1],a, b≥0 andp≥1. (2) The Power-Young inequality
ab≤ a α α +
bβ β ≤
apα α +
bpβ β
p1
for alla, b≥0 andα, β >1 with α1 +β1 = 1 and allp≥1. Lemma 2. (The McCarty inequality). Let A∈B(H)+, then
hAx, xip≤ hApx, xi, p≥1 (2.3)
for any unit vectorx∈H
Lemma 3. Ifa, b >0, andp, q >1 such that 1
p+
1
q = 1, then form= 1,2,3, . . . , (a1pb
1
q)m+rm
0 (a
m
2 −bm2)2≤
ar p +
br q
mr
, r≥1, (2.4)
wherer0=min
n
1
p,
1
q o
. In particular, if p=q= 2, then
(a12b 1
2)m+ 1 2m(a
m
2 −b
m
2)2≤2
−m
r (ar+br) m r .
Form= 1
(a12b 1 2) +1
2(a 1 2 −b
1 2)2≤2
−1
r (ar+br)
1
r.
In 1994, Furuta [11] proved the following generalization of Kato’s inequality (1.3)
D
T|T|α+β−1x, yE
2
≤D|T|2αx, xE D|T|2βy, yE (2.5)
for anyx, y∈H andα, β∈[0,1] withα+β≥1.
The inequality (2.5) was generalized for anyα, β≥0 withα+β≥1 by Dragomir in [8]. Indeed, as noted by Dragomir the conditionα, β∈[0,1] was assumed by Furuta to fit with the Heinz–Kato inequality, which reads:
|hT x, yi| ≤ kAαxk B1−αy
for any x, y ∈ H and α ∈ [0,1] where A and B are positive operators such that kT xk ≤ kAxk and kT∗yk ≤ kBykfor anyx, y∈H.
In the same work [8], Dragomir provides a useful extension of Furuta’s inequality, as follows:
|hDCBAx, yi|2≤DA∗|B|2Ax, xE DD|C∗|2D∗y, yE (2.6)
for anyA, B, C, D∈B(H) and any vectorsx, y∈H. The equality in (2.6) holds iff the vectorsBAxand C∗D∗y are linearly dependent inH. For other closely related version of Kato’s inequality see [2], [14], [15], [18], [19] and [22].
2.1. Basic properties of the generalized Euclidean operator radius. Moslehian et al. [20], mention without proofs the following properties of the generalized Euclidean operator radius:
(1) wp(T1,· · ·, Tn) = 0 if and only ifTk = 0 for eachk= 1,· · ·, n.
(2) wp(λT1,· · ·, λTn) =|λ|wp(T1,· · ·, Tn).
(3) wp(A1+B1,· · ·, An+Bn)≤wp(A1,· · ·, An) +wp(B1,· · · , Bn).
(4) wp(X∗T1X,· · · , X∗TnX) =kXkwp(T1,· · ·, Tn).
for everyTk, Ak, Bk, X∈B(H) (1≤k≤n) and every scalarλ∈C.
LetT1,· · ·, Tn, U ∈ B(H) such that U is a unitary. Then, the following properties of the generalized
Euclidean operator radius holds.
(1) The generalized Euclidean operator radius is weakly unitarily invariant i.e.,
wp(U∗T1U,· · · , U∗TnU) =wp(T1,· · · , Tn). (2) wp(T1,· · ·, Tn) =wp(T1∗,· · ·, Tn∗).
(3) wp(T1∗T1,· · ·, Tn∗Tn) =wp(T1T1∗,· · ·, TnTn∗).
Proof. (1) The first property follows since
wp(U1∗T1U1,· · ·, Un∗TnUn) := sup
kxk=1
n X
i=1
|hUi∗TiUix, xi| p
!1/p
= sup
kxk=1
n X
i=1
|hTiUix, Uixi|p !1/p
= sup
kyk=1
n X
i=1
|hTiy, yi| p
!1/p
=wp(T1,· · · , Tn).
(2) By the definition of the generalized Euclidean operator radius we have
wp(T1,· · ·, Tn) := sup
kxk=1
n X
i=1
|hTix, xi|p !1/p
= sup
kxk=1
n X
i=1
|hx, Ti∗xi|p !1/p
=wp(T1∗,· · · , Tn∗). (3) Similarly, by definition we have
wp(T1T1∗,· · ·, TnT
∗
n) := sup
kxk=1
n X
i=1
|hTiTi∗x, xi|p !1/p
= sup
kxk=1
n X
i=1
|hx, TiTi∗xi|p !1/p
=wp(T1∗T1,· · · , Tn∗Tn). (4) Finally, employing the classical Minlowski inequality, i.e.,
we get
wp(A1+B1,· · ·, An+Bn) =
n X
i=
|h(Ai+Bi)x, xi|p !1p
= n X
i=
(|hAix, xi|+|hBix, xi|)p !1p
≤ n X
i=
|hAix, xi| p
!1p
+ n X
i=
|hBix, xi| p
!1p
=wp(A1,· · · , An) +wp(B1,· · · , Bn).
It remains to provewp(X∗T1X,· · · , X∗TnX) =kXkwp(T1,· · ·, Tn). Form the definition of the
general-ized Euclidean operator radius, we have
wp(X∗T1X,· · ·, X∗TnX) := sup
kxk=1
n X
i=1
|hX∗TiXx, xi|p !1/p
= sup
kxk=1
n X
i=1
|hTiXx, Xxi| p
!1/p
≤ sup
kxk=1
n X
i=1
kXk2|hTix, xi|p !1/p
=kXk2 sup
kxk=1
n X
i=1
|hTix, xi|p !1/p
=kXk2wp(T1∗,· · · , Tn∗).
as required.
Proposition 1. Let T1,· · · , Tn ∈B(H)and f, g are nonnegative continuous functions defined on [0,∞) satisfying thatf(t)g(t) =t(t≥0). Then,
wrr(T1,· · ·, Tn)≤wp(fr(|T1|),· · ·, fr(|Tn|))wq(gr(|T1∗|),· · · , gr(|Tn∗|)) (2.7)
for allr≥2 andp, q >1 with 1p+1q = 1. Proof.
|hTix, xi|r≤ kf(|Ti|)xkrkg(|Ti∗|)xkr
=
f2(|Ti|)x, x r2
g2(|Ti∗|)x, xr2
≤ hfr(|Ti|)x, xi hgr(|Ti∗|)x, xi (by the McCarthy inequality) Taking the sum over allifrom 1 tonwe get
n X
i=1
|hTix, xi| r
≤ n X
I=1
hfr(|Ti|)x, xi hgr(|Ti∗|)x, xi
≤ n X
I=1
hfr(|Ti|)x, xip
!1p n X
I=1
hgr(|Ti∗|)x, xiq !1q
(by the H¨older inequality)
Taking the supremum over all vectorsx∈H such thatkxk= 1, we get the desired result.
Corollary 1. Let T1,· · ·, Tn ∈ B(H) and f, g are nonnegative continuous functions defined on [0,∞)
satisfying thatf(t)g(t) =t(t≥0). Then,
wrr(T1,· · · , Tn)≤we(fr(|T1|),· · ·, fr(|Tn|))we(gr(|T1∗|),· · · , g
r(|T∗
n|)) (2.8)
for allr≥2.
Proof. Settingp=q= 2 in (2.7).
Proposition 2. Let Ai, Bi, Ci, Di∈B(H) (i= 1,· · ·, n). Then, (2.9) we2(D1C1B1A1,· · ·, DnCnBnAn)
≤wpA∗1|B1| 2
A1,· · ·, A∗n|Bn|
2
AnwqD1|C1∗| 2
Proof. Lety=xin (2.6), we get
sup
kxk=1
n X
i=1
|hDiCiBiAix, xi|
2
≤ sup
kxk=1
n X
i=1
D A∗i |Bi|
2
Aix, x E D
Di|Ci∗|
2
Di∗x, xE
≤ sup
kxk=1
n X
i=1
D A∗i|Bi|
2
Aix, xE p!
1
p
sup
kxk=1
n X
i=1
D Di|Ci∗|
2
Di∗x, x Eq
!1q
=wpA∗1|B1|2A1,· · ·, A∗n|Bn|
2
AnwqD1|C1∗| 2
D1∗,· · · , Dn|Cn∗|2D∗n,
which is gives the desired result.
Corollary 2. LetBi∈B(H) (i= 1,· · · , n). Then, we2 B12,· · ·, Bn2≤wp|B1|
2
,· · · ,|Bn|2wq|B1| 2
,· · · ,|Bn|2 (2.10)
for allp, q >1such that 1p+1q = 1. In particular, for p=q= 2 we have we B21,· · ·, B
2
n
≤we
|B1|
2
,· · · ,|Bn|
2 (2.11)
Proof. Setting Ai=Ui, Di =Ui∗ (Ui are unitaries for all i= 1,· · ·, n) andCi =Bi in the previous result. Then
w2e U1∗B12U1,· · · , Un∗B
2
nUn
≤wpU1∗|B1| 2
U1,· · ·, Un∗|Bn|
2
UnwqU1∗|B1∗|2U1,· · ·, Un∗|B
∗
n|
2
Un. Sincewp(·) is weakly unitarily invariant and
wq
|B1∗|2,· · ·,|Bn∗|2=wq
|B1| 2
,· · ·,|Bn|
2
.
Thus, the desired result is obtained.
Corollary 3. LetTi∈B(H) (i= 1,· · ·, n),α, β≥0 such thatα+β ≥1. Then, we2T1|T1|α+β−1,· · ·, Tn|Tn|α+β−1
≤wp|T1|2α,· · ·,|Tn|2α
wq|T1∗|2β,· · ·,|Tn∗|2β (2.12)
for allp, q >1such that 1p+1q = 1.
Proof. LetUi be unitaries for alli= 1,· · · , n, settingDi =Ui,B = 1H, C=|Ti|β andAi =|Ti|α for all α, β≥0 such thatα+β≥1 in (2.9), then we have
DiCiBiAi=Ui|Ti|β|Ti|α=Ui|Ti| |Ti|α+β−1=Ti|Ti|α+β−1,
also, we haveA∗i|Bi|2Ai=|Ti|2α andDi|Ci∗|2D∗i =Ui|Ti|2βUi∗=|Ti|2β for alli= 1,· · ·, n. Corollary 4. LetTi∈B(H) (i= 1,· · ·, n). Then,
we2(T1,· · ·, Tn)≤wp(|T1|,· · · ,|Tn|)wq(|T1∗|,· · ·,|T
∗
n|) (2.13)
for allp, q >1such that 1p+1q = 1.
Proof. Settingα=β= 12 in (2.12).
Corollary 5. LetTi∈B(H) (i= 1,· · ·, n),α, β≥0 such thatα+β ≥1. Then, w2e(T1|T1|,· · · , Tn|Tn|)≤wp
|T1|
2
,· · · ,|Tn|
2
wq
|T1| 2
,· · ·,|Tn|
2
(2.14)
for allp, q >1such that 1p+1q = 1. In particular, for p=q= 2 we have we2(T1|T1|,· · · , Tn|Tn|)≤we2
|T1|
2
,· · ·,|Tn|2 (2.15)
2.2. Inequalities for the generalized Euclidean operator radius. Theorem 1. Let Ai, Bi, Ci, Di∈B(H) (i= 1,· · ·, n). Then,
wR(D1C1B1A1,· · ·, DnCnBnAn)≤
n X
i=1
1 2p
A∗i|Bi|
2
Ai
(1−γ)pr
+A∗i|Bi|
2
Ai γpr (2.16)
+1 2q
Di|Ci∗|
2
D∗i γqr
+Di|Ci∗|
2
D∗i
(1−γ)qr
for allr≥1,p, q >1 andγ∈[0,1]such thatpγ ≥2 and 1
p+
1
q = 1. Proof. Lety=xin (2.6), then we have
n X
i=1
|hDiCiBiAix, xi|
= n X
i=1
D
A∗i|Bi|2Aix, xE 1 2D
Di|Ci∗|2Di∗x, xE 1 2
≤1 2
n X
i=1
D
A∗i |Bi|
2
Aix, xE
1−γD Di|Ci∗|
2
Di∗x, x Eγ
(since √
ab≤ a
γb1−γ+a1−γbγ
2 )
+DA∗i|Bi|
2
Aix, x EγD
Di|Ci∗|
2
D∗ix, xE
1−γ
≤1 2
n X
i=1
( D
A∗i |Bi|
2
Aix, x E(1−γ)p
+DA∗i|Bi|
2
Aix, x Eγp
1
p
( by H¨older inequality)
× D
Di|Ci∗|
2
Di∗x, xE γq
+DDi|Ci∗|
2
D∗ix, xE
(1−γ)q1q)
≤ 1 2p
n X
i=1
A∗i|Bi|
2
Ai
(1−γ)p x, x
+DA∗i|Bi|
2
Ai γp
x, xE
( by AM-GM)
+ 1
2q n X
i=1
D
Di|Ci∗|
2
D∗i γq
x, xE+
Di|Ci∗|
2
D∗i
(1−γ)q x, x
= n X
i=1
1
2p
A∗i|Bi|
2
Ai (1−γ)p
+A∗i |Bi|
2
Ai γp
+1 2q
Di|Ci∗|2D∗i γq
+Di|Ci∗|2Di∗
(1−γ)q x, x
.
Taking the supremum over all unit vectorx∈H we get the required result.
Corollary 6. LetAi, Bi, Ci, Di∈B(H) (i= 1,· · ·, n). Then, (2.17) wR(D1C1B1A1,· · ·, DnCnBnAn)
≤
n X
i=1
1
4
A∗i|Bi|
2
Ai 2(1−γ)
+A∗i|Bi|
2
Ai 2γ
+
Di|Ci∗|
2
D∗i
2γ
+Di|Ci∗|
2
Di∗
2(1−γ)
for allγ∈[0,1]
Proof. Settingp=q= 2 andr= 1 in (2.16).
Corollary 7. LetTi∈B(H) (i= 1,· · ·, n). Then,
(2.18) wR
T1|T1|
α+β−1
,· · · , Tn|Tn| α+β−1
≤
n X
i=1
1
2p h
|Ti|2α(1−γ)pr+|Ti|2αγpri+ 1 2q
h
|Ti∗|2βγqr+|Ti∗|2β(1−γ)qri
for allr≥1,p, q >1 andγ∈[0,1]such thatpγ ≥2 and 1p+1q = 1. In particular, we have
(2.19) wRT1|T1|α+β−1,· · · , Tn|Tn|α+β−1
≤1 4
n X
i=1
nh
|Ti|4α(1−γ)r+|Ti|4αγri+h|Ti∗|4βγr+|Ti∗|4β(1−γ)rio
Proof. The proof is similar to the inequality (2.12) by employing (2.16).
Remark 1. Setting γ= 1,α=β= 12 andr= 2 in (2.19), so that we have
wR(T1,· · · , Tn)≤ 1 4
n X
i=1
nh
1H +|Ti|
4i
+h|Ti∗|4+ 1Hio (2.20)
Corollary 8. LetTi∈B(H) (i= 1,· · ·, n). Then,
wRT1|T1|
α+β−1
,· · ·, Tn|Tn|α+β−1≤1 4
n X
i=1
h
|Ti|83α+|Ti| 4 3α+|T∗
i| 8 3β+|T∗
i| 4 3β
i (2.21)
Proof. Settingp=q= 2,r= 1 andγ=13 in (2.19).
Corollary 9. LetTi∈B(H) (i= 1,· · ·, n). Then,
wRT1|T1|
1
2,· · ·, Tn|Tn|12
≤ 1 4
n X
i=1
|Ti|2+|Ti∗|2+|Ti|+|Ti∗|
Proof. Settingα=β= 34 in (2.21).
Theorem 2. Let Ai, Bi, Ci, Di∈B(H) (i= 1,· · ·, n). Then,
(2.22) wpp(D1C1B1A1,· · · , DnCnBnAn)
≤1 2
n X
i=1
h
(1−γ)A∗i |Bi|
2
Ai+γDi|Ci∗|
2
D∗i ip
+hγA∗i |Bi|
2
Ai+ (1−γ)Di|Ci∗|
2
D∗i ip
for allγ∈[0,1]andp≥1. In particular, we have
wpp(D1C1B1A1,· · ·, DnCnBnAn)≤
1 2p
n X
i=1
A∗i|Bi|
2
Ai+Di|Ci∗|
2
D∗i p
Proof. Lety=xin (2.6), then we have wpp(D1C1B1A1,· · ·, DnCnBnAn)
= n X
i=1
|hDiCiBiAix, xi|p
= n X
i=1
D A∗i|Bi|
2
Aix, x E12D
Di|Ci∗|
2
Di∗x, xE 1 2
p
≤ 1 2p
n X
i=1
D
A∗i|Bi|2Aix, xE
(1−γ)D
Di|Ci∗|2D∗ix, xE γ
(since√ab≤ a
γb1−γ+a1−γbγ
2 )
+
D A∗i|Bi|
2
Aix, xE γD
Di|Ci∗|
2
D∗ix, x
E(1−γ)p
≤1 2
n X
i=1
h
(1−γ)DA∗i|Bi|
2
Aix, xE+γDDi|Ci∗|
2
D∗ix, x Ep
(by AM-GM inequality)
+γDA∗i|Bi|
2
Aix, xE+ (1−γ)DDi|Ci∗|
2
Di∗x, x Epi
=1 2
n X
i=1
Dh
(1−γ)A∗i|Bi|2Ai+γDi|Ci∗|2Di∗ix, xE p
+1 2
n X
i=1
Dh
γA∗i|Bi|2Ai+ (1−γ)Di|Ci∗|2Di∗ix, xE p
≤1 2
n X
i=1
Dh
(1−γ)A∗i|Bi|
2
Ai+γDi|Ci∗|
2
Di∗ ip
x, xE (by McCarthy inequality)
+1 2
n X
i=1
Dh
γA∗i|Bi|
2
Ai+ (1−γ)Di|Ci∗|
2
Di∗i p
x, xE
=1 2
* n X
i=1
nh
(1−γ)A∗i|Bi|
2
Ai+γDi|Ci∗|
2
Di∗ ipo
,
+hγA∗i |Bi|
2
Ai+ (1−γ)Di|Ci∗|
2
D∗i ip
x, xEi.
Taking the supremum over all unit vectorx∈H we get the required result. The particular case is obtained by settingγ=1
2 in (2.22).
Corollary 10. Let Ti∈B(H) (i= 1,· · · , n). Then, for all α, β≥0 such thatα+β ≥1 we have wppT1|T1|
α+β−1
,· · ·, Tn|Tn|α+β−1≤ 1 2p
n X
i=1
|Ti|2α+|Ti∗|2β p
(2.24)
for allp≥1.
Proof. LetUi be unitaries for alli= 1,· · · , n, settingDi =Ui,B = 1H, C=|Ti| β
andAi =|Ti| α
for all α, β≥0 such thatα+β≥1 in (2.23), then we have
DiCiBiAi=Ui|Ti| β
|Ti| α
=Ui|Ti| |Ti| α+β−1
=Ti|Ti| α+β−1
,
also, we haveA∗i|Bi|2Ai=|Ti|2α andDi|Ci∗|2D∗i =Ui|Ti|2βUi∗=|Ti|2β for alli= 1,· · ·, n. Corollary 11. Let Ti∈B(H) (i= 1,· · · , n). Then, we have
wpp(T1|T1|,· · · , Tn|Tn|)≤ 1 2p
n X
i=1
|Ti|
2
+|Ti∗|2p (2.25)
Theorem 3. Let Ai, Bi, Ci, Di∈B(H) (i= 1,· · ·, n). Then, form∈Nandr, p≥m≥1, (2.26) wpp(D1C1B1A1,· · · , DnCnBnAn)
≤ n
1−m r
2mr n X i=1
A∗i |Bi|
2
Ai rpm
+Di|Ci∗|
2
Di∗
rp m m r − inf
kxk=1
ξp,m(x),
where
ξp,m(x) := 2−m n X
i=1
A∗i|Bi|
2 Ai p m x, x m2 − Di|Ci∗|
2
Di∗ mp
x, x m2!
2
.
Proof. Letx∈H be a unit vector. Lety=xin (2.6) then we have n
X
i=1
|hDiCiBiAix, xi|p
= n X
i=1
D A∗i|Bi|
2
Aix, x Ep2D
Di|Ci∗|
2
D∗ix, xE
p 2 = n X i=1 D A∗i|Bi|
2
Aix, x E2pmD
Di|Ci∗|
2
D∗ix, xE
p 2m m ≤ n X i=1
A∗i|Bi|2Ai
p m
x, x 12
Di|Ci∗|2Di∗
p m
x, x 12!
m
(by McCarthy inequality)
≤ n X i=1 1 2
A∗i |Bi|2Ai
p m x, x r +
Di|Ci∗|2D∗i
p m
x, x rmr
(by (2.4))
−2−m n X
i=1
A∗i|Bi|
2 Ai mp x, x m2 − Di|Ci∗|
2
D∗i
p m
x, x m2!
2 ≤ n X i=1 1 2
A∗i |Bi|2Ai
rp m x, x +
Di|Ci∗|2D∗i
rp m
x, x mr
(by McCarthy inequality)
−2−m n X
i=1
A∗i|Bi|2Ai
p m x, x m2 −
Di|Ci∗|2D∗i
p m
x, x m2!
2
≤n
1−m r
2mr
n X i=1 1 2
A∗i |Bi|
2 Ai rpm x, x +
Di|Ci∗|
2
D∗i
rp m
x, x mr
(by concavity oftmr)
−2−m n X
i=1
A∗i|Bi|
2 Ai p m x, x m2 − Di|Ci∗|
2
D∗i mp
x, x m2!
2
Theorem 4. LetTi∈B(H) (i= 1,· · ·, n),α, β≥0 such thatα+β≥1,m∈Nandr, p≥m≥1. Then
wppT1|T1|
α+β−1
,· · ·, Tn|Tn| α+β−1
≤n
1−m r
2mr n X i=1 |Ti|
2αrp m +|T∗
i| 2βrp m m r − inf
kxk=1
ψp,m,α,β(x), (2.27)
where
ψp,m,α,β(x) := 2−m n X
i=1
D |Ti|
2αp m x, x
Em2
−D|Ti∗|2mβpx, x Em2
2
.
Proof. LetUi be unitaries for alli= 1,· · · , n, settingDi =Ui,B = 1H, C=|Ti|β andAi =|Ti|α for all α, β≥0 such thatα+β≥1 in (2.26), then we have
DiCiBiAi=Ui|Ti| β
|Ti| α
=Ui|Ti| |Ti| α+β−1
=Ti|Ti| α+β−1
,
also, we haveA∗i|Bi|2Ai=|Ti|2α andDi|Ci∗|2D∗i =Ui|Ti|2βUi∗=|Ti|2β for alli= 1,· · ·, n. Corollary 12. Let Ti ∈ B(H) (i= 1,· · · , n), α, β ≥0 such that α+β ≥1, m∈ N andr, p ≥m ≥1. Then
wppT1|T1|
α+β−1
,· · · , Tn|Tn| α+β−1
≤ 1 2
n X
i=1
|Ti|
2αp
+|Ti∗|2βp
− inf
kxk=1
ψp,1,α,β(x), (2.28)
where
ψp,1,α,β(x) := 1 2
n X
i=1
D
|Ti|2αpx, xE 1 2
−D|Ti∗|2βpx, xE 1 2
2
.
for allα, β≥0such that α+β≥1.
Proof. Settingm=r= 1 in (2.27).
Corollary 13. Let Ti∈B(H) (i= 1,· · · , n). Then, form∈Nandr, p≥m≥1,
wpp(T1,· · · , Tn)≤ n1−m
r
2mr
n X
i=1
|Ti|
rp m +|T∗
i|
rp m
m
r − inf
kxk=1
ψp,m,1 2,
1 2(x), (2.29)
where
ψp,m,1 2,
1
2(x) := 2
−m n X
i=1
D
|Ti|
p mx, x
Em2
−D|Ti∗|mp x, x Em2
2
.
for allα, β≥0such that α+β≥1. Proof. Settingα=β= 1
2 in (2.27).
Corollary 14. Let Ti∈B(H) (i= 1,· · · , n). Then, form∈Nandr, p≥1,
wpp(T1,· · · , Tn)≤
n1−1r
21r
n X
i=1
(|Ti|rp+|Ti∗|rp) 1
r − inf
kxk=1ψp,1,
1 2,12(x), (2.30)
where
ψp,1,1
2,12(x) := 1 2
n X
i=1
h|Ti|px, xi12 − h|T∗ i|
p
x, xi12
2
.
for allα, β≥0such that α+β≥1. Proof. Settingα=β= 1
2 andm= 1 in (2.29).
Corollary 15. Let Ti∈B(H) (i= 1,· · · , n). Then, form∈Nandp≥m≥1, wpp(T1,· · ·, Tn)≤
1 2
n X
i=1
(|Ti|p+|Ti∗|p)
− inf
kxk=1ψp,m,
1 2,12(x), (2.31)
where
ψp,m,1
2,12(x) := 2
−m n X
i=1
D
|Ti|mp x, x Em2
−D|Ti∗|mp x, x Em2
2
.
Corollary 16. Let Ti∈B(H) (i= 1,· · · , n). Then, form∈Nandr, p≥m≥1, we have wpp(T1|T1|,· · ·, Tn|Tn|)≤
n1−m r
2mr
n X
i=1
|Ti|2mrp +|T∗
i| 2rp
m
m
r − inf
kxk=1
ψp,m,1,1(x),
(2.32)
where
ψp,m,1,1(x) := 2−m
n X
i=1
D
|Ti|2mpx, x Em2
−D|Ti∗| 2p m x, x
Em2
2
.
for allα, β≥0such that α+β≥1.
Proof. Settingα=β= 1 in (2.27).
Corollary 17. Let Bi, Ci, Ui, Vi ∈B(H) (i= 1,· · ·, n)such that Ui, Vi unitaries. Then, for m∈Nand r, p≥m≥1, we have
(2.33) wpp(U1∗C1B1U1,· · · , Un∗CnBnUn)
≤n
1−m r
2mr
n X
i=1
Ui∗|Bi|
2
Ui rpm
+Ui∗|Ci∗|2Ui rpm
m r
− inf
kxk=1
ξp,m(x),
where
ξp,m(x) := 2−m n X
i=1
Ui∗|Bi|
2
Ui mp
x, x m2
−
Ui∗|Ci∗|2Ui mp
x, x m2!
2
.
Proof. SettingAi=Ui andDi=Ui∗ in (2.26) and using the fact that . Corollary 18. Let Ai, Di∈B(H) (i= 1,· · ·, n). Then, form∈Nandr, p≥m≥1,
wpp(D1A1,· · · , DnAn)≤ n1−m
r
2mr
n X
i=1
(A∗iAi)
rp m + (D
iD∗i)
rp m
m
r − inf
kxk=1
ξp,m(x), (2.34)
where
ξp,m(x) := 2−m n X
i=1
D
(A∗iAi)mp x, x Em2
−D(DiD∗i)mp x, x Em2
2
.
Proof. SettingBi=Ci= 1H for alli= 1,· · ·, nin (2.26). Corollary 19. Let Ai, Di∈B(H) (i= 1,· · ·, n). Then, form∈Nandr, p≥m≥1,
wpp(A∗1A1,· · ·, A∗nAn)≤n1−
m r
n X
i=1
(A∗iAi)
rp m m
r
. (2.35)
In particular, form=r
wpp(A∗1A1,· · ·, A∗nAn)≤
n X
i=1
(A∗iAi) p
.
Proof. SettingDi=A∗i for alli= 1,· · · , nin (2.34). 3. Upper and Lower bounds for the generalized Euclidean operator radius
In this section we provide some lower and upper bounds for the product of the generalized Euclidean operator radius. In order to to prove our results we need to recall the the following H¨older type inequality, which reads:
n X
j=1
|xjyj|r
1
r
≤
n X
j=1
|xj|p
1
p
n X
j=1
|yj|q
1
q
(3.1)
Theorem 5. Let Di, Ci, Bi, Ai∈B(H) (i= 1,· · ·, n),r≥1 andp, q≥1 with 1p+1q =1r. Then 1
n2r−1
n X i=1 DiCiBiAi 2r
≤wrpA∗1|B1|2A1,· · · , A∗n|Bn|
2
AnwqrD1|C1∗| 2
D∗1,· · ·, Dn|Cn∗|2D∗n (3.2) ≤max r p, r q n X i=1
A∗i|Bi|2Ai p
+Di|Ci∗|2Di∗ q
− inf
kxk=1λ(x, y),
where,
λ(x, y) := min r p, r q v u u t n X i=1 D
A∗i|Bi|2Aix, yE p − v u u t n X i=1 D
Di|Ci∗|2Di∗x, yE q
2
Proof. Letx, y∈H. Applying inequality (3.1) and the convexity oft2r, we have
1 n2r−1
* n X
i=1
DiCiBiAi ! x, y + 2r = 1
n2r−1
n X i=1
h(DiCiBiAi)x, yi 2r ≤ 1
n2r−1
n X
i=1
|h(DiCiBiAi)x, yi| !2r
≤ n X
i=1
|h(DiCiBiAi)x, yi|2r !
(by Jensen’s inequality)
≤ n X
i=1
D A∗i|Bi|
2
Aix, y E D
Di|Ci∗|
2
D∗ix, yE r!
(by (2.6))
≤ n X
i=1
D
A∗i|Bi|2Aix, yE p! r p n X i=1 D
Di|Ci∗|2D∗ix, yE q! r q (3.3) ≤r p n X i=1 D
A∗i |Bi|2Aix, yE p +r q n X i=1 D
Di|Ci∗|2D∗ix, yE q
by (2.4)
−min r p, r q v u u t n X i=1 D
A∗i|Bi|2Aix, yE p − v u u t n X i=1 D
Di|Ci∗|2D∗ix, yE q 2 ≤r p n X i=1 D A∗i |Bi|
2
Ai p
x, yE+r q
n X
i=1
D Di|Ci∗|
2
Di∗ q
x, yE (by McCarthy inequality)
−min r p, r q v u u t n X i=1 D A∗
i|Bi|
2
Aix, y Ep − v u u t n X i=1 D Di|Ci∗|
2
D∗
ix, y Eq 2 ≤max r p, r q n X i=1 A∗i|Bi|
2
Ai p
+Di|Ci∗|
2
Di∗ q
(properties of max)
−min r p, r q v u u t n X i=1 D A∗
i|Bi|
2
Aix, y Ep − v u u t n X i=1 D Di|Ci∗|
2
D∗
ix, y Eq
Taking the supremum over x, y ∈ H with kxk = kyk = 1, then the left and right hand side follows immediately the middle term of the inequality follows by (3.3), and thus the desired result is obtained.
Corollary 20. Let Di, Ci, Bi, Ai∈B(H) (i= 1,· · · , n),r≥1. Then 1 n n X i=1
DiCiBiAi 2
≤we
A∗1|B1|
2
A1,· · · , A∗n|Bn|
2
Anwe
D1|C1∗|
2
D∗1,· · ·, Dn|Cn∗|
2
D∗n (3.4) ≤ 1 2 n X i=1
A∗i|Bi|
2
Ai 2
+Di|Ci∗|
2
D∗i
2
− inf
kxk=kyk=1
λ(x, y),
where
λ(x, y) :=1 2 v u u t n X i=1 D
A∗i|Bi|2Aix, yE
2 − v u u t n X i=1 D
Di|Ci∗|2Di∗x, yE
2
2
Proof. Settingp=q= 2 andr= 1 in (3.2) we get the desired result. Corollary 21. Let Di, Ai∈B(H) (i= 1,· · ·, n),r≥1 andp, q≥1 with 1p+1q =1r. Then
1 n2r−1
n X i=1 DiAi 2r
≤wpr(A∗1A1,· · · , A∗nAn)w r
q(D1D∗1,· · ·, DnD
∗ n) (3.5) ≤max r p, r q n X i=1
(A∗iAi) p
+ (DiDi∗) q − inf
kxk=1
λ(x, y),
where,
λ(x, y) := min r p, r q v u u t n X i=1 D A∗
i|Bi|
2
Aix, y Ep − v u u t n X i=1 D Di|Ci∗|
2
D∗
ix, y Eq
2
Proof. SettingBi=Ci= 1H in (3.2) we get the required result. Corollary 22. Let Di, Ai∈B(H) (i= 1,· · ·, n),r≥1 andp, q≥1 with 1p+1q =1r. Then
1 n2r−1
n X i=1
A∗iAi 2r
≤wpr(A∗1A1,· · · , A∗nAn)w r q(AiA
∗
i,· · · , AiA
∗ i) (3.6) ≤max r p, r q n X i=1
(A∗iAi)p+ (AiA∗i)q − inf
kxk=1λ(x, y),
where,
λ(x, y) := min r p, r q v u u t n X i=1
hA∗iAix, yip− v u u t n X i=1
hAiA∗ix, yiq
2
Proof. SettingDi=Ai andBi=Ci= 1H in (3.5). Corollary 23. Let Di, Ai∈B(H) (i= 1,· · ·, n),r≥1 andp, q≥1 with 1p+
1
q =
1
r. Then
1 n n X i=1
A2i 2
≤we2(A∗1A1,· · · , A∗nAn)≤ 1 2 n X i=1
(A∗iAi)
2
+ (AiA∗i)
2 − inf
kxk=kyk=1λ(x, y),
where,
λ(x, y) := 1 2 v u u t n X i=1
hA∗iAix, yi2− v u u t n X i=1
hAiA∗ix, yi2
2
Proof. Settingp=q= 2 andr= 1 in (3.6). Theorem 6. Let Ti ∈ B(H) (i = 1,· · ·, n), r ≥ 1, p, q ≥ 1 with p1 +1q = 1r and α, β ≥ 0 such that α+β≥1. Then
1 n2r−1
n X i=1
Ti|Ti|α+β−1 2r
≤wrp|T1| 2α
,· · ·,|Tn|2αwqr|T1∗|2β,· · ·,|Tn∗|2β (3.8) ≤max r p, r q n X i=1
|Ti|2pα+|Ti∗|2qβ − inf
kxk=kyk=1
λ(x, y),
where,
λ(x, y) := min r p, r q v u u t n X i=1 D
|Ti|2αx, yE p − v u u t n X i=1 D
|Ti∗|2βx, yE q
2
Proof. LetUi be unitaries for alli= 1,· · · , n, settingDi =Ui,B = 1H, C=|Ti|β andAi =|Ti|α for all α, β≥0 such thatα+β≥1 in (3.2), then we have
DiCiBiAi=Ui|Ti| β
|Ti| α
=Ui|Ti| |Ti| α+β−1
=Ti|Ti| α+β−1
,
also, we haveA∗i|Bi|2Ai=|Ti|2α andDi|Ci∗|2D∗i =Ui|Ti|2βUi∗=|Ti|2β for alli= 1,· · ·, n.
Corollary 24. Let Ti∈B(H) (i= 1,· · · , n),r≥1 andp, q≥1 with 1p+
1
q =
1
r. Then 1
n2r−1
n X i=1 Ti 2r
≤wrp(|T1|,· · ·,|Tn|)wqr(|T
∗
1|,· · ·,|T
∗
n|) (3.9) ≤max r p, r q n X i=1
|Ti| p
+|Ti∗|q − inf
kxk=kyk=1
λ(x, y),
where,
λ(x, y) := min r p, r q v u u t n X i=1
h|Ti|x, yip− v u u t n X i=1
h|Ti∗|x, yiq
2
Proof. Settingα=β= 12 in (3.8).
Corollary 25. Let Ti∈B(H) (i= 1,· · · , n),α, β≥0such that α+β≥1. Then 1 n n X i=1
Ti|Ti| α+β−1
2
≤we
|T1|
2α
,· · · ,|Tn|
2α we
|T1∗|2β,· · ·,|Tn∗|2β (3.10) ≤ 1 2 n X i=1
|Ti|4α+|Ti∗|4β − inf
kxk=kyk=1λ(x, y),
where,
λ(x, y) := 1 2 v u u t n X i=1 D |Ti|
2α x, yE
2 − v u u t n X i=1 D |T∗
i|
2β x, yE
2
2
Corollary 26. Let Ti∈B(H) (i= 1,· · · , n). Then 1
n
n X
i=1
Ti
2
≤we(|T1|,· · ·,|Tn|)we(|T1∗|,· · · ,|Tn∗|) (3.11)
≤ 1 2
n X
i=1
Ti∗Ti+TiTi∗
− inf
kxk=kyk=1λ(x, y),
where,
λ(x, y) := 1 2
v u u t
n X
i=1
h|Ti|x, yi2− v u u t
n X
i=1
h|Ti∗|x, yi2
2
Proof. Settingα=β= 12 in (3.10).
We note that, in 2005, Kittaneh in [16] proved that
1 4kT
∗T+T T∗k ≤w2(T)
≤1 2kT
∗T +T T∗k
(3.12)
for Hilbert space operator T ∈B(H). These inequalities are sharp. These inequalities were also reformu-lated and generalized in [10] but in terms of the Cartesian decomposition.
In 2009, Popescu [21] proved that
1 2√n
n X
k=1
TkTk∗ 1 2
≤we(T1,· · · , Tn)≤
n X
k=1
TkTk∗ 1 2 (3.13)
As noted in [20], and as a special case of (3.13); ifA=B+iC is the Cartesian decomposition ofA, then w2e(B, C) = sup
kxk=1
n
|hBx, xi|2+|hCx, xi|2o= sup
kxk=1
|hAx, xi|2=w2(A).
But sinceA∗A+AA∗= 2 B2+C2
, then we have
1 16kT
∗T+T T∗k ≤w2(T)≤ 1
2kT
∗T+T T∗k.
(3.14)
It should be noted here that, the case whenn= 2, was also studied by Dragomir in [9] where he obtained some very interesting results regading Euclidean operator radius of two operatorswe(T1, T2).
Next, we give a generalization of (3.12) and refine (indeed improve) (3.13) (and thus (3.14)) to the generalized Euclidean operator radius.
Theorem 7. Let Tk ∈B(H) (k= 1,· · ·, n). Then 1
2p+1np−1
n X
k=1
Tk∗Tk+TkTk∗ p
≤w2pp(T1,· · · , Tn)≤
1 2p
n X
k=1
(Tk∗Tk+TkTk∗) p
(3.15)
for allp≥1.
Proof. LetBk+iCk be the Cartesian decomposition ofTk for allk= 1,· · ·, n. As in the proof of (3.12) in [16], we have
|hTkx, xi|2p=hBkx, xi2+hCkx, xi2p
≥ 1
2p(|hBkx, xi|+|hCkx, xi|)
2p
≥ 1
2p|hBkx, xi+hCkx, xi|
2p
= 1
2p|hBk±Ckx, xi|
Summing overk and then taking the supremum over all unit vectorx∈H, we get
w2pp(T1,· · · , Tn)≥
1 2p sup
kxk=1
n X
k=1
|hBk±Ckx, xi|2p
≥ 1 2p
1 np−1 sup
kxk=1
n X
k=1
|hBk±Ckx, xi|
2
!p
(by Jensen’s inequality)
= 1
2p 1 np−1
n X k=1
(Bk±Ck)2 p . Thus,
2w2pp(T1,· · · , Tn)≥
1 2p
1 np−1
n X k=1
(Bk+Ck)2 p + 1 2p 1 np−1
n X k=1
(Bk−Ck)2 p ≥ 1 2p 1 np−1
n X k=1
(Bk+Ck)2+ n X
k=1
(Bk−Ck)2 p = 1 2p 1 np−1
n X k=1 n
(Bk+Ck)
2
+ (Bk−Ck)
2o p = 1
np−1
n X k=1
Bk2+Ck2 p = 1
np−1
n X k=1
Tk∗Tk+TkTk∗ 2 p = 1
2pnp−1
n X k=1
Tk∗Tk+TkTk∗ p , and hence,
w2pp(T1,· · ·, Tn)≥
1 2p+1np−1
n X k=1
Tk∗Tk+TkTk∗ p ,
which proves the left hand side of the inequality in (3.15).
To prove the second inequality, for every unit vectorx∈H we have n
X
k=1
|hTkx, xi|2p= n X
k=1
hBkx, xi2+hCkx, xi2p
≤ n X
k=1
Bk2x, x
+
Ck2x, xp
= n X
k=1
Bk2+Ck2 x, xp
,
which implies that
sup
kxk=1
n X
k=1
|hTkx, xi|2p=wp2p(T1,· · ·, T1)≤ sup
kxk=1
n X
k=1
Bk2+Ck2 x, xp
= n X k=1
Bk2+Ck2 p = 1 2p n X k=1
which proves the right hand side of (3.15).
Remark 2. Clearly, by settingn= 1 in (3.15)we recapture (3.12). A very interesting case of (3.15) is considered in the following corollary.
Corollary 27. Let T, S∈B(H). Then 1
22pkT
∗T+T T∗+S∗S+SS∗kp
≤wp2p(T, S) (3.16)
≤ 1 2pk(T
∗T+T T∗)p
+ (S∗S+SS∗)pk for allp≥1.
Proof. Settingn= 2 in (3.15).
Remark 3. In particular, setting p= 1 in (3.16)we get 1
4kT
∗T+T T∗+S∗S+SS∗k ≤w
e(T, S)
≤1 2kT
∗T+T T∗+S∗S+SS∗k.
Moreover, if we chooseT =S, then 1 2kT
∗T+T T∗k ≤w
e(T, T)≤ kT∗T+T T∗k.
Remark 4. A lower and upper bounds for the Rhombic numerical radius could be deduced as follows: In (1.4) the inequality holds for anyp≥1. Settingp= 2q, then (1.4)reduces to
w2q(T1,· · · , Tn)≤wR(T1,· · ·, Tn)≤n1−
1
2qw2q(T1,· · ·, Tn).
which implies that
w2qq(T1,· · ·, Tn)≤wRq (T1,· · · , Tn)≤nq−
1 2wq
2q(T1,· · · , Tn).
(3.17)
Combining the inequalities (3.17)with (3.15)we get
1 22q+1nq−1
n X
k=1
Tk∗Tk+TkTk∗ q
≤wq2q(T1,· · ·, Tn)
≤wqR(T1,· · · , Tn)
≤nq−12wq
2q(T1,· · ·, Tn)
≤n q−1
2 2q
n X
k=1
(Tk∗Tk+TkTk∗)q
for any q≥1 2.
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Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, P.O. Box 2600, Irbid, P.C. 21110, Jordan.
