ON SOME INEQUALITIES FOR THE GENERALIZED EUCLIDEAN OPERATOR RADIUS

There are many criterion to generalize the concept of numerical radius; one of the most recent 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.


Introduction
Let B (H ) be the Banach algebra of all bounded linear operators defined on a complex Hilbert space (H ; •, • ) with the identity operator 1 H in B (H ).When H = C n , we identify B (H ) with the algebra M n×n of n-by-n complex matrices.Then, M + n×n is just the cone of n-by-n positive semidefinite matrices.For a bounded linear operator T on a Hilbert space H , the numerical range W (T ) is the image of the unit sphere of H under the quadratic form x → T x, x associated with the operator.More precisely, W (T ) = { T x, x : x ∈ H , x = 1} Also, the numerical radius is defined to be We recall that, the usual operator norm of an operator T is defined to be It is well known that w (•) defines an operator norm on B (H ) which is equivalent to operator norm • .Moreover, we have 1 2 T ≤ w (T ) ≤ T (1.1) for any T ∈ B (H ) and this inequality is sharp.
It is known that w(A) is a norm on B (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 unitary U .Also, let us don't miss the chance to mention the important property that w (T ) = w (T * ) and w (T * T ) = w (T T * ) for every T ∈ B (H ).
Denote |T | = (T * T ) 1/2 the absolute value of the operator T .Then we have It's well known that the numerical radius is not submultiplicative, but it satisfies w(T S) ≤ 4w (T ) w (S) for all T, S ∈ B (H ).In particular if T, S commute, then w(T S) ≤ 2w (T ) w (S) .
Moreover, if T, S are normal then w (•) is submultiplicative, i.e., w(T S) ≤ w (T ) w (S) In 2009, Popsecu [21] introduced the concept of Euclidean operator radius of an n-tuple The Euclidean operator radius of T 1 , • • • , T n is defined by Indeed, the Euclidean operator radius was generalized in [24] as follows: ) is called the Rhombic numerical radius which have been studied in [5].In an interesting case, The Crawford number is defined to be Consequently, we define the generalized Crawford number as: In case p = 1, the generalized Crawford number is called the Rhombic Crawford number and is denoted by We note that in case p = ∞, the generalized Euclidean operator radius is defined as: Thus, the inequality for all p ∈ (1, ∞).This fact follows by Jensen's inequality applied for the function h(p) = w p (T 1 , • • • , T n ), which is log-convex and decreasing for all p > 1.
On the other hand, by employing the Jensen's inequality which holds for every finite positive sequence of real numbers (a k ) n k=1 and p ≥ 1; by setting Taking the supremum over all unit vector x ∈ H , one could get Combining the inequalities (1.2) and (1.3) we get More generally, in the power mean inequality .
Taking the supremum over all unit vector x ∈ H , we get Indeed, one can refine (1.3) by applying the Jensen's inequality which obtained from more general result for superquadratic functions [1].
Thus, by setting Taking the supremum again over all unit vector x ∈ H , we get sup 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].
2.1.Basic properties of the generalized Euclidean operator radius.Moslehian et al. [20], mention without proofs the following properties of the generalized Euclidean operator radius: [20], mentioned the above basic properties of the generalized Euclidean operator radius, but it seems they missed some other important properties, rather than they left these properties without proof.Sometimes, it's nice to elaborate the proof of these elementary facts.Because of that we are going to give a proof of each property.Clearly, the first two properites follows from the definition of the generalized Euclidean operator radius.In what follows, and as the classical sense we have the following properties:

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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., ( Proof. (1) The first property follows since (2) By the definition of the generalized Euclidean operator radius we have (3) Similarly, by definition we have (4) Finally, employing the classical Minlowski inequality, i.e., we get which proves the last property.
Form the definition of the generalized Euclidean operator radius, we have as required.

by the McCarthy inequality)
Taking the sum over all i from 1 to n we get (by the Hölder inequality) Taking the supremum over all vectors x ∈ H such that x = 1, we get the desired result.
for all p, q > 1 such that 1 p + 1 q = 1.In particular, for p = q = 2 we have Since w p (•) is weakly unitarily invariant and Thus, the desired result is obtained.
Proof.Let y = x in (2.6), then we have ( by Hölder inequality) x, x .
Taking the supremum over all unit vector x ∈ H we get the required result.
for all γ ∈ [0, 1] and p ≥ 1.In particular, we have Proof.Let y = x in (2.6), then we have Taking the supremum over all unit vector x ∈ H we get the required result.The particular case is obtained by setting γ = 1 2 in (2.22).Corollary 10.
Proof.Let U i be unitaries for all i = 1,

23), then we have
Then, we have (2.25) where Proof.Let x ∈ H be a unit vector.Let y = x in (2.6) then we have (by McCarthy inequality) for all α, β ≥ 0 such that α + β ≥ 1. Proof.Let U i be unitaries for all i = 1,
Then, for m ∈ N and r, p ≥ 1, .
Then, for m ∈ N and p ≥ m ≥ 1, where Proof.Setting m = r in (2.29).
Then, for m ∈ N and r, p ≥ m ≥ 1, we have where Proof.Setting A i = U i and D i = U * i in (2.26) and using the fact that . where Proof.
Then, for m ∈ N and r, p ≥ m ≥ 1, Proof.

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 Hölder type inequality, which reads: for all complex numbers x j , y j (1 ≤ j ≤ n) and all p, q, r ≥ 1 such that 1 p + 1 q = 1 r .

Preprints
where, Proof.Let x, y ∈ H . Applying inequality (3.1) and the convexity of t 2r , we have x, y (by McCarthy inequality) Taking the supremum over x, y ∈ H with x = y = 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. where Proof.Setting p = q = 2 and r = 1 in (3.2) we get the desired result.
In 2009, Popescu [21] proved that As noted in [20], and as a special case of (3.13); if A = B + iC is the Cartesian decomposition of A, then It should be noted here that, the case when n = 2, was also studied by Dragomir in [9] where he obtained some very interesting results regading Euclidean operator radius of two operators w e (T 1 , T 2 ).
Next, we give a generalization of (3.12) and refine (indeed improve) (3.13) (and thus (3.14)) to the generalized Euclidean operator radius.
for all p ≥ 1.
Proof.Let B k + iC k be the Cartesian decomposition of T k for all k = 1, • • • , n.As in the proof of (3.12) in [16], we have Summing over k and then taking the supremum over all unit vector x ∈ H , we get which implies that  Combining the inequalities (3.17) with (3.15) we get 1 2 2q+1 n q−1 n k=1 q for any q ≥ 1 2 .