INEQUALITIES FOR THE NORM AND NUMERICAL RADIUS FOR HILBERT ????*-MODULE OPERATORS

Автор: Mohsen Shah Hosseini

Probl. Anal. Issues Anal. Vol. 9 (27), No 2, 2020, pp. 87-96 87

DOI: 10.15393/j3.art.2020.7330

UDC 517.98

Mohsen Shah Hosseini, Baharak Moosavi

INEQUALITIES FOR THE NORM AND NUMERICAL RADIUS FOR HILBERT C*-MODULE OPERATORS

Abstract. In this paper, we introduce some inequalities between the operator norm and the numerical radius of adjointable operators on Hilbert C*-module spaces. Moreover, we establish some new refinements of numerical radius inequalities for Hilbert space operators. More precisely, we prove that if T e B(H) and

min ( l|T + T, l|T ) < max ( inf \\\\Tx\\\\2, inf \\\\T*x\\\\2),

V 2 & 2 )< V ||^||=i11 11 |N|=iH 11 /

\\\\T\\\\<V2u(T);

this is a considerable improvement of the classical inequality

\\\\T\\\\< 2w(T).

2010 Mathematical Subject Classification: Primary 47A12; Secondary 47A30

1. Introduction and preliminaries. Let B(H) denote the C*-algebra of all bounded linear operators on a complex Hilbert space H with the inner product (-, •). If dim H = n, we identify B(H) with the space Mn of all n x n matrices with entries in the complex field. For T e B(H), let \\\\T\\ denote the usual operator norm and

u(T) = sup{|(Tx,x)|: 1}

denote the numerical radius of T. It is well known that w(-) is a norm on B(H) and that

H < u(T) < \\\\T\\ (1)

for all T G B(H). The first inequality becomes an equality if T2 = 0 (use the first Kittaneh inequality below). The second inequality becomes an equality if T is normal. Recently, the authors of [13] tried to show that ||T||< V2 u(T) holds, whenever T is an invertible operator. However, Cain [1] constructed some counterexamples. Several numerical radius inequalities improving those in (1) have been recently given in [2-5], [11], and [14]. For instance, Dragomir proved that

u2(T) < 2(u(T2) + ||T||2)

for any T G B(H). And Kittaneh proved that, for any T G B(H),

"(T) < ~(HTM^u2)

||TT* + T*T || 2(m\\ ^ UTT* + T*T ||

< UJ2(T) <

4 - v 7 - 2 Theese inequalities can be found in [9], [10], respectively. Furthermore, Holbrook in [7] showed that, for any R, S G B(H),

u(RS) < 4w(R)u(S), (2)

u(RS) < 2u(R)u(S), (3)

when RS = SR.

See [6] for other results and historical comments on the numerical radius. Now, here is a reminder of the definition of a Hilbert module, according to [12].

Let A be a C*-algebra (not necessarily unital or commutative). An inner-product ^.-module is a linear space E, which is a right ^.-module (with a compatible scalar multiplication: \\(xa) = x(\\a) = (\\x)a for all x G E, a G A and A G C), together with a map (-, •) : E x E —> A, such that

(i) (x,x) ^ 0, meaning it is one of the positive operators in A; (x, x) = 0 iff x = 0,

(ii) (x,\\y + z) = \\(x,y) + (x,z),

(iii) (x,ya) = (x,y)a,

(iv) (x,y) = (y,x)*,

for all x,y,z E E,a E A,\\ E C.

For x E E, we write ||x|| = 1. An inner-product ^-module

that is complete with respect to its norm is called a Hilbert ^-module, or a Hilbert C*-module, over the C*-algebra A. We denote, by L(E), the C*-algebra of all adjointable operators on E (i. e., of all maps T : E —> E, such that there exists a T* : E —> E with the property (T(x),y) = = (x, T*(y)), for all x,y E E) and let L-1(E) denote the set of all invertible operators in L(E).

Definition 1. For T E L(E), let

i(T) =sup{|(Tx,x)| : ||x||= 1}, IT|=sup{|Tx| : HxH=1},

respectively, denote the numerical radius and operator norm of T. Recently, in [15], we have shown that

HT||< 2S(T), (4)

5(TS) < 45(T)5(S). (5)

We are able to improve the inequalities (4) and (5). The results in this paper considerably improve inequalities (1) and (2).

2. Main results. Let T E L(E). For the sake of convenience, we prepare the following notation:

(T) . fHT-T*|2 HT + T*|2x m(l ) = mi^---,---J

M(T) = max ( inf HTx!2, inf *x|2

In order to derive our main results, we need the following lemmas. Lemma 1. If T E L(E) is self-adjoint, then

6(T ) = UT ||. (6)

Proof. First, we show that the result holds for positive operators.

Let G E L(E) be positive. Since L(E) is a C*—algebra, we know that ||G*G||= ||G||2. Then,

||G*G|| = sup H(G*Gx,x)l

Replacing G by \\/G gives

||G|| = sup H(Gx,x)l (7)

Now, let T E L(E) be just self-adjoint. By Proposition 1.1 in [12],

5(T)= sup H{Tx,x)H<HT||. (8)

On the other hand, being self-adjoint, T can be decomposed: T = T+ -T-, such that T+ and T- are both positive and T+T- = T-T+ = 0, and also ||T||= max(||T+||, ||T-||). Note that

sup H(Tlx,x)H= ||Tf||, (by (7)); INI=i

then there exist a sequence [xn] of unit vectors in E, such that

||T_31|= lim H(T3xn,xn)l

Therefore,

sup H(Tx,x)H> (

iMI=I \\ ^HT+XnV HT+XnH/

T+xn

Similarly,

V& r+^J

T+xn

/(T —T )( 1+Xn \\ T+xn \\ V + -)V HT+xnH), HT+xnH/

||(T3xra, xra)|l ||(T3xn,xn)H

HT+xnH

sup H(Tx,x)H>

HT+||3 HT ||2

r H(T3 xn,xn)H lim

sup H(Tx,x)H> ——y. NI=i ||T|2

By (9) and (10),

5(T) = sup \\\\{Tx,x}\\\\> max (^, T^S) = ^\\ (U) IMI=i V \\\\T\\\\2 \\\\T\\\\2 J

The result follows from inequalities (8) and (11). □ Lemma 2. IfT e L(E), then

(a) m(T) < 252(T).

(b) M (T) = , if T is invertible.

Proof. (a) Since T + T* is self-adjoint, from Lemma 1 we have:

\\\\T + T*\\\\= S(T + T *)

\\\\T + T *\\\\2 = (5(T + T *))2 (S(T ) + S(T *))2 = 2 2 2 < 2 ( ). Consequently,

\\\\T + T *\\\\2

< 252{T). (12)

UT + t *|2

Since m(T) < ---, the result follows from (12).

(b) See [8, p. 41]. □ Lemma 3. Let E be a Hilbert C*— module. Then

||(a, a) + (b, b)H< ±(Ua + bf+||a-bf), (13)

for any a,b E E.

Proof. Suppose that a,b E E; then

(a + b,a + b) = (a, a) + (a, b) + (b, a) + (b, b), (a — b,a — b) = (a, a) — (a, b) — (b, a) + (b, b).

(a + b,a + b) + (a — b,a — b) = 2((a, a) + (b, b)).

Therefore,

2U(a,a) + (b, b)H< ||a + bH2+Ha — bU2.

This completes the proof. □ Theorem 1. IfT G L(E) be such that

inf \\\\Tx\\\\2+\\\\T*x\\\\2< \\\\{Tx,Tx) + {T*x,T*x}\\\\ |M|=1

inf \\\\T*x\\\\2+\\\\Tx\\\\2< \\\\{Tx, Tx) + {T*x,T*x}\\\\

for all x e E with \\\\x|| = 1; then

\\\\T\\\\2+M(T) - m(T) < 2ô2(T). (14)

Proof. Suppose that u e E with \\\\u|| = 1. Choose a = Tu,b = T*u in (13) to give

\\\\{Tu,Tu) + {T*u,T*u)\\\\< 1(\\\\Tu + T*u\\\\2+\\\\Tu -T*u\\\\2). (15)

By the assumption, inf \\\\Tx||2+\\\\T*u\\\\2< \\\\{Tu,Tu) + {T*u,T*u)\\\\ gives |M|=1

w= \\\\Tx\\\\2+\\\\T*u\\\\2< 2(||Tu-T*u\\\\2+\\\\Tu + T*u\\\\2). (by (15))

yxy = 1 2\\ J

Taking the supremum over u e E with \\\\u|| = 1 gives 1

...... ■1 ~ 2

Since (T + T*) is self-adjoint, (6) yields

\\ \\ T + T*\\\\< 2S(T).

Therefore,

inf \\\\Tx\\\\2+\\\\T\\\\2< 252(T) + (16)

Similarly, by the assumption,

inf \\\\T*x||2+\\\\Tx\\\\2< \\\\{Tx,Tx) + {T*x,T*x)\\\\,

inf\\Tx\\l 2+\\\\T\\\\2< - (\\\\T-T*\\\\2+\\\\T+T*\\\\2). (since \\\\T\\\\= \\\\T*\\\\)

inf \\\\T*x||2+\\\\T\\\\2< 262(T) + ^

and, so,

||T||2+M(T) - l|T -2T 1,2 < 252(T). (by (16)).

Replacing T by iT in the last inequality gives

||T||2+M(T) - ||r +T*|2 < 252(T).

||T||2+M(T) - min ( |T - T*|2 , |T + T*|2) < 2£2(T),

which is exactly the desired result. □

The following particular case is of interest.

Corollary 1. Let T be as in Theorem 1. If, in addition, T e L-1(E), then

HT||2+^^ - m(T) < 252(T). (17)

Proof. Result follows immediately from Theorem 1 and Lemma 2(b), since T is invertible. □

Our next corollary includes a refinement of the inequality (5).

Corollary. Let R, S be as in Theorem 1. Then

5(RS) < ^(282(R) - M(R) + m(R)) (252(S) - M(S) + m(S)) <

< 4S(R)S(S).

Proof. By Lemma 2(a),

m(R) < 252(R)

||#||< ^252(R) - M(R)+ m(R) < 25(R). (by (14))

Similarly,

US||< V252(S) - M(S) + m(S) < 28(S).

Therefore,

5(RS) < HR\\\\US||<

< ^(282(R) - M (R) + m(R)) (2Ô2(S ) - M (S ) + m(S )) <

< 4S(R)S(S).

The following applications of Theorem 1 improve inequality (4) for some invertible operators.

Corollary. Let R, S E L-i(E) and satisfy the condition of Theorem 1. If m(R) < ||fl-i||-2 and m(S) < ||S-i||-2, then

||fl||< V2S(R), (18)

S(RS) < 2S(R)S(S). (19)

Proof. Inequality (18) follows from Lemma 2(b) and corollary 1. Similarly,

№||< V2s(s). (20)

For inequality (19), observe, using 8(RS) < mRS|| in the first inequality and (18) and (20) in the third, that

S(RS) < !RS||< ||R||||S||< 2S(R)S(S).

This completes the proof. □

3. New inequalities for Hilbert operators. Since a Hilbert space is a Hilbert C*-module, the results in section 2 of this paper hold in B(H).

Theorem 2. If T E B(H) and 0 < ||T f+M(T) - m(T), then

i + m(^ - m(^)

u(T). (21)

Proof. According to Definition 1, we have 5(T) = u(T).

Replacing T by t^L- in (14) gives

||r |2(1+M ( m )- ro( m )) < ^

Since 11 T112+M(T) - m(T) > 0,

| | T11 2<-T-r2-(T),

11 11 < 1 + M(M) -m(M) ( ),

which is exactly the desired result. □

In the next result, we provide some conditions for the inequality | | T11 < V2u(T) to be true.

Corollary. IfT G B(H) and M(T) > m(T), then

11 t 11 <V2u(T).

Acknowledgment. The authors thank the Editorial Board and the referees for their valuable comments that helped to improve the text.

References

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[2] S. S. Dragomir, Rivers inequalities for the numerical radius of pace, Bull. Aust. Math. Soc., 2006, vol. 73, pp. 255-262.

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[3] S. S. Dragomir, Some inequalities of the Gruss type for the Numerical radius of bounded linear operators in Hilbert spaces, J. Inequal. Appl. 2008, Art. ID 763102, 9 p. DOI: https://doi.org/10.1155/2008/763102

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DOI: https://doi .org/10.1515/gmj-2019-2053

Received December 01, 2019. In revised form, June 04, 2020. Accepted June 05, 2020. Published online June 15, 2020.

M. Shah Hosseini

Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran.

E-mail: mohsen_shahhosseini@yahoo.com B. Moosavi

Department of Mathematics, Safadasht Branch, Islamic Azad University, Tehran, Iran.

E-mail: baharak_moosavie@yahoo.com

bounded linear operator hilbert space norm inequality numerical radius

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