CC BY
23Probl. 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 /
then
\\T\\<V2u(T);
this is a considerable improvement of the classical inequality
\\T\\< 2w(T).
Key words: Bounded linear operator, Hilbert space, Norm inequality, Numerical radius
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)
© Petrozavodsk State University, 2020
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),
1
and
"(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)
and
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)
and
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
and
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
IHI=i
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)
N^N=i
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
and so:
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
>
2>
HT ||2
sup H(Tx,x)H>
IMI=i
HT+||3 HT ||2
r H(T3 xn,xn)H lim
HT ||2
HT-||3
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.
\\T T
Proof. (a) Since T + T* is self-adjoint, from Lemma 1 we have:
\\T + T*\\= S(T + T *)
So,
\\T + T *\\2 = (5(T + T *))2 (S(T ) + S(T *))2 = 2 2 2 < 2 ( ). Consequently,
\\T + T *\\2
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).
Thus,
(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
and
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}\\
yxy = 1
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,
T T* 2
inf \\Tx\\2+\\T\\2< 252(T) + (16)
Txy=1 2
Similarly, by the assumption,
inf \\T*x||2+\\Tx\\2< \\{Tx,Tx) + {T*x,T*x)\\,
yxy = 1
inf\Tx\l 2+\\T\\2< - (\\T-T*\\2+\\T+T*\\2). (since \\T\\= \\T*\\)
gives
T T* 2
inf \\T*x||2+\\T\\2< 262(T) + ^
Txy=1 2
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).
Thus,
||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)
and so
||#||< ^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).
T
NT N
Replacing T by t^L- in (14) gives
||r |2(1+M ( m )- ro( m )) < ^
Since 11 T112+M(T) - m(T) > 0,
2
| | 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.
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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: [email protected] B. Moosavi
Department of Mathematics, Safadasht Branch, Islamic Azad University, Tehran, Iran.
E-mail: [email protected]
