BOUNDED LINEAR OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. The main aim of this paper is to establish some connections that
exist between the numerical radiusw(A),operator normkAk and the
semi-inner productshA, Ii_{p,n} and hA, Ii_{p,w} withp ∈ {i, s} that can be naturally
defined on the Banach algebraB(H) of all bounded linear operators defined
on a Hilbert spaceH.Reverse inequalities that provide upper bounds for the
nonnegative quantitieskAk −w(A) andw(A)− hA, Ii_{p,n} under various
as-sumptions for the operatorAare also given.

1. _{Introduction}

In any normed linear space (E,k·k),since the functionf :E→_{R},f(x) = 1_{2}kxk2

is convex, one can introduce the following semi-inner products (see for instance [3]):

(1.1) hx, yi_{i}:= lim
s→0−

ky+sxk2− kyk2

2s , hx, yis:= lim_{t}_{→}_{0}_{−}

ky+txk2− kyk2

2t

where x, yare vectors inE. The mappingsh·,·i_{s} andh·,·i_{i} are called thesuperior
respectively theinferior semi-inner product associated with the normk·k.

In the Banach algebraB(H) of all bounded linear operators defined on the real or complex Hilbert space H we can associate to both the operator norm k·k and the numerical radiusw(·) the following semi-inner products:

(1.2) hA, Bi_{s(i),n}:= lim

t→0+(−)

kB+tAk2− kBk2

2t

and

(1.3) hA, Bi_{s(i),w}:= lim
t→0+(−)

w2_{(}_{B}_{+}_{tA}_{)}_{−}_{w}2_{(}_{B}_{)}
2t

respectively, whereA, B∈B(H).

It is obvious that the semi-inner products h·,·i_{s(i),n(w)} defined above have the
usual properties of such mappings defined on general normed spaces and some
special properties that will be specified in the following.

For the sake of completeness we list here some properties of h·,·i_{s(i),n(w)} that
will be used in the sequel.

We have:

(i) hA, Ai_{s(i),n}=kAk2, hA, Ai_{s(i),w}=w2_{(}_{A}_{) for any}_{A}_{∈}_{B}_{(}_{H}_{) ;}
(ii) hiA, Ai_{p,n(w)}=hA, iAi_{p,n(w)}= 0 for eachA∈B(H) ;

Date: July 27, 2006.

2000Mathematics Subject Classification. 47A12, 47A30, 47A63.

Key words and phrases. Numerical radius, Operator norm, Semi-inner products, Bounded linear operators, Banach algebra.

(iii) hλA, Bi_{p,n(w)} =λhA, Bi_{p,n(w)} =hA, λBi_{p,n(w)} for any λ≥0 and A, B ∈

B(H) ;

(iv) h−A, Bi_{p,n(w)}=hA,−Bi_{p,n(w)}=− hA, Bi_{q,n(w)} for anyA, B∈B(H) ;
(v) hiA, Bi_{p,n(w)}=− hA, iBi_{p,n(w)} for eachA, B∈B(H) ;

(vi) The following Schwarz type inequalities hold true:

(1.4)

hA, Bip,n

≤ kAk kBk and

(1.5)

hA, Bip,w

≤w(A)w(B) for anyA, B∈B(H) ;

(vii) The following identities hold true

(1.6) hαA+B, Ai_{p,n}=αkAk2+hB, Ai_{p,n}

and

(1.7) hαA+B, Ai_{p,w}=αw2(A) +hB, Ai_{p,w}

for eachα∈RandA, B∈B(H) ;

(viii) The following sub(super)-additivity property holds:

(1.8) hA+B, Ci_{s(i),p(w)}≤(≥)hA, Ci_{s(i),p(w)}+hB, Ci_{s(i),p(w)},

where the sign “≤” applies for the superior semi-inner product, while the sign “≥” applies for the inferior one;

(ix) The following continuity properties are valid:

(1.9)

hA+B, Cip,n− hB, Cip,n

≤ kAk kCk and

(1.10)

hA+B, Cip,w− hB, Cip,w

≤w(A)w(C) for eachA, B, C ∈B(H) ;

(x) From the definition we have the inequality

(1.11) hA, Bi_{i,n(w)}≤ hA, Bi_{s,n(w)}

forA, B∈B(H) ;

where everywhere abovep, q∈ {i, s} andp6=q.

As a specific property that follows by the well known inequality between the norm and the numerical radius of an operator, i.e.,w(A)≤ kAkfor eachA∈B(H),we have

(1.12) hA, Ii_{i,n}≤ hA, Ii_{i,w}(≤)hA, Ii_{s,w}≤ hA, Ii_{s,n}

for anyA∈B(H),whereI is the identity operator onH.We also observe that

hA, Ii_{s(i),n}= lim
t→0+(−)

kI+tAk −1 t

and

hA, Ii_{s(i),w}= lim
t→0+(−)

Motivated by the natural connection that exists between the semi-inner products

hA, Ii_{p,n}, hA, Ii_{p,w} with p ∈ {i, s}, the numerical radius w(A) and the operator
normkAkoutlined above, the aim of this paper is to establish deeper relationships
between these concepts. Amongst others, we show, in fact, that the semi-inner
product hA, Ii_{p,n} is equal to hA, Ii_{p,w} forp∈ {i, s} and as a consequence the
nu-merical radiusw(A) is bounded below by the maximum of the quantities

hA, Iii,n

and

hA, Iis,n

. Also, on utilising these quantities various reverse inequalities for the fundamental fact that

1

2kAk ≤w(A)≤ kAk, A∈B(H),

are pointed out, improving the recent results of the author from [4], where some
upper bounds for the nonnegative differenceskAk −w(A) andkAk2−w2_{(}_{A}_{) have}
been established under appropriate conditions for the bounded linear operatorA.

For recent results concerning inequalities between the operator norm and numer-ical radius see the papers [2], [5], [6], [7], [8], [9], the books [1], [10], [11] and the references therein.

2. The Functionals vs(i) onB(H)

The following representation result, which plays a crucial role in deriving various inequalities for numerical radius may be stated:

Theorem 1. For anyA∈B(H), we have:
(2.1) hA, Ii_{s,n}=hA, Ii_{s,w}= sup

kxk=1

RehAx, xi

and

(2.2) hA, Ii_{i,n}=hA, Ii_{i,w} = inf
kxk=1

RehAx, xi.

Proof. Letx∈H,kxk= 1.Then fort >0 we obviously have:

RehAx, xi=Rehx+tAx, xi −1 t

(2.3)

≤|hx+tAx, xi| −1

t ≤

w(I+tA)−1

t .

Taking the supremum overx∈H,kxk= 1,we get

sup kxk=1

RehAx, xi ≤ w(I+tA)−1

t

for eacht >0,which implies, by lettingt→0+ that

(2.4) sup

kxk=1

RehAx, xi ≤ hA, Ii_{s,w},

for eachA∈B(H).

Now, let δ := sup_{k}_{x}_{k}_{=1}RehAx, xi. Ifx ∈H, kxk = 1, then for any α > 0 we
have

k(I−αA)xk ≥Reh(I−αA)x, xi= 1−αRehAx, xi ≥1−αδ.
Therefore, by puttingx= _{k}y_{y}_{k} (y6= 0) withy∈H,we have

for eachα >0 andy∈H.

Now, if in (2.5) we replaceyby (I+αA)z withz∈H,then we get (2.6) I−α2A2z

≥(1−αδ)k(I+αA)zk for each z∈H.

Taking the supremum overkzk= 1 in (2.6), we get the operator norm inequality:

I−α2A2≥(1−αδ)kI+αAk which is equivalent with

(2.7) I−α2A2+αδkI+αAk ≥ kI+αAk, α >0. If we subtract 1 from both sides of (2.7) and divide byα >0,we get

(2.8)

I−α2_{A}2
−1

α +δkI+αAk ≥

kI+αAk −1

α .

Taking the limit overα→0+ in (2.8) and noticing that

lim

α→0+kI+αAk= 1, αlim→0+

kI+αAk −1

α =hA, Iis,n and

lim α→0+

I−α2_{A}2
−1

α = limα→0+α·αlim→0+

I−α2_{A}2
−1

α2 = 0,

then, by (2.8), we have:

(2.9) δ≥ hA, Ii_{s,n}.

Since, by (1.12), we always havehA, Ii_{s,n}≥ hA, Ii_{s,w}, hence (2.4) and (2.9) imply
the desired equality (2.1).

Now, on utilising (iv) and (2.1), we have for allA∈B(H) that

hA, Ii_{i,n} =h−A, Ii_{s,n}=− sup
kxk=1

Reh−Ax, xi= inf

kxk=1RehAx, xi and the identity (2.2) is also obtained.

It is well known that a lower bound for the numerical radius w(A) is 1_{2}kAk.
The following corollary of the above theorem provides the following lower bounds
as well:

Corollary 1. For anyA∈B(H)we have

(2.10) maxn

hA, Iis,n ,

hA, Iii,n o

≤w(A).

Proof. By Schwarz’s inequality for the semi-inner productsh·,·i_{s,w} and h·,·i_{i,w} we
have

hA, Iis,w ,

hA, Iii,w

≤w(A), A∈B(H) which, by (2.1) and (2.2), imply the desired inequality (2.10).

Motivated by the representation Theorem 1, we can introduce the following functionals defined onB(H) :

(2.11) vs(A) := sup kxk=1

RehAx, xi and vi(A) := inf kxk=1

RehAx, xi.

Now, on employing the properties of the semi-inner products outlined above, we can state that:

(aa) vi(A)≥0 for accretive operators onH;

(aaa) vs(i)(A+B)≤(≥)vs(i)(A) +vs(i)(B) for eachA, B∈B(H) ;
(av) vs(i)(A) =hA, Ii_{s(i),n}=hA, Ii_{s(i),w} for anyA∈B(H) ;

(v) vs(i)(A)

≤w(A) for allA∈B(H) ;

(va) vs(i)(A) =vs(i)(αI+A)−αfor any α∈RandA∈B(H) ; (vaa) vs(i)(A+B)−vs(i)(B)

≤w(A) for anyA, B∈B(H). The following inequalities may be stated as well:

Proposition 1. For anyA∈B(H)andλ∈Cwe have

1 2 h

kAk2+|λ|2i

(2.12)

≥vs λA¯ ≥

1 2 h

kAk2+|λ|2i−1

2kA−λIk 2

,

1 4 h

kA+λIk2− kA−λIk2i.

Proof. Letx∈H,kxk= 1.Then, obviously
0≤ kAxk2−2 Re_{¯}

λAx, x

+|λ|2=k(A−λI)xk2≤ kA−λIk2, which is equivalent with

(2.13) 1 2 h

kAxk2+|λ|2i−1

2kA−λIk 2

≤Re¯λAx, x≤1

2 h

kAxk2+|λ|2i,

for anyx∈H,kxk= 1.

Taking the supremum overkxk= 1 we get the first inequality in (2.12) and the one from the first branch in the second.

Forx∈H,kxk= 1 we also have that

(2.14) kAx+λxk2=kAx−λxk2+ 4 ReλAx, x¯ ,

which, on taking the supremum overkxk= 1,will produce the second part of the second inequality in (2.12).

It is well known, in general, that the semi-inner products h·,·i_{s(i)} are not
com-mutative. However, for the Banach algebra B(H) we can point out the
follow-ing connection between hA, Ii_{s(i),n(w)} =vs(i)(A) and the quantities hI, Ai_{s,n} and

hI, Ai_{i,n},whereA∈B(H).

Corollary 2. For anyA∈B(H)we have

(2.15) vi(A)≤ 1

2 h

hI, Ai_{s,n}+hI, Ai_{i,n}i≤vs(A).

Proof. We have from the second part of the second inequality in (2.12) that

(2.16) 1

2 "

kA+tIk2− kAk2

2t −

kA−tIk2− kAk2

2t

#

≤vs(A)

for anyt >0.

Taking the limit overt→0+ and noticing that

lim t→0+

kA−tIk2− kAk2

Now, writing the second inequality in (2.15) for−A,we get

vs(−A)≥ 1

2 h

hI,−Ai_{s,n}+hI,−Ai_{i,n}i

=−1

2 h

hI, Ai_{s,n}+hI, Ai_{i,n}i,

which is equivalent with the first part of (2.15).

Utilising a similar approach for the numerical radius instead of the operator norm, we can state the following result:

Proposition 2. For anyA∈B(H)andλ∈_{C}we have the double inequality:
1

2 h

w2(A) +|λ|2i≥vs ¯λA (2.17)

≥

1 2 h

w2_{(}_{A}_{) +}_{|}_{λ}_{|}2i_{−}1
2w

2_{(}_{A}_{−}_{λI}_{)}_{,}

1 4

w2_{(}_{A}_{+}_{λI}_{)}_{−}_{w}2_{(}_{A}_{−}_{λI}_{)}
.

The above result has the interesting consequence:

Corollary 3. For anyA∈B(H)we have

(2.18) vi(A)≤

1 2 h

hI, Ai_{s,w}+hI, Ai_{i,w}i≤vs(A).

Remark 1. Since w(A) ≥ kAk, hence the first inequality in (2.17) provides a better upper bound for vs λA¯ than the first inequality in (2.12).

3. _{Reverse Inequalities in Terms of Operator Norm}

The following result concerning reverse inequalities for the numerical radius and operator norm may be stated:

Theorem 2. For anyA∈B(H)\ {0} andλ∈C\ {0}we have the inequality:

(3.1) (0≤)kAk −w(A)≤ kAk −vs

_{¯}

λ

|λ|A

≤ 1

2|λ|kA−λIk

2 .

In addition, ifkA−λIk ≤ |λ|, then we have:

(3.2)

s

1−

1 λA−I

2

≤

vs_{|}λ_{λ}¯_{|}A

kAk ≤

w(A)

kAk (≤1)

and

(0≤)kAk2−w2(A)≤ kAk2−vs2
¯_{λ}

|λ|A

(3.3)

≤2

|λ| −

q

|λ|2− kA−λIk2

vs

¯_{λ}

|λ|A

≤2

|λ| −

q

|λ|2− kA−λIk2

w(A)

,

Proof. Utilising the property (v), we have

w(A) =w
¯_{λ}

|λ|A

≥

vs
¯_{λ}

|λ|A

≥vs
¯_{λ}

|λ|A

,

for eachλ∈C\ {0} and the first inequality in (3.1) is proved.

By the arithmetic mean-geometric mean inequality we have

1 2 h

kAk2+|λ|2i≥ |λ| kAk,

which, by (2.12) provides

vs ¯λA

≥ |λ| kAk − 1

2kA−λIk 2

that is equivalent with the second inequality in (3.1).

Utilising the second part of the inequality (2.12) and under the assumption that

kA−λIk ≤ |λ|we can also state that

(3.4) vs λA¯

≥ 1

2 "

kAk2+ q

|λ|2− kA−λIk2

2# .

By the arithmetic mean-geometric mean inequality we have now:

(3.5) 1

2 "

kAk2+ q

|λ|2− kA−λIk2

2#

≥ kAk

q

|λ|2− kA−λIk2,

which, together with (3.4) implies the first inequality in (3.2). The second part of (3.2) follows from (v).

From the proof of Proposition 1 we can state that

(3.6) kAxk2+|λ|2≤2 Re_{¯}
λAx, x

+r2, kxk= 1

where we denotedr:=kA−λIk ≤ |λ|.We also observe, from (3.6), that Re_{¯}
λAx, x
>0 forx∈H,kxk= 1.

Now, if we divide (3.6) by ReD_{|}λ_{λ}¯_{|}Ax, xE>0,we get

(3.7) kAxk 2

ReD_{|}¯λ_{λ}_{|}Ax, xE

≤2|λ|+ r 2

ReD_{|}¯λ_{λ}_{|}Ax, xE

− |λ|

2

ReD_{|}¯λ_{λ}_{|}Ax, xE

If in this inequality we subtract from both sides the quantity ReD_{|}¯λ_{λ}_{|}Ax, xE,then
we get

kAxk2

ReD_{|}¯λ_{λ}_{|}Ax, xE

−Re

_{¯}

λ

|λ|Ax, x

≤2|λ|+ r
2_{− |}_{λ}_{|}2

ReD_{|}¯λ_{λ}_{|}Ax, xE

−Re
¯_{λ}

|λ|Ax, x

= 2|λ| −

q

|λ|2−r2 r

ReD ¯λ |λ|Ax, x

E

−

s

Re
¯_{λ}

|λ|Ax, x

2 −2 q

|λ|2−r2

≤2

|λ| −

q

|λ|2−r2

,

which obviously implies that

(3.8) kAxk2≤

Re

_{λ}¯

|λ|Ax, x

2

+ 2

|λ| −

q

|λ|2−r2

Re
_{λ}¯

|λ|Ax, x

for anyx∈H,kxk= 1.

Now, taking the supremum in (3.8) overx∈H,kxk= 1,we deduce the second inequality in (3.3). The other inequalities are obvious and the theorem is proved.

The following lemma is of interest in itself.

Lemma 1. For anyA∈B(H)andγ,Γ∈Kwe have:

(3.9) vi[(A∗−γI¯ ) (ΓI−A)] = 1 4|Γ−γ|

2 −

A−γ+ Γ

2 I 2 .

Proof. We observe that, for anyu, v, y∈H we have:

(3.10) Rehu−y, y−vi=1

4ku−vk 2 −

y−u+v

2 2 .

Now, choosingu= Γx, y=Ax, v=γxwithx∈H,kxk= 1 we get

RehΓx−Ax, Ax−γxi= 1 4|Γ−γ|

2 −

Ax−γ+ Γ

2 x 2 , giving inf kxk=1

Reh(A∗−γI¯ ) (ΓI−A)x, xi= 1 4|Γ−γ|

2

− sup kxk=1

Ax−γ+ Γ

2 x 2 ,

which is equivalent with (3.9).

We recall that the bounded linear operatorB∈B(H) is calledstronglym−accretive (with m > 0) if RehBy, yi ≥ m for any y ∈ H, kyk = 1. For m = 0 the opera-tor is called accretive. In general, we then can call the operator m−accretive for m∈[0,∞).

Lemma 2. For A ∈ B(H), γ,Γ ∈ K with Γ 6= γ and q ∈ R, the following

statements are equivalent:

(i) The operator(A∗−γI¯ ) (ΓI−A) isq2_{−}_{accretive;}
(ii) We have the norm inequality:

(3.11)

A−γ+ Γ

2 I

2

≤1

4|Γ−γ| 2

−q2.

The proof is obvious by Lemma 1 and the details are omitted.

Remark 2. Since the self-adjoint operatorsB satisfying the condition B≥mI in the operator partial under “≥”, are m−accretive, then, a sufficient condition for Cγ,Γ(A) := (A∗−γI¯ ) (ΓI−A) to be q2−accretive is that Cγ,Γ(A) is self-adjoint andCγ,Γ(A)≥q2I.

Corollary 4. Let A∈ B(H), γ,Γ ∈Kwith Γ 6=±γ andq ∈R. If the operator

Cγ,Γ(A)isq2−accretive, then

(0≤)kAk −w(A)≤ kAk −vs

_{Γ + ¯}_{¯} _{γ}

|Γ +γ|A

(3.12)

≤ 1 |γ+ Γ|

_{1}

4|Γ−γ| 2

−q2

.

Remark 3. If M, m are positive real numbers with M > m and the operator Cm,M(A) = (A∗−mI) (M I−A)isq2−accretive, then

(0≤)kAk −w(A)≤ kAk −vs(A) (3.13)

≤ 1

M +m
_{1}

4(M −m) 2

−q2

.

Remark 4. We observe that for q= 0, i.e., ifCγ,Γ(A)respectivelyCm,M(A)are accretive, then we obtain from (3.12) and (3.13) the inequality:

(0≤)kAk −w(A)≤ kAk −vs

_{Γ + ¯}_{¯} _{γ}

|Γ +γ|A

(3.14)

≤ |Γ−γ|

2

4|Γ +γ|

and

(3.15) (0≤)kAk −w(A)≤ kAk −vs(A)≤ (M−m)

2

4 (M+m)

respectively, which provide refinements of the corresponding inequalities (2.7) and (2.34) in[4].

Remark 5. For any bounded linear operatorA we know that w(A)_{k}_{A}_{k} ≥ 1

2,therefore (3.2) would produce a useful result only if

1 2 ≤

s

1−

1 λA−I

2

,

which is equivalent with

(3.16) kA−λIk ≤

√

In conclusion, forA∈B(H)\ {0}and λ∈C\ {0} satisfying the condition (3.16),

the inequality (3.2) provides a refinement of the classical result:

(3.17) 1

2 ≤ w(A)

kAk , A∈B(H). Corollary 5. IfkA−λIk ≤ |λ|, then we have

(0≤)kAk2−w2(A)≤ kAk2−v_{s}2
¯_{λ}

|λ|A

(3.18)

≤ kAk

2

kA−λIk2 |λ|2 .

The proof follows by the inequality (3.2). The details are omitted.

The following corollary providing a sufficient condition in terms ofq2_{−}_{accretive}
property may be stated as well:

Corollary 6. Let A∈B(H)\ {0} andγ,Γ∈K,Γ6=−γ, q∈Rso thatRe (Γ¯γ) +

q2≥0. If Cγ,Γ(A)is q2−accretive, then

(3.19) 2

p

Re (Γ¯γ) +q2

|Γ +γ| ≤

vs_{|}Γ+¯_{Γ+γ}¯ γ_{|}A

kAk ≤

w(A)

kAk (≤1)

and

(0≤)kAk2−w2(A)≤ kAk2−v2_{s}

_{Γ + ¯}¯ _{γ}

|Γ +γ|A

(3.20)

≤ 2kAk

2

|Γ +γ|

_{1}

4|Γ−γ| 2

−q2

≤kAk

2

|Γ−γ|2

2|Γ +γ|

!

.

Proof. By Lemma 2, the fact thatCγ,Γ(A) isq2−accretive implies that

(3.21)

A−γ+ Γ

2 I 2 ≤1

4|Γ−γ| 2

−q2.

Since, obviously

1 4|Γ +γ|

2

−

_{1}

4|Γ−γ| 2

−q2

= Re (Γ¯γ) +q2≥0,

hence A− γ+Γ 2 I ≤ 1

2|Γ +γ| and we can apply the inequality (3.2) forλ= γ+Γ 2 to get: (3.22) r γ+Γ 2 2 − A− γ+Γ 2 I 2 γ+Γ 2 ≤

vs_{|}¯Γ+¯_{Γ+γ}γ_{|}A

kAk ,

and since, by (3.21),

γ+ Γ 2 2 −

A−γ+ Γ

2 I 2 ≥ γ+ Γ 2 2 −1

4|Γ−γ| 2

+q2

hence by (3.22) we deduce the desired inequality in (3.19). The inequality (3.20) follows from (3.19). We omit the details.

Remark 6. If γ,Γandqare such that|Γ +γ| ≤4pRe (Γ¯γ) +q2_{,} _{then (3.19) will}
provide a refinement of the classical result (3.17).

Remark 7. If M > m≥0 and the operatorCm,M(A)isq2−accretive, then

(3.23) 2

p

M m+q2

m+M ≤

vs(A)

kAk ≤

w(A)

kAk (≤1)

and

(0≤)kAk2−w2(A)≤ kAk2−v_{s}2(A)
(3.24)

≤ 2kAk

2

m+M
_{1}

4(M−m) 2

−q2

.

Remark 8. We also observe that, forq= 0,i.e., ifCγ,Γ(A)respectivelyCm,M(A) are accretive, then we obtain:

(3.25) 2

p Re (Γ¯γ)

|Γ +γ| ≤

vs_{|}Γ+¯_{Γ+γ}¯ γ_{|}A

kAk ≤

w(A)

kAk ,

(3.26) 2

√

M m

m+M ≤

vs(A)

kAk ≤

w(A)

kAk ,

(0≤)kAk2−w2(A)≤ kAk2−v_{s}2

_{¯}

Γ + ¯γ

|Γ +γ|A

(3.27)

≤ kAk

2

|Γ−γ|2

2|Γ +γ|

and

(0≤)kAk2−w2(A)≤ kAk2−v2s(A) (3.28)

≤kAk

2

(M −m)2 2 (m+M)

respectively, which provides refinements of the inequalities (2.17), (2.31) and (2.20) in[4], respectively. The inequality between the first and the last term in (3.28) was not stated in[4].

Corollary 7. Let A∈B(H), γ,Γ∈_{K},Γ6=−γ, q∈_{R} so that Re (Γ¯γ) +q2_{≥}_{0}_{.}
If Cγ,Γ(H)isq2−accretive, then

(0≤)kAk2−w2(A)≤ kAk2−v2_{s}

_{Γ + ¯}¯ _{γ}

|Γ +γ|A

(3.29)

≤|Γ +γ| −2pRe (Γ¯γ) +q2_{vs}

_{¯}

Γ + ¯γ

|Γ +γ|A

≤|Γ +γ| −2pRe (Γ¯γ) +q2_{w}_{(}_{A}_{)}_{.}

Remark 9. If M > m≥0 and the operatorCm,M(A)isq2−accretive, then
(0≤)kAk2−w2(A)≤ kAk2−v2_{s}(A)

(3.30)

≤M+m−2pM m+q2_{v}
s(A)

≤M+m−2pM m+q2_{w}_{(}_{A}_{)}_{.}

Remark 10. Finally, for q= 0, i.e., ifCγ,Γ(A) respectivelyCm,M(A)are accre-tive, then we obtain from (3.29) and (3.30) some refinements of the inequalities (2.29) and (2.33) from[4].

4. _{Reverse Inequalities in Terms of Numerical Radius}

In Section 2 we established amongst others the following lower bound for the numerical radiusw(A)

(4.1) vs(i)(A)

≤w(A) for anyAa bounded linear operator, where

(4.2) vs(i)(A) =hA, Ii_{s(i)}= sup
kxk=1

inf kxk=1

RehAx, xi.

It is then a natural problem to investigate how far the left side of (4.1) is from the numerical radiusw(A).

We start with the following result:

Theorem 3. For anyA∈B(H)\ {0} andλ∈_{C}\ {0}we have
(0≤)w(A)−

vs
¯_{λ}

|λ|A

≤w(A)−vs
_{λ}¯

|λ|A

(4.3)

≤ 1

2|λ|w

2_{(}_{A}_{−}_{λI}_{)}

≤ 1

2|λ|kA−λIk

2 .

Moreover, if w(A−AI)≤ |λ|,then we have:

(4.4)

s

1−w2
_{1}

λA−I

≤

vs_{|}λ_{λ}¯_{|}A

w(A) ≤ vs

_{¯}
λ
|λ|A

w(A) (≤1) and

(0≤)w2(A)−v2_{s}
_{λ}¯

|λ|A

(4.5)

≤2

|λ| −

q

|λ|2−w2_{(}_{A}_{−}_{λI}_{)}

vs

_{¯}

λ

|λ|A

≤2

|λ| −

q

|λ|2−w2_{(}_{A}_{−}_{λI}_{)}

w(A)

,

respectively

Proof. From (4.1), we obviously have that

w(A) =w
_{λ}¯

|λ|A

≥

vs
_{λ}¯

|λ|A

≥vs
¯_{λ}

|λ|A

Now, utilising the inequality (2.17) and the elementary arithmetic-geometric mean inequality, we have

vs λA¯

≥ 1

2 h

w2(A) +|λ|2i−1

2w

2_{(}_{A}_{−}_{λI}_{)}

≥ |λ|w(A)−1

2w

2_{(}_{A}_{−}_{λI}_{)}

which is clearly equivalent with the third inequality in (4.3).

Under the assumption thatw(A−AI)≤ |λ|,on making use of (2.17) and the arithmetic mean-geometric mean inequality, we can also state that

vs λA¯

≥ 1

2 "

w2(A) + q

|λ|2−w2_{(}_{A}_{−}_{λI}_{)}
2#

≥w(A) q

|λ|2−w2_{(}_{A}_{−}_{λI}_{)}_{,}
which is clearly equivalent to (4.4).

Let us denoteρ:=w(A−λI)≤ |λ|. Then for anyx∈H,kxk= 1 we have
ρ2≥ |hAx, xi −λ|2=|hAx, xi|2−2 Re_{¯}

λhAx, xi +|λ|2

which yields that

(4.6) |hAx, xi|2+|λ|2≤2 Re_{¯}

λhAx, xi +ρ2

for anyx∈H,kxk= 1.

Making use of an argument similar to that in the proof of Theorem 2, we can get out of (4.6) the following inequality:

(4.7) |hAx, xi|2≤

Re

_{λ}¯

|λ|Ax, x

2

+ 2

|λ| −

q

|λ|2−ρ2

Re
_{λ}¯

|λ|Ax, x

for anyx∈H,kxk= 1.

Taking the supremum overx∈H,kxk= 1, we deduce the desired inequality in (4.5).

Then following lemma is of interest in itself.

Lemma 3. For anyA∈B(H)andγ,Γ∈Kwe have

(4.8) inf kxk=1

Re [h(ΓI−A)x, xi hx,(A−γI)xi] = 1 4|Γ−γ|

2

−w2

A−γ+ Γ

2 I

.

Proof. We observe that for any u, v, y complex numbers, we have the elementary identity:

(4.9) Re [(u−y) (¯y−v¯)] = 1 4|u−v|

2

−

y−u+v

2 2

.

If we choose in (4.9)u= Γ, y=hHx, xiand v=γ with x∈H, kxk= 1, then by (4.9) we have:

(4.10) Re [h(ΓI−A)x, xi hx,(A−γI)xi] = 1 4|Γ−γ|

2

−

A−γ+ Γ

2 I

x, x

2

for eachx∈H,kxk= 1.

Remark 11. We observe that for anyx∈H,kxk= 1we have µ(A;γ,Γ) (x) := Re [h(ΓI−A)x, xi hx,(A−γI)xi]

= (Re Γ−RehAx, xi) (RehAx, xi −Reγ)

+ (Im Γ−ImhAx, xi) (ImhAx, xi −Imγ)

and therefore a sufficient condition for µ(A;γ,Γ) (x) to be nonnegative for each x∈H,kxk= 1is that:

(4.11)

Re Γ≥RehAx, xi ≥Reγ

Im Γ≥ImhAx, xi ≥Imγ

, x∈H, kxk= 1.

Now, if we denote by µ_{i}(A;γ,Γ) := infkxk=1µ(A;γ,Γ) (x), then we can state
the following lemma.

Lemma 4. ForA∈B(H), φ,Φ∈_{K}, the following statements are equivalent:
(i) µ_{i}(A;φ; Φ)≥0;

(ii) wA−φ+Φ_{2} I≤ 1

2|Φ−φ|.

Utilising the above results we can provide now some particular reverse inequali-ties that are of interest.

Corollary 8. Let A∈ B(H) andφ,Φ∈ Kwith Φ6=±φ such that either (i) or

(ii) of Lemma 4 holds true. Then

(0≤)w(A)−

vs

¯_{φ}_{+ ¯}_{Φ}

|φ+ Φ|A

≤w(A)−vs

_{φ}¯_{+ ¯}_{Φ}

|φ+ Φ|A

(4.12)

≤1

4 ·

|Φ−φ|2 |Φ +φ|.

Remark 12. IfN > n >0are such that eitherµi(A;n, N)≥0orw A−n+N2 I

≤

1

2(N−n)for a given operator A∈B(A),then

(4.13) (0≤)w(A)− |vs(A)| ≤w(A)−vs(A)≤ 1

4·

(N−n)2 N+n .

From a different perspective, we can state the following multiplicative reverse of the inequality (4.1).

An equivalent additive version of (4.4) is incorporated in the following:

Corollary 9. Ifw(A−λI)≤ |λ|, then we have

(0≤)w2(A)−w_{s}2

_{¯}

λ

|λ|A

≤w

2_{(}_{A}_{)}_{w}2_{(}_{A}_{−}_{λI}_{)}

|λ|2

(4.14)

≤

kAk2_{w}2_{(A}_{−}_{λI}_{)}
|λ|2

w2(A)kA−λIk2 |λ|2

≤ kAk

2

kA−λIk2 |λ|2

.

Corollary 10. Let A∈B(H)\ {0} andφ,Φ∈Kwith Φ6=−φ. If Re Φ¯φ>0

and either the statement (i) or equivalently (ii) from Lemma 4 holds true, then:

(4.15)

2 q

Re Φ¯φ

|φ+ Φ| ≤

vs
_{¯}

φ+ ¯Φ |φ+Φ|A

w(A) ≤ vs

_{¯}
φ+ ¯Φ
|φ+Φ|A

w(A) (≤1) and

(0≤)w2(A)−v_{s}2

_{φ}¯_{+ ¯}_{Φ}

|φ+ Φ|A

(4.16)

≤ 1

4·

|Φ−φ|2

Re Φ¯φv 2 s

_{φ}¯_{+ ¯}_{Φ}

|φ+ Φ|A

≤1

4 ·

|Φ−φ|2

Re Φ¯φw

2_{(}_{A}_{)}_{≤}1
4 ·

|Φ−φ|2

Re Φ¯φkAk 2

!

.

The proof follows by Theorem 3 on utilising a similar argument to the one incorporated in the proof of Corollary 6 and the details are omitted.

Remark 13. IfN > n >0 are such that eitherµ_{i}(A;n, N)≥0 or, equivalently

(4.17) w

A−n+N

2

≤1

2(N−n), then

(4.18) 2

√

nN n+N ≤

vs(A) w(A) (≤1),

(4.19) (0≤)w(A)−vs(A)≤ √

N−√n 2

2√nN vs(A)

≤

√

N−√n 2

2√nN w(A)

and

(4.20) (0≤)w2(A)−v_{s}2(A)≤ (N−n)

2

4nN v 2

s(A) ≤

(N−n)2 4nN w

2_{(}_{A}_{)}
!

.

Finally, we can state the following result as well:

Corollary 11. Let A ∈ B(H), φ,Φ ∈ K such that Re Φ¯φ > 0. If either

µ_{i}(A;φ,Φ)≥0 or, equivalently

w

A−Φ +φ

2 I

≤ 1

2|Φ−φ|, then

(0≤)w2(A)−v2s

_{¯}

φ+ ¯Φ

|φ+ Φ|A

(4.21)

≤

|φ+ Φ| −2 q

Re Φ¯φ

vs

_{φ}¯_{+ ¯}_{Φ}

|φ+ Φ|A

≤

|φ+ Φ| −2 q

Re Φ¯φ

w(A)

Remark 14. IfN > n >0 are as in Remark 13, then we have the inequality:
(4.22) (0≤)w2(A)−v2_{s}(A)≤√N−√n

2 vs(A)

≤√N−√n 2

w(A)

.

References

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[2] R. BHATIA and F. KITTANEH, Norm inequalities for positive operators,Lett. Math. Phys.,

43(3) (1998), 205-231.

[3] S.S. DRAGOMIR, Semi Inner Products and Applications, Nova Science Publishers Inc., N.Y., 2004.

[4] S.S. DRAGOMIR, Reverse inequalities for the numerical radius of linear operators in Hilbert spaces,Bull. Austral. Math. Soc.,73(2006), 255-262.

[5] S.S. DRAGOMIR, Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces,Lin. Alg. Appl.,in press.

[6] S.S. DRAGOMIR, Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces, Preprint available inRGMIA Res. Rep. Coll.,8(2005), Supplement, Article 10. [ONLINEhttp://rgmia.vu.edu.au/v8(E).html].

[7] S.S. DRAGOMIR, Inequalities for the norm and the numerical radius of composite operators in Hilbert spaces, Preprint available inRGMIA Res. Rep. Coll.,8(2005), Supplement, Article 11. [ONLINEhttp://rgmia.vu.edu.au/v8(E).html].

[8] F. KITTANEH, Numerical radius inequalities for Hilbert space operators, Studia Math.,

168(1) (2005), 73-80.

[9] F. KITTANEH, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix,Studia Math.,158(1) (2003), 11-17.

[10] K.E. GUSTAFSON and D.K.M. RAO,Numerical Range, Springer Verlag, New York, 1997. [11] P.R. HALMOS,A Hilbert Space Problem Book,Second Ed., Springer-Verlag, New York, 1982. [12] J.K. MERIKOSKI and P. KUMAR, Lower bounds for the numerical radius,Lin. Alg. Appl.,

410(2005), 135-142.

School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, VIC, Australia. 8001

E-mail address:sever.dragomir@vu.edu.au