PDF Reverse Triangle Inequality for Hilbert C*-Modules

Reverse Triangle Inequality for Hilbert C*-Modules

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Khosravi, Mariam, Mahyar, Hakimeh and Moslehian, Mohammad Sal (2009) Reverse Triangle Inequality for Hilbert C*-Modules. Research report collection, 12 (1).

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M. KHOSRAVI¹, H. MAHYAR² AND M. S. MOSLEHIAN³

Abstract. We prove several versions of reverse triangle inequality in HilbertC^∗-modules based on some works of S. S. Dragomir. In particular, we show that if e1,· · · , em are vectors in a Hilbert moduleX over aC^∗-algebraA such thathe_i, e_ji= 0 (1≤i6=j ≤m) andkeik= 1 (1≤i≤m), and alsork, ρk ∈R(1≤k≤m) andx1,· · ·, xn∈Xsatisfy

0≤r²_kkxjk ≤Rehrkek, xji, 0≤ρ²_kkxjk ≤Imhρkek, xji, then

"_m X

k=1

(r_k²+ρ²_k)

#¹_{2 n} X

j=1

kxjk ≤

n

X

j=1

x_j ,

and the equality holds if and only if

n

X

j=1

xj=

n

X

j=1

kxjk

m

X

k=1

(rk+iρk)ek.

1. Introduction and preliminaries

The triangle inequality is one of the most fundamental inequalities in mathematics. Several mathematician have been investigated its generalizations and reverses.

Petrovitch [15] in 1917, proved that for complex number z1,· · · , zn, we have

|

n

X

j=1

zj| ≥cosθ

n

X

j=1

|zj|,

where 0< θ < ^π₂ and α−θ < arg z_j < α+θ (1≤j ≤n) for a real number α.

This inequality can be found also in Karamata’s book [8]. The first generalization of the triangle inequality in Hilbert space was given by Diaz and Matcalf [4]. They proved that for x₁,· · · , x_n in a Hilbert space H, if e be a unit vector H such that 0≤r ≤ ^Rehx_kx^j,ei

jk for some r ∈R and each 1≤j ≤n, then

r

n

X

j=1

kx_jk ≤ k

n

X

j=1

x_jk,

2000 Mathematics Subject Classification. Primary 46L08; Secondary 15A39, 26D15, 46L05, 51M16.

Key words and phrases. Triangle inequality, Reverse inequality, HilbertC^∗-module.

1

and the equality holds if and only if Pn

j=1xj =rPn

j=1kxjke.

Recently, a number of mathematicians have represented several refinements of the reverse triangle inequality in Hilbert spaces and normed spaces. See [1, 2, 3, 6, 7, 9, 12, 14].

Our aim is to give some generalizations of results in Hilbert spaces to the framework of Hilbert C^∗-modules. For this purpose, first we recall some fundamental definitions in the theory of Hilbert modules. We also use the elementary C^∗-algebra theory, in particular we utilize this property that if a ≤ b then a^1/2 ≤ b^1/2, where a, b are positive elements of a C^∗-algebra A. We also repeatedly apply the following known relation:

1

2(aa^∗+a^∗a) = (Rea)²+ (Ima)², () whereais an arbitrary element ofA. For details onC^∗-algebra theory we referred the readers to [13].

Suppose that A is a C^∗-algebra and X is a linear space which is an algebraic right A- module. The space X is called a pre-Hilbert A-module (or an inner product A-module) if there exists an A-valued inner product h., .i:X×X→Awith the following properties:

(i) hx, xi ≥0 and hx, xi= 0 if and only if x= 0 (ii) hx, λy+zi=λhx, yi+hx, zi

(iii) hx, yai=hx, yia (iv) hx, yi^∗ =hy, xi

for all x, y, z ∈ X, a ∈ A, λ ∈ C. By (ii) and (iv), h., .i is conjogate linear in the first variable. Using the Cauchy–Schwartz inequality hy, xihx, yi ≤ khx, xikhy, yi [10, Page 5], it follows that kxk = khx, xik¹² is a norm on X making it into a right normed module. The pre-Hilbert module X is called a Hilbert A-module if it is complete with respect to this norm. Notice that the inner structure of a C^∗-algebra is essentially more complicated than complex numbers. For instance, the notations such as orthogonality and theorems such as Riesz’ representation in the complex Hilbert space theory cannot simply be generalized or transferred to the theory of Hilbert C^∗-modules.

One may define an “A-valued norm” |.| by |x|= hx, xi^1/2. Clearly, k |x| k=kxk for each x ∈ X. It is known that |.| does not satisfy the triangle inequality in general. See [10, 11]

for more information on Hilbert C^∗-modules.

2. Main results

Utilizing some C^∗-algebraic techniques we present our first result as a generalization of [6, Theorem 2.3].

Theorem 2.1. Let A be a C^∗-algebra with unit 1, let X be a Hilbert A-module and let x₁,· · · , x_n ∈X. If there exist real numbers k₁, k₂ ≥0 with

0≤k1kxjk ≤Rehe, xji, 0≤k2kxjk ≤Imhe, xji, for some e∈X with |e| ≤1 and all 1≤j ≤n, then

(k²₁+k₂²)¹²

n

X

j=1

kxjk ≤

n

X

j=1

xj

. (2.1)

Proof. Applying the Cauchy–Schwarz inequality, we get

|he,

n

X

j=1

x_ji|² ≤ kek²

n

X

j=1

x_j

2

≤

n

X

j=1

x_j

2

,

and

|h

n

X

j=1

x_j, ei|² ≤

n

X

j=1

x_j

2

|e|² ≤

n

X

j=1

x_j

2

,

whence

Pn j=1x_j

2

≥ ¹₂

|he,Pn

j=1x_ji|²+|hPn

j=1x_j, ei|²

= ¹₂ he,Pn

j=1x_ji^∗he,Pn

j=1x_ji+hPn

j=1x_j, ei^∗hPn

j=1x_j, ei

= (Rehe,Pn

j=1x_ji)²+ (Imhe,Pn

j=1x_ji)² by ()

= (RePn

j=1he, x_ji)²+ (ImPn

j=1he, x_ji)²

≥k₁²(Pn

j=1kx_jk)²+k²₂(Pn

j=1kx_jk)²

= (k²₁+k₂²)(Pn

j=1kxjk)².

Using the same argument as in the proof of Theorem 2.1 one can obtain the following result, where k₁, k₂ are hermitian elements of A.

Theorem 2.2. If the vectors x₁,· · · , x_n∈X satisfy the conditions

0≤k²₁kx_jk² ≤(Rehe, x_ji)², 0≤k₂²kx_jk² ≤(Imhe, x_ji)²,

for some hermitian elements k₁, k₂ in A, some e ∈ X with |e| ≤ 1 and all 1 ≤ j ≤ n then the inequality 2.1 holds.

One may observe an integral version of inequality (2.1) as follows:

Corollary 2.3. LetXbe a Hilbert module over aC^∗-algebraAwith unit 1and letf : [a, b]→ X be strongly measurable such that the Lebesgue integral Rb

a kf(t)kdt exist and be finite. If there exist self-adjoint elements a₁, a₂ in A with

a²₁kf(t)k² ≤Rehf(t), ei², a²₂kf(t)k² ≤Imhf(t), ei² ( a.e. t∈[a, b]), where e∈X with |e| ≤1, then

(a²₁+a²₂)¹² Z b

a

kf(t)kdt≤ k Z b

a

f(t)dtk.

Now we prove a useful lemma which is frequently applied in the next theorems.

Lemma 2.4. Let X be a Hilbert A-module and let x, y ∈X. If |hx, yi|=kxkkyk, then y= xhx, yi

kxk² . Proof. For x, y ∈X we have

0≤

y− ^xhx,yi_kxk2

2

=hy−^xhx,yi_kxk2 , y−^xhx,yi_kxk2 i

=hy, yi −_kxk¹2hy, xihx, yi+ _kxk¹4hy, xihx, xihx, yi − _kxk¹2hy, xihx, yi

≤ |y|²− _kxk¹2|hx, yi|² =|y|²− _kxk¹2kxk²kyk²

=|y|²− kyk² ≤0, whence

y− ^xhx,yi_kxk2

= 0. Hence y= ^xhx,yi_kxk2 .

Using the Cauchy-Schwarz inequality, we have the following theorem for Hilbert modules, which is similar to [1, Theorem 2.5].

Theorem 2.5. Let e1,· · ·, em be a family of vectors in a Hilbert C^∗-module X such that he_i, e_ji= 0 (1≤i6=j ≤m)and ke_ik= 1 (1≤i≤m). Suppose thatr_k, ρ_k∈R (1≤k≤m) and that the vectors x₁,· · · , x_n ∈X satisfy

0≤r²_kkx_jk ≤Rehr_ke_k, x_ji, 0≤ρ²_kkx_jk ≤Imhρ_ke_k, x_ji, Then

" _m X

k=1

(r²_k+ρ²_k)

#¹_{2 n} X

j=1

kx_jk ≤

n

X

j=1

x_j

, (2.2)

and the equality holds if and only if

n

X

j=1

x_j =

n

X

j=1

kx_jk

m

X

k=1

(r_k+iρ_k)e_k. (2.3)

Proof. There is nothing to prove if Pm

k=1(r_k² +ρ²_k) = 0. Assume that Pm

k=1(r²_k+ρ²_k) 6= 0.

From the hypothesis we have Pm

k=1(r²_k+ρ²_k)2 Pn

j=1kxjk2

≤ RehPm

k=1rkek,Pn

j=1xji+ ImhPm

k=1ρkek,Pn

j=1xji2

= RehPn

j=1x_j,Pm

k=1(r_k+iρ_k)e_ki2

by Im(a) = Re(ia^∗),Re(a^∗) = Re(a) (a∈A)

≤ ¹₂|hPn

j=1x_j,Pm

k=1(r_k+iρ_k)e_ki|² +|hPm

k=1(r_k+iρ_k)e_k,Pn

j=1x_ji|² by ()

≤ ¹₂kPn

j=1xjk²|Pm

k=1(rk+iρk)ek|² +kPm

k=1(r_k+iρ_k)e_kk²|Pn j=1x_j|²

≤ kPn

j=1x_jk²kPm

k=1(r_k+iρ_k)e_kk² by|a| ≤ kak (a∈A)

=kPn

j=1x_jk²khPm

k=1(r_k+iρ_k)e_k,Pm

k=1(r_k+iρ_k)e_kik

=kPn

j=1xjk²Pm

k=1|rk+iρk|²kekk²

=kPn

j=1x_jk²Pm

k=1(r²_k+ρ²_k). Hence

" _m X

k=1

(r_k²+ρ²_k)

# (

n

X

j=1

kx_jk)² ≤

n

X

j=1

x_j

2

.

By taking square roots the desired result follows.

Clearly we have equality in 2.2 if condition 2.3 holds. To see the converse, first note that if equality holds in 2.2, then all inequalities in the above relations should be equality. Therefore

r_k²kx_jk=Rehr_ke_k, x_ji, ρ²_kkx_jk=Imhρ_ke_k, x_ji,

Reh

n

X

j=1

xj,

m

X

k=1

(rk+iρk)eki=h

n

X

j=1

xj,

m

X

k=1

(rk+iρk)eki, and also

|h

m

X

k=1

(r_k+iρ_k)e_k,

n

X

j=1

x_ji|=k

n

X

j=1

x_jkk

m

X

k=1

(r_k+iρ_k)e_kk.

From Lemma 2.4 and these equalities we have Pn

j=1xj =

Pm

k=1(r_k+iρ_k)e_k kPm

k=1(rk+iρk)ekk²hPm

k=1(rk+iρk)ek,Pn j=1xji

=

Pm

k=1(r_k+iρ_k)e_k Pm

k=1(r²_k+ρ²_k) RehPm

k=1(rk+iρk)ek,Pn j=1xji

=

Pm

k=1(r_k+iρ_k)e_k Pm

k=1(r²_k+ρ²_k)

Pm k=1

Pn

j=1(r_k²kx_jk+ρ²_kkx_jk)

=Pn

j=1kx_jkPm

k=1(r_k+iρ_k)e_k,

which is the desired result.

From now on we assume that X is a right Hilbert module over a C^∗-algebra A, which is an algebraic left A-module subject to

hx, ayi=ahx, yi (x, y ∈X, a∈A). (†) For example if A is a C^∗-algebra and Ibe a commutative right ideal of A, thenI is a right Hilbert module over A and

hx, ayi=x^∗(ay) =ax^∗y =ahx, yi (x, y ∈I, a∈A).

The next theorem is a refinement of [6, Theorem 2.1]. To prove it we need the following lemma.

Lemma 2.6. Let X be a Hilbert A-module and e1,· · · , en ∈ X be a family of vectors such that he_i, e_ji= 0 (i6=j) and ke_ik= 1. If x∈X, then

|x|² ≥

n

X

k=1

|hek, xi|² and |x|² ≥

n

X

k=1

|hx, eki|².

Proof. The first result follows from the following inequality:

0≤ |x−Pn

k=1e_khe_k, xi|² =hx−Pn

k=1e_khe_k, xi, x−Pn

j=1e_jhe_j, xii

=hx, xi+Pn k=1

Pn

j=1he_k, xi^∗he_k, e_jihe_j, xi −2Pn

k=1|he_k, xi|²

=hx, xi+Pn

k=1he_k, xi^∗he_k, e_kihe_k, xi −2Pn

k=1|he_k, xi|²

≤ |x|²+Pn

k=1he_k, xi^∗he_k, xi −2Pn

k=1|he_k, xi|²

=|x|²−Pn

k=1|hek, xi|². By considering |x−Pn

k=1he_k, xie_k|², similarly, we have the second one.

Now we are able to prove the next theorem without using the Cauchy–Schwarz inequality.

Theorem 2.7. Let e1,· · · , em ∈ X be a family of vectors with hei, eji= 0 (1 ≤i 6=j ≤m) and ke_ik= 1 (1≤i≤m). If the vectors x₁,· · ·, x_n ∈X satisfy the conditions

0≤r_kkx_jk ≤Rehe_k, x_ji, 0≤ρ_kkx_jk ≤Imhe_k, x_ji (1≤j ≤n,1≤k ≤m), (2.4) where rk, ρk ∈[0,∞) (1≤k ≤m), then

" _m X

k=1

(r²_k+ρ²_k)

#¹_{2 n} X

j=1

kx_jk ≤

n

X

j=1

x_j

. (2.5)

Proof. Applying the previous lemma for x=Pn

j=1x_j, we obtain

Pn j=1x_j

2

≥ ¹₂ Pm

k=1|he_k,Pn

j=1x_ji|²+Pm

k=1|hPn

j=1x_j, e_ki|²

=Pm k=1

1 2

he_k,Pn

j=1x_ji^∗he_k,Pn

j=1x_ji+hPn

j=1x_j, e_ki^∗hPn

j=1x_j, e_ki

=Pm

k=1(Rehe_k,Pn

j=1x_ji)²+ (Imhe_k,Pn

j=1x_ji)² by ()

=Pm

k=1(RePn

j=1he_k, x_ji)²+ (ImPn

j=1he_k, x_ji)²

≥Pm

k=1(r_k²(Pn

j=1kx_jk)²+ρ²_k(Pn

j=1kx_jk)²) by (2.4)

=Pm

k=1(r_k²+ρ²_k)(Pn

j=1kx_jk)².

Proposition 2.8. In Theorem 2.7, if he_k, e_ki = 1, then the equality holds in (2.5) if and only if

n

X

j=1

x_j = (

n

X

j=1

kx_jk)

m

X

k=1

(r_k+iρ_k)e_k. (2.6)

Proof. If (2.6) holds, then inequality in (2.5) turns trivially into equality.

Next, assume that equality holds in (2.5). Then two inequalities in the proof of Theorem 2.7 should be equality. Hence

|

n

X

j=1

x_j|² =

m

X

k=1

|he_k,

n

X

j=1

x_ji|² and |

n

X

j=1

x_j|² =

m

X

k=1

|h

n

X

j=1

x_j, e_ki|²,

which is equivalent to

n

X

j=1

x_j =

m

X

k=1 n

X

j=1

e_khe_k, x_ji=

m

X

k=1 n

X

j=1

he_k, x_jie_k,

and also

rkkxjk= Rehek, xji, ρkkxjk= Imhek, xji.

So

n

X

j=1

x_j =

m

X

k=1 n

X

j=1

e_khe_k, x_ji=

m

X

k=1 n

X

j=1

e_k(r_k+iρ_k)kx_jk= (

n

X

j=1

kx_jk)

m

X

k=1

(r_k+iρ_k)e_k. There are some versions of additive reverse of triangle inequality. In [5], S. S. Dragomir established the following theorem:

Theorem 2.9. Let {e_k}^m_k=1 be a family of orthonormal vectors in Hilbert space H and M_jk ≥0 (1≤i≤n,1≤k ≤m) such that

kx_jk −Rehe_k, x_ji ≤M_jk, for each 1≤i≤n and 1≤k ≤m. Then

n

X

j=1

kx_jk ≤ 1

√mk

n

X

j=1

x_jk+ 1 m

n

X

j=1 m

X

k=1

M_jk; and the equality holds if and only if

n

X

j=1

kx_jk ≥ 1 m

n

X

j=1 m

X

k=1

M_jk,

and

n

X

j=1

x_j =

n

X

j=1

kx_jk − 1 m

n

X

j=1 m

X

k=1

M_jk

! _m X

k=1

e_k.

Now we extend this theorem for Hilbert modules.

Theorem 2.10. Let {e_k}^m_k=1 be a family of vectors in Hilbert A-module Xwith|e_k| ≤1 (1≤ k ≤ m) and he_i, e_ji = 0 (1 ≤ i 6= j ≤ m) and x_j ∈ X(1 ≤ i ≤ n). If for some scalars M_jk ≥0 (1≤i≤n, 1≤k ≤m),

kx_jk −Rehe_k, x_ji ≤M_jk (1≤i≤n,1≤k ≤m), (2.7) then

n

X

j=1

kx_jk ≤ 1

√mk

n

X

j=1

x_jk+ 1 m

n

X

j=1 m

X

k=1

M_jk. (2.8)

Moreover, if |e_k|= 1 (1≤k≤m), then the equality in (2.8) holds if and only if

n

X

j=1

kx_jk ≥ 1 m

n

X

j=1 m

X

k=1

M_jk, (2.9)

and

n

X

j=1

x_j =

n

X

j=1

kx_jk − 1 m

n

X

j=1 m

X

k=1

M_jk

! _m X

k=1

e_k. (2.10)

Proof. Taking the summation in (2.7) over ifrom 1 to n, we obtain

n

X

j=1

kx_jk ≤Rehe_k,

n

X

j=1

x_ji+

n

X

j=1

M_jk,

for each k∈ {1,· · · , m}. Summing these inequalities over k from 1 to m, we deduce

n

X

j=1

kxjk ≤ 1 mReh

m

X

k=1

ek,

n

X

j=1

xji+ 1 m

m

X

k=1 n

X

j=1

Mjk. (2.11)

Using the Cauchy–Schwarz we obtain RehPm

k=1e_k,Pn

j=1x_ji2

≤ ¹₂(|hPm

k=1e_k,Pn

j=1x_ji|²+|hPm

k=1e_k,Pn

j=1x_ji^∗|²) by ()

≤ ¹₂(kPm

k=1e_kk²|Pn

j=1x_j|²+|Pm

k=1e_k|²kPn

j=1x_jk²)

≤ kPm

k=1e_kk²kPn j=1x_jk²

≤mkPn

j=1x_jk²,

(2.12)

since

k

m

X

k=1

e_kk² =kh

m

X

k=1

e_k,

m

X

k=1

e_kik=k

m

X

k=1 m

X

l=1

he_k, e_lik=k

m

X

k=1

|e_k|²k ≤m .

Using (2.12) in (2.11), we deduce the desired inequality.

If (2.9) and (2.10) hold, then

√1 m

Pn j=1x_j

= ^√1_m Pn

j=1kx_jk −_m¹ Pn j=1

Pm

k=1M_jk kPm

k=1e_kk

=Pn

j=1kx_jk −_m¹ Pn j=1

Pm

k=1M_jk, and then the equality in 2.8 holds true.

Conversely, if the equality holds in (2.8), then obviously (2.9) is valid and we have equalities all over in the above proof. This means that

kx_jk −Rehe_k, x_ji=M_jk, Reh

m

X

k=1

e_k,

n

X

j=1

x_ji=h

m

X

k=1

e_k,

n

X

j=1

x_ji,

and

|h

m

X

k=1

e_k,

n

X

j=1

x_ji|=k

m

X

k=1

e_kkk

n

X

j=1

x_jk.

Now from Lemma 2.4 and these relations, we have Pn

j=1x_j =

Pm k=1ek

kPm

k=1e_kk²hPm

k=1e_k,Pn j=1x_ji

=

Pm k=1ek

m RehPm

k=1e_k,Pn j=1x_ji

=

Pm k=1e_k

m

Pm k=1

Pn

j=1(kxjk −Mjk)

= Pn

j=1kx_jk − _m¹ Pn j=1

Pm

k=1M_jk Pm

k=1e_k.

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[13] J. G. Murphy, C^∗-Algebras and Operator Theory, Academic Press, Boston, 1990.

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[15] M. Petrovich, Module d’une somme, L’ Ensignement Math.,19(1917), 53–56.

Maryam Khosravi: Department of Mathematics, Teacher Training University, Tahran, Iran.

E-mail address: khosravi−m@saba.tmu.ac.ir

Hakimeh Mahyar: Department of Mathematics, Teacher Training University, Tahran, Iran.

E-mail address: mahyar@saba.tmu.ac.ir

Mohammad Sal Moslehian: Department of Mathematics, Ferdowsi University, P. O. Box 1159, Mashhad 91775, Iran;

Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University, Iran.

E-mail address: moslehian@ferdowsi.um.ac.ir and moslehian@ams.org