Volume 17, Number 1 (2019), 1-13
URL:https://doi.org/10.28924/2291-8639
DOI:10.28924/2291-8639-17-2019-1
CONTROLLED ∗-G-FRAMES AND ∗-G-MULTIPLIERS IN HILBERT PRO-C∗-MODULES
ZAHRA AHMADI MOOSAVI1,∗ AND AKBAR NAZARI2
1Department of Mathematics Faculty of Mathematics and computer Shahid Bahonar University of Kerman,
76169-14111, Kerman, Iran
2Department of Mathematics Faculty of Mathematics and computer Shahid Bahonar University of Kerman,
76169-14111, Kerman, Iran
∗Corresponding author: [email protected]
Abstract. A generalization of multiplier, controlled g-frames and g-Bessel sequences to ∗-g-frames and ∗-g-Bessel sequences in Hilbert pro-C∗-modules is presented. It is demonstrated that controlled∗-g-frames
are equivalent to∗-g-frames in Hilbert pro-C∗-modules.
1. Introduction
Frame theory is an application of harmonic analysis. This theory has been rapidly generalized to Hilbert
spaces and Hilbert C∗-modules. In 2005, Sun [22] introduced the notion of g-frames as a generalization of
frames for bounded operators on Hilbert spaces. Frank-Larson [5] have extended the theory for elements of
C∗-algebras and (finitely or countably generated) HilbertC∗-modules have been considered in [1].
It is well known that HilbertC∗-modules are a generalization of Hilbert spaces where the inner product
takes values in aC∗-algebra rather than in the field of complex numbers. The theory of HilbertC∗-modules
Received 2018-04-13; accepted 2018-06-22; published 2019-01-04. 2010Mathematics Subject Classification. 42C15, 46L08.
Key words and phrases. Hilbert pro-C∗-modules;∗-g-frames;∗-g-Bessel sequences; controlled∗-g-frames; (C, C0)-controlled
∗-g-frames.
c
2019 Authors retain the copyrights
of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
has applications in the study of locally compact quantum groups, complete maps betweenC∗-algebras,
non-commutative geometry and KK-theory. Not all properties of Hilbert spaces hold in HilbertC∗-modules. For
instance, the Riesz representation theorem for continuous linear functionals on Hilbert spaces can not be
extended to HilbertC∗-modules [23] and there exist closed subspaces in HilbertC∗-modules that have no
orthogonal complement [16]. Moreover, as known, every bounded operator on a Hilbert space has an adjoint
whereas there are bounded operators on HilbertC∗-modules which do not have this property [17]. So, it is
to be expected that frames and∗-frames in HilbertC∗-modules are more complicated than those in Hilbert
spaces. The properties of g-frames for HilbertC∗-modules have been widely investigated in the literature (
see [1,5,12,25], and the references therein).
The paper is organized as follows.In the next section, we give a brief survey of the fundamental definitions
and notations of Hilbert pro-C∗-modules.
Section 3 is devoted to investigating ∗-g-frames with A-valued bounds and analyzing their elementary
properties. In Section 4 we define the concept of controlled∗-g-frames and we show that a controlled
∗-g-frame is equivalent to a ∗-g-frame in Hilbert pro-C∗-modules. Finally, in section 5 we define multipliers of
controlled∗-g-frame operators in Hilbert pro-C∗-modules.
2. Preliminaries
In this section, we recall some of the basic definitions and properties of pro-C∗-algebras and Hilbert
modules over them [7,15,18].
A pro-C∗-algebra is a complete Hausdorff complex topological∗-algebraAwhose topology is determined
by its continuousC∗-seminorms in the sense that a net{aλ} converges to 0 iffρ(aλ)→0 for any continuous
C∗-seminormρonAand we have:
(1) ρ(ab)≤ρ(a)ρ(b);
(2) ρ(a∗a) =ρ(a)2;
for allC∗-seminormsρonAanda, b∈ A.
If the topology of pro-C∗-algebra is determined by only countably manyC∗-seminorms, then it is called
aσ-C∗-algebra.
Let A be a unital pro-C∗-algebra with unit 1A and let a ∈ A . Then spectrum sp(a) of a ∈ A is the
set {λ∈C: λ1A−ais not invertible}. If Ais not unital, then the spectrum is taken with respect to its
unitization ˜A.
IfA+ denotes the set of all positive elements of A, thenA+ is a closed convexC∗-seminorms onA. We
denote byS(A), the set of all continuousC∗-seminorms onA.
Example 2.2. A sub-closed ∗-algebra of a pro-C∗-algebra is a pro-C∗-algebra.
Proposition 2.1 ( [6]). Let Abe a unital pro-C∗-algebra with an identity 1A.Then for anyρ∈S(A), we
have:
(1) ρ(a) =ρ(a∗) for alla∈A;
(2) ρ(1A) = 1;
(3) If a, b∈ A+ anda≤b, thenρ(a)≤ρ(b);
(4) If 1A≤b, thenb is invertible andb−1≤1A;
(5) If a, b∈ A+ are invertible and0≤a≤b, then0≤b−1≤a−1;
(6) If a, b, c∈ Aanda≤b thenc∗ac≤c∗bc;
(7) If a, b∈ A+ anda2≤b2, then0≤a≤b.
Definition 2.1. A pre-Hilbert module over pro-C∗-algebra A, is a complex vector space E which is also
a left A-module compatible with the complex algebra structure, equipped with an A-valued inner product
h., .i:E×E→ A which isC-andA-linear in its first variable and satisfies the following conditions:
(1) hx, yi∗=hy, xi;
(2) hx, xi ≥0;
(3) hx, xi= 0 iffx= 0;
for everyx, y∈E. We sayE is a HilbertA-module (or Hilbert pro-C∗-module overA) IfE is complete with
respect to the topology determined by the family of seminorms
ρE(x) =pρ(hx, xi) x∈E, ρ∈S(A).
Let E be a pre-Hilbert A-module.By [6], for ρ ∈ S(A) and for all x, y ∈ E, the following
Cauchy-Bunyakovskii inequality holds:
ρ(hx, yi)2≤ρ(hx, xi)ρ(hy, yi).
Consequently, for eachρ∈S(A), we have:
ρE(ax)≤ρ(a)ρ(x), a∈ A, x∈E.
LetA be a pro-C∗-algebra and E and F be two Hilbert A-modules. An A-module mapT : E → F is
said to bounded if for eachρ∈S(A), there isCρ>0 such that :
ρF(T x)≤Cρ. ρE(x) (x∈E),
whereρE, respectivelyρF, are continuous seminorms onE, respectivelyF. A boundedA-module map from
EtoF is called an operators fromEtoF . We denote the set of all operators fromEtoF byHomA(E, F),
Proposition 2.2. Let T∗ ∈HomA(E, F). We say T is adjointable if there exists an operator T∗ ∈ T ∈
HomA(F, E)such that:
hT x, yi=hx, T∗yi
holds for allx∈E, y∈F.
We denote byHom∗A(E, F), the set of all adjointable operator fromEtoFandEnd∗A(E) =Hom∗A(E, E)
Proposition 2.3 ( [6]). Let T :E→F andT∗:F→E be two maps such that the equality
hT x, yi=hx, T∗yi
holds for allx∈E, y∈F.ThenT ∈Hom∗
A(E, F).
It is easy to see that for anyρ∈S(A), the map defined by:
ˆ
ρE,F(T) = sup{ρF(T(x) : x∈E, ρE(x)≤1}, T∈HomA(E, F),
is a seminorm onHomA(E, F).
Definition 2.2. Let E andF be two Hilbert modules over pro-C∗-algebraA. Then the operatorT :E→F
is called uniformly bounded (below), if there exists C >0 such that:
ρF(T x)≤C ρE(x). (2.1)
(C ρE(x)≤ρF(T x)) (2.2)
The numberC in (2.1) is called an upper bound forT and we set :
kTk∞= inf{C:Cis an upper bound forT}.
Clearly, in this case we have:
ˆ
ρ(T)≤ kTk∞, ∀ρ∈S(A).
LetT be an invertible element inEnd∗A(E) such that both are uniformly bounded. Then by [2, Proposition
3.2], for eachx∈E we have the inequality
kT−1k−2
∞hx, xi ≤ hT x, T xi ≤ kTk2∞hx, xi. (2.3)
The following proposition will be used in the next section.
Proposition 2.4( [6]).LetTbe an uniformly bounded below operator inHomA(E, F). then T is closed(range)
3. ∗-G-frames in Hilbert pro-C∗-modules
Throughout this sectionA is a pro-C∗-algebra, U and V are two Hilbert A-modules. also{Vj}j∈J is a
countable sequence of closed submodules ofV.
Definition 3.1. A sequence Λ = {Λj ∈ Hom∗A(U, Vj)}j∈J is called a ∗- g-frame for U with respect to
{Vj}j∈J if
Chf, fiC∗≤P
j∈JhΛjf,Λjfi ≤Dhf, fiD∗
for allf ∈U and strictly nonzero elementsC, D∈ A.
The number C and D are called ∗-g-frame bounds for Λ. The ∗-g-frame is called tight if C =D and a
Parseval ifC=D= 1. If in the above we only have the upper bound, thenΛis called a∗-g-Bessel sequence.
Also if for each j∈J, Vj =V, we callΛ a∗-g-frame for U with respect toV.
We mentioned that the set of all g-frames in Hilbert pro-C∗-modules are a subset of the family of
∗-g-frames. To illustrate this, let Λ = {Λj}j∈J be a g-frame for U with respect to {Vj}j∈J. Note that for
f ∈U,
(√C)1Ahf, fi( √
C)1A≤Pj∈JhΛjf,Λjfi(
√
D)1Ahf, fi( √
D)1A
Therefore, every g-frame for U with real bounds C amd D is a∗-g-frame for U with A-valued ∗-g-frame
bounds (√C)1A and ( √
D)1A.
Example 3.1. Let `2(A) be the set of all sequences (an)n∈N of elements of a pro-C∗-algebraA such that
the seriesP
i∈Naia∗i is convergent inA. Then, by [2, Example 3.2],`2(A) is a Hilbert module overAwith respect to pointwise operations and inner product defined by:
h(ai)i∈N,(bi)i∈Ni=
X
i∈N aib∗i.
Let a = (ai)i∈N and b = (bi)i∈N in `2(A). We define ab ={aibi}i∈N and ρ(a) =
p
ρ(ha, ai) and a∗ :=
{ai}i∈N andha, bi=ab∗=Pi∈Naib∗i.
Now, let j∈J :=N and definefj ∈`2(A)byfj={fj
i}i∈N such that
fij=
1
i1A i=j;
0 i6=j, ∀j ∈N.
Set Λj:`2(A)→ A byΛfj(U) =hU, fji for anyU ∈`
2(A). We see that
P
j∈JhΛfj(U),Λfj(U)i ≤ hU, Ui.
Definition 3.2. Let Λ ={Λj ∈End∗A(U, Vj)}j∈J be a∗-g-frame forU with respect to{Vj}j∈J with bounds
C andD. We define the corresponding∗-g-frame transform as follows:
TΛ:U →Lj∈JVj , TΛf ={Λjf : j ∈J}, for all f ∈U.
Since Λis a∗-g-frame, hence for eachf ∈U we have:
C hf, fiC∗≤P
j∈JhΛjf,Λjfi ≤D hf, fi D
∗,
SoTΛ is well-defined. Also for any ρ∈S(A) andf ∈U the following inequality is obtained:
ρ(C)2 ρ
U(f)≤ρL
jVj(TΛf)≤ρ(D) 2 ρ
U(f).
From the above, it follows that the∗-g-frame transform is an uniformly bounded below operator inEndA(U,Lj∈JVj).
Thus by Proposition 2.4,TΛ is closed and injective.
Now, we define the synthesis operator for∗-g-frameΛ as follows:
TΛ∗:M
j∈j
Vj →U, TΛ∗({yj}j) =
X
j∈J
Λ∗j(yj), (3.1)
whereΛ∗j is the adjoint operator ofΛj.
Proposition 3.1. The synthesis operator defined by (3.1)is well-defined, uniformly bounded and the adjoint
of the transform operator.
Proof. Since Λ ={Λj :j∈J}is a∗-g-frame forU with respect to{Vj}j∈J, there existC, D∈ A such that
for anyf ∈U,
C hf, fiC∗≤P
j∈JhΛjf,Λjfi ≤D hf, fiD∗.
LetI be an arbitrary finite subset ofJ. Using the Cauchy-Bunyakovskii inequality and [24, Lemma 2.2], for
anyρ∈S(A) and (yj)j ∈Lj∈JVj we have:
ρ(X
j∈I
Λ∗j(yj)) = sup{ρhX
j∈I
Λ∗j(yj), fi: f ∈U , ρ(f)≤1}
= sup{ρ(X
j∈I
hyj,Λjfi) : f ∈U , ρ(f)≤1}
≤ sup
ρ(f)≤1
ρ( X
j∈I
hyj, yji)
1/2
ρ( X
j∈I
hΛjf,Λjfi)
1/2
≤ sup
ρ(f)≤1
ρ(DD∗)1/2ρ(f)(ρX j∈I
hyj, yji)1/2
≤
ρ(D) (ρ X
j∈I
hyj, yji)1/2
Now, since the seriesP
j∈Jhyj, yjiconverges inA, the above inequality shows that
P
j∈JΛ
∗
j(yj) is convergent.
HenceTΛ∗ is well-defined. On the other hand, for anyf ∈U and (yj)j ∈Lj∈JVj, we have:
hTΛ(f),(yj)ji=h(Λjf)j,(yj)ji
=X
j∈J
hΛjf, yji
=X
j∈J
hf,Λ∗jyji
=hf,X j∈J
Λ∗jyji
=hf, TΛ∗(yj)j∈Ji.
Therefore by Proposition2.2it follows that the synthesis operator is the adjoint of the transform operator.
Also, for anyρ∈S(A) we have:
ρU(TΛ∗(y))≤ρ(D)ρL
j∈J Vj (y), y= (yj)j∈
L
j∈JVj.
Hence the synthesis operator is uniformly bounded.
Let Λ ={Λj , j ∈J} be a ∗-g-frame forU with repect to {Vj}j∈J. Define the corresponding∗-g-frame
operatorSΛ as follows:
SΛ=TΛ∗TΛ:U →U SΛ(f) =Pj∈JΛ∗jΛjf.
SinceSΛ is a combination of two bounded operators, it is a bounded operator.
Theorem 3.1. Let Λ ={Λj}j∈J be a∗-g-frame forU with respect to{Vj}j∈J and with bounds C, D. Then
SΛ is an invertible positive operator. Also it is a self-adjoint operator such that:
CIUC∗≤SΛ≤DIUD∗. (3.2)
Here IU is the identity function onU.
Proof. According to the definition of the transform operator, for anyf ∈U we can write:
hTΛ(f), TΛ(f)i=h{Λjf}j∈J,{Λjf}j∈Ji=Pj∈JhΛjf,Λjfi.
Since Λ is a∗-g-frame for U with boundsC andD, for eachf ∈U it follows that
Chf, fiC∗≤ hT
Λ(f), TΛ(f)i ≤Dhf, fiD∗.
On the other hand,
hSΛ(f), fi=hTΛ∗TΛ(f), fi=hTΛ(f), TΛ(f)i=hf, TΛ∗TΛ(f)i=hf, SΛ(f)i.
Chf, fiC∗≤ hSΛ(f), fi ≤Dhf, fiD∗.
It follows that∗-g-frame operator is positive and (3.2) also holds. Moreover, sinceSΛ is one-to-one it follows
thatSΛis invertible.
According to (3.2) and Proposition2.1we have the following Lemma
Lemma 3.1.
D−1IU(D−1)∗≤SΛ−1≤C−1IU(C−1)∗.
Hence the ∗-g-frame operator and its inverse belong toEnd∗A(U).
Theorem 3.2. Let {Λj ∈ End∗A(U, Vj)}j∈J and Pj∈JhΛjf,Λjfi converge in the semi-norm for f ∈ U.
ThenΛ ={Λj}j∈J is a∗-g-frame for U with respect to{Vj}j∈J if and only if there are two strictly nonzero
elementsC, D∈ Asuch that for everyf ∈U,
ρ(C−1)−1ρ(hf, fi)ρ(C∗−1)−1≤ρ(X
j∈J
hΛjf,Λjfi)
≤ρ(D)ρ(hf, fi)ρ(D∗). (3.3)
Proof. If{Λj∈End∗A(U, Vj)}j∈J is a∗-g-frame forU with respect to {Vj}j∈J, then
(hf, fi)≤C−1(X
j∈J
hΛjf,Λjfi)(C∗)−1)
and
(X
j∈J
hΛjf,Λjfi)≤Dhf, fiD∗.
Therefore, by Proposition2.1,
ρ(C−1)−1ρ(hf, fi)ρ(C∗−1)−1≤ρ(X
j∈J
hΛjf,Λjfi)
≤ρ(D)ρ(hf, fi)ρ(D∗). (3.4)
For the converse, let (3.3) hold. Then we define a linear operator as follows:
M :U →M
j∈J
Vj, M(f) ={Λjf}j∈J, ∀f ∈U,
hM f, M fi=X
j∈J
hΛjf,Λjfi, ∀f ∈U.
Hence, by (3.3), we have
ρU(M(f))≤ ρ(D)12 ρ
U(f)ρ(D
This shows that M is uniformly bounded. We write M∗M =K. ThenhM(f), M(f)i =hM∗M(f), fi=
hK(f), fi. Therefore,K is positive. As,K∗= (M∗M), K is self-adjoint. On the other hand,
hK12f, K 1
2fi=hKf, fi=X
j∈J
hΛjf,Λjfi.
Now, according to Proposition 2.4 and (3.3), K12 is invertible and uniformly bounded; therefore, by [2,
Proposition 3.2], we have:
kK−12k−1
∞hf, fikK− 1 2k−1
∞ ∗
≤ hK12(f), K12(f)i ≤ kK12k
∞hf, fikK 1 2k
∞
Hence{Λj}j∈J is a∗-g-frame.
4. Controlled ∗-G-frames in Hilbert pro-C∗-modules
In this section, we define the concept of multipliers for∗-g-Bessel sequences and we show that controlled
∗-g-frames are equivalent to∗-g-frames.
LetAbe a pro-C∗-algebra,U andV be two HilbertA-modules. also, let{Vj}j∈J be a countable sequence
of closed submodules ofV, L(U, V) andL(U) the collection of all bounded linear operators fromU into V
andU respectively. gl(U) the set of all bounded operators with a bounded inverse andgl+(U) be the set of
positive operators ingl(U).
Proposition 4.1. LetΛ ={Λj∈L(U, Vj) : j∈J}andθ={θj∈L(U, Vj) : j∈J}be∗-g-Bessel sequences
with bounds BΛ andBθ. If form={mj}j⊆`∞(R), the operator
M =Mm,Λ,θ:U →U
M(f) =X
j
mjΛ∗jθjf , (4.1)
is well-defined, then M is called the∗-g-multiplier ofΛ, θandm.
Proof. LetI be an arbitrary finite subset ofJ. Using the Cauchy-Bunyakovskii inequality and [24, Lemma
2.2], for anyρ∈S(A) andf ∈U we have:
ρ(X
j∈I
mjΛ∗jθjf) = sup{ρh
X
j∈I
mjΛ∗jθjf, gi: g∈U , ρ(g)≤1}
= sup{ρ(X
j∈I
hmjθjf,Λjgi) : g∈U , ρ(g)≤1}
≤ sup
ρ(g)≤1
ρ( X
j∈I
hmjθjf, mjθjfi)
1/2
ρ( X
j∈I
hΛjg,Λjgi)
1/2
Since
X
j
hmjθjf, mjθjfi=
X
j
mjhθjf, θjfim∗j
=X
j
(ρ(mj))2hθjf, θjfi
≤ kmk2∞Bθhf, fiBθ∗,
so by Proposition2.1we have:
ρ(P
jhmjθjf, mjθjfi)≤ kmk
2
∞(ρ(f))2ρ(Bθ)2.
Hence we have:
ρ(X
j∈I
mjΛ∗jθjf)≤ kmk∞ρ(f)ρ(Bθ)ρ(BΛ)
Definition 4.1. LetC, C0∈gl+(U). The family Λ ={Λ
j ∈L(U, Vj) : j∈J} is called a(C, C0)-controlled
∗-g-frame for U with respect to{Vj}j∈J, if Λis a∗-g-Bessel sequence and
Ahf, fiA∗≤X
j∈J
hΛjCf,ΛjC0fi ≤Bhf, fiB∗, (4.2)
for allf ∈U and strictly nonzero elementsA, B∈ A.
A, B are called controlled ∗-g-frame bounds. If C0 =I, we call Λ ={Λj}j a C-controlled ∗-g-frame for
U with bounds A, B. If only the second part of the above inequality holds, it is called a (C, C0)-controlled
∗-g-Bessel sequence with bound B.
Lemma 4.1 ( [2]). Let X be a Hilbert module over C∗-algebra B, S ≥ 0, i.e. this element is positive in
C∗-algebraL(U). Then for eachx∈X,
hSx, xi ≤ kSkhx, xi.
Proposition 4.2. Let C∈gl+(H). The family
Λ ={Λj∈L(U, Vj) : j∈J}
is a∗-g-frame if and only if Λis a C2- controlled∗-g-frame.
Proof. Let Λ be aC2- controlled ∗-g-frame with boundsA, B. Then
Ahf, fiA∗≤X
j∈J
hΛjCf,ΛjCfi ≤Bhf, fiB∗, forf ∈U.
Ahf, fiA∗=AhCC−1f, CC−1fiA∗≤AkCk2hC−1f, C−1fiA∗≤ kCk2X
j∈J
Hence
AkCk−1hf, fiA∗kCk−1≤X
j∈J
hΛjf,Λjfi.
On the other hand for everyf ∈U
X
j∈J
hΛjf,Λjfi=
X
j∈J
hΛjCC−1f, CC−1fi
≤BhC−1f, C−1fiB∗
≤BkC−1k2hf, fiB∗.
These inequalities yield that Λ is a∗-g-frame with boundsAkC−1k, BkC−1k. Conversely assume that Λ is
a∗-g-frame with boundsA0, B0. Then for allf ∈U,
A0hf, fiA0∗≤X
j∈J
hΛjf,Λjfi ≤B0hf, fiB0∗.
So forf ∈U,
X
j∈J
hΛjCf,ΛjCfi ≤B0hCf, CfiB0 ∗
≤B0kCk2B0∗.
For the lower bound, since Λ is∗-g-frame for anyf ∈U,
A0hf, fiA0∗=A0hC−1Cf, C−1CfiA0∗
≤A0kC−1k2hCf, CfiA0∗
≤ kC−1k2X
j∈J
hΛjCf,ΛjCfi.
Therefor Λ is aC2-controlled ∗-g-frame with bounds A0kC−1k, B0kC−1k
5. Multipliers of controlled∗-G-frames in Hilbert pro-C∗-modules
In this section, we define the multiplier of a controlled ∗-g-frame for C-controlled ∗-g-frames in Hilbert
pro-C∗-modules. The definition of general case (C, C0)-controlled∗-g-frames is similar.
Lemma 5.1. Let C, C0 ∈gl+(U)andΛ ={Λ
j ∈L(U, Vj) : j ∈J}, θ={θj∈L(U, Vj) : j∈J} beC02 and
C2-controlled ∗-g-Bessel sequences forU, respectively. Letm=`∞ . Then
Mm,C,θ,Λ,C0 :U →U,
defined by
Mm,C,θ,Λ,C0f :=
X
j∈J
mjCθ∗jΛjC0f,
Proof. Let Λ ={Λj ∈L(U, Vj) : j ∈J}, θ={θj ∈L(U, Vj) : j ∈J} be C02 and C2-controlled ∗-g-Bessel
sequences forU, with boundsB, B0, respectively. For anyf, g∈U and finite subsetI⊆J,
ρ(X
j∈I
mjCθ∗jΛjC0f)≤sup{ρh
X
j∈I
mjCθ∗jΛjC0f, gi: g∈U , ρ(g)≤1}
= sup{ρ(X
j∈I
hmjΛjC0f, θjC∗gi) : g∈U , ρ(g)≤1}
≤ sup
ρ(g)≤1
ρ( X
j∈I
hmjΛjC0f, mjΛjC0fi)
1/2
ρ( X
j∈I
hθjC∗g, θjC∗gi)
1/2
,
since
X
j
hmjΛjC0f, mjΛjC0fi=X
j
mjhΛjC0f,ΛjC0fim∗j
=X
j
(ρ(mj))2hΛjC0f,ΛjC0fi
≤ kmk2
∞Bhf, fiB∗.
So by Proposition2.1we have:
ρ(X
j
hmjΛjC0f, mjΛjC0fi) =ρ(X
j
mjhΛjC0f,ΛjC0fim∗j)
≤ kmk2∞(ρ(f))2ρ(B)2.
Hence
ρ(X
j∈I
mjCθ∗jΛjC0f)≤ kmk∞ ρ(f)ρ(B)ρ(B)
0.
This shows thatMm,C,θ,Λ,C0 is well-defined and
ρ(Mm,C,θ,Λ,C0)≤ kmk∞ ρ(B)ρ(B)0.
The above Lemma provides a motivation for the following definition.
Definition 5.1. Let C, C0 ∈gl+(U)andΛ ={Λ
j ∈L(U, Vj) : j∈J}, θ={θj ∈L(U, Vj) : j∈J} beC02
andC2-controlled ∗-g-Bessel sequences forU, respectively. Letm=`∞ . The operator
Mm,C,θ,Λ,C0 :U →U,
defined by
Mm,C,θ,Λ,C0f :=
X
j∈J
mjCθ∗jΛjC0f,
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