The continuous g-frames in Hilbert C*-modules were introduced and investigated by Kouchi and Nazari (2011). They also studied the continuous g-Riesz basis and a characterization for it was presented by using the synthesis operator. However, we found that there is an error in the proof. The purpose of this paper is to improve their result by introducing the so-called modular continuous g-Riesz basis.

Kouchi and Nazari in [1] introduced the continuous g-frames in Hilbert C*-modules and investigated some of their properties. The following lemma is a useful tool in their study.

Lemma 1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let A be a C*-algebra, 𝒰 and 𝒱 two Hilbert A-modules, and T∈
End
A*(𝒰,𝒱). The following statements are equivalent:

T is surjective;

T* is bounded below with respect to norm, that is, there is m>0 such that ∥T*f∥≥m∥f∥ for all f∈𝒱;

T* is bounded below with respect to inner product, that is, there is m′>0 such that 〈T*f,T*f〉≥m′〈f,f〉 for all f∈𝒱.

The authors also defined the continuous g-Riesz basis in Hilbert C*-modules as follows.

Definition 2.

A continuous g-frame {Λm∈EndA*(𝒰,Vm):m∈ℳ} for Hilbert C*-module 𝒰 with respect to {Vm:m∈ℳ} is said to be a continuous g-Riesz basis if it satisfies the following:

Λm≠0 for any m∈ℳ;

if ∫m∈𝒦Λm*gmdμ(m)=0, then Λm*gm is equal to zero for each m∈ℳ, where {gm}m∈𝒦∈⊕m∈ℳVm and 𝒦 is a measurable subset of ℳ.

By using the synthesis operator TΛ for a sequence {Λm∈EndA*(𝒰,Vm):m∈ℳ} defined by
(1)TΛ(g)=∫m∈ℳΛm*gmdμ(m),∀g={gm}∈⨁m∈ℳVm,

they gave a characterization of continuous g-Riesz basis [1, Theorem 4.6].

Theorem 3.

A family {Λm∈
End
A*(𝒰,Vm):m∈ℳ} is a continuous g-Riesz basis for 𝒰 with respect to {Vm:m∈ℳ} if and only if the synthesis operator TΛ is a homeomorphism.

We note, however, that in the proof of the above theorem, they said that “Λm*fm=0 for any m∈ℳ, and Λm≠0, so fm=0”, which is not true, because if Λm has a dense range, then Λm* is one-to-one. We can improve their result by introducing the following modular continuous g-Riesz basis.

Definition 4.

One calls a family {Λm∈EndA*(𝒰,Vm):m∈ℳ} in Hilbert C*-module 𝒰 a modular continuous g-Riesz basis if

{f∈𝒰:Λmf=0,m∈ℳ}={0};

there exist constants A,B>0 such that for any g={gm}∈⊕m∈ℳVm,
(⋆)A∥g∥2≤∥∫m∈ℳΛm*gmdμ(m)∥2≤B∥g∥2.

Theorem 5.

A sequence {Λm∈
End
A*(𝒰,Vm):m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with respect to {Vm:m∈ℳ} if and only if the synthesis operator TΛ is a homeomorphism.

Proof.

Suppose first that {Λm∈EndA*(𝒰,Vm):m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with synthesis operator TΛ. Then (⋆) turns to be
(2)A∥g∥2≤∥TΛ(g)∥2≤B∥g∥2,∀g={gm}∈⨁m∈ℳVm,

showing that TΛ is bounded below with respect to norm. Hence, by Lemma 1, its adjoint operator TΛ* is surjective. Since the condition (1) in Definition 4 implies that TΛ* is injective, it follows that TΛ* is invertible and so TΛ is invertible.

Conversely, let TΛ be a homeomorphism. Then TΛ is surjective, and again by Lemma 1, TΛ* is injective. So the condition (1) in Definition 4 holds. Now for any g={gm}∈⊕m∈ℳVm,
(3)∥TΛ-1∥-2∥g∥2≤∥∫m∈ℳΛm*gmdμ(m)∥2∥TΛ-1∥-2∥g∥2=∥TΛ(g)∥2≤∥TΛ∥2∥g∥2.

Therefore, {Λm∈EndA*(𝒰,Vm):m∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with respect to {Vm:m∈ℳ}.

KouchiM. R.NazariA.Continuous g-frame in Hilbert C*-modulesArambašićL.On frames for countably generated Hilbert C*-modules