Comment on “Continuous g-Frame in Hilbert 𝐶^∗-Modules”

Abstract

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.

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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 [2]). Let 𝐴 be a 𝐶^∗-algebra, 𝒰 and 𝒱 two Hilbert 𝐴-modules, and 𝑇∈End^∗_𝐴(𝒰,𝒱). The following statements are equivalent:(1)𝑇 is surjective;(2)𝑇^∗ is bounded below with respect to norm, that is, there is 𝑚>0 such that ‖𝑇^∗𝑓‖≥𝑚‖𝑓‖ for all 𝑓∈𝒱;(3)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 {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} for Hilbert 𝐶^∗-module 𝒰 with respect to {𝑉_𝑚∶𝑚∈ℳ} is said to be a continuous g-Riesz basis if it satisfies the following:(1)Λ_𝑚≠0 for any 𝑚∈ℳ;(2)if ∫_𝑚∈𝒦Λ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚)=0, then Λ^∗_𝑚𝑔_𝑚 is equal to zero for each 𝑚∈ℳ, where {𝑔_𝑚}_𝑚∈𝒦∈⊕_𝑚∈ℳ𝑉_𝑚 and 𝒦 is a measurable subset of ℳ.

By using the synthesis operator 𝑇_Λ for a sequence {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} defined by 𝑇_Λ𝑔=_𝑚∈ℳΛ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚),∀𝑔=𝑔_𝑚∈_𝑚∈ℳ𝑉_𝑚, (1)

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

Theorem 3. A family {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} is a continuous g-Riesz basis for 𝒰 with respect to {𝑉_𝑚∶𝑚∈ℳ} if and only if the synthesis operator 𝑇_Λ is a homeomorphism.

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

Definition 4. One calls a family {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} in Hilbert 𝐶^∗-module 𝒰 a modular continuous g-Riesz basis if(1) {𝑓∈𝒰∶Λ_𝑚𝑓=0,𝑚∈ℳ}={0};(2)there exist constants 𝐴,𝐵>0 such that for any 𝑔={𝑔_𝑚}∈⊕_𝑚∈ℳ𝑉_𝑚, 𝐴𝑔²≤_𝑚∈ℳΛ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚)²≤𝐵𝑔². (⋆)

Theorem 5. A sequence {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with respect to {𝑉_𝑚∶𝑚∈ℳ} if and only if the synthesis operator 𝑇_Λ is a homeomorphism.

Proof. Suppose first that {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with synthesis operator 𝑇_Λ. Then (⋆) turns to be 𝐴𝑔²≤𝑇_Λ𝑔²≤𝐵𝑔²,∀𝑔=𝑔_𝑚∈_𝑚∈ℳ𝑉_𝑚, (2)
showing that 𝑇_Λ is bounded below with respect to norm. Hence, by Lemma 1, its adjoint operator 𝑇^∗_Λ is surjective. Since the condition (1) in Definition 4 implies that 𝑇^∗_Λ is injective, it follows that 𝑇^∗_Λ is invertible and so 𝑇_Λ is invertible.
Conversely, let 𝑇_Λ be a homeomorphism. Then 𝑇_Λ is surjective, and again by Lemma 1, 𝑇^∗_Λ is injective. So the condition (1) in Definition 4 holds. Now for any 𝑔={𝑔_𝑚}∈⊕_𝑚∈ℳ𝑉_𝑚, 𝑇_Λ⁻¹⁻²𝑔²≤_𝑚∈ℳΛ^∗_𝑚𝑔_𝑚𝑑𝜇(𝑚)²=𝑇_Λ𝑔²≤𝑇_Λ²𝑔². (3)
Therefore, {Λ_𝑚∈End^∗_𝐴(𝒰,𝑉_𝑚)∶𝑚∈ℳ} is a modular continuous g-Riesz basis for 𝒰 with respect to {𝑉_𝑚∶𝑚∈ℳ}.