𝑔-Bases in Hilbert Spaces

Abstract

The concept of g-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results about g-bases are proved. In particular, we characterize the g-bases and g-orthonormal bases. And the dual g-bases are also discussed. We also consider the equivalent relations of g-bases and g-orthonormal bases. And the property of g-minimal of g-bases is studied as well. Our results show that, in some cases, g-bases share many useful properties of Schauder bases in Hilbert spaces.

1. Introduction

In 1946, Gabor [1] introduced a fundamental approach to signal decomposition in terms of elementary signals. In 1952, Duffin and Schaeffer [2] abstracted Gabor’s method to define frames in Hilbert spaces. Frame was reintroduced by Daubechies et al. [3] in 1986. Today, frame theory is a central tool in many areas such as characterizing function spaces and signal analysis. We refer to [4–10] for an introduction to frame theory and its applications. The following are the standard definitions on frames in Hilbert spaces. A sequence {𝑓_𝑖}_𝑖∈𝑁 of elements of a Hilbert space 𝐻 is called a frame for 𝐻 if there are constants 𝐴,𝐵>0 so that 𝐴𝑓²≤_𝑖∈𝑁𝑓,𝑓_𝑖²≤𝐵𝑓². (1.1) The numbers 𝐴,𝐵 are called the lower (resp., upper) frame bounds. The frame is a tight frame if 𝐴=𝐵 and a normalized tight frame if 𝐴=𝐵=1.

In [11], Sun raised the concept of g-frame as follows, which generalized the concept of frame extensively. A sequence {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is called a g-frame for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, which is a sequence of closed subspaces of a Hilbert space 𝑉, if there exist two positive constants 𝐴 and 𝐵 such that for any 𝑓∈𝐻𝐴𝑓²≤_𝑖∈𝑁Λ_𝑖𝑓²≤𝐵𝑓². (1.2) We simply call {Λ_𝑖∶𝑖∈𝑁} a 𝑔-frame for 𝐻 whenever the space sequence {𝐻_𝑖∶𝑖∈𝑁} is clear. The tight 𝑔-frame, normalized tight 𝑔-frame, 𝑔-Riesz basis are defined similarly. We call {Λ_𝑖∶𝑖∈𝑁} a 𝑔-frame sequence, if it is a 𝑔-frame for span{Λ^∗_𝑖(𝐻_𝑖)}_𝑖∈𝑁. We call {Λ_𝑖∶𝑖∈𝑁} a 𝑔-Bessel sequence, if only the right inequality is satisfied. Recently, 𝑔-frames in Hilbert spaces have been studied intensively; for more details see [12–17] and the references therein.

It is well known that frames are generalizations of bases in Hilbert spaces. So it is natural to view 𝑔-frames as generalizations of the so-called 𝑔-bases in Hilbert spaces, which will be defined in the following section. And that is the main object which will be studied in this paper. In Section 2, we will give the definitions and lemmas. In Section 3, we characterize the 𝑔-bases. In Section 4, we discuss the equivalent relations of 𝑔-bases and 𝑔-orthonormal bases. In Section 5, we study the property of 𝑔-minimal of 𝑔-bases. Throughout this paper, we use 𝑁 to denote the set of all natural numbers, 𝑍 to denote the set of all integer numbers, and 𝐶 to denote the field of complex numbers. The sequence of {𝐻_𝑗∶𝑗∈𝑁} always means a sequence of closed subspace of some Hilbert space 𝑉.

2. Definitions and Lemmas

In this section, we introduce the definitions and lemmas which will be needed in this paper.

Definition 2.1. For each Hilbert space sequence {𝐻_𝑖}_𝑖∈𝑁, we define the space 𝑙²(⊕𝐻_𝑖) by 𝑙²⊕𝐻_𝑖=𝑓_{𝑖𝑖∈𝑁}∶𝑓_𝑖∈𝐻_𝑖,𝑖∈𝑁,^+∞_𝑖=1𝑓_𝑖²<+∞. (2.1) With the inner product defined by ⟨{𝑓_𝑖},{𝑔_𝑖}⟩=∑^+∞_𝑖=1⟨𝑓_𝑖,𝑔_𝑖⟩, it is easy to see that 𝑙²(⊕𝐻_𝑖) is a Hilbert space.

Definition 2.2. {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 is called g-complete with respect to {𝐻_𝑗} if {𝑓∶Λ_𝑗𝑓=0,forall𝑗}={0}.

Definition 2.3. {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 is called g-linearly independent with respect to {𝐻_𝑗} if ∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗=0, then 𝑔_𝑗=0, where 𝑔_𝑗∈𝐻_𝑗(𝑗=1,2,…).

Definition 2.4. {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 is called g-minimal with respect to {𝐻_𝑗} if for any sequence {𝑔_𝑗∶𝑗∈𝑁} with 𝑔_𝑗∈𝐻_𝑗 and any 𝑚∈𝑁 with 𝑔_𝑚≠0, one has Λ^∗_𝑚𝑔_𝑚∉span_𝑖≠𝑚{Λ^∗_𝑖𝑔_𝑖}.

Definition 2.5. {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 and {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 are called g-biorthonormal with respect to {𝐻_𝑗}, if Λ^∗_𝑗𝑔_𝑗,Γ^∗_𝑖𝑔_𝑖=𝛿_𝑗,𝑖𝑔_𝑗,𝑔_𝑖,∀𝑗,𝑖∈𝑁,𝑔_𝑗∈𝐻_𝑗,𝑔_𝑖∈𝐻_𝑖. (2.2)

Definition 2.6. We say {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 is g-orthonormal basis for 𝐻 with respect to {𝐻_𝑗}, if it is g-biorthonormal with itself and for any 𝑓∈𝐻 one has _𝑗∈𝑁Λ_𝑗𝑓²=𝑓². (2.3)

Definition 2.7. We call {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 a g-basis for 𝐻 with respect to {𝐻_𝑗} if for any 𝑥∈𝐻 there is a unique sequence {𝑔_𝑗} with 𝑔_𝑗∈𝐻_𝑗 such that 𝑥=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗.

The following result is about pesudoinverse, which plays an important role in some proofs.

Lemma 2.8 (see [5]). Suppose that 𝑇∶𝐾→𝐻 is a bounded surjective operator. Then there exists a bounded operator (called the pseudoinverse of T) 𝑇^†∶𝐻→𝐾 for which 𝑇𝑇^†𝑓=𝑓,∀𝑓∈𝐻. (2.4)

The following lemmas characterize g-frame sequence and g-Bessel sequence in terms of synthesis operators.

Lemma 2.9 (see [14]). A sequence {Λ_𝑗∶𝑗∈𝑁} is a g-frame sequence for H with respect to {𝐻_𝑗∶𝑗∈𝑁} if and only if Q∶𝑔_𝑗∶𝑗∈𝑁→_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 (2.5) is a well-defined bounded linear operator from 𝑙²(⊕H_𝑗) into H with closed range.

Lemma 2.10 (see [12]). A sequence {Λ_𝑗∶𝑗∈𝑁} is a g-Bessel sequence for H with respect to {𝐻_𝑗∶𝑗∈𝑁} if and only if Q∶𝑔_𝑗∶𝑗∈𝑁→_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 (2.6) is a well-defined bounded linear operator from 𝑙²(⊕H_𝑗) into H.

The following is a simple property about g-basis, which gives a necessary condition for g-basis in terms of g-complete and g-linearly independent.

Lemma 2.11. If {Λ_𝑗∶𝑗∈𝑁} is a g-basis for H with respect to {𝐻_𝑗∶𝑗∈𝑁}, then {Λ_𝑗∶𝑗∈𝑁} is g-complete and g-linearly independent with respect to {𝐻_𝑗∶𝑗∈𝑁}.

Proof. Suppose Λ_𝑖𝑓=0 for each 𝑖. Then for each 𝑔_𝑖∈𝐻_𝑖, we have ⟨Λ_𝑖𝑓,𝑔_𝑖⟩=⟨𝑓,Λ^∗_𝑖𝑔_𝑖⟩=0. Hence 𝑓⟂span{Λ_𝑖(𝐻𝑖)∶𝑖∈𝑁}. Therefore 𝑓⟂span{Λ_𝑖(𝐻_𝑖)∶𝑖∈𝑁}=𝐻. So 𝑓=0. So {Λ_𝑖∶𝑖∈𝑁} is g-complete. Now suppose ∑^+∞_𝑖=1Λ^∗_𝑖𝑔_𝑖=0. Since ∑^+∞_𝑖=1Λ^∗_𝑖0=0 and {Λ_𝑖∶𝑖∈𝑁} is a g-basis, so 𝑔_𝑖=0 for each 𝑖. Hence {Λ_𝑖∶𝑖∈𝑁} is g-linearly independent.

The following remark tells us that g-basis is indeed a generalization of Schauder basis of Hilbert space.

Remark 2.12. If {𝑥_𝑖}_𝑖∈𝑁 is a Schauder basis of Hilbert space 𝐻, then it induces a g-basis {Λ_𝑥𝑖∶𝑖∈𝑁} of 𝐻 with respect to the complex number field 𝐶, where Λ_𝑥𝑖 is defined by Λ_𝑥𝑖𝑓=⟨𝑓,𝑥_𝑖⟩. In fact, it is easy to see that Λ^∗_𝑥𝑖𝑐=𝑐⋅𝑥_𝑖 for any 𝑐∈𝐶, so for any 𝑥∈𝐻, there exists a unique sequence of constants {𝑎_𝑛∶𝑛∈𝑁} such that 𝑥=∑_𝑖∈𝑁𝑎_𝑖𝑥_𝑖=∑_𝑖∈𝑁Λ^∗_𝑖𝑎_𝑖.

Definition 2.13. Suppose {Λ_𝑗∶𝑗∈𝑁} is a g-Riesz basis of 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} is a g-Riesz basis of 𝑌 with respect to {𝐻_𝑗∶𝑗∈𝑁}. If there is a homomorphism 𝑆∶𝐻→𝑌 such that Γ_𝑗=Λ_𝑗𝑆^∗ for each 𝑗∈𝑁, then we say that {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are equivalent.

Definition 2.14. If {Λ_𝑗} is a g-basis of 𝐻 with respect to {𝐻_𝑗}, then for any 𝑥∈𝐻, there exists a unique sequence {𝑔_𝑗∶𝑗∈𝑁} such that 𝑔_𝑗∈𝐻_𝑗 and 𝑥=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗. We define a map Γ_𝑗∶𝐻→𝐻_𝑗, by Λ_𝑗𝑥=𝑔_𝑗 for each 𝑗. Then {Γ_𝑗} is well defined. We call it the dual sequence of {Λ_𝑗}, in case that {Γ_𝑗} is also a g-basis, we call it the dual g-basis of {Λ_𝑗}.

The following results link g-Riesz basis with g-basis.

Lemma 2.15 (see [11]). A g-Riesz basis {Λ_𝑗∶𝑗∈𝑁} is an exact g-frame. Moreover, it is g-biorthonormal with respect to its dual {Λ_𝑗∶𝑗∈𝑁}.

Lemma 2.16. Let Λ_𝑗∈𝐵(𝐻,𝐻𝑗),𝑗∈𝑁. Then the following statements are equivalent.(1)The sequence {Λ_𝑗}_𝑗∈𝑁 is a g-Riesz basis for H with respect to {𝐻_𝑗}_𝑗∈𝑁.(2)The sequence {Λ_𝑗}_𝑗∈𝑁 is a g-frame for H with respect to {𝐻_𝑗}_𝑗∈𝑁 and {Λ_𝑗}_𝑗∈𝑁 is g-linearly independent.(3)The sequence {Λ_𝑗}_𝑗∈𝑁 is a g-basis and a g-frame with respect to {𝐻_𝑗}_𝑗∈𝑁.

Proof. The equivalent between statements (1) and (2) is shown in Theorem2.8 of [12]. By Lemma 2.11, we know that if {Λ_𝑗}_𝑗∈𝑁 is a g-basis, then it is g-linearly independent, so (3) implies (2). If {Λ_𝑗}_𝑗∈𝑁 is a g-frame for 𝐻, then for every 𝑥∈𝐻, 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗Λ_𝑗𝑥, where {Λ_𝑗}_𝑗∈𝑁 is the canonical dual g-frame of {Λ_𝑗}_𝑗∈𝑁. Hence for every 𝑥∈𝐻, there exists a sequence {𝑔_𝑗∶𝑗∈𝑁}, 𝑔_𝑗∈𝐻_𝑗, such that 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗. Since {Λ_𝑗}_𝑗∈𝑁 is g-linearly independent, the sequence is unique. Hence {Λ_𝑗}_𝑗∈𝑁 is a g-basis for 𝐻. So (2) implies (3).

From Lemma 2.16, it is easy to get the following well-known result, which is proved more directly.

Corollary 2.17. Suppose {Λ_𝑗}_𝑗∈𝑁 is a g-Riesz basis for 𝐻, then {Λ_𝑗}_𝑗∈𝑁 has a unique dual g-frame.

Proof. It has been shown that every g-frame has a dual g-frame in [11], so it suffices to show the uniqueness of dual g-frame for g-Riesz bases. Suppose {Γ_𝑗∶𝑗∈𝑁} and {𝜂_𝑗∶𝑗∈𝑁} are dual g-frames of {Λ_𝑗}_𝑗∈𝑁. Then for every 𝑥∈𝐻, we have 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗Γ_𝑗𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝜂_𝑗𝑥. Hence ∑_𝑗∈𝑁Λ^∗_𝑗(Γ_𝑗−𝜂_𝑗)𝑥=0. But {Λ_𝑗}_𝑗∈𝑁 is g-linearly independent by Lemma 2.16, so (Γ_𝑗−𝜂_𝑗)𝑥=0, that is, Γ_𝑗𝑥=𝜂_𝑗𝑥 for each 𝑗∈𝑁. Thus, Γ_𝑗=𝜂_𝑗 for each 𝑗∈𝑁, which implies that the dual g-frame of {Λ_𝑗}_𝑗∈𝑁 is unique.

The following lemma generalizes the similar result in frames to g-frames.

Lemma 2.18. Suppose {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are both g-Bessel sequences for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁}. Then the following statements are equivalent.(1)For any 𝑥∈𝐻, 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗Γ_𝑗𝑥.(2)For any 𝑥∈𝐻, x=∑_𝑗∈𝑁Γ^∗_jΛ_𝑗𝑥.(3)For any 𝑥,𝑦∈𝐻, ⟨𝑥,𝑦⟩=∑_𝑗∈𝑁⟨Λ_𝑗𝑥,Γ_𝑗𝑥⟩.
Moreover, any of the above statements implies that {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are dual g-frames for each other.

Proof. (1)⇒(2): Since {Λ_𝑗∶𝑗∈𝑁} is a g-Bessel sequence, {Λ_𝑗𝑥}_𝑗∈𝑁∈𝑙²(⊕𝐻_𝑗) for any 𝑥∈𝐻. Since {Γ_𝑗∶𝑗∈𝑁} is a g-Bessel sequence, the series ∑_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥 is convergent by Lemma 2.10. Let 𝑥=∑_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥. Then for any 𝑦∈𝐻, we have 𝑥,𝑦=𝑥,_𝑗∈𝑁Λ^∗_𝑗Γ_𝑗𝑦=_𝑗∈𝑁𝑥,Λ^∗_𝑗Γ_𝑗𝑦=_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗=𝑥,𝑦_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥,𝑦=𝑥,𝑦. (2.7) So 𝑥=𝑥, that is, (2) is established.
(2)⇒(3): Since for any 𝑥∈𝐻, we have 𝑥=∑_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥; hence for any 𝑥,𝑦∈𝐻, 𝑥,𝑦=_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥,𝑦=_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥,𝑦=_𝑗∈𝑁Λ_𝑗𝑥,Γ_𝑗𝑥. (2.8)
(3)⇒(1): From the proof of (1)⇒(2), we know that for any 𝑥∈𝐻, ∑_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥 is convergent. Let 𝑥=∑_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥, then for any 𝑦∈𝐻, we have ⟨𝑦,𝑥⟩=_𝑗∈𝑁⟨Λ_𝑗𝑦,Γ_𝑗𝑥⟩=⟨𝑦,_𝑗∈𝑁Γ^∗_𝑗Λ_𝑗𝑥⟩=⟨𝑦,𝑥⟩. (2.9) So 𝑥=𝑥, that is, (1) is true.
If any one of the three statements is true, then for any x  H, we have ‖𝑥‖²=_𝑗∈𝑁Λ^∗_𝑗Γ_𝑗𝑥,𝑥=_𝑗∈𝑁⟨Λ_𝑗𝑥,Γ_𝑗𝑥⟩≤_𝑗∈𝑁Λ_𝑗𝑥Γ_𝑗𝑥≤_𝑗∈𝑁Λ_𝑗𝑥^21/2_𝑗∈𝑁Γ_𝑗𝑥^21/2≤𝐵₁‖𝑥‖^21/2_𝑗∈𝑁Γ_𝑗𝑥^21/2, (2.10) where 𝐵₁ is the bound for the g-Bessel sequence {Λ_𝑗∶𝑗∈𝑁}. So _𝑗∈𝑁Γ_𝑗𝑥²≥1𝐵₁‖𝑥‖², (2.11) which implies that the g-Bessel sequence {Γ_𝑗∶𝑗∈𝑁} is a g-frame. Similarly, {Λ_𝑗∶𝑗∈𝑁} is also a g-frame. And that they are dual g-frames for each other is obvious by the equality that for any 𝑥∈𝐻, 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗Γ_𝑗𝑥.

3. Characterizations of 𝑔-Bases

In this section, we characterized g-bases.

Theorem 3.1. Suppose that {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 is a g-frame sequence with respect to {𝐻_𝑗} and it is g-linearly independent with respect to {𝐻_𝑗}. Let 𝑌={{𝑔_𝑗}^∞_𝑗=1∣𝑔_𝑗∈𝐻_𝑗,∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗isconvergent}. If for any {𝑔_𝑗}∈𝑌, set ‖{𝑔_𝑗}‖_𝑌=Sup_𝑁‖∑^𝑁_𝑗=1𝑔_𝑗‖, then(1)𝑌 is a Banach space,(2)when {Λ_𝑗} is a g-basis with respect to {𝐻_𝑗} as well, 𝑆∶𝑌→𝐻,𝑆({g_𝑗})=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗 is a linear bounded and invertible operator, that is, S is a homeomorphism between 𝐻 and 𝑌.

Proof. (1) Let {𝑔_𝑗}∈𝑌, then ∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗 is convergent as 𝑁→∞. Hence {∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗}^∞_𝑁=1 is a convergent sequence, so it is bounded. So ‖{𝑔_𝑗}‖_𝑌<∞. It is obvious that for 𝑎∈𝐶, {𝑔_𝑗}, {ℎ_𝑗}∈𝑌, we have ‖{𝑔_𝑗}+{ℎ_𝑗}‖_𝑌≤‖{𝑔_𝑗}‖_𝑌+‖{ℎ_𝑗}‖_𝑌 and ‖𝑎⋅{𝑔_𝑗}‖_𝑌=|𝑎|⋅‖{𝑔_𝑗}‖_𝑌. If ‖{𝑔_𝑗}‖_𝑌=0, then for any 𝑁, ‖∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗‖=0, which implies that ∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗=0. Since {Λ_𝑗} is g-linearly independent with respect to {𝐻_𝑗}, we get that 𝑔_𝑗=0, for 𝑗=1,2,…,𝑁. Since 𝑁 is arbitrary, so {𝑔_𝑗}={0}. Thus ‖⋅‖_𝑌 is a norm on 𝑌. Suppose {𝐺^𝑘}^∞_𝑘=1∈𝑌 is a Cauchy sequence, where 𝐺^𝑘={𝑔^𝑘_𝑗}^∞_𝑗=1. Then lim_{𝑘,𝑙→∞}𝐺^𝑘−𝐺^𝑙_𝑌=lim_{𝑘,𝑙→∞}𝑔^𝑘_𝑗−𝑔^𝑙_𝑗𝑌,lim_{𝑘,𝑙→∞}Sup_𝑁^𝑁_𝑗=1Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗=0. (3.1)(3.2) For any fixed 𝑗, we have Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗=^𝑗_𝑡=1Λ^∗_𝑡𝑔^𝑘_𝑡−𝑔^𝑙_𝑡−^𝑗−1_𝑡=1Λ^∗_𝑡𝑔^𝑘_𝑡−𝑔^𝑙_𝑡≤^𝑗_𝑡=1Λ^∗_𝑡𝑔^𝑘_𝑡−𝑔^𝑙_𝑡+^𝑗−1_𝑡=1Λ^∗_𝑡𝑔^𝑘_𝑡−𝑔^𝑙_𝑡≤2𝐺^𝑘−𝐺^𝑙_𝑌. (3.3) Now let 𝑇∶𝑙²(⊕𝐻_𝑗)→𝐻,𝑇({𝑔_𝑗})=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗. Since {Λ_𝑗} is a g-frame sequence with respect to {𝐻_𝑗}, 𝑇 is a well-defined linear bounded operator with closed range by Lemma 2.9. Since {Λ_𝑗} is g-linearly independent with respect to {𝐻_𝑗}, 𝑇 is injective. Hence 𝑇^∗∶𝐻→𝑙²(⊕𝐻_𝑗) is surjective. So by Lemma 2.8, there is a bounded operator 𝐿, the pseudoinverse of 𝑇^∗, such that 𝑇^∗𝐿=𝐼_𝑙²_{(⊕𝐻𝑗)}, which implies that 𝐿^∗𝑇=𝐼_𝑙²_{(⊕𝐻𝑗)}. Let {𝛿_𝑗} denote the canonical basis of 𝑙²(𝑁), then for any 𝑔_𝑗∈𝐻_𝑗,{𝑔_𝑗𝛿_𝑗}∈𝑙²(⊕𝐻_𝑗). So {𝑔_𝑗𝛿_𝑗}=𝐿^∗𝑇({𝑔_𝑗𝛿_𝑗})=𝐿^∗(Λ^∗_𝑗𝑔_𝑗); hence 𝑔_𝑗=𝑔_𝑗𝛿_𝑗=𝐿^∗Λ^∗_𝑗𝑔_𝑗≤‖𝐿‖Λ^∗_𝑗𝑔_𝑗. (3.4) So by inequalities (3.3), we get 𝑔^𝑘_𝑗−𝑔^𝑙_𝑗≤‖𝐿‖Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗≤2‖𝐿‖𝐺^𝑘−𝐺^𝑙_𝑌. (3.5) So for any fixed 𝑗, {𝑔^𝑘_𝑗}^∞_𝑗=1 is a Cauchy sequence. Suppose lim_𝑘→∞𝑔^𝑘_𝑗=𝑔_𝑗. From (3.2), we know that, for any 𝜀>0, there exists 𝐿₀>0, such that whenever 𝑘,𝑙≥𝐿₀, we have Sup_𝑄^𝑄_𝑗=1Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗<𝜀. (3.6) Fix 𝑙≥𝐿₀, since lim_𝑘→∞𝑔^𝑘_𝑗=𝑔_𝑗, so whenever 𝑙≥𝐿₀, Sup_𝑄^𝑄_𝑗=1Λ^∗_𝑗𝑔_𝑗−𝑔^𝑙_𝑗≤𝜀. (3.7) Since 𝐺^𝐿0={𝑔^𝐿0_𝑗}^∞_𝑗=1∈𝑌,∑^∞_𝑗=1Λ^∗_𝑗𝑔^𝐿0_𝑗 is convergent. So there exists 𝐾₀>0, such that whenever 𝑀>𝑃≥𝐾₀, we have ‖∑^𝑀_𝑗=𝑃+1Λ^∗_𝑗𝑔^𝐿0_𝑗‖<𝜀. So when 𝑀>𝑃>max{𝐿₀,𝐾₀}, we have ^𝑀_𝑗=𝑃+1Λ^∗_𝑗𝑔_𝑗=^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗−𝑔^𝐿0_𝑗−^𝑃_𝑗=1Λ^∗_𝑗𝑔_𝑗−𝑔^𝐿0_𝑗+^𝑀_𝑗=𝑃+1Λ^∗_𝑗𝑔^𝐿0_𝑗≤^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗−𝑔^𝐿0_𝑗+^𝑃_𝑗=1Λ^∗_𝑗𝑔_𝑗−𝑔^𝐿0_𝑗+^𝑀_𝑗=𝑃+1Λ^∗_𝑗𝑔^𝐿0_𝑗≤3𝜀. (3.8) So ∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗 is convergent, thus 𝐺={𝑔_𝑗}∈𝑌. Let 𝑙→∞ in (3.7), we get that ‖𝐺_𝑙−𝐺‖_𝑌→0. Hence 𝑌 is a complete normed space, that is, 𝑌 is a Banach space.
(2) If {Λ_𝑗} is a g-basis, then it is g-complete and g-linearly independent with respect to {𝐻_𝑗} by the Lemma 2.11, then the operator 𝑆∶𝑌→𝐻,𝑆({𝑔_𝑗})=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗 not only is well defined but also is one to one and onto. And for any {𝑔_𝑗}∈𝑌, we have 𝑆𝑔_𝑗=^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗=lim_𝑁→∞^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗≤Sup_𝑄^𝑄_𝑗=1Λ^∗_𝑗𝑔_𝑗=𝑔_𝑗𝑌. (3.9) So 𝑆 is bounded operator. Since 𝑌 is a Banach space, by the Open Mapping Theorem, we get that 𝑆 is a homeomorphism.

Theorem 3.2. Suppose {Λ_𝑗∶𝑗∈𝑁} is a 𝑔-basis of 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} is its dual sequence. If {Λ_𝑗∶𝑗∈𝑁} is also a 𝑔-frame sequence of 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁}, then(1)∀x∈𝐻, let 𝑆_𝑁𝑥=∑^𝑁_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥, then Sup_𝑁‖𝑆_𝑁𝑥‖<∞,(2)𝐶=Sup_𝑁‖𝑆_𝑁‖<∞,(3)|||𝑥|||=Sup_𝑁‖𝑆_𝑁𝑥‖ is a norm on 𝐻 and ‖⋅‖≤|||⋅|||≤𝐶‖⋅‖.

Proof. (1) Let 𝑌 and 𝑆 be as defined in Theorem 3.1. Then for any 𝑥∈𝐻, 𝑆⁻¹𝑥={Γ_𝑗𝑥∶𝑗∈𝑁}. So Sup_𝑁𝑆_𝑁𝑥=Sup_𝑁^𝑁_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥=Γ_𝑗𝑥_𝑌=𝑆⁻¹𝑥≤𝑆⁻¹‖𝑥‖<∞. (3.10)
(2) Since ‖𝑆_𝑁𝑥‖≤Sup_𝑄‖𝑆_𝑄𝑥‖≤‖𝑆⁻¹‖‖𝑥‖, ‖𝑆_𝑁‖≤‖𝑆⁻¹‖. Thus 𝐶=Sup_𝑁‖𝑆_𝑁‖<∞.
(3) It is obvious that |||⋅||| is a seminorm. It is sufficient to show that ‖⋅‖≤|||⋅|||≤𝐶‖⋅‖. For any 𝑥∈𝐻, we have |||𝑥|||=Sup_𝑁𝑆_𝑁𝑥≤Sup_𝑁𝑆_𝑁‖𝑥‖=𝐶‖𝑥‖. (3.11) On the other hand, ‖𝑥‖=^∞_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥=lim_𝑁→∞^𝑁_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥≤Sup_𝑄^𝑄_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥=Sup_𝑄𝑆_𝑄𝑥=|||𝑥|||. (3.12)

Theorem 3.3. Suppose {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝑁} is a g-frame with respect to {𝐻_𝑗∶𝑗∈𝑁}. Then {Λ_𝑗∶𝑗∈𝑁} is a g basis with respect to {𝐻_𝑗∶𝑗∈𝑁} if and only if there exists a constant 𝐶 such that for any 𝑔_𝑗∈𝐻_𝑗, any 𝑚,𝑛∈𝑁 and 𝑚≤𝑛, one has ^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝐶⋅^𝑛_𝑗=1Λ^∗_𝑗𝑔_𝑗. (3.13)

Proof. ⇒∶ Suppose {Λ_𝑗∶𝑗∈𝑁} is a g-basis with respect to {𝐻_𝑗∶𝑗∈𝑁}. Then for any 𝑥∈𝐻, there exists a unique sequence {𝑔_𝑗∶𝑗∈𝑁} with 𝑔_𝑗∈𝐻_𝑗 for each 𝑗∈𝑁 such that 𝑥=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗. Let |||𝑥|||=Sup_𝑛‖∑^𝑛_𝑗=1Λ^∗_𝑗𝑔_𝑗‖. Then by Theorem 3.2, |||⋅||| is a norm on 𝐻 and it is equivalent to ‖⋅‖. So there exists a constant 𝐶 such that, for any 𝑥∈𝐻, |||𝑥|||≤𝐶⋅‖𝑥‖. Hence for any 𝑛∈𝑁, any 𝑔_𝑗∈𝐻_𝑗, 𝑗=1,2,…,𝑛, we choose 𝑥=∑^𝑛_𝑗=1Λ^∗_𝑗𝑔_𝑗, then for any 𝑚≤𝑛, we have ^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝐶⋅^𝑛_𝑗=1Λ^∗_𝑗𝑔_𝑗. (3.14)
⇐∶ Let 𝔸={∑_𝑘∈𝑁Λ^∗_𝑘𝑔_𝑘,𝑔_𝑘∈𝐻_𝑘,and∑_𝑘∈𝑁Λ^∗_𝑘𝑔_𝑘is𝑐𝑜𝑣𝑒𝑟𝑔𝑒𝑛𝑡}. First, we show that 𝔸=𝐻. Since {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝑁}⊂𝐵(𝐻,𝐻_𝑗) is a g-frame, 𝔸 is dense in 𝐻. It is sufficient to show that 𝔸 is closed. Suppose {𝑦_𝑘}⊂𝔸 and lim_𝑘→∞𝑦_𝑘=𝑦. Denote 𝑦_𝑘=∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗^(𝑘). Then for any 𝑗∈𝑁 and any 𝑛≤𝑚≤𝑗, we have, for any 𝑘,𝑙∈𝑁, Λ^∗_𝑗𝑔_𝑗^(𝑘)−Λ^∗_𝑗𝑔_𝑗^(𝑙)≤2𝐶⋅^𝑚_𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠^(𝑙)≤2𝐶²⋅^𝑛_𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠^(𝑙)≤2𝐶²⋅^𝑛_𝑠=1Λ^∗_𝑗𝑔_𝑗^(𝑘)−𝑦_𝑘+𝑦_𝑘−𝑦+2𝐶²⋅𝑦−𝑦_l+𝑦_l−^𝑛_𝑗=1Λ^∗_𝑠𝑔^(l)_𝑠. (3.15) Since lim_𝑘→∞𝑦_𝑘=𝑦, so for any 𝜀>0, there exists 𝑀>0, such that whenever 𝑘≥𝑀, we have ‖𝑦−𝑦_𝑘‖≤𝜀/2𝐶². In the above inequality, let 𝑛→∞, we get Λ^∗_𝑗𝑔_𝑗^(𝑘)−Λ^∗_𝑗𝑔_𝑗^(𝑙)≤𝜀forany𝑗∈𝑁and𝑘,𝑙≥𝑀,_𝑚𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠^(𝑙)≤𝜀2𝐶forany𝑚∈𝑁andany𝑘,𝑙≥𝑀. (3.16) Since {Λ_𝑗∶𝑗∈𝑁} is a g-frame sequence, by inequality (3.4), we have that ‖𝑔_𝑗^(𝑘)−𝑔_𝑗^(𝑙)‖≤‖𝐿‖‖Λ^∗_𝑗(𝑔_𝑗^(𝑘)−𝑔_𝑗^(𝑙))‖≤‖𝐿‖𝜀 for any 𝑗∈𝑁 and any 𝑘,𝑙≥𝑀. So {𝑔_𝑗^(𝑘)}_𝑘∈𝑁 is convergent for each 𝑗∈𝑁. Suppose lim_𝑘→∞𝑔_𝑗^(𝑘)=𝑔_𝑗. Then Λ^∗_𝑗𝑔_𝑗^(𝑘)−𝑔_𝑗≤𝜀forany𝑗∈𝑁and𝑘≥𝑀,_𝑚𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠≤𝜀2𝐶forany𝑚∈𝑁andany𝑘≥𝑀. (3.17) Since 𝑦−^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝑦−𝑦_𝑘+𝑦_𝑘−^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗^(𝑘)+^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗^(𝑘)−^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗, (3.18) so ∑^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗 converges to 𝑦, which implies that 𝑦∈𝔸. Thus 𝔸 is a closed set. Now we will show that {Λ_𝑗∶𝑗∈𝑁} is g-linearly independent. Suppose that ∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗=0, where 𝑔_𝑗∈𝐻_𝑗 for each 𝑗∈𝑁. Since for any 𝑛∈𝑁 and any 𝑗≤𝑛, we have ‖Λ^∗_𝑗𝑔_𝑗‖≤𝐶⋅‖∑^𝑛_𝑠=1Λ^∗_𝑠𝑔_𝑠‖, hence ‖Λ^∗_𝑗𝑔_𝑗‖=0 for any 𝑗≤𝑛. But from inequality (3.4), we have ‖𝑔_𝑗‖≤‖𝐿‖‖Λ^∗_𝑗𝑔_𝑗‖. So for each 𝑗≤𝑛,𝑔_𝑗=0. Since 𝑛 is arbitrary, 𝑔_𝑗=0 for any 𝑗∈𝑁. Thus {Λ_𝑗∶𝑗∈𝑁} is g-linearly independent. So {Λ_𝑗∶𝑗∈𝑁} is a g-basis.

4. Equivalent Relations of 𝑔-Bases

In this section, the equivalent relations of g-bases were discussed.

Theorem 4.1. Suppose that {Λ_𝑗∶𝑗∈𝑁} is a g-basis of 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁}, and 𝑆∶𝐻→𝑌 is a homeomorphism. Then {Λ_𝑗𝑆^∗∶𝑗∈𝑁} is a g-basis of 𝑌 with respect to {𝐻_𝑗}.

Proof. For any 𝑦∈𝑌, 𝑆⁻¹𝑦∈𝐻. Since {Λ_𝑗∶𝑗∈𝑁} is a g-basis of 𝐻, there exists a unique sequence {𝑔_𝑗∶𝑗∈𝑁} and 𝑔_𝑗∈𝐻_𝑗 for each 𝑗∈𝑁 such that 𝑆⁻¹𝑦=∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗. So 𝑦=∑_𝑗∈𝑁𝑆Λ^∗_𝑗𝑔_𝑗=∑_𝑗∈𝑁(Λ_𝑗𝑆^∗)^∗𝑔_𝑗. Suppose there is another sequence {ℎ_𝑗∶𝑗∈𝑁} and ℎ_𝑗∈𝐻_𝑗 for each 𝑗∈𝑁 such that 𝑦=∑_𝑗∈𝑁(Λ_𝑗𝑆^∗)^∗ℎ_𝑗, then 𝑦=∑_𝑗∈𝑁𝑆Λ^∗_𝑗ℎ_𝑗. So 𝑆⁻¹𝑦=∑_𝑗∈𝑁Λ^∗_𝑗ℎ_𝑗. But the expansion for 𝑆⁻¹𝑦 is unique, so ℎ_𝑗=𝑔_𝑗 for each 𝑗∈𝑁. Hence {Λ_𝑗𝑆^∗} is a g-basis of 𝑌 with respect to {𝐻_𝑗}.

Theorem 4.2. Suppose {Λ_𝑗∶𝑗∈𝑁} is a 𝑔-basis of Hilbert space 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁}, {Γ_𝑗∶𝑗∈𝑁} is a 𝑔-basis of Hilbert space 𝑌 with respect to {𝐻_𝑗∶𝑗∈𝑁}, and {𝐺_𝑗∶𝑗∈𝑁}, and {𝐿_𝑗∶𝑗∈𝑁} are dual sequences of {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁}, respectively. If {𝐺_𝑗∶𝑗∈𝑁} or {𝐿_𝑗∶𝑗∈𝑁} is a g-basis, then the following statements are equivalent.(1){Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are equivalent.(2)∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 is convergent if and only if ∑_𝑗∈𝑁Γ^∗_𝑗𝑔_𝑗 is convergent, where 𝑔_𝑗∈𝐻_𝑗 for each 𝑗∈𝑁.
Moreover, any one of the above statements implies that both {𝐺_𝑗∶𝑗∈𝑁} and {𝐿_𝑗∶𝑗∈𝑁} are g-bases and they are also equivalent.

Proof. (1)⇒(2): Suppose there is an invertible bounded operator 𝑆∶𝐻→𝑌 such that Λ_𝑗=Γ_𝑗𝑆^∗ for each 𝑗∈𝑁 and ∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 is convergent. Then ∑_𝑗∈𝑁𝑆Γ^∗_𝑗𝑔_𝑗 is convergent. So 𝑆⁻¹(∑_𝑗∈𝑁𝑆Γ^∗_𝑗𝑔_𝑗)=∑_𝑗∈𝑁Γ^∗_𝑗𝑔_𝑗 is convergent. Conversely, if ∑_𝑗∈𝑁Γ^∗_𝑗𝑔_𝑗 is convergent, then ∑_𝑗∈𝑁𝑆⁻¹Λ^∗_𝑗𝑔_𝑗 is convergent. So 𝑆(∑_𝑗∈𝑁𝑆⁻¹Λ^∗_𝑗𝑔_𝑗)=∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 is convergent.
(2)⇒(1): Without loss of generality, suppose {𝐺_𝑗∶𝑗∈𝑁} is a g-basis of 𝐻. Then for any 𝑥∈𝐻, we have 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐺_𝑗𝑥, which is convergent in 𝐻, so ∑_𝑗∈𝑁Γ^∗_𝑗𝐺_𝑗𝑥 is convergent in 𝑌. Define operator 𝑆∶𝐻→𝑌 by 𝑆𝑥=∑_𝑗∈𝑁Γ^∗_𝑗𝐺_𝑗𝑥. Then 𝑆 is well defined and linear. If 𝑆𝑥=0, that is, ∑_𝑗∈𝑁Γ^∗_𝑗𝐺_𝑗𝑥=0, then 𝐺_𝑗𝑥=0 for each 𝑗∈𝑁. So 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐺_𝑗𝑥=0. Hence 𝑆 is injective. For any 𝑦∈𝑌, 𝑦=∑_𝑗∈𝑁Γ^∗_𝑗𝐿_𝑗𝑦, which is convergent in 𝑌. Then ∑_𝑗∈𝑁Λ^∗_𝑗𝐿_𝑗𝑦 is convergent in 𝐻. Suppose 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐿_𝑗𝑦, but we know that 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐺_𝑗𝑥 and {Λ_𝑗∶𝑗∈𝑁} is a g-basis, so 𝐺_𝑗𝑥=𝐿_𝑗𝑦 for each 𝑗∈𝑁. Hence 𝑆𝑥=∑_𝑗∈𝑁Γ^∗_𝑗𝐺_𝑗𝑥=∑_𝑗∈𝑁Γ^∗_𝑗𝐿_𝑗𝑦=𝑦, which implies that 𝑆 is surjective. Next, we want to verify that S is bounded.
For any 𝑘∈𝑁, let 𝑇_𝑘∶𝐻→𝑌 be defined by 𝑇_𝑘𝑥=∑^𝑘_𝑗=1Γ^∗_𝑗𝐺_𝑗𝑥. Then it is obvious that 𝑇_𝑘 is well defined and linear. Since 𝑇_𝑘𝑥=^𝑘_𝑗=1Γ^∗_𝑗𝐺_𝑗𝑥≤^𝑘_𝑗=1Γ^∗_𝑗𝐺_𝑗𝑥≤^𝑘_𝑗=1Γ^∗_𝑗𝐺_𝑗‖𝑥‖, (4.1) thus 𝑇_𝑘 is a bounded operator. It is easy to see that for any 𝑥∈𝐻, 𝑇_𝑘𝑥→𝑆𝑥(𝑘→∞). So for any 𝜀>0, there exists 𝑘₀, such that whenever 𝑘≥𝑘₀, we have that ‖𝑇_𝑘𝑥−𝑆𝑥‖<𝜀. Since for any 𝑘∈𝑁, we have ‖𝑇_𝑘𝑥‖≤‖𝑆𝑥‖+‖𝑇_𝑘𝑥−𝑆𝑥‖, so for any 𝑘∈𝑁, we have 𝑇_𝑘𝑥≤Sup𝑇_𝑗𝑥,‖𝑆𝑥‖+𝜀∣𝑗=1,2,…,𝑘₀−1<∞. (4.2) Hence, by the Banach-Steinhaus Theorem, Sup_𝑘‖𝑇_𝑘‖<∞. Since for any 𝑥∈𝐻, 𝑇_𝑘𝑥→𝑆𝑥(𝑘→∞), we have ‖𝑆𝑥‖≤Sup_𝑘𝑇_𝑘𝑥≤Sup_𝑘𝑇_𝑘‖𝑥‖. (4.3) So 𝑆 is bounded. Hence 𝑆 is a bounded invertible operator from 𝐻 onto 𝑌. Since for any 𝑥∈𝐻, we have ∑_𝑗∈𝑁𝑆Λ^∗_𝑗𝐺_𝑗𝑥=∑_𝑗∈𝑁Γ^∗_𝑗𝐺_𝑗𝑥, that is, ∑_𝑗∈𝑁𝑆Λ^∗_𝑗𝐺_𝑗=∑_𝑗∈𝑁Γ^∗_𝑗𝐺_𝑗, so ∑_𝑗∈𝑁𝐺^∗_𝑗Λ_𝑗𝑆^∗=∑_𝑗∈𝑁𝐺^∗_𝑗Γ_𝑗. Hence for any 𝑦∈𝑌, we have ∑_𝑗∈𝑁𝐺^∗_𝑗Λ_𝑗𝑆^∗𝑦=∑_𝑗∈𝑁𝐺^∗_𝑗Γ_𝑗𝑦. Since {𝐺_𝑗∶𝑗∈𝑁} is a g-basis, so Λ_𝑗𝑆^∗=Γ_𝑗 for each 𝑗∈𝑁, which implies that {Λ_𝑗∶𝑗∈𝑁} and {𝐺_𝑗∶𝑗∈𝑁} are equivalent. In the case that any of the two statements is true, then there is an invertible operator 𝑆∶𝐻→𝑌 such that Γ_𝑗=Λ_𝑗𝑆^∗ for each 𝑗∈𝑁. Hence, for any 𝑥∈𝐻, 𝑥=_𝑗∈𝑁Γ^∗_𝑗𝐿_𝑗𝑥=_𝑗∈𝑁𝑆Λ^∗_𝑗𝐿_𝑗𝑥=𝑆_𝑗∈𝑁Λ^∗_𝑗𝐿_𝑗𝑥. (4.4) So 𝑆⁻¹𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐿_𝑗𝑥. Thus 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐿_𝑗𝑆𝑥, but 𝑥=∑_𝑗∈𝑁Λ^∗_𝑗𝐺_𝑗𝑥, and {Λ_𝑗} is a g-basis of 𝐻, it follows that 𝐿_𝑗𝑆=𝐺_𝑗 for each 𝑗∈𝑁. By Theorem 4.1, we know that {𝐿_𝑗∶𝑗∈𝑁} is also a g-basis and {𝐺_𝑗∶𝑗∈𝑁} and {𝐿_𝑗∶𝑗∈𝑁} are equivalent.

Theorem 4.3. Suppose {Λ_𝑗∶𝑗∈𝑁} is a g-orthonormal basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁}. Then {Λ_𝑗∶𝑗∈𝑁} is a g-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝑁} and it is self-dual.

Proof. By the definition of g-orthonormal bases, we know that {Λ_𝑗∶𝑗∈𝑁} is a normalized tight g-frame. So for any 𝑓∈𝐻, 𝑓=∑_𝑗∈𝑁Λ^∗_𝑗Λ_𝑗𝑓. Since for any 𝑖≠𝑗 and for any 𝑔_𝑗∈𝐻_𝑗,ℎ_𝑖∈𝐻_𝑖, we have Λ_𝑖Λ^∗_𝑗𝑔_𝑗,ℎ_𝑖=Λ^∗_𝑗𝑔_𝑗,Λ^∗_𝑖ℎ_𝑖=0, (4.5) hence Λ_𝑖Λ^∗_𝑗𝑔_𝑗=0 for 𝑖≠𝑗. For 𝑖=𝑗 and for any 𝑔_𝑖,ℎ_𝑖∈𝐻_𝑖, we have Λ_𝑖Λ^∗_𝑖𝑔_𝑖,ℎ_𝑖=Λ^∗_𝑖𝑔_𝑖,Λ^∗_𝑖ℎ_𝑖=𝑔_𝑖,ℎ_𝑖, (4.6) so for any 𝑔_𝑖∈𝐻_𝑖, Λ_𝑖Λ^∗_𝑖𝑔_𝑖=𝑔_𝑖. Thus if 𝑓=∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗, then for any 𝑖∈𝑁, we have Λ_𝑖𝑓=∑_𝑗∈𝑁Λ_𝑖Λ^∗_𝑗𝑔_𝑗=Λ_𝑖Λ^∗_𝑖𝑔_𝑖=𝑔_𝑖. Thus for any 𝑓∈𝐻, there is a unique sequence {𝑔_𝑗∶𝑗∈𝑁} such that 𝑔_𝑗∈𝐻_𝑗 for any 𝑗∈𝑁 and 𝑓=∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗. So {Λ_𝑗∶𝑗∈𝑁} is a g-basis. It is obvious that {Λ_𝑗∶𝑗∈𝑁} is self-dual.

Theorem 4.4. Suppose {Λ_𝑗∶𝑗∈𝑁} is a g-orthonormal basis with respect to {𝐻_𝑗∶𝑗∈𝑁}. Then ∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 is convergent if and only if {𝑔_𝑗∶𝑗∈𝑁}∈𝑙²(⊕𝐻_𝑗).

Proof. Since {Λ_𝑗∶𝑗∈𝑁} is a g-orthonormal basis, by Theorem 4.3, {Λ_𝑗∶𝑗∈𝑁} is a g-basis and a g-frame. So {Λ_𝑗∶𝑗∈𝑁} is a g-Riesz basis. Thus there exist constants 𝐴,𝐵>0, such that for any integer 𝑛, we have 𝐴^𝑛_𝑗=1𝑔_𝑗²≤^𝑛_𝑗=1Λ^∗_𝑗𝑔_𝑗²≤𝐵^𝑛_𝑗=1𝑔_𝑗². (4.7) So ∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗 is convergent if and only if {𝑔_𝑗∶𝑗∈𝑁}∈𝑙²(⊕𝐻_𝑗).

From Theorems 4.2, 4.3, and 4.4, the following corollary is obvious.

Corollary 4.5. Any two g-orthonormal bases are equivalent.

5. The Property of 𝑔-Minimal of 𝑔-Bases

In this section, we studied the property of g-minimal of g-bases.

Theorem 5.1. Suppose {Λ_𝑗∶𝑗∈𝑁} is a g-frame sequence. Then(1)if {Λ_𝑗∶𝑗∈𝑁} is a g-basis, then {Λ_𝑗∶𝑗∈𝑁} is g-minimal;(2)if {Λ_𝑗∶𝑗∈𝑁} is g-minimal, then {Λ_𝑗∶𝑗∈𝑁} is g-linearly independent.

Proof. (1). Since {Λ_𝑗} is a g-basis and it is also a g-frame sequence, it is easy to see that {Λ_𝑗∶𝑗∈𝑁} is a g-frame. Hence {Λ_𝑗∶𝑗∈𝑁} is a g-Riesz basis by (3) of Lemma 2.16. Suppose {Λ_𝑗∶𝑗∈𝑁} is the unique dual g-frame of {Λ_𝑗∶𝑗∈𝑁}. By Lemma 2.15, we know that {Λ_𝑗} and {Λ_𝑗∶𝑗∈𝑁} are g-biorthonormal, that is, ⟨Λ^∗_𝑗𝑔_𝑗,Λ^∗_𝑗𝑔_𝑖⟩=𝛿_𝑖𝑗⟨𝑔_𝑗,𝑔_𝑖⟩, where 𝑔_𝑗∈𝐻_𝑗,𝑔_𝑖∈𝐻_𝑖. For any 𝑚∈𝑁 and any sequence {𝑔_𝑗∶𝑗∈𝑁} with 𝑔_𝑗∈𝐻_𝑗 and 𝑔_𝑚≠0, let 𝐸_𝑚=span_𝑖≠𝑚{Λ^∗_𝑖𝑔_𝑖}. Then for any 𝑥∈𝐸_𝑚, ⟨𝑥,Λ^∗_𝑚𝑔_𝑚⟩=0, but ⟨Λ^∗_𝑚𝑔_𝑚,Λ^∗_𝑚𝑔_𝑚⟩=⟨𝑔_𝑚,𝑔_𝑚⟩≠0, so Λ^∗_𝑚𝑔_𝑚∉𝐸_𝑚. Hence {Λ_𝑗∶𝑗∈𝑁} is g-minimal.
(2). Suppose {Λ_𝑗∶𝑗∈𝑁} is g-minimal. If ∑_𝑗∈𝑁Λ^∗_𝑗𝑔_𝑗=0, where 𝑔_𝑗∈𝐻_𝑗 for each 𝑗∈𝑁, then 𝑔_𝑗=0 for any 𝑗∈𝑁. In fact, if there exists 𝑚∈𝑁 such that 𝑔_𝑚≠0, then ‖Λ^∗_𝑚𝑔_𝑚‖≥(1/𝐿)‖𝑔_𝑚‖>0 by inequality (3.4), which implies that Λ^∗_𝑚𝑔_𝑚≠0. Since Λ^∗_𝑚𝑔_𝑚=−∑_𝑗≠𝑚Λ^∗_𝑗𝑔_𝑗, Λ^∗_𝑚𝑔_𝑚∈span_𝑗≠𝑚{Λ^∗_𝑗𝑔_𝑗}, which contradicts with the fact that {Λ_𝑗∶𝑗∈𝑁} is 𝑔-minimal.

Theorem 5.2. Given sequence {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝑁}.(1)If there exists a sequence {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝑁}, such that {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are biorthonormal, then {Λ_𝑗∶𝑗∈𝑁} is g-minimal.(2)If there exists a unique sequence {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝑁} such that {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are biorthonormal, then {Λ_𝑗∶𝑗∈𝑁} is 𝑔-minimal and g-complete.

Proof. The proof of (1) is similar to the proof of (1) of Theorem 5.1, we omit the details. Now we prove (2): By (1) we know that {Λ_𝑗∶𝑗∈𝑁} is g-minimal. So it only needs to show that {Λ_𝑗∶𝑗∈𝑁} is g-complete. Suppose that 𝑥∈𝐻 and ⟨𝑥,Λ^∗_𝑗𝑔_𝑗⟩=0 for any 𝑗∈𝑁 and any 𝑔_𝑗∈𝐻_𝑗. Since ⟨Λ^∗_𝑗𝑔_𝑗,Γ^∗_𝑗𝑔_𝑖⟩=𝛿_𝑖𝑗⟨𝑔_𝑗,𝑔_𝑖⟩, so ⟨Λ^∗_𝑗𝑔_𝑗,𝑥+Γ_𝑗𝑔_𝑖⟩=𝛿_𝑖𝑗⟨𝑔_𝑗,𝑔_𝑖⟩, which implies that {𝛿_𝑥+Γ_𝑗∶𝑗∈𝑁} and {Λ_𝑗∶𝑗∈𝑁} are biorthonormal, where 𝛿^∗_𝑥∈𝐵(𝑉,𝐻_𝑗) for each 𝑗∈𝑁 defined by 𝛿^∗_𝑥(𝑓)=𝑥 for any 𝑓∈𝑉. But it is assumed that there exists a unique sequence {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝑁} such that {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are biorthonormal, so 𝛿_𝑥=0, hence 𝑥=𝛿^∗_𝑥(𝑓)=0, which implies that span{Λ^∗_𝑗𝑔_𝑗}=𝐻. So {Λ_𝑗∶𝑗∈𝑁} is g-complete.

Acknowledgment

This work was partially supported by SWUFE's Key Subjects Construction Items Funds of 211 Project.