Some New Characterizations and 𝑔-Minimality and Stability of 𝑔-Bases in the Hilbert Spaces

Abstract

The concept of 𝑔-basis in the Hilbert spaces is introduced by Guo (2012) who generalizes the Schauder basis in the Hilbert spaces. 𝑔-basis plays the similar role in 𝑔-frame theory to that the Schauder basis plays in frame theory. In this paper, we establish some important properties of 𝑔-bases in the Hilbert spaces. In particular, we obtain a simple condition under which some important properties established in Guo (2012) are still true. With these conditions, we also establish some new interesting properties of 𝑔-bases which are related to 𝑔-minimality. Finally, we obtain a perturbation result about 𝑔-bases.

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 the 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 the 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) 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 generalizes 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. (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. Recently, 𝑔-frames in the Hilbert spaces have been studied intensively; for more details see [12–16] and the references therein.

In the history of the Hilbert space theory, the concept of basis was introduced more earlier than the concept of frame. And frames are generalizations of bases. When we look back the development of 𝑔-frame theory, it is well known that 𝑔-frames are generalizations of frames. Then it is naturally to ask what are the generalizations of bases in the similar manner? It is the motivation of our paper [17]. In [17], we put up with the concept of 𝑔-basis in the Hilbert space. We also find that 𝑔-basis really fills in the gap in the 𝑔-frame theory. 𝑔-bases play the similar role in 𝑔-frame theory to that the bases play in the frame theory. But we also find that the properties of basis cannot be adjusted trivially so that they are hold by 𝑔-bases. For more details, please see [17]. In this paper, we will do some study on 𝑔-bases in the Hilbert spaces further. And some important properties of 𝑔-bases are established. In Section 2, we will give the definitions and lemmas. In Section 3, we give some characterizations of 𝑔-bases. In Section 4, we study the property of 𝑔-minimal of 𝑔-bases. In Section 5, we discuss the stability of 𝑔-bases under perturbations.

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. And all Hilbert spaces are complex separable Hilbert spaces.

2. Definitions and Lemmas

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

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

Definition 2. {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 3. {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 is called 𝑔-minimal with respect to {𝐻_𝑗} if, for any 𝑚∈𝒩 and any 𝑔_𝑚∈𝐻_𝑚 such that Λ^∗_𝑚𝑔_𝑚≠0, we have Λ^∗_𝑚𝑔_𝑚∉span_𝑖≠𝑚{Λ^∗_𝑖(𝐻_𝑖)}.

Definition 4. Let 𝑀(𝐻,𝐾) denote the set of all maps from 𝐻 to 𝐾. {𝑇_𝑖∈𝑀(𝐻_𝑖,𝐻)∶𝑖∈𝒩} and {𝐿_𝑖∈𝑀(𝐻_𝑖,𝐻)∶𝑖∈𝒩} are called biorthonormal if 𝑇_𝑗𝑔_𝑗,𝐿_𝑖𝑔_𝑖=𝛿_𝑗,𝑖𝑔_𝑗,𝑔_𝑖,∀𝑗,𝑖∈𝒩,𝑔_𝑗∈𝐻_𝑗,𝑔_𝑖∈𝐻_𝑖. (3){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^∗_𝑗∶𝑗∈𝒩} and {Γ^∗_𝑗∶𝑗∈𝒩} are biorthonormal; that is, Λ^∗_𝑗𝑔_𝑗,Γ^∗_𝑖𝑔_𝑖=𝛿_𝑖,𝑗𝑔_𝑗,𝑔_𝑖,∀𝑖,𝑗∈𝒩,𝑔_𝑖∈𝐻_𝑖,𝑔_𝑗∈𝐻_𝑗. (4)

Definition 5. We call {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)}^∞_𝑗=1 a 𝑔-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩} if for any 𝑥∈𝐻 there is a unique sequence {𝑔_𝑗∶𝑗∈𝒩} with 𝑔_𝑗∈𝐻_𝑗 such that 𝑥=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗. The constant 𝐶=sup^𝑚_𝑘=1Λ^∗_𝑘𝑔_𝑘∣^𝑛_𝑘=1Λ^∗_𝑘𝑔_𝑘=1,𝑚≤𝑛,𝑔_𝑘∈𝐻_𝑘 (5) is called the basis constant associated with {Λ_𝑗∶𝑗∈𝒩}.

Remark 6. 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 7. If {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis of 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩}, 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∶𝑗∈𝒩} is well defined. We call it the 𝑔-dual sequence of {Λ_𝑗∶𝑗∈𝒩}; in case that {Γ_𝑗∶𝑗∈𝒩} is also a 𝑔-basis, we call it the dual 𝑔-basis of {Λ_𝑗∶𝑗∈𝒩}.

Lemma 8 (see [13]). {Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete if and only if span{Λ^∗_𝑗(𝐻_𝑗)∶𝑗∈𝒩}=𝐻.

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

Lemma 9 (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. (6)

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 10 (see [17]). 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}.

3. Characterizations of 𝑔-Bases

In this section, we characterize 𝑔-bases under the condition that the sequence is bounded surjective operators.

Theorem 11. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective for each 𝑗∈𝒩. Let 𝑌={{𝑔_𝑗}^∞_𝑗=1∣𝑔_𝑗∈𝐻_𝑗,∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗 is convergent}. If, for any {𝑔_𝑗}^∞_𝑗=1∈𝑌, set ‖{𝑔_𝑗}‖_𝑌=Sup_𝑁‖∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗‖, then(1)𝑌 is a Banach space,(2) when {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis with respect to {𝐻_𝑗∶𝑗∈𝒩} as well, 𝑆∶𝑌→𝐻, 𝑆({𝑔_𝑗}^∞_𝑗=1)=∑^∞_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}^∞_𝑗=1∈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. In particular, ‖Λ^∗₁𝑔₁‖=0. Since Λ₁ is surjective, there exists a bounded operator Λ^†₁∈𝐵(𝐻₁,𝐻) such that Λ₁Λ^†₁=𝐼_𝐻1 by Lemma 9. So (Λ^†₁)^∗Λ^∗₁=𝐼_𝐻1, which implies that 𝑔₁=(Λ^†₁)^∗Λ^∗₁𝑔₁. It implies that ‖𝑔₁‖≤‖(Λ^†₁)^∗‖‖Λ^∗₁𝑔₁‖=0. So 𝑔₁=0. Since ‖∑²_𝑗=1Λ^∗_𝑗𝑔_𝑗‖=‖Λ^∗₂𝑔₂‖=0, similarly, we have 𝑔₂=0. Keep doing in this way; then we get 𝑔_𝑗=0 for all 𝑗=1,…,𝑁. But 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_𝑗}_𝑌=lim_{𝑘,𝑙→∞}Sup_𝑁^𝑁_𝑗=1Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗=0. (7)(8) For any fixed 𝑗, we have that 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. (9) Since Λ_𝑗 is surjective, Λ_𝑗Λ^†_𝑗=𝐼_𝐻𝑗 by Lemma 9. So (Λ^†_𝑗)^∗Λ^∗_𝑗=𝐼_𝐻𝑗. It implies that 𝑔^𝑘_𝑗−𝑔^𝑙_𝑗=Λ^†_𝑗^∗Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗≤Λ^†_𝑗^∗⋅Λ^∗_𝑗𝑔^𝑘_𝑗−𝑔^𝑙_𝑗≤2⋅Λ^†_𝑗^∗⋅𝐺^𝑘−𝐺^𝑙_𝑌. (10)(11) So, for any fixed 𝑗, {𝑔^𝑘_𝑗}^∞_𝑘=1 is a Cauchy sequence. Suppose lim_𝑘→∞𝑔^𝑘_𝑗=𝑔_𝑗. From (8), 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. (12) 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. (13) 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. (14) So ∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗 is convergent; thus 𝐺={𝑔_𝑗}∈𝑌. Let 𝑙→∞ in inequality (13), 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∶𝑗∈𝒩} is a 𝑔-basis, then it is 𝑔-complete and 𝑔-linearly independent with respect to {𝐻_𝑗∶𝑗∈𝒩} by Lemma 10; then the operator 𝑆∶𝑌→𝐻,𝑆({𝑔_𝑗}^∞_𝑗=1)=∑^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗 is not only well defined but also is one to one and onto. And, for any {𝑔_𝑗}^∞_𝑗=1∈𝑌, we have 𝑆𝑔_𝑗^∞_𝑗=1=^∞_𝑗=1Λ^∗_𝑗𝑔_𝑗=lim_𝑁→∞^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗≤Sup_𝑄^𝑄_𝑗=1Λ^∗_𝑗𝑔_𝑗=𝑔_𝑗𝑌. (15) So 𝑆 is bounded operator. Since 𝑌 is a Banach space, by the open mapping theorem we get that 𝑆 is a homeomorphism.

Theorem 12. Suppose, for each 𝑗∈𝒩, Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis of 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} is its 𝑔-dual sequence. Then(1) for all 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 11. 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‖<∞. (16)
(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‖. (17) 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|‖. (18)

Theorem 13. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} is the 𝑔-dual sequence of {Λ_𝑗∶𝑗∈𝒩}. Then Γ_𝑗 is linear and bounded; that is, Γ_𝑗∈𝐵(𝐻,𝐻_𝑗) for each 𝑗∈𝒩.

Proof. Suppose that 𝑥=∑_𝑗∈𝒩Λ^∗_𝑗𝑔_𝑗 and 𝑦=∑_𝑗∈𝒩Λ^∗_𝑗ℎ_𝑗; then Γ_𝑗𝑥=𝑔_𝑗 and Γ_𝑗𝑦=ℎ_𝑗 by Definition 7. Since 𝑥+𝑦=∑_𝑗∈𝒩Λ^∗_𝑗(𝑔_𝑗+ℎ_𝑗), Γ_𝑗(𝑥+𝑦)=𝑔_𝑗+ℎ_𝑗=Γ_𝑗𝑥+Γ_𝑗𝑦 and Γ_𝑗(𝑎𝑥)=𝑎𝑔_𝑗=𝑎Γ_𝑗𝑥. It implies that Γ_𝑗 is linear, since, for any 𝑥∈𝐻, we have Λ^∗_𝑗Γ_𝑗𝑥=^𝑗_𝑖=1Λ^∗_𝑖Γ_𝑖𝑥−^𝑗−1_𝑖=1Λ^∗_𝑖Γ_𝑖𝑥≤^𝑗_𝑖=1Λ^∗_𝑖Γ_𝑖𝑥+^𝑗−1_𝑖=1Λ^∗_𝑖Γ_𝑖𝑥=𝑆_𝑗𝑥+𝑆_𝑗−1𝑥≤2𝐶⋅‖𝑥‖. (19) Since Λ_𝑗 is surjective, ‖Γ_𝑗𝑥‖≤‖(Λ^†_𝑗)^∗‖⋅‖Λ^∗_𝑗Γ_𝑗𝑥‖. It follows that Γ_𝑗𝑥≤Λ^†_𝑗^∗⋅2𝐶⋅‖𝑥‖. (20) So Γ_𝑗 is bounded for each 𝑗∈𝒩.

Theorem 14. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} is the dual sequence of {Λ_𝑗∶𝑗∈𝒩}. Then {Γ_𝑗∶𝑗∈𝒩} is also a 𝑔-basis for 𝐻 and {Λ_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal.

Proof. By Theorem 13, we know that Γ_𝑗∈𝐵(𝐻,𝐻_𝑗). Since {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} is the dual sequence of {Λ_𝑗∶𝑗∈𝒩}, 𝑥=∑_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗𝑥, for all 𝑥∈𝐻. In particular, for each 𝑖∈𝒩, 𝑔_𝑖∈𝐻_𝑖, we have Λ^∗_𝑖𝑔_𝑖=_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗Λ^∗_𝑖𝑔_𝑖=Λ^∗_𝑖Γ_𝑖Λ^∗_𝑖𝑔_𝑖+_𝑗≠𝑖Λ^∗_𝑗Γ_𝑗Λ^∗_𝑖𝑔_𝑖. (21) Since the representation is unique, Γ_𝑖Λ^∗_𝑖𝑔_𝑖=𝑔_𝑖 and Γ_𝑗Λ^∗_𝑖𝑔_𝑖=0 for 𝑗≠𝑖. It implies that Λ^∗_𝑖𝑔_𝑖,Γ^∗_𝑗𝑔_𝑗=Γ_𝑗Λ^∗_𝑖𝑔_𝑖,𝑔_𝑗=𝛿_𝑖,𝑗⟨𝑔_𝑖,𝑔_𝑗⟩. (22) So, {Λ_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal. Since ∑_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗=𝐼, ∑_𝑗∈𝒩Γ^∗_𝑗Λ_𝑗=𝐼. So for every 𝑥∈𝐻, we have that 𝑥=∑_𝑗∈𝒩Γ^∗_𝑗Λ_𝑗𝑥. To prove that {Γ_𝑗∶𝑗∈𝒩} is a 𝑔-basis, it is sufficient to show that the representation is unique. It is sufficient to prove that {Γ_𝑗∶𝑗∈𝒩} is 𝑔-linearly independent. Suppose that ∑_𝑗∈𝒩Γ^∗_𝑗𝑔_𝑗=0; then, for each 𝑖∈𝒩, we have 𝑔_𝑖=Λ_{𝑖𝑗∈𝒩}Γ^∗_𝑗𝑔_𝑗=0. (23) It implies that {Γ_𝑗∶𝑗∈𝒩} is 𝑔-linearly independent.

Theorem 15. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} is the dual 𝑔-basis of {Λ_𝑗∶𝑗∈𝒩}. 𝑆_𝑁𝑥=∑^𝑁_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥; that is, 𝑆_𝑁 is the partial sum operator associated with ({Λ_𝑗},{Γ_𝑗}). Then under the norm ‖|⋅|‖ defined in Theorem 12(3), ‖𝑆_𝑁‖=1, for all 𝑁>0.

Proof. Since {Λ_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal by Theorem 14, if 𝐾≤𝐿, then 𝑆_𝐾𝑆_𝐿𝑥=^𝐾_𝑗=1Λ−𝑗^∗Γ_𝑗𝑆_𝐿𝑥=^𝐾_𝑗=1Λ^∗_𝑗Γ_𝑗^𝐿_𝑖=1Λ^∗_𝑖Γ_𝑖𝑥=^𝐾𝐿_𝑗=1𝑖=1Λ^∗_𝑗Γ_𝑗Λ^∗_𝑖Γ_𝑖𝑥=^𝐾_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥=𝑆_𝐾𝑥; (24) if 𝐾≥𝐿, then 𝑆_𝐾𝑆_𝐿𝑥=^𝐾_𝑗=1Λ−𝑗^∗Γ_𝑗𝑆_𝐿𝑥=^𝐾_𝑗=1Λ^∗_𝑗Γ_𝑗^𝐿_𝑖=1Λ^∗_𝑖Γ_𝑖𝑥=^𝐾𝐿_𝑗=1𝑖=1Λ^∗_𝑗Γ_𝑗Λ^∗_𝑖Γ_𝑖𝑥=^𝐿_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥=𝑆_𝐿𝑥. (25) Hence 𝑆_𝐿𝑥=sup_𝐾𝑆_𝐾𝑆_𝐿𝑥=sup𝑆₁𝑥,…,𝑆_𝐿𝑥. (26) So, for any 𝐿∈𝒩, ‖|𝑆_𝐿𝑥|‖≥‖𝑆_𝐿𝑥‖, which follows that sup_𝐿𝑆_𝐿𝑥≥sup_𝐿𝑆_𝐿𝑥=‖|𝑥|‖. (27) On the other hand, for any 𝐿∈𝒩, from (26), we also have that 𝑆_𝐿𝑥=sup_𝐾≤𝐿𝑆_𝐾𝑥≤sup_𝐾∈𝒩𝑆_𝐾𝑥=‖|𝑥|‖, (28) which follows that sup_𝐿‖|𝑆_𝐿𝑥|‖≤‖|𝑥|‖. So sup_𝐿‖|𝑆_𝐿𝑥|‖=‖|𝑥|‖. So 𝑆_𝐿=sup_{‖|𝑥|‖=1}𝑆_𝐿𝑥=sup_{‖|𝑥|‖=1}‖|𝑥|‖=1. (29)

Theorem 16. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. Then the following statements are equivalent:(1){Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with respect to {𝐻_𝑗∶𝑗∈𝒩};(2) there exists {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝒩} such that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-biorthonormal with {Γ_𝑗∶𝑗∈𝒩} and ∀𝑥∈𝐻,𝑥=_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗𝑥, (30)(3){Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete and there exists {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝒩} such that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-biorthonormal with {Γ_𝑗∶𝑗∈𝒩} and ∀𝑥∈𝐻,sup_𝑁𝑆_𝑁𝑥<∞, (31)(4){Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete and there exists {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝒩} such that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-biorthonormal with {Γ_𝑗∶𝑗∈𝒩} and sup_𝑁𝑆_𝑁<∞. (32)

Proof. (1)⇒(2): It is obvious that the 𝑔-dual basis {Γ_𝑗∶𝑗∈𝒩} is 𝑔-biorthonormal with {Λ_𝑗∶𝑗∈𝒩} by Theorem 13 and, for each 𝑥∈𝐻, 𝑥=∑_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗𝑥.
(2)⇒(1): It is only needed, to show that the representation for 𝑥 is unique. It is sufficient to show that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-linearly independent. Suppose that ∑_𝑗∈𝒩Λ^∗_𝑗𝑔_𝑗=0, where 𝑔_𝑗∈𝐻_𝑗, for all 𝑗∈𝒩. Then 𝑔_𝑖=Γ_𝑖∑_𝑗∈𝒩Λ^∗_𝑗𝑔_𝑗=0, for each 𝑖∈𝒩, since {Γ_𝑗∶𝑗∈𝒩} and {Λ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal. So {Λ_𝑗∶𝑗∈𝒩} is 𝑔-linearly independent.
(2)⇒(3): It is obvious that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete. It is only needed to prove that, for all 𝑥∈𝐻, sup_𝑁‖𝑆_𝑁𝑥‖<∞. Since 𝑥=∑_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗𝑥, 𝑥=lim_𝑁→∞𝑆_𝑁𝑥. It implies that {𝑆_𝑁𝑥}^∞_𝑁=1 is convergent for each 𝑥∈𝐻. Thus {𝑆_𝑁𝑥}^∞_𝑁=1 is a bounded sequence for each 𝑥∈𝐻. Hence sup_𝑁‖𝑆_𝑁𝑥‖<∞.
(3)⇒(4): It only needed, to show that sup_𝑁‖𝑆_𝑁‖<∞. Since {𝑆_𝑁}^∞_𝑁=1 is a sequence of linear bounded operators on 𝐻 and, for each 𝑥∈𝐻, sup_𝑁‖𝑆_𝑁𝑥‖<∞. So sup_𝑁‖𝑆_𝑁‖<∞ by the Banach-Steinhaus theorem.
(4)⇒(2): For all 𝑥∈span{Λ^∗_𝑗(𝐻_𝑗)∶𝑗∈𝒩}, suppose that 𝑥=∑^𝑀_𝑛=1Λ^∗_𝑛𝑔_𝑛, 𝑔_𝑛∈𝐻_𝑛. Since 𝑆_𝑁 is linear and {Λ_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal, for 𝑁≥𝑀, we have 𝑆_𝑁𝑥=^𝑀_𝑛=1𝑆_𝑁Λ^∗_𝑛𝑔_𝑛=^𝑀𝑁_𝑛=1𝑖=1Λ^∗_𝑖Γ_𝑖Λ^∗_𝑛𝑔_𝑛=^𝑀_𝑛=1Λ^∗_𝑛𝑔_𝑛=𝑥. (33) It implies that, for any 𝑥∈span{Λ^∗_𝑗(𝐻_𝑗)∶𝑗∈𝒩}, 𝑥=lim_𝑁→∞𝑆_𝑁𝑥=∑_𝑗∈𝒩Λ^∗_𝑗Γ_𝑗𝑥. Next we prove that, for all 𝑥∈𝐻, it is also correct. Since {Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete, span{Λ^∗_𝑗(𝐻_𝑗)∶𝑗∈𝒩}=𝐻 by Lemma 8. For any 𝜀>0, there exists 𝑦∈span{Λ_𝑗(𝐻_𝑗)∶𝑗∈𝒩} such that ‖𝑥−𝑦‖<𝜀(1+sup_𝑁‖𝑆_𝑁‖)⁻¹. Since 𝑦∈span{Λ_𝑗(𝐻_𝑗)∶𝑗∈𝒩}, there exist 𝑁 such that 𝑆_𝑁𝑦=𝑦. Since 𝑥−𝑆_𝑁𝑥≤𝑥−𝑦+𝑦−𝑆_𝑁𝑦+𝑆_𝑁𝑦−𝑆_𝑁𝑥≤𝑥−𝑦+0+𝑆_𝑁≤𝑥−𝑦1+sup_𝑁𝑆_𝑁𝑥−𝑦<𝜀, (34) thus lim_𝑁→∞𝑆_𝑁𝑥=𝑥.

Theorem 17. Suppose, for each 𝑗∈𝒩, Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. {Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete 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, we have ^𝑚_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝐶⋅^𝑛_𝑗=1Λ^∗_𝑗𝑔_𝑗. (35)

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 12, ‖|⋅|‖ 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. (36)⇐: Let 𝒜={∑_𝑘∈𝒩Λ^∗_𝑘𝑔_𝑘,𝑔_𝑘∈𝐻_𝑘, and ∑_𝑘∈𝒩Λ^∗_𝑘𝑔_𝑘 is convergent}. First, we show that 𝒜=𝐻. Since {Λ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝒩} is 𝑔-complete, 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)≤𝐶⋅^𝑚_𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠^(𝑙)≤𝐶²⋅^𝑛_𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠^(𝑙)≤𝐶²⋅^𝑛_𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑦_𝑘+𝑦_𝑘−𝑦+𝐶²⋅𝑦−𝑦_𝑙+𝑦_𝑙−^𝑛_𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑙). (37) 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∈𝒩,𝑘,𝑙≥𝑀,_𝑚𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠^(𝑙)≤𝜀𝐶forany𝑚∈𝒩andany𝑘,𝑙≥𝑀. (38) Since Λ_𝑗 is surjective, by inequality (10), we have that ‖𝑔_𝑗^(𝑘)−𝑔_𝑗^(𝑙)‖≤‖(Λ^†_𝑗)^∗‖⋅‖Λ^∗_𝑗(𝑔_𝑗^(𝑘)−𝑔_𝑗^(𝑙))‖≤‖(Λ^†_𝑗)^∗‖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∈𝒩,𝑘≥𝑀,_𝑚𝑠=1Λ^∗_𝑠𝑔_𝑠^(𝑘)−𝑔_𝑠≤𝜀𝐶forany𝑚∈𝒩andany𝑘≥𝑀. (39) 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, (40) 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 since Λ_𝑗 is surjective, ‖𝑔_𝑗‖≤‖(Λ^†_𝑗)^∗‖⋅‖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. The Property of 𝑔-Minimal of 𝑔-Bases

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

Theorem 18. Suppose Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective. Then(1) if {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis, then {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal and 𝑔-complete;(2) if {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal, then {Λ_𝑗∶𝑗∈𝒩} is 𝑔-linearly independent.

Proof. (1) Since {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis, it is obvious that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete. We only need to prove that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal. Suppose that {Γ_𝑗∶𝑗∈𝒩} is the unique dual 𝑔-basis of {Λ_𝑗∶𝑗∈𝒩}. Then {Λ_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal by Theorem 16; that is, ⟨Λ^∗_𝑗𝑔_𝑗,Γ^∗_𝑖𝑔_𝑖⟩=𝛿_𝑖,𝑗⟨𝑔_𝑗,𝑔_𝑖⟩ for any 𝑖,𝑗∈𝒩 and 𝑔_𝑗∈𝐻_𝑗,𝑔_𝑖∈𝐻_𝑖. For any 𝑚∈𝒩 and any 𝑔_𝑚∈𝐻_𝑚 such that Λ^∗_𝑚𝑔_𝑚≠0, let 𝐸_𝑚=span_𝑖≠𝑚{Λ^∗_𝑖𝑔_𝑖}. Then, for any 𝑥∈𝐸_𝑚, ⟨𝑥,Γ^∗_𝑚𝑔_𝑚⟩=0, but ⟨Λ^∗_𝑚𝑔_𝑚,Γ^∗_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 and ∑_𝑗∈𝒩Λ^∗_𝑗𝑔_𝑗=0. Since Λ_𝑗 is surjective for any 𝑗∈𝒩, then ‖𝑔_𝑗‖≤‖(Λ^†_𝑗)^∗‖⋅‖Λ^∗_𝑗𝑔_𝑗‖ for any 𝑗∈𝒩 and 𝑔_𝑗∈𝐻_𝑗. If there exists 𝑗₀∈𝒩 such that 𝑔_𝑗0≠0, then ‖Λ^∗_𝑗0𝑔_𝑗0‖>0. It follows that Λ^∗_𝑗0𝑔_𝑗0≠0. Since Λ^∗_𝑗0𝑔_𝑗0=−∑_𝑗≠𝑗0Λ^∗_𝑗𝑔_𝑗, Λ^∗_𝑗0𝑔_𝑗0∈span_𝑗≠𝑗0{Λ^∗_𝑗𝑔_𝑗}, which contradicts with the fact that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal.

Theorem 19. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective for each 𝑗∈𝒩. Then(1) there exists a sequence {Γ_𝑗∈𝑀(𝐻_𝑗,𝐻)∶𝑗∈𝒩}, such that {Λ^∗_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are biorthonormal if and only if {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal,(2) there exists a unique sequence {Γ_𝑗∈𝑀(𝐻_𝑗,𝐻)∶𝑗∈𝒩} such that {Λ^∗_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are biorthonormal, if and only if {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal and 𝑔-complete.

Proof. (1)⇒: Suppose that {Γ_𝑗∶𝑗∈𝒩} is biorthonormal with {Λ^∗_𝑗∶𝑗∈𝒩}; that is, ⟨Λ^∗_𝑗𝑔_𝑗,Γ_𝑖𝑔_𝑖⟩=𝛿_𝑖,𝑗⟨𝑔_𝑗,𝑔_𝑖⟩ for any 𝑖,𝑗∈𝒩 and 𝑔_𝑗∈𝐻_𝑗,𝑔_𝑖∈𝐻_𝑖. For any 𝑚∈𝒩 and any 𝑔_𝑚∈𝐻_𝑚 such that Λ^∗_𝑚𝑔_𝑚≠0, let 𝐸_𝑚=span_𝑖≠𝑚{Λ^∗_𝑖(𝐻_𝑖)}. Then, for any 𝑥∈𝐸_𝑚, ⟨𝑥,Γ_𝑚𝑔_𝑚⟩=0, but ⟨Λ^∗_𝑚𝑔_𝑚,Γ_𝑚𝑔_𝑚⟩=⟨𝑔_𝑚,𝑔_𝑚⟩≠0, so Λ^∗_𝑚𝑔_𝑚∉𝐸_𝑚. Hence {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal.
⇐: Suppose that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal. For all 𝑚∈𝒩, for all 𝑔_𝑚∈𝐻_𝑚 such that 𝑔_𝑚≠0, let 𝐸_𝑚=span_𝑗≠𝑚{Λ^∗_𝑗(𝐻_𝑗)}. Since Λ_𝑗 is surjective, ‖(Λ^†_𝑗)^∗‖⋅‖Λ^∗_𝑚𝑔_𝑚‖≥‖𝑔_𝑚‖>0. Since {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal, Λ^∗_𝑚𝑔_𝑚∉𝐸_𝑚. So there exists 𝑥_𝑚∈𝐻 such that ⟨Λ^∗_𝑚𝑔_𝑚,𝑥_𝑚⟩=‖𝑔_𝑚‖² and ⟨𝑦,𝑥_𝑚⟩=0, for all 𝑦∈𝐸_𝑚. Keep in mind that the notation 𝑀(𝐻_𝑗,𝐻) in statement (1) means just a general map with no linearity and bounded requirements, defining a map Γ_𝑚∶𝐻_𝑚→𝐻 by Γ_𝑚𝑔_𝑚=𝑥_𝑚 if 𝑔_𝑚≠0, Γ_𝑚𝑔_𝑚=0 if 𝑔_𝑚=0. Then {Γ_𝑗∶𝑗∈𝒩} is biorthonormal with {Λ^∗_𝑗∶𝑗∈𝒩}, since ⟨Λ^∗_𝑗𝑔_𝑗,Γ_𝑖𝑔_𝑖⟩=𝛿_𝑖,𝑗⟨𝑔_𝑗,𝑔_𝑖⟩ for every 𝑖,𝑗∈𝒩 and 𝑔_𝑗∈𝐻_𝑗,𝑔_𝑖∈𝐻_𝑖.
(2)⇒: By (1) we know that {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal. So it is only needed 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_𝑖𝑔_𝑖⟩=𝛿_𝑖𝑗⟨𝑔_𝑗,𝑔_𝑖⟩, so ⟨Λ^∗_𝑗𝑔_𝑗,𝑥+Γ_𝑖𝑔_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(𝐻_𝑗,𝐻) for each 𝑗∈𝒩 defined by 𝛿_𝑥(𝑓)=𝑥 for any 𝑓∈𝐻_𝑗. But it is assumed that there exists a unique sequence {Γ_𝑗∈𝑀(𝐻_𝑗,𝐻)∶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.
⇐: By (1), we know that if {Λ_𝑗∶𝑗∈𝒩} is 𝑔-minimal, then there exists a sequence {Γ_𝑗∈𝑀(𝐻_𝑗,𝐻)∶𝑗∈𝒩} such that {Λ^∗_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are biorthonormal. Now we only need to show that such sequence is unique. If there is another sequence {𝑇_𝑗∈𝑀(𝐻_𝑗,𝐻)∶𝑗∈𝒩} such that {Λ^∗_𝑗∶𝑗∈𝒩} and {𝑇_𝑗∶𝑗∈𝒩} are biorthonormal; then ⟨Λ^∗_𝑗𝑔_𝑗,(Γ_𝑖−𝑇_𝑖)𝑔_𝑖⟩=0 for any 𝑖,𝑗∈𝒩 and 𝑔_𝑖∈𝐻_𝑖,𝑔_𝑗∈𝐻_𝑗. Since {Λ_𝑗∶𝑗∈𝒩} is 𝑔-complete, span{Λ^∗_𝑗(𝐻_𝑗)∶𝑗∈𝒩}=𝐻 by Lemma 8. Thus (Γ_𝑖−𝑇_𝑖)𝑔_𝑖=0 for every 𝑖∈𝒩 and 𝑔_𝑖∈𝐻_𝑖. It follows that 𝑇_𝑖=Γ_𝑖 for every 𝑖∈𝒩. So the sequence which is biorthonormal with {Λ^∗_𝑗∶𝑗∈𝒩} is unique.

Theorem 20. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective for each 𝑗∈𝒩. Then the following statements are equivalent:(1) there exists a sequence {Γ_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝒩} which is 𝑔-biorthonormal with {Λ_𝑗∶𝑗∈𝒩},(2) for all 𝑀>0, there exists a constant 𝐶_𝑀≥1 such that, for any 𝑁≥𝑀 and 𝑔_𝑗∈𝐻_𝑗(𝑗=1,…,𝑁), we have ^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝐶_𝑀^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗. (41)

Proof. (1)⇒(2): For any 𝑁>0, any 𝑥∈𝐻, let 𝑆_𝑁𝑥=∑^𝑁_𝑗=1Λ^∗_𝑗Γ_𝑗𝑥. Then for any 𝑁≥𝑀 and any 𝑔_𝑗∈𝐻_𝑗(𝑗=1,…,𝑁), we have 𝑆_𝑀^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗=^𝑁_𝑗=1𝑆_𝑀Λ^∗_𝑗𝑔_𝑗=^𝑁𝑀_𝑗=1𝑖=1Λ^∗_𝑖Γ_𝑖Λ^∗_𝑗𝑔_𝑗=^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗. (42) Hence ^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗=𝑆_𝑀^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝑆_𝑀^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗. (43) Then ‖𝑆_𝑀‖≥1. So 𝐶_𝑀=‖𝑆_𝑀‖ satisfies the requirement.
(2)⇒(1): Let 𝐸=span{Λ^∗_𝑗(𝐻_𝑗)∶𝑗∈𝒩} and 𝑥∈𝐸. Suppose that 𝑥=∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗 and 𝑀≤𝑁. Since 𝑔_𝑀≤Λ^†_𝑀^∗⋅Λ^∗_𝑀𝑔_𝑀≤Λ^†_𝑀^∗𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗+^𝑀−1_𝑗=1Λ^∗_𝑗𝑔_𝑗≤Λ^†_𝑀^∗𝐶_𝑀^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗+𝐶_𝑀−1^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗=Λ^†_𝑀^∗⋅𝐶_𝑀+𝐶_𝑀−1‖𝑥‖, (44) so, if 𝑥=0, then 𝑔_𝑗=0 for 𝑗=1,…,𝑁, which follows that the vectors 𝑔_𝑗,𝑗=1,…,𝑁 are uniquely defined by 𝑥=∑^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗. Now, for each 𝑚∈𝒩, we define operator Γ_𝑚∶𝐸→𝐻_𝑚 by Γ_𝑚^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗=𝑔_𝑚,𝑚≤𝑁,0,𝑚>𝑁. (45) Then Γ_𝑚 is linear and Γ_𝑚^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝑔_𝑚≤Λ^†_𝑚^∗⋅𝐶_𝑚+𝐶_𝑚−1^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗. (46) So Γ_𝑚 is bounded on 𝐸; hence it can be linearly extended to 𝐸. Let Γ_𝑚=0 on 𝐸^⟂; then Γ_𝑚∈𝐵(𝐻,𝐻_𝑚) for each 𝑚∈𝒩 and Λ^∗_𝑗𝑔_𝑗,Γ^∗_𝑖𝑔_𝑖=Γ_𝑖Λ^∗_𝑗𝑔_𝑗,𝑔_𝑖=𝛿_𝑖,𝑗𝑔_𝑗,𝑔_𝑖; (47) that is, {Γ_𝑗∶𝑗∈𝒩} is 𝑔-biorthonormal with {Λ_𝑗∶𝑗∈𝒩}.

5. Perturbation of 𝑔-Bases

In this section, we give one result about the stability of 𝑔-bases under certain perturbation.

Theorem 21. Suppose that Λ_𝑗∈𝐵(𝐻,𝐻_𝑗) and Λ_𝑗 is surjective for each 𝑗∈𝒩. {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with {Γ_𝑗∶𝑗∈𝑁} as its 𝑔-dual basis and its basis constant is 𝐶. If {𝐿_𝑗∈𝐵(𝐻,𝐻_𝑗)∶𝑗∈𝒩} is a sequence such that 𝑅=∑_𝑗∈𝒩‖Λ^∗_𝑗−𝐿^∗_𝑗‖⋅‖Γ_𝑗‖<1, then {𝐿_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻 with basis constant 𝐶₁≤((1+𝑅)/(1−𝑅))𝐶.

Proof. Suppose 𝑥=∑^𝑛_𝑗=𝑚Λ^∗_𝑗𝑔_𝑗, 𝑔_𝑗∈𝐻_𝑗(𝑗=𝑚,…,𝑛). Then 𝑔_𝑗=Γ_𝑗𝑥(𝑗=𝑚,…,𝑛), since {Λ_𝑗∶𝑗∈𝒩} and {Γ_𝑗∶𝑗∈𝒩} are 𝑔-biorthonormal by Theorem 16. Since ^𝑛_𝑗=𝑚Λ^∗_𝑗−𝐿^∗_𝑗𝑔_𝑗≤^𝑛_𝑗=𝑚Λ^∗_𝑗−𝐿^∗_𝑗⋅𝑔_𝑗=^𝑛_𝑗=𝑚Λ^∗_𝑗−𝐿^∗_𝑗⋅Γ_𝑗𝑥≤^𝑛_𝑗=𝑚Λ^∗_𝑗−𝐿^∗_𝑗⋅Γ_𝑗⋅‖𝑥‖≤𝑅⋅‖𝑥‖=𝑅⋅^𝑛_𝑗=𝑚Λ^∗_𝑗𝑔_𝑗, (48) which implies that if ∑_𝑗∈𝒩Λ^∗_𝑗𝑔_𝑗 is convergent, then ∑_𝑗∈𝒩(Λ^∗_𝑗−𝐿^∗_𝑗)𝑔_𝑗 is convergent, the following operator is well defined: 𝑇∶𝐻⟶𝐻,𝑇_𝑗∈𝒩Λ^∗_𝑗𝑔_𝑗=_𝑗∈𝒩Λ^∗_𝑗−𝐿^∗_𝑗𝑔_𝑗. (49) It is obvious that 𝑇 is linear and ‖𝑇‖≤𝑅≤1. It implies that 𝐼−𝑇 is invertible. Since, for any 𝑗∈𝒩 and 𝑔_𝑗∈𝐻_𝑗, we have (𝐼−𝑇)Λ^∗_𝑗𝑔_𝑗=Λ^∗_𝑗𝑔_𝑗−𝑇Λ^∗_𝑗𝑔_𝑗=Λ^∗_𝑗𝑔_𝑗−Λ^∗_𝑗−𝐿^∗_𝑗𝑔_𝑗=𝐿^∗_𝑗𝑔_𝑗, (50) so 𝐿_𝑗=Λ_𝑗(𝐼−𝑇)^∗ for any 𝑗∈𝒩. Since (𝐼−𝑇)^∗ is invertible, it is easy to see that {𝐿_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻. Now we start to show the basis constants inequality. For any 𝑁≥𝑀, we have ^𝑀_𝑗=1𝐿^∗_𝑗𝑔_𝑗≤^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗+^𝑀_𝑗=1𝐿^∗_𝑗−Λ^∗_𝑗𝑔_𝑗≤^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗+𝑅⋅^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗=(1+𝑅)⋅^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗. (51) On the other hand, we have ^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗≤^𝑁_𝑗=1𝐿^∗_𝑗𝑔_𝑗+^𝑁_𝑗=1Λ^∗_𝑗−𝐿^∗_𝑗𝑔_𝑗≤^𝑁_𝑗=1𝐿^∗_𝑗𝑔_𝑗+𝑅⋅^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗. (52) It follows that ^𝑁_𝑗=1𝐿^∗_𝑗𝑔_𝑗≥(1−𝑅)⋅^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗. (53) Since {Λ_𝑗∶𝑗∈𝒩} is a 𝑔-basis for 𝐻, by Theorem 17, we have that ^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗≤𝐶^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗. (54) So, by inequalities (51), (53) and (54), we have ^𝑀_𝑗=1𝐿^∗_𝑗𝑔_𝑗≤(1+𝑅)⋅^𝑀_𝑗=1Λ^∗_𝑗𝑔_𝑗≤(1+𝑅)𝐶⋅^𝑁_𝑗=1Λ^∗_𝑗𝑔_𝑗≤1+𝑅1−𝑅𝐶⋅^𝑁_𝑗=1𝐿^∗_𝑗𝑔_𝑗. (55)

Acknowledgment

This work was partially supported by SWUFE’s Key Subjects Construction Items Funds 211 Projects.