Operator Characterizations and Some Properties of 𝑔-Frames on Hilbert Spaces

Abstract

Given the 𝑔-orthonormal basis for Hilbert space 𝐻, we characterize the 𝑔-frames, normalized tight 𝑔-frames, and 𝑔-Riesz bases in terms of the 𝑔-preframe operators. Then we consider the transformations of 𝑔-frames, normalized tight 𝑔-frames, and 𝑔-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and 𝑔-dual frames of 𝑔-frames by applying the results of characterizations.

1. Introduction

A sequence (𝑓_𝑖)_𝑖∈𝐼 of elements of a Hilbert space 𝐻 is called a frame for 𝐻 if there are constants 𝐴,𝐵>0 so that 𝐴𝑓²≤_𝑖∈𝐼𝑓,𝑓_𝑖²≤𝐵𝑓²,∀𝑓∈𝐻. (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.

The concept of frame first appeared in the late 40s and early 50s (see, [1–3]). The development and study of wavelet theory during the last decades also brought new ideas and attentions to frames because of their close connections. There are many related references on this topic, see; [4–8].

In [9], 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 𝐾, 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 and normalized tight 𝑔-frame 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 [10–14] and the references therein.

Constructing frames and g-frames is an interesting problem in frame theory and it is also useful in applications. In this respect, many mathematicians considered the algebraic operations among frames, which allows us to construct a large number of new frames from existing frames. For more details, see [15–18] and the references therein. In this paper, we will consider the algebraic operations among 𝑔-frames; in particular, we will consider the direct sums of 𝑔-frames and general sums of 𝑔-frames. We also will consider the transforms of 𝑔-frames and their dual 𝑔-frames.

2. 𝑔-Preframe Operators and Transformations of 𝑔-Frames

In this section, we introduce the 𝑔-preframe operators and use them to characterize 𝑔-frames. We also discuss the transformations of 𝑔-frames. Since any countable infinite sequence can be made as one indexed by natural number 𝑁, so all infinite sequences in this paper are assumed to be indexed by 𝑁. For each sequence {𝐻_𝑖∶𝑖∈𝑁}, we define the space (∑_𝑖∈𝑁⊕𝐻_𝑖)_𝑙2 by _𝑖∈𝑁⊕𝐻_𝑖𝑙2=𝑓_{𝑖𝑖∈𝑁}∶𝑓_𝑖∈𝐻_𝑖,𝑖∈𝑁,_𝑖∈𝑁𝑓_𝑖²<+∞, (3) with the inner product defined by 𝑓_𝑖,𝑔_𝑖=_𝑖∈𝑁𝑓_𝑖,𝑔_𝑖. (4)

Definition 1 (see, [9]). Let Λ_𝑖∈𝐵(𝐻,𝐻_𝑖), 𝑖∈𝑁.(i)If {𝑓∶Λ_𝑖𝑓=0;𝑖∈𝑁}={0}, then we say that {Λ_𝑖∶𝑖∈𝑁} is 𝑔-complete.(ii)If {Λ_𝑖∶𝑖∈𝑁} is 𝑔-complete and there are positive constants 𝐴 and 𝐵 such that for any finite subset 𝐽⊂𝑁 and 𝑔_𝑗∈𝐻_𝑗, 𝑗∈𝐽, 𝐴_𝑗∈𝐽𝑔_𝑗²≤_𝑗∈𝐽Λ^∗_𝑗𝑔_𝑗²≤𝐵_𝑗∈𝐽𝑔_𝑗², (5) then we say that {Λ_𝑖∶𝑖∈𝑁} is a 𝑔-Riesz basis for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}.(iii)We say {Λ_𝑖∶𝑖∈𝑁} is a 𝑔-orthonormal basis for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁} if it satisfies the following: Λ^∗_𝑖𝑔_𝑖,Λ^∗_𝑗𝑔_𝑗=𝛿_𝑖𝑗𝑔_𝑖,𝑔_𝑗,∀𝑖,𝑗∈𝑁,𝑔_𝑖∈𝐻_𝑖,𝑔_𝑗∈𝐻_𝑗,_𝑗∈𝑁Λ_𝑗𝑓²=𝑓²,∀𝑓∈𝐻. (6)

Remark 2. It is obvious that any 𝑔-frame {Λ_𝑖∶𝑖∈𝑁} is 𝑔-complete and any 𝑔-orthonormal basis is a normalized tight 𝑔-frame.

Lemma 3 (see [11]). {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖), 𝑖∈𝑁} is g-complete if and only if  span{Λ^∗_𝑖(𝐻_𝑖)}_𝑖∈𝑁=𝐻.

Lemma 4 (see, [11]). {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖), 𝑖∈𝑁} is a 𝑔-Bessel sequence for 𝐻 with bound 𝐵, if and only if the operator 𝑇∶_𝑖∈𝑁⊕𝐻_𝑖𝑙2⟶𝐻 (7) defined by 𝑇({𝑔_𝑖})=∑_𝑗∈𝑁Λ^∗_𝑗(𝑔_𝑗) is a well-defined bounded operator with ‖𝑇‖≤√𝐵.

Lemma 5. Let {𝜃_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} be a 𝑔-orthonormal basis for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}; then the sequence {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence with respect to {𝐻_𝑖∶𝑖∈𝑁} if and only if there is a unique bounded operator 𝑉∶𝐻→𝐻 such that Λ_𝑖=𝜃_𝑖𝑉^∗, for all 𝑖∈𝑁.

Proof. ⇒∶ Since {𝜃_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis for 𝐻, {𝜃_𝑖𝑓∶𝑖∈𝑁}∈(∑_𝑖∈𝑁⊕𝐻_𝑖)_𝑙2, for any 𝑓∈𝐻. If {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence, then the operator 𝑉∶𝐻⟶𝐻,𝑉𝑓=_𝑖∈𝑁Λ^∗_𝑖𝜃_𝑖𝑓 (8) is well-defined and bounded operator by Lemma 5. Also by the definition of 𝑔-orthonormal basis, it is easy to see that 𝜃_𝑖𝜃^∗_𝑗𝑔=𝛿_𝑖𝑗𝑔. So 𝑉𝜃^∗_𝑗𝑔=_𝑖∈𝑁Λ^∗_𝑖𝜃_𝑖𝜃^∗_𝑗𝑔=Λ^∗_𝑗𝜃_𝑗𝜃^∗_𝑗𝑔=Λ^∗_𝑗𝑔 (9) for all 𝑔∈𝐻_𝑗, 𝑗∈𝑁. Hence 𝑉𝜃^∗_𝑗=Λ^∗_𝑗, which implies that 𝜃_𝑗𝑉^∗=Λ_𝑗, 𝑗∈𝑁. Suppose 𝑉₁,𝑉₂∈𝐵(𝐻) and 𝜃_𝑖𝑉^∗₁=𝜃_𝑖𝑉^∗₂=Λ_𝑖 for any 𝑖∈𝑁. Then for any 𝑓∈𝐻, 𝑔_𝑖∈𝐻_𝑖, we have ⟨𝜃_𝑖𝑉^∗₁𝑓,𝑔_𝑖⟩=⟨𝜃_𝑖𝑉^∗₂𝑓,𝑔_𝑖⟩, that is, ⟨𝑉^∗₁𝑓,𝜃^∗_𝑖𝑔_𝑖⟩=⟨𝑉^∗₂𝑓,𝜃^∗_𝑖𝑔_𝑖⟩ Since span{𝜃^∗_𝑖(𝐻_𝑖)}_𝑖∈𝑁=𝐻 by Lemma 3, 𝑉^∗₁𝑓=𝑉^∗₂𝑓, which means that 𝑉₁=𝑉₂. Thus the operator 𝑉 is unique.
⇐∶ Since Λ_𝑖=𝜃_𝑖𝑉^∗, for all 𝑖∈𝑁, for any 𝑓∈𝐻, we have _𝑖∈𝑁Λ_𝑖𝑓²=_𝑖∈𝑁𝜃_𝑖𝑉^∗𝑓²=𝑉^∗𝑓²≤𝑉^∗2𝑓². (10) So {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence with respect to {𝐻_𝑖∶𝑖∈𝑁}.

Definition 6. Let {𝜃_𝑛∶𝑛∈𝑁} be a 𝑔-orthonormal basis for 𝐻, {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, and the 𝑔-preframe operator associated with {Λ_𝑖∶𝑖∈𝑁} is the operator 𝑉∈𝐵(𝐻) such that 𝜃_𝑛𝑉^∗=Λ_𝑛 for any 𝑛∈𝑁. The 𝑔-frame operator 𝑆 associated with {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is defined by 𝑆𝑓=∑_𝑖∈𝑁Λ^∗_𝑖Λ_𝑖𝑓.

Remark 7. It is well known that if {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-frame, then 𝑆 is a well-defined positive invertible operator on 𝐻 and {Λ_𝑖=Λ_𝑖𝑆⁻¹∶𝑖∈𝑁} is the canonical dual 𝑔-frame of {Λ_𝑖∶𝑖∈𝑁}; for details see [9]. In this case, for any 𝑓∈𝐻, we have the Fourier series type decomposition: 𝑓=∑_𝑖∈𝑁Λ^∗_𝑖Λ_𝑖𝑓=∑_𝑖∈𝑁Λ^∗_𝑖Λ_𝑖𝑓. In general, there are many 𝑔-frames {Γ_𝑖∶𝑖∈𝑁} for which the formula 𝑓=∑_𝑖∈𝑁Λ^∗_𝑖Γ_𝑖𝑓=∑_𝑖∈𝑁Γ^∗_𝑖Λ^∗_𝑖𝑓 holds. All such 𝑔-frames are called alternate dual 𝑔-frames of {Λ_𝑖∶𝑖∈𝑁}.

Lemma 8. Suppose {𝜃_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis for 𝐻, {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence for 𝐻, and 𝑉 and 𝑆 are the 𝑔-preframe operator and 𝑔-frame operator associated with {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁}, respectively. Then 𝑆=𝑉𝑉^∗.

Proof. Since Λ_𝑖=𝜃_𝑖𝑉^∗ for all 𝑖∈𝑁, for any 𝑓∈𝐻 we have 𝑆𝑓=_𝑖∈𝑁Λ^∗_𝑖Λ_𝑖𝑓=_𝑖∈𝑁𝑉𝜃^∗_𝑖𝜃_𝑖𝑉^∗𝑓=𝑉_𝑖∈𝑁𝜃^∗_𝑖𝜃_𝑖𝑉^∗𝑓=𝑉𝑉^∗𝑓. (11) So 𝑆=𝑉𝑉^∗.

Lemma 9 (see, [11]). A sequence {𝜃_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-frame for 𝐻 if and only if 𝑆∶𝑓⟶_𝑖∈𝑁Λ^∗_𝑖Λ_𝑖𝑓 (12) is a well-defined and bounded mapping from 𝐻 onto 𝐻.

Remark 10. Since 𝑆 is adjoint, the above condition can be replaced with 𝑆 is a well-defined and bounded invertible mapping.

Lemma 11 (see, [19]). 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. (13)

Theorem 12. Suppose {𝜃_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis for 𝐻, {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence for 𝐻, and 𝑉 and 𝑆 are the 𝑔-preframe operator and 𝑔-frame operator associated with {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁}, respectively. Then(1){Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-frame if and only if 𝑉 is onto.(2){Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a normalized tight 𝑔-frame if and only if 𝑉 is a coisometry.(3){Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Riesz basis if and only if 𝑉 is invertible.(4){Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis if and only if 𝑉 is unitary.

Proof. (1) ⇒∶ By Lemma 9, if {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-frame, then 𝑆 is invertible, but 𝑆=𝑉𝑉^∗ by Lemma 8, so 𝑉 is onto. ⇐∶ If 𝑉 is onto, then {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Bessel sequence by Lemma 5. So we just need to show the existence of lower frame bound. Since 𝑉 is onto, 𝑉𝑉^†=𝐼 by Lemma 11. Hence (𝑉^†)^∗𝑉^∗=𝐼, so for any 𝑓∈𝐻, ‖𝑓‖²=‖(𝑉^†)^∗𝑉^∗𝑓‖²≤‖(𝑉^†)^∗‖²‖𝑉^∗𝑓‖². So we have _𝑖∈𝑁Λ_𝑖𝑓²=_𝑖∈𝑁𝜃_𝑖𝑉^∗𝑓²=𝑉^∗𝑓²≥1(𝑉^†)^∗2𝑓². (14) So {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-frame.
(2) Since {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a normalized tight 𝑔-frame if and only if 𝑆=𝐼, but 𝑆=𝑉𝑉^∗ by Lemma 8, so 𝑆=𝐼 if and only 𝑉 is coisometry.
(3) If {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Riesz basis then 𝑉 is invertible by [11, Corollary 2.18]. Conversely, suppose 𝑉 is invertible. If for any 𝑖∈𝑁, Λ_𝑖𝑓=𝜃_𝑖𝑉^∗𝑓=0, since {𝜃_𝑖∶𝑖∈𝑁} is 𝑔-complete, then 𝑉^∗𝑓=0; hence 𝑓=0. It follows that {Λ_𝑖∶𝑖∈𝑁} is 𝑔-complete. Now for any finite set 𝐽⊂𝑁, and 𝑔_𝑗∈𝐻_𝑗,𝑗∈𝐽, we have _𝑗∈𝐽Λ^∗_𝑗𝑔_𝑗²=_𝑗∈𝐽𝑉𝜃^∗_𝑗𝑔_𝑗²=𝑉_𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗². (15) Since _𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗²=𝑉𝑉⁻¹_𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗²≤𝑉⁻¹²𝑉_𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗²,𝑉_𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗²≤‖𝑉‖²_𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗²,_𝑗∈𝐽𝜃^∗_𝑗𝑔_𝑗²=_𝑗∈𝐽𝑔_𝑗². (16) So we have 1𝑉⁻¹²_𝑗∈𝐽𝑔_𝑗²≤_𝑗∈𝐽Λ^∗_𝑗𝑔_𝑗²≤‖𝑉‖²_𝑗∈𝐽𝑔_𝑗². (17) Hence {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-Riesz basis.
(4) If {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis, then 𝑉 is unitary by [11, Corollary 2.19]. Conversely, suppose that 𝑉 is unitary. Then for 𝑖,𝑗∈𝑁, and 𝑔_𝑖∈𝐻_𝑖, 𝑔_𝑗∈𝐻_𝑗 we have Λ^∗_𝑖𝑔_𝑖,Λ^∗_𝑗𝑔_𝑗=𝑉𝜃^∗_𝑖𝑔_𝑖,𝑉𝜃^∗_𝑗𝑔_𝑗=𝜃^∗_𝑖𝑔_𝑖,𝜃^∗_𝑗𝑔_𝑗=𝛿_𝑖𝑗𝑔_𝑖,𝑔_𝑗,_𝑖∈𝑁Λ^∗_𝑖𝑓²=_𝑖∈𝑁𝜃_𝑖𝑉^∗𝑓²=𝑉^∗𝑓²=𝑓². (18) So {Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis.

Lemma 13. (1) Suppose that 𝑇,𝑉∈𝐵(𝐻) and 𝑉 is onto. Then 𝑇𝑉 is onto if and only if 𝑇 is onto.
(2) Suppose that 𝑇,𝑉∈𝐵(𝐻) and 𝑉 is a coisometry. Then 𝑇𝑉 is a coisometry if and only if 𝑇 is a coisometry.
(3) Suppose that 𝑇,𝑉∈𝐵(𝐻) and 𝑉 is invertible. Then 𝑇𝑉 is invertible if and only if 𝑇 is invertible.
(4) Suppose that 𝑇,𝑉∈𝐵(𝐻) and 𝑉 is a unitary. Then 𝑇𝑉 is unitary if and only if 𝑇 is unitary.

Proof. (1) If 𝑇𝑉 is onto, then for any 𝑦∈𝐻, there exists 𝑥∈𝐻 such that 𝑦=𝑇𝑉𝑥=𝑇(𝑉𝑥), so 𝑇 is onto. Conversely, if 𝑇 is onto, then for any 𝑦∈𝐻 there exists 𝑥∈𝐻 such that 𝑦=𝑇𝑥. Since 𝑉 is onto, there exists 𝑧∈𝐻 such that 𝑥=𝑉𝑧. So 𝑦=𝑇𝑉𝑧, which implies that 𝑇𝑉 is onto.
(2) If 𝑇𝑉 is a coisometry, then (𝑇𝑉)(𝑇𝑉)^∗=𝐼. Since 𝑉 is a coisometry, 𝑉𝑉^∗=𝐼. So 𝑇𝑇^∗=𝑇(𝑉𝑉^∗)𝑇^∗=(𝑇𝑉)(𝑇𝑉)^∗=𝐼, which implies that 𝑇 is a coisometry. Conversely, if 𝑇 is a coisometry, then (𝑇𝑉)(𝑇𝑉)^∗=𝑇𝑉𝑉^∗𝑇^∗=𝑇(𝑉𝑉^∗)𝑇^∗=𝑇𝑇^∗=𝐼. Hence 𝑇𝑉 is a coisometry.
(3) If 𝑇𝑉 is invertible, then 𝑇=(𝑇𝑉)𝑉⁻¹, so 𝑇 is invertible. Conversely, if 𝑇 is invertible, since 𝑉 is invertible, then 𝑇𝑉 is invertible.
(4) If 𝑇𝑉 is unitary, then 𝑇=𝑇𝑉𝑉^∗=(𝑇𝑉)𝑉^∗, so 𝑇 is unitary. Conversely, if 𝑇 is unitary, since 𝑉 is unitary, so 𝑇𝑉 is unitary.

By Theorem 12 and Lemma 13, it is easy to get the following results. We leave the details to the readers.

Theorem 14. Suppose that {𝜃_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁} is a 𝑔-orthonormal basis for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, 𝑇∈𝐵(𝐻). Define the transformation Φ_𝑇∶{Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁}→{Λ_𝑖𝑇^∗∶𝑖∈𝑁}, then:(1)It transforms 𝑔-frames to 𝑔-frames if and only if 𝑇 is onto.(2)Ittransforms normalized tight 𝑔-frames to normalized tight 𝑔-frames if and only if 𝑇 is a coisometry.(3)Ittransforms 𝑔-Riesz bases to 𝑔-Riesz bases if and only if 𝑇 is invertible.(4)Ittransforms 𝑔-orthonormal bases to 𝑔-orthonormal bases if and only if 𝑇 is unitary.

Remark 15. The above results are established under the constraint that 𝐻 has a 𝑔-orthonormal basis with respect to {𝐻_𝑖∶𝑖∈𝑁}. But in general, the statement (1) in the above Theorem is still true.

Theorem 16. Given 𝑇∈𝐵(𝐻). The transformation Φ_𝑇∶{Λ_𝑖∈𝐵(𝐻,𝐻_𝑖)∶𝑖∈𝑁}→{Λ_𝑖𝑇^∗∶𝑖∈𝑁}, transforms 𝑔-frames to 𝑔-frames if and only if 𝑇 is onto.

Proof. ⇒∶ If Φ_𝑇 transforms 𝑔-frames to 𝑔-frames, then {Λ_𝑖𝑇^∗∶𝑖∈𝑁} is a 𝑔-frame whenever {Λ_𝑖∶𝑖∈𝑁} is a 𝑔-frame. Suppose that 𝐴₁,𝐵₁ and 𝐴₂,𝐵₂ are the lower and upper frame bounds of {Λ_𝑖𝑇^∗∶𝑖∈𝑁} and {Λ_𝑖∶𝑖∈𝑁} respectively. Then for any 𝑓∈𝐻, we have 𝐴₁𝑓²≤_𝑖∈𝑁Λ_𝑖𝑇^∗𝑓²≤𝐵₁𝑓². (19) So if 𝑇^∗𝑓=0, then 𝑓=0, which implies that 𝑇^∗ is one to one. If 𝑇^∗𝑓_𝑛→𝑦(𝑛→∞), then {𝑓_𝑛} is convergent; that is, there exists 𝑦₀∈𝐻 such that 𝑓_𝑛→𝑦₀. Hence 𝑇^∗𝑓_𝑛→𝑇^∗𝑦₀, so 𝑦=𝑇^∗𝑦₀, that is, 𝑦∈range(𝑇^∗), which implies that 𝑇^∗ has closed range. So 𝑇 has closed range. Since range(𝑇)=range(𝑇)=(ker𝑇^∗)^⟂=𝐻, 𝑇 is onto.
⇐∶ If 𝑇 is onto and {Λ_𝑖∶𝑖∈𝑁} is a 𝑔-frame for 𝐻 with frame bounds 𝐴 and 𝐵, then for any 𝑓∈𝐻, we have _𝑖∈𝑁Λ_𝑖𝑇^∗𝑓²≤𝐵𝑇^∗𝑓²≤𝐵𝑇^∗2𝑓² (20) and since 𝑇𝑇^†=𝐼 by Lemma 11,(𝑇^†)^∗𝑇^∗=𝐼. So _𝑖∈𝑁Λ_𝑖𝑇^∗𝑓²≥𝐴𝑇^∗𝑓²≥𝐴(𝑇^†)^∗2𝑓². (21) So {Λ_𝑖𝑇^∗∶𝑖∈𝑁} is a 𝑔-frame for 𝐻, which implies that Φ_𝑇 transforms 𝑔-frames to 𝑔-frames.

3. Sums of 𝑔-Frames

In this section, we will consider the direct sums and usual sums of 𝑔-frames, which generate 𝑔-frames for direct sum of Hilbert spaces and sum of Hilbert spaces, respectively.

Definition 17. Suppose 𝑇,𝐿∈𝐵(𝐻,𝐾), then 𝑇⊕𝐿∈𝐵(𝐻⊕𝐻,𝐾) is defined by (𝑇⊕𝐿)(𝑥⊕𝑦)=𝑇𝑥+𝐿𝑦.

Remark 18. Since for any scalars 𝑎 and 𝑏, any 𝑥,𝑦∈𝐻, we have (𝑇⊕𝐿)𝑎𝑥₁⊕𝑦₁+𝑏𝑥₂⊕𝑦₂=(𝑇⊕𝐿)𝑎𝑥₁+𝑏𝑥₂⊕𝑎𝑦₁+𝑏𝑦₂=𝑇𝑎𝑥₁+𝑏𝑥₂+𝐿𝑎𝑦₁+𝑏𝑦₂=𝑎𝑇𝑥₁+𝑏𝑇𝑥₂+𝑎𝐿𝑦₁+𝑏𝐿𝑦₂=𝑎𝑇𝑥₁+𝐿𝑦₁+𝑏𝑇𝑥₂+𝐿𝑦₂=𝑎(𝑇⊕𝐿)𝑥₁⊕𝑦₁+𝑏(𝑇⊕𝐿)𝑥₂⊕𝑦₂. (22) So 𝑇⊕𝐿 is linear. Since for any 𝑥,𝑦∈𝐻, we have )=≤‖‖+≤‖𝑇𝑥‖+‖𝐿𝑦𝑥‖+𝑦√(𝑇⊕𝐿𝑥⊕𝑦𝑇𝑥+𝐿𝑦𝑇𝑥𝐿𝑦‖‖‖≤𝑀‖≤𝑀2‖𝑥‖²+𝑦²=√2𝑀𝑥⊕𝑦, (23) where 𝑀=max{‖𝑇‖,‖𝐿‖}. So 𝑇⊕𝐿 is bounded. It is easy to see that (𝑇₁⊕𝑇₂)^∗𝑧=𝑇^∗₁𝑧⊕𝑇^∗₂𝑧 for any 𝑧∈𝐻_𝑖. In fact, for any 𝑥,𝑦∈𝐻, we have 𝑇₁⊕𝑇₂𝑥⊕𝑦,𝑧=𝑥⊕𝑦,𝑇₁⊕𝑇₂^∗𝑧,𝑇₁⊕𝑇₂𝑥⊕𝑦,𝑧=𝑇₁𝑥+𝑇₂=𝑇𝑦,𝑧₁𝑥,𝑧+𝑇₂=𝑦,𝑧𝑥,𝑇^∗₁𝑧+𝑦,𝑇^∗₂𝑧=𝑥⊕𝑦,𝑇^∗₁𝑧⊕𝑇^∗₂𝑧. (24) So (𝑇₁⊕𝑇₂)^∗𝑧=𝑇^∗₁𝑧⊕𝑇^∗₂𝑧. The finite direct sum of operators have the same meaning.

Definition 19. Let {Λ_𝑗∶𝑗∈𝑁}, {Γ_𝑗∶𝑗∈𝑁} be normalized tight 𝑔-frames for Hilbert space 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, if {Λ_𝑗⊕Γ_𝑗∶𝑗∈𝑁} is a normalized tight 𝑔-frame for 𝐻⊕𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, then we call them strongly disjoint. More generally, if {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are 𝑔-frames with respect to {𝐻_𝑖∶𝑖∈𝑁} such that {Λ_𝑗⊕Γ_𝑗∶𝑗∈𝑁} is a 𝑔-frame for 𝐻⊕𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, then they are called disjoint.
In general, let {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} be normalized tight 𝑔-frames for Hilbert space 𝐻, if {Λ_1𝑗⊕⋯⊕Λ_𝑘𝑗∶𝑗∈𝑁} is a normalized 𝑔-frame for 𝐻⊕𝐻⊕⋯⊕𝐻, which is the inner direct sum of 𝑘 copies of 𝐻, then we call them strongly disjoint. If the 𝑘 number of sequences are 𝑔-frames and their inner direct sum is a 𝑔-frame for the inner direct sum of 𝑘 copies of 𝐻, then we call them disjoint.

Definition 20. Let {Λ_𝑗∶𝑗∈𝑁}, {Γ_𝑗∶𝑗∈𝑁} be normalized tight 𝑔-frames for Hilbert space 𝐻, if {Λ_𝑗⊕Γ_𝑗∶𝑗∈𝑁} is a 𝑔-orthonormal basis of 𝐻⊕𝐻, then we call them strongly complementary.
In general, let {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} be normalized tight 𝑔-frames for Hilbert space 𝐻, if {Λ_1𝑗⊕⋯⊕Λ_𝑘𝑗∶𝑗∈𝑁} is a 𝑔-orthonormal basis for the inner direct sum of 𝑘 copies of 𝐻, then we call them strongly complementary.
In [8], the authors characterize sums of frames by the topological and algebraic properties of 𝑃(𝐻), the set of orthogonal projections on 𝐻. In this section, we always assume that there exists a 𝑔-orthonormal basis {𝜃_𝑖∶𝑖∈𝑁} for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁} and we will characterize all kinds of sums of 𝑔-frames through the properties of 𝑔-preframe operators.

Theorem 21. Suppose {Λ_𝑗∶𝑗∈𝑁}, {Γ_𝑗∶𝑗∈𝑁} are normalized tight 𝑔-frames for Hilbert space 𝐻, 𝑇₁ and 𝑇₂ are the 𝑔-preframe operators associated with {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁}, respectively; then {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are strongly disjoint if and only if 𝑇₂𝑇^∗₁=0.

Proof. ⇒∶ Since {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are strongly disjoint, so {Λ_𝑗⊕Γ_𝑗∶𝑗∈𝑁} is a normalized tight 𝑔-frame for 𝐻⊕𝐻. Since 𝑇₁, 𝑇₂ are 𝑔-preframe operators associated with {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁}, respectively, so 𝑇₁ and 𝑇₂ are coisometries by Theorem 12 and Λ_𝑛=𝜃_𝑛𝑇^∗₁, Γ_𝑛=𝜃_𝑛𝑇^∗₂ for any 𝑛∈𝑁. So for any 𝑥,𝑦∈𝐻, we have ‖𝑥‖²+𝑦²=𝑥⊕𝑦²=_𝑖∈𝑁Λ_𝑖⊕Γ_𝑖𝑥⊕𝑦²=_𝑖∈𝑁Λ_𝑖𝑥+Γ_𝑖𝑦²=_𝑖∈𝑁𝜃_𝑖𝑇^∗₁𝑥+𝜃_𝑖𝑇^∗₂𝑦²=_𝑖∈𝑁𝜃_𝑖𝑇^∗₁𝑥+𝑇^∗₂𝑦²=𝑇^∗₁𝑥+𝑇^∗₂𝑦²=‖𝑥‖²+𝑦²+2Re𝑇₂𝑇^∗₁𝑥,𝑦. (25) So Re⟨𝑇₂𝑇^∗₁𝑥,𝑦⟩=0. Replace 𝑦 by 𝑖𝑦 in the above argument; then we get Im⟨𝑇₂𝑇^∗₁𝑥,𝑦⟩=0. Hence ⟨𝑇₂𝑇^∗₁𝑥,𝑦⟩=0 for any 𝑥,𝑦∈𝐻. Therefore 𝑇₂𝑇^∗₁=0.
⇐∶ Suppose 𝑇₂𝑇^∗₁=0; then for any 𝑥,𝑦∈𝐻, we have _𝑖∈𝑁Λ_𝑖⊕Γ_𝑖𝑥⊕𝑦²=_𝑖∈𝑁Λ_𝑖𝑥+Γ_𝑖𝑦²=_𝑖∈𝑁𝜃_𝑖𝑇^∗₁𝑥+𝜃_𝑖𝑇^∗₂𝑦²=_𝑖∈𝑁𝜃_𝑖𝑇^∗₁𝑥+𝑇^∗₂𝑦²=𝑇^∗₁𝑥+𝑇^∗₂𝑦²=𝑇^∗₁𝑥²+𝑇^∗₂𝑦²+2Re𝑇^∗₁𝑥,𝑇^∗₂𝑦=‖𝑥‖²+𝑦²+2Re𝑇₂𝑇^∗₁=‖𝑥‖𝑥,𝑦₂+𝑦²=𝑥⊕𝑦². (26) So {Λ_𝑗⊕Γ_𝑗∶𝑗∈𝑁} is a normalized tight 𝑔-frame for 𝐻⊕𝐻; that is, {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are strongly disjoint.

It is easy to see that for any finite number of normalized tight 𝑔-frames, we have the following result.

Corollary 22. Suppose {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are normalized tight 𝑔-frames for Hilbert space 𝐻, 𝑇₁,…,𝑇_𝑘 are 𝑔-preframe operators associated with {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁}, respectively; then {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are strongly disjoint if and only if 𝑇_𝑚𝑇^∗_𝑛=0, 𝑚,𝑛=1,…,𝑘.

Theorem 23. Suppose {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are 𝑔-frames for Hilbert space 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝑁}, 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁}, respectively. If 𝑇₂𝑇^∗₁=0, then {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are disjoint.

Proof. Since 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with 𝑔-frames {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁}, respectively, so 𝑇₁ and 𝑇₂ are onto operators by Theorem 12. So there exist 𝑇^†₁ and 𝑇^†₂ such that 𝑇₁𝑇^†₁=𝐼 and 𝑇₂𝑇^†₂=𝐼 by Lemma 11. So (𝑇^†₁)^∗𝑇^∗₁=𝐼 and (𝑇^†₂)^∗𝑇^∗₂=𝐼. Hence ‖𝑥‖=‖(𝑇^†₁)^∗𝑇^∗₁𝑥‖≤‖(𝑇^†₁)^∗‖‖𝑇^∗₁𝑥‖, so ‖𝑇^∗₁𝑥‖≥(1/‖(𝑇^†₁)^∗‖)‖𝑥‖. Similarly, ‖𝑇^∗₂𝑥‖≥(1/‖(𝑇^†₂)^∗‖)‖𝑥‖. Since 𝑇₂𝑇^∗₁=0, we have _𝑖∈𝑁Λ_𝑖⊕Γ_𝑖𝑥⊕𝑦²=_𝑖∈𝑁Λ_𝑖𝑥+Γ_𝑖𝑦²=_𝑖∈𝑁𝜃_𝑖𝑇^∗₁𝑥+𝜃_𝑖𝑇^∗₂𝑦²=_𝑖∈𝑁𝜃_𝑖𝑇^∗₁𝑥+𝑇^∗₂𝑦²=𝑇^∗₁𝑥+𝑇^∗₂𝑦²=𝑇^∗₁𝑥²+𝑇^∗₂𝑦²+2Re𝑇^∗₁𝑥,𝑇^∗₂𝑦=𝑇^∗₁𝑥²+𝑇^∗₂𝑦²+2Re𝑇₂𝑇^∗₁=𝑇𝑥,𝑦_∗1𝑥²+𝑇^∗₂𝑦². (27) Since 𝑇^∗₁𝑥²+𝑇^∗₂𝑦²≤𝑇^∗₁²⋅‖𝑥‖²+𝑇^∗₂²⋅‖𝑥‖²≤𝐵⋅‖𝑥‖²+𝑦²=𝐵⋅𝑥⊕𝑦², (28) where 𝐵=max{‖𝑇^∗₁‖²,‖𝑇^∗₂‖²}. Hence ∑_𝑖∈𝑁‖(Λ_𝑖⊕Γ_𝑖)(𝑥⊕𝑦)‖²≤𝐵⋅‖𝑥⊕𝑦‖². Since 𝑇^∗₁𝑥²+𝑇^∗₂𝑦²≥1𝑇^†₁^∗2‖𝑥‖²+1𝑇^†₂^∗2𝑦²≥𝐴⋅‖𝑥‖²+𝑦²=𝐴⋅𝑥⊕𝑦², (29) where 𝐴=min{1/‖(𝑇^†₁)^∗‖²,1/‖(𝑇^†₂)^∗‖²}. Hence _𝑖∈𝑁Λ_𝑖⊕Γ_𝑖𝑥⊕𝑦²≥𝐴⋅𝑥⊕𝑦². (30) So {Λ_𝑛⊕Γ_𝑛∶𝑛∈𝑁} is a 𝑔-frame for 𝐻⊕𝐻; that is, {Λ_𝑗∶𝑗∈𝑁} and {Γ_𝑗∶𝑗∈𝑁} are disjoint.

Similarly, for any finite number of 𝑔-frames, we have the following.

Corollary 24. Suppose {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are 𝑔-frames for Hilbert space 𝐻, 𝑇₁,…,𝑇_𝑘 are 𝑔-preframe operators of {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁}, respectively; If 𝑇_𝑚𝑇^∗_𝑛=0,𝑚,𝑛=1,…,𝑘, then {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are disjoint.

Theorem 25. Let {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} be normalized tight 𝑔-frames for 𝐻, 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively. If {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are strongly complementary, then 𝑇₂𝑇^∗₁=0 and 𝑇^∗₁𝑇₁+𝑇^∗₂𝑇₂=𝐼.

Proof. If {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are strongly complementary, then {Λ_𝑛⊕Γ_𝑛∶𝑛∈𝑁} is a 𝑔-orthonormal basis for 𝐻⊕𝐻. So they are of course strongly disjoint, therefore 𝑇₂𝑇^∗₁=0 by Theorem 21. Since {𝜃_𝑖∶𝑖∈𝑁} is a 𝑔-orthonormal basis for 𝐻, for any 𝑛,𝑘∈𝑁 and 𝑔_𝑛∈𝐻_𝑛, 𝑔_𝑘∈𝐻_𝑘, we have Λ_𝑛⊕Γ_𝑛^∗𝑔_𝑛,Λ_𝑘⊕Γ_𝑘^∗𝑔_𝑘=𝛿_𝑛𝑘𝑔_𝑛,𝑔_𝑘=𝜃^∗_𝑛𝑔_𝑛,𝜃^∗_𝑘𝑔_𝑘. (31) But Λ_𝑛⊕Γ_𝑛^∗𝑔_𝑛,Λ_𝑘⊕Γ_𝑘^∗𝑔_𝑘=Λ^∗_𝑛𝑔_𝑛⊕Γ^∗_𝑛𝑔_𝑛,Λ^∗_𝑘𝑔_𝑘⊕Γ^∗_𝑘𝑔_𝑘=𝑇₁𝜃^∗_𝑛𝑔_𝑛⊕𝑇₂𝜃^∗_𝑛𝑔_𝑛,𝑇₁𝜃^∗_𝑘𝑔_𝑘⊕𝑇₂𝜃^∗_𝑘𝑔_𝑘=𝑇₁𝜃^∗_𝑛𝑔_𝑛,𝑇₁𝜃^∗_𝑘𝑔_𝑘+𝑇₂𝜃^∗_𝑛𝑔_𝑛,𝑇₂𝜃^∗_𝑘𝑔_𝑘=𝑇^∗₁𝑇₁𝜃^∗_𝑛𝑔_𝑛,𝜃^∗_𝑘𝑔_𝑘+𝑇^∗₂𝑇₂𝜃^∗_𝑛𝑔_𝑛,𝜃^∗_𝑘𝑔_𝑘=𝑇^∗₁𝑇₁+𝑇^∗₂𝑇₂𝜃^∗_𝑛𝑔_𝑛,𝜃^∗_𝑘𝑔_𝑘. (32) Since span{𝜃^∗_𝑖𝑔_𝑖∶𝑖∈𝑁}=𝐻, we have that 𝑇^∗₁𝑇₁+𝑇^∗₂𝑇₂=𝐼.

Similarly, for any finite number of normalized tight 𝑔-frames, we have

Corollary 26. Suppose {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are normalized tight 𝑔-frames for Hilbert space 𝐻, 𝑇₁,…,𝑇_𝑘 are 𝑔-preframe operators associated with {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁}, respectively, if {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are strongly complementary, then 𝑇_𝑚𝑇^∗_𝑛=0,𝑚,𝑛=1,…,𝑘, and ∑^𝑘_𝑖=1𝑇^∗_𝑖𝑇_𝑖=𝐼.

The above theorems are related with the inner direct sums of 𝑔-frames or normalized tight 𝑔-frames. In the following, we will consider the usual sums of 𝑔-frames or normalized tight 𝑔-frames.

Theorem 27. Suppose that {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are 𝑔-frames for Hilbert space 𝐻, 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively. If 𝑇₂𝑇^∗₁=0, then {Λ_𝑛+Γ_𝑛∶𝑛∈𝑁} is a 𝑔-frame for 𝐻. Moreover, if {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are normalized tight 𝑔-frames and 𝑇₂𝑇^∗₁=0, then {Λ_𝑛+Γ_𝑛∶𝑛∈𝑁} is a tight 𝑔-frame for 𝐻 with bounds 2.

Proof. Since 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with 𝑔-frames {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively, so Λ_𝑛=𝜃_𝑛𝑇^∗₁ and Γ_𝑛=𝜃_𝑛𝑇^∗₂ for any 𝑛∈𝑁, where {𝜃_𝑛∶𝑛∈𝑁} is a 𝑔-orthonormal basis for 𝐻. Hence Λ_𝑛+Γ_𝑛=𝜃_𝑛(𝑇₁+𝑇₂)^∗ for any 𝑛∈𝑁. To show {Λ_𝑛+Γ_𝑛∶𝑛∈𝑁} is a 𝑔-frame, it is sufficient to show that 𝑇₁+𝑇₂ is onto by Theorem 12. Since 𝑇₂𝑇^∗₁=0, so 𝑇₁+𝑇₂𝑇^∗₁=𝑇₁𝑇^∗₁+𝑇₂𝑇^∗₁=𝑇₁𝑇^∗₁. (33) Since 𝑇₁𝑇^∗₁ is invertible, so for any 𝑧∈𝐻, there exists 𝑥=𝑇^∗₁(𝑇₁𝑇^∗₁)⁻¹𝑧∈𝐻 such that 𝑇₁+𝑇₂𝑥=𝑇₁+𝑇₂𝑇^∗₁𝑇₁𝑇^∗₁⁻¹𝑧=𝑇₁𝑇^∗₁𝑇₁𝑇^∗₁⁻¹𝑧=𝑧. (34) So 𝑇₁+𝑇₂ is onto.
If {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are normalized tight 𝑔-frames and 𝑇₂𝑇^∗₁=0, then {Λ_𝑛+Γ_𝑛∶𝑛∈𝑁} is a 𝑔-frame and for any 𝑥∈𝐻, we have _𝑛∈𝑁Λ_𝑛+Γ_𝑛𝑥²=_𝑛∈𝑁Λ_𝑛𝑥²+_𝑛∈𝑁Γ_𝑛𝑥²+_𝑛∈𝑁Λ_𝑛𝑥,Γ_𝑛𝑥+_𝑛∈𝑁Γ_𝑛𝑥,Λ_𝑛𝑥=2‖𝑥‖²+_𝑛∈𝑁𝜃_𝑛𝑇^∗₁𝑥,𝜃_𝑛𝑇^∗₂𝑥+_𝑛∈𝑁𝜃_𝑛𝑇^∗₂𝑥,𝜃_𝑛𝑇^∗₁𝑥=2‖𝑥‖²+_𝑛∈𝑁𝑇₂𝜃^∗_𝑛𝜃_𝑛𝑇^∗₁𝑥,𝑥+_𝑛∈𝑁𝑇₁𝜃^∗_𝑛𝜃_𝑛𝑇^∗₂‖𝑥‖𝑥,𝑥=2²+𝑇_2𝑛∈𝑁𝜃^∗_𝑛𝜃_𝑛𝑇^∗₁+𝑇𝑥,𝑥_1𝑛∈𝑁𝜃^∗_𝑛𝜃_𝑛𝑇^∗₂‖𝑥‖𝑥,𝑥=2²+𝑇₂𝑇^∗₁𝑥,𝑥+𝑇₁𝑇^∗₂‖𝑥‖𝑥,𝑥=2². (35) So {Λ_𝑛+Γ_𝑛∶𝑛∈𝑁} is a tight 𝑔-frame with bound 2.

Similarly, for any finite number of 𝑔-frames, we have the following.

Corollary 28. Suppose {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are 𝑔-frames for Hilbert space 𝐻, 𝑇₁,⋯,𝑇_𝑘 are 𝑔-preframe operators associated with {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁}, respectively. If 𝑇_𝑚𝑇^∗_𝑛=0,𝑚,𝑛=1,…,𝑘, then {Λ_1𝑗+⋯+Λ_𝑘𝑗∶𝑗∈𝑁} is a 𝑔-frame for 𝐻. Moreover, if {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are normalized tight 𝑔-frames for Hilbert space 𝐻 and 𝑇_𝑚𝑇^∗_𝑛=0,𝑚,𝑛=1,…,𝑘, then {Λ_1𝑗+⋯+Λ_𝑘𝑗∶𝑗∈𝑁} is a tight 𝑔-frame for 𝐻 with bound 𝑘.

The above theorem can be generalized further as follows.

Theorem 29. Suppose that {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are 𝑔-frames for Hilbert space 𝐻, 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively, such that 𝑇₂𝑇^∗₁=0 and 𝐴,𝐵∈𝐵(𝐻). If either 𝐴 or 𝐵 is onto, then {Λ_𝑛𝐴^∗+Γ_𝑛𝐵^∗∶𝑛∈𝑁} is a 𝑔-frame for 𝐻.

Proof. Since 𝑇₁ and 𝑇₂ are 𝑔-preframe operators associated with {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively, Λ_𝑛=𝜃_𝑛𝑇^∗₁ and Γ_𝑛=𝜃_𝑛𝑇^∗₂. So Λ_𝑛𝐴^∗+Γ_𝑛𝐵^∗=𝜃_𝑛𝑇^∗₁𝐴^∗+𝜃_𝑛𝑇^∗₂𝐵^∗=𝜃_𝑛𝑇^∗₁𝐴^∗+𝑇^∗₂𝐵^∗=𝜃_𝑛𝐴𝑇₁+𝐵𝑇₂^∗. (36) To show {Λ_𝑛𝐴^∗+Γ_𝑛𝐵^∗∶𝑛∈𝑁} is a 𝑔-frame it is sufficient to show that 𝐴𝑇₁+𝐵𝑇₂ is onto by Theorem 12. It is well known that 𝑇₁𝑇^∗₁ is invertible. If 𝐴 is onto, then for any 𝑧∈𝐻, there exists 𝑥∈𝐻 such that 𝐴𝑥=𝑧. So for any 𝑧∈𝐻, there exists 𝑓=𝑇^∗₁(𝑇₁𝑇^∗₁)⁻¹𝑥∈𝐻 such that 𝐴𝑇₁+𝐵𝑇₂𝑓=𝐴𝑇₁+𝐵𝑇₂𝑇^∗₁𝑇₁𝑇^∗₁⁻¹𝑥=𝐴𝑇₁𝑇^∗₁𝑇₁𝑇^∗₁⁻¹𝑥=𝐴𝑥=𝑧. (37) It follows that 𝐴𝑇₁+𝐵𝑇₂ is onto. In the case that 𝐵 is onto, the result can be proved similarly.

Similarly, for any finite number of 𝑔-frames, we have the following.

Corollary 30. Suppose {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁} are 𝑔-frames for Hilbert space 𝐻, and 𝑇₁,…,𝑇_𝑘 are 𝑔-preframe operators associated with {Λ_1𝑗∶𝑗∈𝑁},…,{Λ_𝑘𝑗∶𝑗∈𝑁}, respectively, and 𝑇_𝑛𝑇^∗_𝑚=0, 𝑚,𝑛=1,2,…,𝑘, 𝐴_𝑖∈𝐵(𝐻), 𝑖=1,…,𝑘. If there exists 𝑖∈{1,2,…,𝑘} such that 𝐴_𝑖 is surjective, then {∑^𝑘_𝑖=1Λ_𝑖𝑗𝐴^∗_𝑖} is a 𝑔-frame for 𝐻.

4. Dual 𝑔-Frames of 𝑔-Frames

In this section, we study the dual 𝑔-frames by using the properties of 𝑔-preframe operators.

Theorem 31. Suppose 𝑔-frames {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for Hilbert space 𝐻, 𝑇 is a coisometry in 𝐵(𝐻). Then {Λ_𝑛𝑇^∗} and {Γ_𝑛𝑇^∗∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for 𝐻.

Proof. Since {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are 𝑔-frames for 𝐻 and 𝑇 is a coisometry, so {Λ_𝑛𝑇^∗} and {Γ_𝑛𝑇^∗∶𝑛∈𝑁} are also 𝑔-frames for 𝐻 by Theorem 14. Since {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are alternate dual 𝑔-frame pair, we have, for any 𝑥∈𝐻𝑥=_𝑛∈𝑁Λ^∗_𝑛Γ_𝑛𝑥. (38) So 𝑇𝑥=_𝑛∈𝑁𝑇Λ^∗_𝑛Γ_𝑛𝑥=_𝑛∈𝑁Λ_𝑛𝑇^∗∗Γ_𝑛𝑥. (39) Replacing 𝑥 by 𝑇^∗𝑥 and applying 𝑇𝑇^∗=𝐼, we have 𝑥=𝑇𝑇^∗𝑥=_𝑛∈𝑁Λ_𝑛𝑇^∗∗Γ_𝑛𝑇^∗𝑥. (40) So {Λ_𝑛𝑇^∗} and {Γ_𝑛𝑇^∗∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for 𝐻.

More general, we have the following result.

Theorem 32. Suppose {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for Hilbert space 𝐻, 𝑇 is a surjective operator in 𝐵(𝐻). Then {Λ_𝑛𝑇^∗∶𝑛∈𝑁} and {Γ_𝑛𝑇^†∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for 𝐻. Similarly, {Λ_𝑛𝑇^†} and {Γ_𝑛𝑇^∗∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for 𝐻.

Proof. Since 𝑇 is onto, 𝑇𝑇^†=𝐼 by Lemma 11, so (𝑇^†)^∗𝑇^∗=𝐼. It follows that (𝑇^†)^∗ is onto. Since {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are 𝑔-frames for 𝐻, so {Λ_𝑛𝑇^∗∶𝑛∈𝑁}, {Γ_𝑛𝑇^∗∶𝑛∈𝑁}, {Λ_𝑛𝑇^†∶𝑛∈𝑁} and {Γ_𝑛𝑇^†∶𝑛∈𝑁} are 𝑔-frames for 𝐻 by Theorem 14. Since {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are alternate dual 𝑔-frame pair, we have, for any 𝑥∈𝐻𝑥=_𝑛∈𝑁Λ^∗_𝑛Γ_𝑛𝑥. (41) So 𝑇𝑥=_𝑛∈𝑁𝑇Λ^∗_𝑛Γ_𝑛𝑥=_𝑛∈𝑁Λ_𝑛𝑇^∗∗Γ_𝑛𝑥. (42) Replaced 𝑥 by 𝑇^†𝑥 and applying 𝑇𝑇^†=𝐼, we have 𝑥=𝑇𝑇^†𝑥=_𝑛∈𝑁Λ_𝑛𝑇^∗∗Γ_𝑛𝑇^†𝑥. (43) So {Λ_𝑛𝑇^∗} and {Γ_𝑛𝑇^†∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for 𝐻. Interchanging Λ_𝑛 and Γ_𝑛 for any 𝑛∈𝑁, we get that {Λ_𝑛𝑇^†} and {Γ_𝑛𝑇^∗∶𝑛∈𝑁} are alternate dual 𝑔-frame pair for 𝐻.

Theorem 33. Let {Λ_𝑛∶𝑛∈𝑁} be a 𝑔-frame for 𝐻. Let {Λ_𝑛∶𝑛∈𝑁} denote the canonical dual 𝑔-frame of {Λ_𝑛∶𝑛∈𝑁} and 𝑇_𝑥 denote the the 𝑔-preframe operator associated with {Λ_𝑛∶𝑛∈𝑁}. If 𝑃 is an orthogonal projection with 𝑃𝑇_𝑥=𝑇_𝑥𝑃, then {Λ_𝑛𝑃∶𝑛∈𝑁} is a 𝑔-frame for 𝑃𝐻 and the canonical dual 𝑔-frame for {Λ_𝑛𝑃∶𝑛∈𝑁} is {Λ_𝑛𝑃∶𝑛∈𝑁}.

Proof. It is easy to see that {Λ_𝑛𝑃∶𝑛∈𝑁} is a 𝑔-frame for 𝑃𝐻 and {𝜃_𝑛𝑃∶𝑛∈𝑁} is a 𝑔-orthonormal basis for 𝑃𝐻 with respect to {𝐻_𝑖∶𝑛∈𝑁}. We only need to prove the second part. By definition, the canonical dual 𝑔-frame for {Λ_𝑛∶𝑛∈𝑁} is {Λ_𝑛𝑆⁻¹∶𝑛∈𝑁}, where 𝑆=𝑇_𝑥𝑇^∗_𝑥. So Λ_𝑛=Λ_𝑛𝑆⁻¹. Since 𝑃𝑇_𝑥=𝑇_𝑥𝑃, so 𝑃𝑇^∗_𝑥=𝑇^∗_𝑥𝑃. Since Λ_𝑛=𝜃_𝑛𝑇^∗_𝑥, so Λ_𝑛𝑃=𝜃_𝑛𝑇^∗_𝑥𝑃=𝜃_𝑛𝑃𝑇^∗_𝑥. Hence the 𝑔-preframe operator associated with {Λ_𝑛𝑃∶𝑛∈𝑁} is 𝑇_𝑥 as well. So the canonical dual 𝑔-frame for {Λ_𝑛𝑃} is {Λ_𝑛𝑃(𝑇_𝑥𝑇^∗_𝑥)⁻¹={Λ_𝑛(𝑇_𝑥𝑇^∗_𝑥)⁻¹𝑃}={Λ_𝑛𝑃∶𝑛∈𝑁}.

The following theorem characterizes the alternate dual 𝑔-frames in terms of 𝑔-preframe operators.

Theorem 34. Let {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} be 𝑔-frames for Hilbert space 𝐻, and let 𝑇_𝑥 and 𝑇_𝑦 be the 𝑔-preframe operators associated with {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively. Then {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁} are alternate dual 𝑔-frame pair if and only if 𝑇_𝑦𝑇^∗_𝑥=𝐼 or 𝑇_𝑥𝑇^∗_𝑦=𝐼.

Proof. Since 𝑇_𝑥 and 𝑇_𝑦 are 𝑔-preframe operators associated with {Λ_𝑛∶𝑛∈𝑁} and {Γ_𝑛∶𝑛∈𝑁}, respectively, so 𝜃_𝑛𝑇^∗_𝑥=Λ_𝑛 and 𝜃_𝑛𝑇^∗_𝑦=Γ_𝑛 for any 𝑛∈𝑁. Hence for any 𝑥∈𝐻, _𝑛∈𝑁Λ^∗_𝑛Γ_𝑛𝑥=_𝑛∈𝑁𝑇_𝑥𝜃^∗_𝑛𝜃_𝑛𝑇^∗_𝑦𝑥=𝑇_𝑥𝑇^∗_𝑦𝑥. (44) Similarly, ∑_𝑛∈𝑁Γ^∗_𝑛Λ_𝑛𝑥=𝑇_𝑦𝑇^∗_𝑥𝑥. It follows that {Λ_𝑛∶𝑛∈𝑁 and {Γ_𝑛∶𝑛∈𝑁} are alternate dual 𝑔-frame pair if and only if 𝑇_𝑦𝑇^∗_𝑥=𝐼 or 𝑇_𝑥𝑇^∗_𝑦=𝐼.

Remark 35. Throughout this paper we assume that there exists 𝑔-orthonormal basis for Hilbert space 𝐻. It is well known that any Hilbert space has orthonormal basis. A natural question is that given Hilbert space 𝐻, 𝐾 and sequence of Hilbert space {𝐻_𝑖∶𝑖∈𝐽}⊂𝐾, Do the 𝑔-orthonormal bases for 𝐻 with respect to {𝐻_𝑖∶𝑖∈𝐽} always exist? if not, what is the condition such that the 𝑔-orthonormal bases exist? These interesting questions will be our next research subjects.

Acknowledgment

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