# TightK-g-Frame and Its Novel Characterizations via Atomic Systems

Authors: Yongdong Huang; Dingli Hua
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

K
-
g-frame is a generalization of g-frame. We generalize the tight g-frame to K-g-frame via atomic systems. In this paper, the definition of tight K-g-frame is put forward; equivalent characterizations and necessary conditions of tight K-g-frame are given. In particular, the necessary and sufficient condition for tight K-g-frame being tight g-frame is obtained. Finally, by means of methods and techniques of frame theory, several properties of tight K-g-frame are given.


---

## Body

## 1. Introduction

Frame in Hilbert space was first introduced in 1952 by Duffin and Schaeffer [1] to deal with nonharmonic Fourier series and reintroduced in 1986 by Daubechies et al. [2]. Since then the frame theory began to be more wildly studied. Today, frame theory has been widely used in filter theory [3], image processing [4], numerical analysis, and other areas. We refer to [5–8] for an introduction to frame theory in Hilbert space and its application.With the deepening of research on frame theory, various generalizations of frames have been proposed; see [9–12]. Atomic systems for subspaces were first introduced by Feichtinger and Werther in [13] based on examples arising in sampling theory. In 2011, Găvruţa [14] introduced K-frame in Hilbert spaces to study atomic decomposition systems and discussed some properties of them. In [15–18], some conclusions of K-frame were given. With the extensive research of K-frame and g-frame in Hilbert space, Zhu et al. [19, 20] began to study K-g-frame, which was limited to the range of a bounded linear operator in Hilbert space and had gained greater flexibility in practical application relative to g-frame. K-g-frame, as a more general frame than g-frame and K-frame, has become one of the most active fields in frame theory in recent years. In [20, 21], several properties and characterizations of K-g-frame were obtained. However, many problems of K-g-frame have not been studied. Based on these important results of K-g-frame, we extend tight g-frame to K-g-frame and put forward the concept of tight K-g-frame. In this paper, we give equivalent characterizations and necessary conditions of tight K-g-frame for Hilbert space. We also obtain the necessary and sufficient condition of tight K-g-frame to be tight g-frame. Finally, we present several properties of tight K-g-frame for Hilbert space.Throughout this paper,H is separable Hilbert space and I is the identity operator. L
(
H
1
,
H
2
) is a collection of all bounded linear operators from H
1 to H
2, where H
1 and H
2 are two Hilbert spaces. In particular, L
(
H
) is a collection of all bounded linear operators from H to H. For any T
∈
L
(
H
1
,
H
2
), R
T is the range of T and T
∗ is the adjoint operator of T. {
H
j
:
j
∈
J
} is a sequence of closed subspaces of H, where J is a subset of integers Z. l
2
H
j
j
∈
J is defined by (1)
l
2
H
j
j
∈
J
=
a
j
j
∈
J
:
a
j
∈
H
j
,
j
∈
J
,
∑
j
∈
J
a
j
2
<
+
∞
,with the inner product given by (2)
a
j
j
∈
J
,
b
j
j
∈
J
=
∑
j
∈
J
a
j
,
b
j
H
j
.It is clear that l
2
H
j
j
∈
J is a complex Hilbert space.


## 2. Preliminaries

In this section, some necessary definitions and lemmas are introduced.Definition 1 (see [9, Definition 1.1]).
A sequenceΛ
j
∈
L
H
,
H
j
:
j
∈
J is called a g-frame for H with respect to H
j
j
∈
J if there exist two positive constants A and B such that, for all f
∈
H,(3)
A
f
2
≤
∑
j
∈
J
Λ
j
f
2
≤
B
f
2
.The constants A and B are called the lower and upper bounds of g-frame, respectively. If the right inequality is satisfied, then Λ
j
j
∈
J is said to be a g-Bessel sequence for H with respect to H
j
j
∈
J. If A
=
B, we call this g-frame a tight g-frame, and if A
=
B
=
1, it is called a Parseval g-frame.
For ag-Bessel sequence Λ
j
j
∈
J, T
:
l
2
H
j
j
∈
J
→
H defines a bounded linear operator, that is, (4)
T
g
j
j
∈
J
=
∑
j
∈
J
Λ
j
∗
g
j
,
∀
g
j
j
∈
J
∈
l
2
H
j
j
∈
J
.The adjoint operator T
∗
:
H
→
l
2
H
j
j
∈
J is given by (5)
T
∗
f
=
Λ
j
f
j
∈
J
,
∀
f
∈
H
.By composing T with its adjoint T
∗, we obtain the bounded linear operator (6)
S
:
H
⟶
H
,
S
f
=
T
T
∗
f
=
T
Λ
j
f
j
∈
J
=
∑
j
∈
J
Λ
j
∗
Λ
j
f
,
∀
f
∈
H
.We call T, T
∗, and S the preframe operator, analysis operator, and frame operator of g-Bessel sequence, respectively.Definition 2 (see [22, Definition 2.6]).
We say{
Λ
j
∈
L
(
H
,
H
j
)
}
j
∈
J is g-orthonormal basis for H with respect to H
j
j
∈
J, if it is g-biorthonormal with itself, that is, Λ
j
∗
g
j
,
Λ
i
∗
g
i
=
δ
j
,
i
〈
g
j
,
g
i
〉, ∀
j
,
i
∈
J, g
j
∈
H
j, g
i
∈
H
i, and for any f
∈
H one has ∑
j
∈
J
Λ
j
f
2
=
f
2.Definition 3 (see [21, Theorem 2.5]).
LetK
∈
L
H and Λ
j
∈
L
H
,
H
j for any j
∈
J. A sequence Λ
j
j
∈
J is called a K-g-frame for H with respect to H
j
j
∈
J if there exist constants A
,
B
>
0 such that (7)
A
K
∗
f
2
≤
∑
j
∈
J
Λ
j
f
2
≤
B
f
2
,
∀
f
∈
H
.The constants A and B are called the lower and upper bounds of K-g-frame, respectively.Remark 4.
EveryK-g-frame is a g-Bessel sequence for H with respect to H
j
j
∈
J. If K
=
I, then K-g-frame is just the ordinary g-frame.Motivated by the definition of tightg-frame, we give the following definition of tight K-g-frame.Definition 5.
LetK
∈
L
H and Λ
j
∈
L
H
,
H
j for any j
∈
J. A sequence Λ
j
j
∈
J is called a tight K-g-frame for H with respect to H
j
j
∈
J if there exists constant A
>
0 such that (8)
A
K
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.The constant A is called the bound of tight K-g-frame. If A
=
1, we call this tight K-g-frame a Parseval K-g-frame for H with respect to H
j
j
∈
J.Remark 6.
IfK
=
I, then tight K-g-frame and Parseval K-g-frame are tight g-frame and Parseval g-frame, respectively.Definition 7.
LetK
∈
L
(
H
). An operator K is said to be left-invertible if there exists an operator P
∈
L
(
H
) such that P
K
=
I. The operator P is called a left-inverse of K; that is, K
l
-
1
=
P. Similarly, an operator K is said to be right-invertible if there exists an operator P
∈
L
(
H
) such that K
P
=
I. The operator P is called a right-inverse of K; that is, K
r
-
1
=
P. If there exists P
∈
L
(
H
) such that K
P
=
P
K
=
I, then we say that K is invertible and P is an inverse of K; that is, K
-
1
=
P.Lemma 8 (see [23, Theorem 1]).
LetT
1
∈
L
H
1
,
H and T
2
∈
L
H
2
,
H. The following conditions are equivalent:(1)
R
T
1
⊂
R
T
2.
(2)
There existsλ
>
0 such that T
1
T
1
∗
≤
λ
T
2
T
2
∗.
(3)
There exists a bounded operatorX
∈
L
H
1
,
H
2 so that T
1
=
T
2
X.Lemma 9 (see [21, Theorem 2.5]).
LetK
∈
L
H. Then the following statements are equivalent:(1)
Λ
j
j
∈
J is a K-g-frame for H with respect to H
j
j
∈
J.
(2)
Λ
j
j
∈
J is a g-Bessel sequence for H with respect to H
j
j
∈
J and there exists a g-Bessel sequence for H with respect to H
j
j
∈
J such that (9)
K
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
f
,
∀
f
∈
H
.


## 3. Properties of TightK-g-Frame for Hilbert Space

In this section, we first give characterizations of tightK-g-frame and then give several properties of tight K-g-frame.Theorem 10.
LetK
∈
L
(
H
); Λ
j
∈
L
(
H
,
H
j
), and let T be the preframe operator of {
Λ
j
}
j
∈
J. Then the following statements are equivalent:(1)
Λ
j
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A.
(2)
There exists constantA
>
0 such that A
K
∗
f
2
=
T
∗
f
2 for any f
∈
H.
(3)
There exists constantA
>
0 such that A
K
K
∗
=
T
T
∗.Proof.
(
1
)
⇒
(
2
). Suppose that {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A. By the definition of tight K-g-frame, we get (10)
A
K
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.Since T is the preframe operator of {
Λ
j
}
j
∈
J, we have (11)
∑
j
∈
J
Λ
j
f
2
=
∑
j
∈
J
Λ
j
∗
Λ
j
f
,
f
=
T
T
∗
f
,
f
=
T
∗
f
2
,
∀
f
∈
H
.This implies that A
K
∗
f
2
=
T
∗
f
2 for any f
∈
H.
(
2
)
⇒
(
3
). If there exists constant A
>
0 such that A
K
∗
f
2
=
T
∗
f
2 for any f
∈
H, then we obtain (12)
A
K
K
∗
f
,
f
=
A
K
K
∗
f
,
f
=
A
K
∗
f
2
=
T
∗
f
2
=
T
T
∗
f
,
f
,
∀
f
∈
H
.Hence A
K
K
∗
=
T
T
∗.
(
3
)
⇒
(
1
). If there exists constant A
>
0 such that A
K
K
∗
=
T
T
∗, then 〈
A
K
K
∗
f
,
f
〉
=
〈
T
T
∗
f
,
f
〉, ∀
f
∈
H. That is, (13)
A
K
∗
f
2
=
T
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.Therefore, {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J with bound A. The proof of Theorem 10 is completed.Corollary 11.
Suppose that{
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J with bound A, T is the preframe operator of {
Λ
j
}
j
∈
J, and S is the frame operator of {
Λ
j
}
j
∈
J; then(1)
R
(
K
)
=
R
(
T
);
(2)
S
=
A
K
K
∗;
(3)
T
=
A
K.Proof.
Theorem10 together with Lemma 8 shows that (1) and (2) are satisfied. We only need to prove that (3) holds. Assume that {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J with bound A. By Theorem 10, we have A
K
∗
f
2
=
T
∗
f
2, ∀
f
∈
H. Therefore,(14)
T
=
T
∗
=
sup
f
=
1
,
f
∈
H
⁡
T
∗
f
=
sup
f
=
1
,
f
∈
H
⁡
A
K
∗
f
=
A
K
.The proof of Corollary 11 is completed.Lemma9 gives an equivalent characterization of K-g-frame; does the tight K-g-frame have the similar characterization? Clearly, if Λ
j
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J, then there exists a g-Bessel sequence Γ
j
j
∈
J for H with respect to {
H
j
}
j
∈
J such that K
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
f for any f
∈
H. The theorem below gives a necessary condition of tight K-g-frame.Theorem 12.
LetK
∈
L
(
H
). Suppose that {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A. Then there exists a g-Bessel sequence Γ
j
j
∈
J for H with respect to H
j
j
∈
J with bound B such that K
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
f, ∀
f
∈
H, and A
B
≥
1.Proof.
Let{
Λ
j
}
j
∈
J be a tight K-g-frame for H with respect to H
j
j
∈
J. By Lemma 9, there exists a g-Bessel sequence Γ
j
j
∈
J for H with respect to H
j
j
∈
J such that K
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
f, ∀
f
∈
H. For any f
,
g
∈
H, we have (15)
K
∗
f
,
g
=
f
,
K
g
=
f
,
∑
j
∈
J
Λ
j
∗
Γ
j
g
=
∑
j
∈
J
Γ
j
∗
Λ
j
f
,
g
.Via (15), (16)
K
∗
f
=
∑
j
∈
J
Γ
j
∗
Λ
j
f
,
∀
f
∈
H
.Since {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A, we get (17)
A
K
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.Furthermore, (18)
∑
j
∈
J
Λ
j
f
2
=
A
K
∗
f
2
=
A
∑
j
∈
J
Γ
j
∗
Λ
j
f
2
=
A
s
u
p
g
=
1
,
g
∈
H
∑
j
∈
J
Γ
j
∗
Λ
j
f
,
g
2
=
A
sup
g
=
1
,
g
∈
H
⁡
∑
j
∈
J
Λ
j
f
,
Γ
j
g
2
≤
A
sup
g
=
1
,
g
∈
H
⁡
∑
j
∈
J
Λ
j
f
2
∑
j
∈
J
Γ
j
g
2
≤
A
s
u
p
g
=
1
,
g
∈
H
∑
j
∈
J
Λ
j
f
2
B
g
2
=
A
B
∑
j
∈
J
Λ
j
f
2
.Therefore, A
B
≥
1. The proof of Theorem 12 is completed.Note that whenK
=
I, tight K-g-frame is tight g-frame. One may wonder whether K
=
I when tight K-g-frame is tight g-frame as well. In fact, the answer is negative. The following example demonstrates this.Example 13.
Suppose thatH
=
R
3; J
=
1,2
,
3. Let e
j
j
∈
J be an orthonormal basis of H, and let H
j
=
s
p
a
n
¯
e
j. Now define {
Λ
j
}
j
∈
J and {
Γ
j
}
j
∈
J as follows:(19)
Λ
j
:
H
⟶
H
j
,
Λ
1
f
=
f
,
e
1
e
1
,
Λ
2
f
=
f
,
e
2
e
2
,
Λ
3
f
=
f
,
e
3
e
3
;
Γ
j
:
H
⟶
H
j
,
Γ
1
f
=
1
2
f
,
e
1
+
f
,
e
2
e
1
,
Γ
2
f
=
1
2
f
,
e
1
-
f
,
e
2
e
2
,
Γ
3
f
=
f
,
e
3
e
3
.By a simple calculation, we have(20)
Λ
1
∗
:
H
1
⟶
H
,
Λ
1
∗
a
1
e
1
=
a
1
e
1
,
Λ
2
∗
:
H
2
⟶
H
,
Λ
2
∗
a
2
e
2
=
a
2
e
2
,
Λ
3
∗
:
H
3
⟶
H
,
Λ
3
∗
a
3
e
3
=
a
3
e
3
;
Γ
1
∗
:
H
1
⟶
H
,
Γ
1
∗
a
1
e
1
=
1
2
a
1
e
1
+
1
2
a
1
e
2
,
Γ
2
∗
:
H
2
⟶
H
,
Γ
2
∗
a
2
e
2
=
1
2
a
2
e
1
-
1
2
a
2
e
2
,
Γ
3
∗
:
H
3
⟶
H
,
Γ
3
∗
a
3
e
3
=
a
3
e
3
.For any f
=
c
1
,
c
2
,
c
3
∈
H, we have (21)
∑
j
∈
J
Λ
j
f
2
=
c
1
2
+
c
2
2
+
c
3
2
=
f
2
;
∑
j
∈
J
Γ
j
f
2
=
1
2
c
1
+
c
2
2
+
1
2
c
1
-
c
2
2
+
c
3
2
=
f
2
.Obviously, {
Λ
j
}
j
∈
J and {
Γ
j
}
j
∈
J are Parseval g-frames for H with respect to {
H
j
}
j
∈
J.
Define the bounded linear operatorK as follows: (22)
K
:
H
→
H
,
K
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
f
,
∀
f
∈
H
.Now we prove that {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J. For any f
,
g
∈
H, we have (23)
K
∗
f
,
g
=
f
,
K
g
=
f
,
∑
j
∈
J
Λ
j
∗
Γ
j
g
=
∑
j
∈
J
Γ
j
∗
Λ
j
f
,
g
.Hence, K
∗
f
=
∑
j
∈
J
Γ
j
∗
Λ
j
f. It follows that (24)
K
∗
f
2
=
∑
j
∈
J
Γ
j
∗
Λ
j
f
2
=
Γ
1
∗
c
1
e
1
+
Γ
2
∗
c
2
e
2
+
Γ
3
∗
c
3
e
3
2
=
1
2
c
1
e
1
+
e
2
+
1
2
c
2
e
1
-
e
2
+
c
3
e
3
2
=
f
2
,
∀
f
∈
H
.Therefore, for any f
∈
H, we have ∑
j
∈
J
Λ
j
f
2
=
f
2
=
K
∗
f
2. Via the definition of tight K-g-frame, {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J. For any f
=
(
c
1
,
c
2
,
c
3
)
∈
H, we have K
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
f
=
(
1
/
2
)
(
c
1
+
c
2
)
,
(
1
/
2
)
(
c
1
-
c
2
)
,
c
3
≠
f.
Example13 shows that if a tight K-g-frame {
Λ
j
}
j
∈
J is a tight g-frame, then K cannot be I. In the following theorem, we state a necessary and sufficient condition for a tight K-g-frame being a tight g-frame.Theorem 14.
LetK
∈
L
(
H
) and A
1
,
A
2
>
0. Suppose that {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J with bound A
1. Then {
Λ
j
}
j
∈
J is a tight g-frame for H with respect to {
H
j
}
j
∈
J with bound A
2 if and only if K is right-invertible and the right-invertible operator is K
r
-
1
=
(
A
1
/
A
2
)
K
∗.Proof.
First, we prove the sufficient condition. Since{
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J with bound A
1, we have (25)
A
1
K
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.Assume that {
Λ
j
}
j
∈
J is a tight g-frame for H with respect to {
H
j
}
j
∈
J with bound A
2. Then, for any f
∈
H, we get (26)
∑
j
∈
J
Λ
j
f
2
=
A
2
f
2
.By (25) and (26), A
1
K
∗
f
2
=
A
2
f
2 for any f
∈
H, implying that K
∗
f
2
=
(
A
2
/
A
1
)
f
2. Then, for any f
∈
H, we have 〈
K
K
∗
f
,
f
〉
=
(
A
2
/
A
1
)
f
,
f. This implies that K
(
A
1
/
A
2
)
K
∗
=
I. So K is right-invertible and the right-invertible operator is K
r
-
1
=
(
A
1
/
A
2
)
K
∗.
Next, we prove the necessary condition. Suppose thatK is right-invertible and the right-invertible operator is K
r
-
1
=
(
A
1
/
A
2
)
K
∗. Then K
K
r
-
1
=
K
(
A
1
/
A
2
)
K
∗
=
I; that is, K
K
∗
=
(
A
2
/
A
1
)
I. So (27)
K
K
∗
f
,
f
=
A
2
A
1
f
,
f
,
∀
f
∈
H
.That is, (28)
K
∗
f
2
=
A
2
A
1
f
2
.Since {
Λ
j
}
j
∈
J is a tight K-g-frame for H with respect to {
H
j
}
j
∈
J with bound A
1, we have (29)
∑
j
∈
J
Λ
j
f
2
=
A
1
K
∗
f
2
=
A
2
f
2
.This implies that {
Λ
j
}
j
∈
J is a tight g-frame for H with respect to {
H
j
}
j
∈
J with bound A
2. The proof of Theorem 14 is completed.In the following, we will verify whether theK
r
-
1 in Example 13 is equal to K
∗. For any f
=
(
c
1
,
c
2
,
c
3
)
∈
H, we have (30)
K
K
∗
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
K
∗
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
∑
j
∈
J
Γ
j
∗
Λ
j
f
=
∑
j
∈
J
Λ
j
∗
Γ
j
1
2
c
1
+
c
2
,
1
2
c
1
-
c
2
,
c
3
=
Γ
1
∗
1
2
c
1
+
c
2
+
1
2
c
1
-
c
2
e
1
+
Γ
2
∗
1
2
c
1
+
c
2
-
1
2
c
1
-
c
2
e
2
+
Γ
3
∗
c
3
e
3
=
Γ
1
∗
c
1
e
1
+
Γ
2
∗
c
2
e
2
+
Γ
j
∗
c
3
e
3
=
f
.It follows that K
K
∗
=
I. This implies that K is right-invertible and the right-invertible operator is K
r
-
1
=
K
∗.Corollary 15.
LetK
∈
L
(
H
). Suppose that {
Λ
j
}
j
∈
J is a Parseval K-g-frame for H with respect to {
H
j
}
j
∈
J; then {
Λ
j
}
j
∈
J is a Parseval g-frame for H with respect to {
H
j
}
j
∈
J if and only if K is right-invertible and the right-invertible operator is K
r
-
1
=
K
∗.To enrich the theory of atomic systems, we give several important properties of tightK-g-frame in the following.Theorem 16.
LetK
∈
L
(
H
). If Λ
j
j
∈
J is a tight g-frame for H with respect to H
j
j
∈
J with bound A, then Λ
j
K
∗
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A.Proof.
SinceΛ
j
j
∈
J is a tight g-frame for H with respect to H
j
j
∈
J with bound A, we have (31)
∑
j
∈
J
Λ
j
f
2
=
A
f
2
,
∀
f
∈
H
.Again, for any f
∈
H, we have K
∗
f
∈
H; then (32)
∑
j
∈
J
Λ
j
K
∗
f
2
=
A
K
∗
f
2
,
∀
f
∈
H
.Therefore, Λ
j
K
∗
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A. The proof of Theorem 16 is completed.Corollary 17.
LetK
∈
L
(
H
). If Λ
j
j
∈
J is a g-orthonormal basis for H with respect to H
j
j
∈
J, then Λ
j
K
∗
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J.Theorem 18.
LetT
,
K
∈
L
H. If Λ
j
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A, then Λ
j
T
∗
j
∈
J is a tight T
K-g-frame for H with respect to H
j
j
∈
J with bound A.Proof.
SinceΛ
j
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J with bound A, we have (33)
A
K
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.And, for any f
∈
H, we have T
∗
f
∈
H; then (34)
∑
j
∈
J
Λ
j
T
∗
f
2
=
A
K
∗
T
∗
f
2
=
A
T
K
∗
f
2
,
∀
f
∈
H
.So Λ
j
T
∗
j
∈
J is a tight T
K-g-frame for H with respect to H
j
j
∈
J with bound A. The proof of Theorem 18 is completed.Corollary 19.
LetK
∈
L
(
H
). If Λ
j
j
∈
J is a tight K-g-frame for H with respect to H
j
j
∈
J, then Λ
j
K
∗
N
j
∈
J is a tight K
N
+
1-g-frame for H with respect to H
j
j
∈
J, where N is a given positive integer.Theorem 20.
LetK
1
,
K
2
∈
L
(
H
), and let g
F
K
(
H
) be the collection of all tight K-g-frames for H with respect to H
j
j
∈
J. Then g
F
K
1
(
H
)
⊂
g
F
K
2
(
H
) if and only if there exists A
>
0 such that K
1
K
1
∗
=
A
K
2
K
2
∗.Proof.
⇒. If {
Λ
j
}
j
∈
J is a g-orthonormal basis for H with respect to H
j
j
∈
J, by Corollary 17, we get that Λ
j
K
1
∗
j
∈
J is a tight K
1-g-frame for H with respect to H
j
j
∈
J. Since g
F
K
1
(
H
)
⊂
g
F
K
2
(
H
), we have that Λ
j
K
1
∗
j
∈
J is a tight K
2-g-frame for H with respect to H
j
j
∈
J. Assume that the bound of tight K
2-g-frame Λ
j
K
1
∗
j
∈
J is A. By the definition of tight K-g-frame, we obtain (35)
∑
j
∈
J
Λ
j
K
1
∗
f
2
=
A
K
2
∗
f
2
,
∀
f
∈
H
.Since {
Λ
j
}
j
∈
J is a g-orthonormal basis for H with respect to H
j
j
∈
J, we have K
∗
f
∈
H for any f
∈
H. By the definition of g-orthonormal basis, we get (36)
∑
j
∈
J
Λ
j
K
1
∗
f
2
=
K
1
∗
f
2
,
∀
f
∈
H
.By (35) and (36), we get K
1
∗
f
2
=
A
K
2
∗
f
2 for any f
∈
H. So K
1
K
1
∗
=
A
K
2
K
2
∗.
⇐. Suppose that {
Λ
j
}
j
∈
J is a tight K
1-g-frame for H with respect to {
H
j
}
j
∈
J with bound A
1; then (37)
∑
j
∈
J
Λ
j
f
2
=
A
1
K
1
∗
f
2
,
∀
f
∈
H
.Under this assumption, there exists A
>
0 such that K
1
K
1
∗
=
A
K
2
K
2
∗. So for any f
∈
H, we have K
1
∗
f
2
=
A
K
2
∗
f
2. Hence (38)
A
1
A
K
2
∗
f
2
=
A
1
K
1
∗
f
2
=
∑
j
∈
J
Λ
j
f
2
,
∀
f
∈
H
.Therefore, {
Λ
j
}
j
∈
J is a tight K
2-g-frame for H with respect to {
H
j
}
j
∈
J with bound A
1
A. The proof of Theorem 20 is completed.Corollary 21.
LetK
1
,
K
2
∈
L
(
H
) and let P
g
F
K
(
H
) be the collection of all Parseval K-g-frames for H with respect to H
j
j
∈
J. Then P
g
F
K
1
(
H
)
=
P
g
F
K
2
(
H
) if and only if K
1
K
1
∗
=
K
2
K
2
∗.



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*Source: 3783456-2016-01-26.xml*