We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in Hilbert

Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [

The theory of frames was rapidly generalized and, until 2006, various generalizations consisting of vectors in Hilbert spaces were developed. In 2006, Sun introduced the concept of g-frame in a Hilbert space in [

On the other hand, the concept of frames especially the g-frames was introduced in Hilbert

The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaiser [

The continuous g-frames in Hilbert

The paper is organized in the following manner. We continue this introductory section with a review of the basic definitions and notations about Hilbert

Let us recall the definitions and some basic properties of Hilbert

For

Let

Let

the operator

there exists a constant

Let

Let

For any

Throughout this paper,

In this section, we recall some basic properties of continuous g-frames in Hilbert

We call a family of adjointable

for any

there is a pair of constants

We have the following equivalent definition for continuous g-Bessel sequences in Hilbert

Let

“⇒”. It is obvious.

“⇐”. Define a linear operator

The following proposition gives an equivalent condition for a continuous g-Bessel sequence to be a continuous g-frame.

Let

“⇒”. It is straightforward.

“⇐”. We define a linear operator as follows:

Using the above equivalent definition of continuous g-frames we can easily prove the following result that will be used in the proof of Lemma

Let

Let us denote the Bessel bound of

Let

We can characterize the continuous g-frames in Hilbert

Let

We are now ready to present a necessary and sufficient condition for a Hilbert

A Hilbert

“⇒”. Assume that

“⇐”. Suppose that

It is easy to see that a continuous g-Bessel sequence

Our next result is a generalization of Lemma 2.1 in [

Let

More generally, whenever

We begin with showing the first statement. Since

Now, suppose that

A continuous g-frame

if

By using the synthesis operator, Kouchi and Nazari gave a characterization for continuous g-Riesz basis as follows.

A family of adjointable

We note, however, that in the proof of the above theorem, they said that “

We call a family

there exist constants

A sequence

Suppose first that

Conversely, let

The following is an immediate consequence of Theorem

Let

Let

Let

For any

The definitions of similar and unitary equivalent frames give rise to definitions of similar and unitary equivalent continuous g-frames in Hilbert

Let

They are said to be similar or equivalent if there is an adjointable and invertible operator

They are said to be unitary equivalent if there exists an adjointable and unitary linear operator

Let

there is an adjointable and invertible operator

there exists a constant

On the other hand,

For the last statement, the assumptions implies that

To complete this section, we generalize the results in [

Let

(1) “⇒”. Assume that

“⇐”. It is obvious.

For the second part of (1), since

(2) Suppose that

For the general case, we have the following proposition.

Let

(1) “⇒”. Assume that

“⇐”. It is straightforward.

(2) “⇒”. If

“⇐”. It is obvious.

The stability of frames is important in practice and is therefore studied widely by many authors. The stability of dual frames is also needed in practice. However, most of the known results on this topic are stated about canonical dual; see [

Let

Assume first that

Conversely, let

Let

Let us denote by

Let

If we take

This work was supported by the National Natural Science Foundation of China (Grant no. 11271148). The author thanks the referee(s) and the editor(s) for their valuable comments and suggestions which improved the quality of the paper.