G-frames and g-Riesz frames as generalized frames in Hilbert spaces have been studied by many authors in recent years. The super Hilbert space has a certain advantage compared with the Hilbert space in the field of studying quantum mechanics. In this paper, for super Hilbert space

Frame as generalized basis in Hilbert space was first introduced by Duffin and Schaeffer [

Sun [

Throughout this paper,

The literature [

Super Hilbert space is a direct sum that

In this section, some necessary definitions and lemmas are introduced.

A sequence

One says that

A sequence

Let

Let

Let

The literature [

With the definition of g-complete of g-frame for Hilbert space, we give the definition of g-complete of g-frame for super Hilbert space as follows.

In this section, we first give the concept and the characterization of g-Riesz frame for super Hilbert space

Before giving the characterization of g-Riesz frames for super Hilbert space

Suppose that

Suppose that

For the above

Suppose that for every

Denote

On the other hand, suppose that

Therefore,

Let

For any

Inspired by the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert space.

Let

Suppose that

We prove Lemma

By Lemmas

Obviously,

Suppose

Based on this, we can obtain a characterization of g-Riesz frame for super Hilbert space

Let

(1)

(2) There exists

There are two cases to prove that

Again by (

Now we prove

Let

Like (

For every

Let

In terms of the concept of minimal g-complete of g-frame for Hilbert space, we give the definition of minimal g-complete of g-frame for super Hilbert space

Let

Suppose that

The proof is similar to proof of Lemma

By above lemma, we can get a characterization of g-Riesz frame for super Hilbert space

Let

For any nonempty subset

The proof is analogous to proof of Theorem

In this section, we use the characterization of g-Riesz frame for super Hilbert space

The stability of g-frames is important in practice which is wildly studied by many authors; for example, see [

Suppose that

Example

Suppose that

First, we prove that

For any

Next, we prove that

If there exists

Let

Now we prove that

The proof is by contradiction. Suppose that

On the other hand, for any

From Example

Let

Since

For any finite subset

If

Then for any

For any

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (Grants nos. 61261043 and 10961001), Natural Science Foundation of Ningxia (Grant no. NZ13084), and Innovation Project for Graduate of Beifang University of Nationalities.