On the Riesz fusion bases in Hilbert spaces

Some properties of alternate duals and approximate alternate duals of fusion frames

“…} i∈I,j∈Ji is a dual for {ω i e i,j } i∈I,j∈Ji . On the other hand, the sequence {ω i e i,j } i∈I,j∈Ji is a Riesz basis for H by Theorem 3.6 of [2]. Using the fact that discrete Riesz bases have only one dual, we obtain…”

“…Q-duals are useful tools for establishing the reconstruction formula. For more information on fusion frames, we refer the reader to [2,4,5].…”

“…Suppose that {W i } i∈I is a Riesz fusion basis. By Theorem 3.9 in [2], there exists an orthonormal fusion basis {U i } i∈I and a bounded bijective linear operator T : H → H for which T U i = W i for all i ∈ I . Therefore, the canonical dual of a Riesz fusion basis is also a Riesz fusion basis.…”

“…It is clear that every Riesz fusion basis is a 1 -uniform fusion frame for H , and also a fusion frame is a Riesz basis if and only if it is a Riesz decomposition for H ; see [2,4].…”

Some properties of dual and approximate dual of fusion frames

“…It is clear that every Riesz fusion basis is a 1-uniform fusion frame for H, and also a fusion frame is a Riesz basis if and only if it is a Riesz decomposition for H, see [3,5].…”