Sum and direct sum of frame sequences

Abstract

Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In particular, we find a necessary and sufficient condition for the sum of two strongly disjoint (orthogonal) frame sequences (in the same Hilbert space) to be a frame sequence, and thereby show that the sum of two strongly disjoint frame sequences may not be a frame sequence. We also show that the closedness of the sum of the synthesis operators of two frame sequences and that of the sum of the frame operators of the same frame sequences are not related. Other observations are also included.

Keywords:

• frame
• frame sequence
• sum of frame sequences
• direct sum of frame sequences
• operator with closed range

AMS Subject Classifications::

• 42C15
• 47A05

Acknowledgments

The authors thank the referee for his/her careful suggestions that helped them to improve this manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0008917).