Fusion Frames and G-frames in Tensor Product and Direct Sum of Hilbert Spaces

Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [10] in 1952 to study some problems in nonharmonic Fourier series, reintroduced in 1986 by Daubechies, Grossmann and Meyer [8]. Frames are very useful in characterization of function spaces and other fields of applications such as filter bank theory, sigmadelta quantization, signal and image processing and wireless communications. Fusion frame is a generalization of frame which was introduced in [5] and investigated in [2, 6, 21]. Fusion frames have important applications e.g., in sensor networks and packet encoding. Sun in [23] introduced g-frame as a generalization of frame. He showed that oblique frames, pseudo frames and fusion frames are special cases of g-frames. Note that fusion frames and g-frames have been introduced in Hilbert C∗modules and Banach spaces, see [18, 19]. Let H be a Hilbert space and let I be a finite or countable index set. A family {fi}i∈I ⊆ H is a frame for H, if there exist 0 < A ≤ B <∞, such that A‖f‖ ≤ ∑