Frames in Quaternionic Hilbert Spaces

Duality and Sum on frames in Q-Hilbert spaces

“…In this section we arrange some notions and results of frames in Q-Hilbert spaces (see [9][10][11][12][13][14][15][16][17][18] for details), which are necessary for below. Q denotes a noncommutative quaternion field, and J is an index set.…”

“…In [18], Sharma and Goel extended the concept of frame in Hilbert space to Q-Hilbert space, as described next.…”

“…Note that both real field and complex field are associative and commutative, while quaternion field only constitutes a non-commutative associative algebra, this key characteristic greatly limited mathematicians to establish a complete theory of functional analysis in Q-Hilbert spaces [14,15], which affected the development of quantum physics in Q-Hilbert space. Luckily, the study on quaternion field has been developed from the mathematical point of view, especially frames in Q-Hilbert space have been obtained some achievements in recent years [16][17][18][19][20][21][22].…”

Woven Frames in Quaternionic Hilbert Spaces

“…Khokulan et al [15] introduced and studied frames for finite-dimensional quaternionic Hilbert spaces, which were further studied in [16]. Recently, Sharma and Goel [17] introduced and studied frames for separable quaternionic Hilbert spaces. Frames in separable right quaternionic Hilbert spaces H R (Q) are defined as follows.…”

“…Definition 3 (see [17]). Let H R (Q) be a right quaternionic Hilbert space and u i i∈N be a sequence in H R (Q).…”