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g-frames and g-frame sequences in Hilbert spaces. (English) Zbl 1198.42032

Let U and V be two complex Hilbert spaces, let {V j } jJ be a sequence of closed subspaces of V, where J, and let L(U,V j ) be the collection of all bounded linear spaces from U to V j · For a g-frame {Λ j :Λ j L(U,V j )} jJ for V with respect to {V j } jJ (i.e. there exist A,B>0 such that

Af 2 jJ Λ j f 2 Bf 2

for all fV) the authors find some relations between operators

S:f jJ Λ j * Λ j f,Q:{g j } jJ jJ Λ j * g j

and A,B; moreover necessary and sufficient conditions for a g-frame in terms of Q are given.

Further the authors define a g-frame sequence {Λ j } jJ for U as a g-frame for

W={ jJ 1 Λ j * g j foranyfiniteJ 1 Jandanyg j V j ,jJ} ¯·

They discuss that definition and consider the stability of a g-frame sequence under perturbation.

MSC:
42C15General harmonic expansions, frames
46C99Inner product spaces, Hilbert spaces
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