Dergiler / Turkish Journal of Mathematics / 2011 / Cilt: 35 - Sayı: 4
Some results on g-frames in Hilbert spaces
- Sayfa
- 695–704
- DOI
- —
Özet
In this paper we show that every g-frame for a Hilbert space H can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. We also show that every g-frame can be written as a sum of two tight g-frames with g-frame bounds one or a sum of a g-orthonormal basis and a g-Riesz basis for H. We further give necessary and sufficient conditions on g-Bessel sequences {Li \in L (H,Hi) : i \in J} and {Gi \in L(H,Hi): i \in J} and operators L1, L2 on H so that {LiL1+GiL2: i \in J} is a g-frame for H. We next show that a g-frame can be added to any of its canonical dual g-frame to yield a new g-frame.
Abstract
In this paper we show that every g-frame for a Hilbert space H can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. We also show that every g-frame can be written as a sum of two tight g-frames with g-frame bounds one or a sum of a g-orthonormal basis and a g-Riesz basis for H. We further give necessary and sufficient conditions on g-Bessel sequences {Li \in L (H,Hi) : i \in J} and {Gi \in L(H,Hi): i \in J} and operators L1, L2 on H so that {LiL1+GiL2: i \in J} is a g-frame for H. We next show that a g-frame can be added to any of its canonical dual g-frame to yield a new g-frame.