Matrix Representation of Operators Using Frames
نویسنده
چکیده
In this paper it is investigated how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs). Therefore it is useful to extend the known method of matrix representation by using these sequences instead of ONBs for these application areas. We will give basic definitions of the functions connecting infinite matrices defining bounded operators on l2 and operators on H. We will show some structural results and give some examples. Furthermore in the case of Riesz bases we prove that those functions are isomorphisms. Finally we are going to apply this idea to the connection of Hilbert-Schmidt operators and Frobenius matrices.
منابع مشابه
G-Frames, g-orthonormal bases and g-Riesz bases
G-Frames in Hilbert spaces are a redundant set of operators which yield a representation for each vector in the space. In this paper we investigate the connection between g-frames, g-orthonormal bases and g-Riesz bases. We show that a family of bounded operators is a g-Bessel sequences if and only if the Gram matrix associated to its denes a bounded operator.
متن کاملCoherent Frames
Frames which can be generated by the action of some operators (e.g. translation, dilation, modulation, ...) on a single element $f$ in a Hilbert space, called coherent frames. In this paper, we introduce a class of continuous frames in a Hilbert space $mathcal{H}$ which is indexed by some locally compact group $G$, equipped with its left Haar measure. These frames are obtained as the orbits of ...
متن کامل(C; C\')-Controlled g-Fusion Frames in Hilbert Spaces
Controlled frames in Hilbert spaces have been recently introduced by P. Balazs and etc. for improving the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper we develop a theory based on g-fusion frames on Hilbert spaces, which provides exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In part...
متن کاملThe Interplay of Frames and Operators
This paper will look at the connection of operators and frames respectively related sequences from different viewpoints. We will give a survey of the classification of frames and similar sequences by the properties of the associated operators. Furthermore an important part will be the investigation of how to find a matrix representation of operators with Bessel sequences, frames and Riesz bases...
متن کاملOn Some Characterization of Generalized Representation Wave-Packet Frames Based on Some Dilation Group
In this paper we consider (extended) metaplectic representation of the semidirect product $G_{mathbb{J}}=mathbb{R}^{2d}timesmathbb{J}$ where $mathbb{J}$ is a closed subgroup of $Sp(d,mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Fina...
متن کامل