%0 Journal Article
%T Operator Characterizations and Some Properties of -Frames on Hilbert Spaces
%A Xunxiang Guo
%J Journal of Function Spaces
%D 2013
%R 10.1155/2013/931367
%X Given the -orthonormal basis for Hilbert space , we characterize the -frames, normalized tight -frames, and -Riesz bases in terms of the -preframe operators. Then we consider the transformations of -frames, normalized tight -frames, and -Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and -dual frames of -frames by applying the results of characterizations. 1. Introduction A sequence of elements of a Hilbert space is called a frame for if there are constants so that The numbers ; are called the lower (resp., upper) frame bounds. The frame is a tight frame if and a normalized tight frame if . The concept of frame first appeared in the late 40s and early 50s (see, [1¨C3]). The development and study of wavelet theory during the last decades also brought new ideas and attentions to frames because of their close connections. There are many related references on this topic, see; [4¨C8]. In [9], Sun raised the concept of -frame as follows, which generalized the concept of frame extensively. A sequence is called a -frame for with respect to , which is a sequence of closed subspaces of a Hilbert space , if there exist two positive constants and such that for any We simply call a -frame for whenever the space sequence is clear. The tight -frame and normalized tight -frame are defined similarly. We call a -frame sequence, if it is a -frame for . We call a -Bessel sequence, if only the right inequality is satisfied. Recently, -frames in Hilbert spaces have been studied intensively; for more details see [10¨C14] and the references therein. Constructing frames and g-frames is an interesting problem in frame theory and it is also useful in applications. In this respect, many mathematicians considered the algebraic operations among frames, which allows us to construct a large number of new frames from existing frames. For more details, see [15¨C18] and the references therein. In this paper, we will consider the algebraic operations among -frames; in particular, we will consider the direct sums of -frames and general sums of -frames. We also will consider the transforms of -frames and their dual -frames. 2. -Preframe Operators and Transformations of -Frames In this section, we introduce the -preframe operators and use them to characterize -frames. We also discuss the transformations of -frames. Since any countable infinite sequence can be made as one indexed by natural number , so all infinite sequences in this paper are assumed to be indexed by . For each sequence , we define the space by with the
%U http://www.hindawi.com/journals/jfs/2013/931367/