| The theory of frame and atomic decomposition for Banach spaces is a new and important branch in the theory of frame.In 1991,Grochnig [3] generalized Hilbert frame to Banach spaces and called them atomic decomposition.He also gave the definition of Banach frame.Now The theory of frame and atomic decomposition for Banach spaces are completely developed but still exists some open problems to need to be dealed with. This paper mainly explores the stability of Banach frame and atomic decomposition,totality and judgment and so on.This paper is composed of five parts. The first part is basic knowledge,which introduces several definitions involved in this paper. The second part talks about the stability of Banach frame and atomic decomposition. Enlightened by [14] and [15], this paper further study the stability of Banach frame, improve and generalize the results of the original paper. At the same time, this paper talks about the stability of atomic decomposition of Banach spaces and come to a series of conclusions.The main results:Theorem 2.9:Let X and Y be Banach space and Let Xd be a BK-space.((j||n)ne||, S) is a Banach frame for X with respect to Xd and A and B are frame bounds. Let U : X ?> Y be a linear homeomorphism.Let (j/n)neN be a sequence of vectors in X' and exist (3 > 0, M > 0 such that satisfy the inequalityII (< y,yn >)neN \\xd< 0 || (< V~ly,xn >)neN ||xd +M II y ||Y,for Vy e Y. If 0B + M || U ||< A and (< y,yn >)n N N(S) for Vy € y.then (((U~1)'xn + yn)n,US) is a Banach frame of Y with respect to X& with bounds (A - (33} || U H-1 -M and (1 + ft)B \\ U~l \\ +M.Theorem 2.17:Let X and Y be Banach space and Let Xd be a BK-space. ((?/n)neyv, (#n)nejv) is a atomic decomposition of X with resoect to Xd and A and B are frame bounds. Let U : X > Y be a linear homeomorphism.Let (zn)neN be asequence of vectors in X' and exist 0 > 0, M > 0 such that satisfy the inequalityfor Vy Y. If /3B + M \\ U \\< A and (xn)n N and (yn)nN are strongly disjoint,then is a atomic decomposition of y with respect to Xj with bounds -M and (1 + P)B Some significant corollaries are abtained from two theorem above.The third part gives a judgment of Banach frame and atomic decomposition and generalizes the theory of Hilbert frame to banach spaces, thus talking about the property of Banach frame, such as : totality and dilation and so on.At the same time, it concludes that some finite vectors may be a Banach frame if they satisfy some conditions. The main results:Lemma 3.11:Let X be weakly sequentially complete Banach space. (yn)neN ls a Bessel sequence of X with respect to Xj with bounds D.Let (yn)newbe norm densely in X' and X^be BK-space satisfying the property (*).If U : X ?>?X^ is given by U(x) = (< x,yn >)neN then U is a bounded linear operator , RanU is a close subspace of X^ and kerU = {0}.Theorem 3.13:Let X be weakly sequentially complete Banach space.(yn)nejv is a Bessel sequence of X with respect to Xj with bounds D.Let (yn)neN be norm densely in X' and Xj, be BK-space satisfying poroperty (*).If X = RanU (& RanU , then there is a bounded linear operater S : X,i X so that ((yn}neN,S) is a Banach frame of X with respect to X&.Theorem 3.19:Let X be Banach space and (j/i)"=1 be finite vectors of X'.Let M and M be a closed subspace.Let M satisfyM = (()1 = {x e X : yi(x) = 0,i = 1,and X = M M.Then there is a close subspace X of X , BK-space Xj and a bounded linear operater S : Xd t X so that (() is a Banach frame of Xwith respect to X.Theorem 3.20:Let X be Banach space and Xd be BK-space satisfying the property (**). () is a Banach frame of X with respect to Xd with bounds A and B.Then for Vj 6 N, (yn)n^j isn't either total set or there is BK-space Xd and S : Xd X with S) so that (yn) is a Banach frame of X with respect toXd.Theorem 3.24:Let X be weakly sequentially Banach space .Let Xd be BK-space satisfying the property (*)and ((xn)nN, (yn)new) be a weak framing of X.lf (yn)n is a Bessel sequence of X with respect to Xd with bound D and (yn)nN is norm densely in X', t... |
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