含Riesz基框架的性质

【作者】 郭燕妮；

【导师】 周家云；

【作者基本信息】 曲阜师范大学 ， 基础数学， 2002， 硕士

【摘要】 本文主要在Hilbert空间上探讨含Riesz基框架的一般性质、稳定性，不相交性及其交错对偶的情况。受文献[11]的启发，探讨了Besselian框架的一般性质、摄动、交错对偶，得到了一系列结果．在[5]，[9]的基础上推广了Risez基的稳定性结果，得出了Riesz框架满足摄动的条件。此外，研究了含Riesz基框架的交错对偶及其不交性质，讨论了各种含Riesz基框架的相互关系。本文共分四节.第一节阐述Hilbert空间一般框架理论，所谓Hilbert空间H上的框架是指H中的一列元素{fi正N满足：3A，B>0，V／正"，有CX，AtIftl’≤∑(<f，fi>1’三BIIf2／II2．i=1通过算子来研究框架是框架研究的主要方法，为此我们引入预框架算子和框架算子，并列出各种不同框架和其相应预框架算了之间的对应关系，为后文需要我们还列出了框架的相似，弱不交，不交和强不交的概念。在一般框架的性质探讨中，框架序列不相交时，得出在一定条件下，框架序列的和是其并的线性闭包的框架。在本节的结尾，给出了框架最一般形式的一个稳定性结果。第二节研究含Riesz基框架的相互关系，探讨各种框架和投影法可行框架的关系及在各种算子作用下的稳定性。首先给出Riesz基、Bcsselian框架、Riesz框架，投影法可行框架的定义和相应等价说法，在投影法可行框架是条件Riesz框架([14])和强投影法可行框架一定是投影法可行框架的基础上得到：(ⅰ)Riesz基，Riesz框架是投影法可行框架，但反之不成立。(ⅱ)Besselian框架不必是投影法可行框架，投影法可行框架也不必是Besselian框架，并例示．在各种含Riesz基框架的相互关系中得到： （i）Riesz基是 Riesz框架，同时也是 Besselian框架，反之不成立． 门）R的sz框架不必是Besselian框架，反之亦然． wesz框架和投影法可行框架在线性有界逆存在且连续算子作用下具有稳 定性．特别，Riesz基、Besselian框架、Riesz框架，投影法可行框架在相似 意义下具有稳定性． 第三节研究含Riesz基框架的摄动． （一）利用一般框架摄动中的不等式： I二心（人一＊川卜）川二c注人I 人【I二c8了蓉I且 川二卜J会 （＊） 2＝二 左。If＝1 土＝l （i）oesz基摄动．推广了仰，［20］中的摄动结果，即： 若uh。N仅是颐人入。N的R二sz基，切｝。厂不必含于厕人h。N； 则在（）式下可得沮 入。N是棘。小。N的RlesZ 基． （h）＊iesz框架摄动．在一般框架摄动条件maxu十…八7，人）＜1和尸＊）式 下，引入条件O＋AZ）厄／刀＋p／刀 ＜1 得到 Ri22框架的摄动结果如下： 定理 3．6 设｛fiL叩是 H的 Riesz框架，界为人B．｛gi｝；。。二 H，如果存 十、、＿．．－1 休l、；卜、而；0－1 口。＿＿＿。，尸儿，。T＼体口 在人l，人2，P＞0仅《1＋人2）羌＋为＜＊且V＊1，＊2，…＊＊ Eq＊ E w）俩尾。 I二c＜（人一＊云）【巴 入＊【二～人11十｝I二C＜刃《I冲 以【二卜 门叁， i＝h＝h＝h＝1 则｛J。｝；。＊是 H的 Riesz脚，且…1一～）‘，B（1＋一）’． （iii）Bessclian框架的摄动．利用 Besselian框架和其预框架算子是 iedholm 算子的—一对应关系得到主要结果： 定理 3＊ 设｛fi｝；＊＊是 H的 Besselian框架，切小＊＊ C H，如果 二s1b一人＊ ＜①，则｛ ｝。N是顽 ｝。厂的＊SSell。n框架． （Th pJ用不等式 11 Z？Hlqg；ti S圳ZL。qfill＋顺｛二义＊蚓i得到形如 巳人＋U29小。。的框架的摄动结果（命题 3二0）． 在命题3＊ 下或对其改进条件后得到iesz基、Besselian框架，Riesz框 tv 架的摄动结果定理3二8，3．20，并得到常用的形如（人＋M小。。的框架摄动的 结论一推论3．17，3．19，3．ZI．列之如下： 定理 3．18 设｛n｝；。＊是 H的 Bessllan框架，界为＊B．切｝。＊ G H， UI：H一 K是可逆算子，UZ：H＋ K是有界线性算子．如果 SM＞0，q三0】使（oa 干顶I队＊卯ID＜刀且 ＊q｝E妒歹有S I二cz了2115 圳二＊人【诅 而（二冲IV 2＝11＝h＝l 贝（UI人＋U2＋小。。是 K的 Besselian框架． 定理 3．20 设｛人｝t。。是 H的 Riesz框架，下上界分别为人B．切小。。G H， q：H一凡的：H一K有界线性算子．如果三MZ0，尸三0，使 阳屈十顶）UZ川IU；hl＜刀 且 V｛C土｝ C E＊有二更多还原

【Abstract】 We consider frames which contain a Riesz basis in Hilbert space and focus our attention on those general characteristics,stability,disjoint and alternate dual. Inspired by the work of James R.Holub on [11], we come to some conclusions for the general traits, perturbation ,alternatc dual of Bcsselian frame ,besides generalize some of stability result on Riesz basis in view of [5],[9].We show some perturbation results of Riesz frame by adding a slightly strong condition to the ordinary ones .Also,we study alternate dual and disjoint for frames which contain a Riesz basis and discuss their relations.There are four sections in this paper.The basic elements of our approach to frames are contain in section one .A sequence of victors on a Hilbert space H is called a frame if there are constants , such thatAs it’s a important way to frame study by operators, we introduce preframe operator and frame operator associated with a frame {fi}i N and show corresponding relations between several frames and their preframe operators.Also, we introduce some definitions such as similarity of frames ,weak disjoint ,disjoint and strong disjoint.In the study of frame’s traits,we find :If H are frame sequence disjoint with each other ,then is a frame for with an added condition .Finally,we show a general perturbation result for frame as an end to this section .In second section ,we study relations among several frames and their stability .To begin with ,we show what do Riesz basis ,Besselian frame, Riesz frame . A frame which satisfied projection method mean . With the contribution in [14],we reach the following :vn(i) Riesz basis and Riesz frame are frames which satisfied projection method,The inverse is not true.(ii) Besselian frame needn’t satisfied projection method .(iii) Riesz basis is both Riesz frame and Besselian frame. The inverse is not true.(iv) Riesz frame needn’t a Besselian frame .The inverse is exactly the same.Among others ,we get the results:(i) Besselian frame is still the same kind suppose it is acted by a bound linear operator.(ii) Riesz frame and satisfied method frame are the same sort respectively if they are acted by a linear bound operator with it’s inverse existed and bound.In particular , The stability for Riesz basis ,Bessclian frame ,Riesz frame is provable if they are acted by a invertible opetator respectively.In third section, we study perturbations for frames which contain a Riesz ba-sis.There are two ways to approach it:(I) using the inequality of frame perturbation :(i)The perturbation for Riesz basis ,we extend the results in [9],[20]. e.g.If is a Riesz basis for only isn’t contained in span ,then is a Riesz basis for span by using ().(ii) The perturbation for Riesz frame .We refer to the theorem 3.C for a review.Theorem3.6 If is a Riesz frame for Hilbert space H with frame bound A,B. H is a vector sequence .If there exists , which satisfiedVlllthen is aRiesz frame for H with bound A(iii) The perturbation for Besselian frame. We obtain the main result theorem 3.12 by using the relation between Besselian frame and it’s preframe operator .ThcorcmS. 12 If is a Besselian frame for Hilbert space H ,v C H, Given IS a Besselian frame for span(II)Using the inequality :and the perturbation results for frame formed into(see Proposition 3.16)With the above we gain :Theorems. 18 If {fi}i^N is a Besselian frame for H with frame bound A,B.{gj}i€^ C H and U\ : H ?i K is a invertible operator ,t/2 : H ?t K is a linear bound operator . If 3M > 0,/3 < 0, such thatthen is a Besselian frame for K.Theorem3. 20 If {/ijieN is a Riesz frame for H with frame bound A,B. H , are linear bound operators.Surpose 0 such that :then frame for K.In fourth section, we discuss the characterization of alternate dual and disjoint.IXThe principle result concerned general frame is Theorem 4.6:If are frames for H, then is a frame for H with the assumption in proposition 4.5. As for Besselian fr更多还原