Including The Nature Of The Riesz Basis Framework

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...