In this work, we consider some relationships between a closed range operator T and a fusion frame W=(Wi,wi)i∈I for a Hilbert space H that provides that the sequence (T(Wi)‾,vi)i∈I is a fusion frame sequence for H, if we consider a suitable family of weights {vi}i∈I. This (sufficient) condition generalizes some previous work in the subject.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Calderón, Pablo Luis

Ruiz, Mariano Andrés

Journal of Mathematical Analysis and Applications

English

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arXiv:1709.06872v1  [math.FA]  20 Sep 2017A note on perturbations of Fusion FramesM. Ruiz and P. Caldero´nOctober 6, 2018AbstractIn this work, we consider some relationships between a closed range operator Tand a fusion frame W = (Wi, wi)i∈I for a Hilbert space H that provides that thesequence (T (Wi), vi)i∈I is a fusion frame sequence for H, if we consider a suitablefamily of weights {vi}i∈I . This (sufficient) condition generalizes some previous work inthe subject.1 IntroductionLet H be a separable Hilbert space. Let {Wi}i∈I be a family of closed subspaces of H and{wi}i∈I be a family of positive weights. The pair W = (Wi, wi)i∈I is a fusion frame for Hif there exists A,B > 0 which satisfy thatA‖f‖2 ≤∑i∈Iw2i ‖PWif‖2 ≤ B‖f‖2 for every f ∈ H (1)where by PW we denote the orthogonal projection onto W . We say that W = (Wi, wi)i∈I isa fusion frame sequence for H if it is a fusion frame for a closed proper subspace of H.Fusion frames were introduced by P. Casazza and G. Kutyniok in [3], under the name offrame of subspaces. They were motivated by the problem of finding a way to join togetherlocal frames in order to get global frames. These authors were also studying methods todecompose a frame into a family of frame sequences. Both problems lead them to definethe concept of frame of subspaces, which clearly generalizes classical vector frames. Namely,given a frame {fi}i∈I for H, then (span{fi}, ‖fi‖)i∈I constitutes a fusion frame (of one-dimensional subspaces) for H.During the last decade, fusion frame theory has been a fast-growing area of research,driven by several applications such as sensor networks [13], neurology [14], coding theory[15], among others.As well as for vector frames, there are some bounded operators associated to a fusionframe. First, we set the Hilbert space KW :=⊕i∈I Wi (endowed with the ℓ2 norm) and wedefine the synthesis operator TW : KW → H and the analysis operator T ∗W : H → KWof (Wi, wi)i∈I given by:TW(g) =∑i∈Iwi gi and T∗W(f) = {wi PWif}i∈I .1It is clear that some properties and problems in classical vector frames need to be treatedin a completely different way if we are in the fusion frame setting. For example, problemsof design of frames, such as the existence and construction of finite frames with prescribednorms need to be attacked with different tools if they are posed for fusion frames (see[2, 4, 17]).Also, the equivalence between frames and epimorphisms (see [7]), useful in abstract frametheory, is not longer true for fusion frames (see for example [18]). Related to this, there isanother interesting issue to study in the fusion frame setting: linear perturbation of fusionframes. That is, the goal is to find conditions on a bounded linear operator T in H anda fusion frame W = (Wi, wi)i∈I in order that the perturbed sequence (T (Wi), wi)i∈I is afusion frame for T (H). In [3], [11], [9] and [18] it is shown that for an invertible operatorT it is always true that (T (Wi), wi)i∈I is a fusion frame for H. Later, in [10] and [12] theauthors proved that the perturbed sequence is a fusion frame sequence (or fusion frame ifT is surjective), by adding some extra conditions (such as T ∗T (Wi) ⊂ Wi, for example).As it is noticed by the authors in [11], these hypothesis can be described in a geometricalway by means of the gap between T ∗T (Wi) (or PR(T ∗)(Wi)) and Wi. The gap between twosubspaces M and N , denoted by δ(M,N) is the supremum among the distances d(x,N),where x is in M and ‖x‖ = 1 (see [16] for results and properties of the gap). Namely, in[11], the authors show that if the gaps δ(T ∗T (Wi),Wi) are uniformely bounded above by aconstant c < 1 then the sequence (T (Wi), wi)i∈I is a fusion frame sequence for H. In thesame article, the authors presents alternative sufficient conditions which are related to theprevious one, replacing the Grammian operator T ∗T by the orthogonal projection onto therange of T ∗ (see [11, Cor 2.14]).In this note, we follow the techniques and tools used in [18] to establish a sufficientcondition that slightly generalizes those found in [11] using the notion of angle betweensubspaces.2 PreliminariesWe shall consider different notions that provides some information about the geometricalposition between two subspaces. They are the Friedrichs and Dixmier angles (angle andminimal angle respectively) and the gap defined by Kato. They will play a central role inthe study of perturbation of fusion frames.Definition 2.1. Given two closed subspaces M and N of a Hilbert space H, the anglebetween M and N is the angle in [0, π/2] whose cosine is defined byc (M, N ) = sup{ | 〈x, y〉 | : x ∈M ⊖N, y ∈ N ⊖M and ‖x‖ = ‖y‖ = 1}.We get the minimal angle between M and N as the angle in [0, π/2] whose cosine isdefined byc0 (M, N ) = sup{ | 〈x, y〉 | : x ∈M, y ∈ N and ‖x‖ = ‖y‖ = 1}.We list below some of the properties that satisfy these constants:Proposition 2.2. Let M and N be closed subspaces of H. Then21. c (M, N ) = c (N, M ) = c(M⊥, N⊥)2. c (M, N ) = c (M, N ⊖M ) = c0 (M, N ⊖M )3. c (M, N ) = ‖PMPNP(M∩N)⊥‖4. c0 (M, N ) = ‖PMPN‖The minimal angle is also related with the gap between closed subspaces of H:Definition 2.3. Let M and N be closed subspaces of H. The gap between M and N isdefined asδ(M,N) = supx∈M, ‖x‖=1dist(x,N) = ‖PN⊥PM‖ = c0(N⊥, M).The following result shows the closed connection between angles and closed range oper-ators.Proposition 2.4. Let A,B ∈ L(H) be closed range operators. Then, AB has closed rangeiff c (R(B), N(A) ) < 1.For more details and properties we refer the reader to the work by Deutsch [8].For an operator T ∈ L(H) we define its reduced minimum modulus byγ(T ) := inf{‖Tx‖ | ‖x‖ = 1 , x ∈ N(T )⊥}. (2)It is well known that γ(T ) = γ(T ∗) = γ(T ∗T )1/2 = ‖T †‖−1, where T † is the Moore-Penrosepseudoinverse. Also, it can be shown that an operator T has closed range if and only ifγ(T ) > 0.The following proposition will be useful for our results. It relates the notion of angle andreduced minimum modulus of a closed range operator:Proposition 2.5. (Prop. 4.14, [1]) Let T ∈ L(H) a closed range operator in H, W a closedsubspace of H and PM its orthogonal projection. Suppose that c := c (N(T ), W ) < 1. Then,γ(T )(1− c2)1/2 ≤ γ(TPW ) ≤ ‖T‖(1− c2)1/2. (3)Our next goal is to obtain inequalities that relate the gap δ(A(W ),W ) and the cosine ofthe Friedrichs’ angle between N(A) and W , where A is a closed range semidefinite positiveoperator and W is a closed subspace of H.Theorem 2.6. Let A ∈ L(H) be a semidefinite positive and closed range operator, W be aclosed subspace of H such that c (N(A), W ) < 1. Thenc (N(A), W ) ≤ c (A−1(W⊥), W ) ≤ [1− γ(A1/2)4‖A‖2 (1− c (N(A),W )2)]1/2.3Proof. Notice that A−1(W⊥) ∩W = N(A) ∩W . Hence, since N(A) ⊂ A−1(W⊥) we havethat N(A) ⊖ (N(A) ∩W ) ⊂ A−1(W⊥) ⊖ (A−1(W⊥) ∩W ) so the first inequality follows byDefinition 2.1.On the other hand, notice that A1/2PW has closed range, since N(A1/2) = N(A) andc (N(A), W ) < 1 by hypothesis. Therefore, by Prop. 2.5,γ(PWAPW ) = γ(A1/2PW )2 ≥ γ(A1/2)2 (1− c2 (N(A),W )) . (4)We also have that, by Prop. 2.5,γ(PWAPW ) ≤ ‖PWA‖(1− c (A−1(W⊥),W )2)1/2 ≤ ‖A‖(1− c (A−1(W⊥),W )2)1/2 (5)since N(PWA) = A−1(W⊥).Combining eqs. (4) and (5) we deduce thatc(A−1(W⊥), W) ≤ [1− γ(A1/2)4‖A‖2 (1− c (N(A),W )2)]1/2.Corollary 2.7. Let A ∈ L(H)+ be a closed range operator and {Wi}i∈I be a sequence ofclosed subspaces of H. Then, the following are equivalent:1. inf i∈I γ(APWi) > 0.2. supi∈I c (N(A), Wi ) < 1.3. supi∈I δ(A(Wi),Wi)< 1.Proof. The equivalence between 1. and 2. follows from Prop. 2.5. In order to prove 2. ⇔3. we claim that any of them implies that δ(A(Wi),Wi)= c(A−1(W⊥i ), Wi), for all i inI. Then, is clear that the rest of the proof follows by Thm. 2.6.Suppose first that c (N(A), Wi ) < 1. Therefore A(Wi) = A(Wi), which impliesδ(A(Wi),Wi)= ‖PW⊥iPA(Wi)‖ = ‖PW⊥iPA(Wi)P(W⊥i∩A(Wi))⊥‖= c(W⊥i , A(Wi))= c(Wi, A−1(W⊥i ))= c(A−1(W⊥i ), Wi).On the other hand, if δ(A(Wi),Wi)< 1 then A(Wi) ∩W⊥i = {0}, thereforeδ(A(Wi),Wi)= ‖PW⊥iPA(Wi)‖ = c0(A(Wi), W⊥i)= c(A(Wi), W⊥i)= c(Wi, A−1(W⊥i )).43 Fusion frame perturbationIn this section we apply the results stated previously to operator perturbations of fusionframes to obtain a sufficient condition that generalizes the results stated in [11] (see Cor.2.10 and 2.12 and Thm. 3.4). We show that these conditions can be reduced to check acompatibility relationship between N(T ) and the family {Wi}i∈I .We need the following theorem, stated in [18], which can be seen as a result concerningoperator perturbation of orthonormal basis of subspaces. In order to make the paper clearand self-contained, we state it in a general form, using perturbations by closed range opera-tors instead of surjective operators. It is worth to mention that the proof follows the samelines as the proof of the cited result. Notice that the family of weights of the perturbedsequence can vary.Theorem 3.1. [18] Let E = {Ei}i∈I be an orthonormal basis of subspaces for a Hilbert spaceK and let T ∈ L(K,H) be a closed range operator. Suppose that 0 < c := inf i∈I γ(TPEi )2‖TPEi‖2 .Let 0 < A ≤ B < ∞ such that, AB≤ c for all i ∈ I. Let w = {wi}i∈I ∈ ℓ∞+ (I) such that‖TPEi‖2B≤ w2i ≤ γ(TPEi)2Afor all i ∈ I . Then (T (Ei), wi)i∈I is a fusion frame sequence for H.Moreover, the optimal frame bounds for the sequence satisfy the inequalitiesγ(T )2B≤ AT (E) ≤ γ(T )2Aand‖T‖2B≤ BT (E) ≤ ‖T‖2A.From this result it can be easily deduced the following:Theorem 3.2. Let W = (Wi, wi)i∈I be a fusion frame for H and T ∈ L(H) a closed rangeoperator such that0 < c := infi∈Iγ(TPWi)2‖TPWi‖2. (6)Let 0 < A ≤ B < ∞ such that AB≤ c and suppose that the family of weights v = {vi}i∈I ∈ℓ∞+ (I) satisfyw2i‖TPWi‖2B≤ v2i ≤ w2iγ(TPWi)2Afor all i ∈ I ,then (T (Wi), vi)i∈I is a fusion frame sequence for K.Proof. Denote by Ej to the isometric copy of Wj in KW = ⊕i∈IWi, hence {Ej}j∈I is anorthonormal basis of subspaces of KW . Consider the composition of the synthesis operatorTW : KW →H with T :F (g) := TTW(g) =∑i∈Iwi T (gi), g ∈ KW .Notice that F ∈ L(KW , H) is a closed range operator, since T is closed range and TTW issurjective. It is clear that‖FPEi‖ = wi‖TPWi‖for all i ∈ I. (7)5On the other side, we claim that γ(FPEi) = wiγ(TPWi), i ∈ I. Indeed, from the equalityN(FPEi)⊥ = ((N(F ) ∩ Ei)⊕E⊥i )⊥ = Ei ∩ (N(F ) ∩ Ei)⊥it follows that every unitary vector x ∈ N(FPEi)⊥ has the form x = (0, . . . , xi, . . . , 0, . . .)with ||xi|| = 1 and xi ∈ (N(T ) ∩Wi)⊥.Then, if x ∈ N(FPEi)⊥ with ‖x‖ = 1, ‖FPEix‖ = ‖wiTxi‖ = wi‖TPWixi‖, thereforeγ(FPEi) ≥ wiγ(TPWi). (8)The reverse inequality can be proven in a similar way: we take x ∈ (N(T ) ∩Wi)⊥ andthe corresponding x˜ = (0, . . . , x, . . . , 0, . . .), then x˜ ∈ (N(F ) ∩ Ei)⊥. Then we have that‖TPWix‖ = 1/wi‖F x˜‖ = 1/wi‖FPEix˜‖ ≥ 1/wiγ(FPEi). (9)From (8) and (9) we have the equalityγ(FPEi) = wiγ(TPWi). (10)The result follows by Thm.3.1, applied to E = {Ei}i∈I and F ∈ L(KW ,H).Notice that in both results the equality T (Ei) = T (Ei) (resp. T (Wi) = T (Wi)) isguaranteed for every i ∈ I since γ(TPEi) > 0, ∀i ∈ I (resp. γ(TPWi) > 0, ∀i ∈ I).Remark 3.3. Suppose that W = (Wi, wi)i∈I is a fusion frame for H and T ∈ L(H) is aclosed range operator. Then, using Corollary 2.7 and the fact that ‖TPWi‖ ≤ ‖T‖, ∀i ∈ I,we have that any of the following conditions1. 0 < inf i∈I γ(TPWi)2. supi∈I c (N(T ), Wi ) < 1.3. supi∈I δ(T ∗T (Wi),Wi)< 1.implies Eq. (6). In particular, Theorem 3.2 generalizes all the sufficient conditions provedin [11].It is easy to construct examples in which Eq. (6) holds but inf i∈N γ(TPWi) = 0. Noticethat in such cases Thm. 3.2 implies, by considering a different family of weights (whichtends to 0), that the perturbed sequence is a fusion frame sequence.The following example shows that Eq. (6) is not neccesary to produce a fusion framesequence via an operator perturbation of a fusion frame for H:Example 3.4. Let {ei}i∈N be an orthonormal basis for H. Let {θk}∞k=0 be a sequence ofpositive numbers such that θ0 <pi4and limk→∞ θk = 0. For k = 0, 1, 2, . . ., letEk = span{e3k+1 , sin(θk)e3k+2 + cos(θk)e3(k+1)}andFk = span{e3k+1, 1√2e3k+2 +1√2e3(k+1)}.6Then, the family {Wi}i∈N of closed subspaces of H given by W2k+1 = Ek and W2(k+1) = Fkdetermines a fusion frame for H by considering the weights wi = 1, ∀i ∈ N (Thm. 3.2 [3]).Indeed, this claim follows by the fact that the union of the orthonormal basis that generateseach Ek and Fk make a frame of vectors of H. To be more precise, for each k, the family ofvectorsf1,k = e3k+1f2,k = sin(θk)e3k+2 + cos(θk)e3(k+1)f3,k = e3k+1f4,k =1√2e3k+2 +1√2e3(k+1)is a frame for span{e3k+1 , e3k+2 , e3(k+1)}, with frame bounds Ak = 1−√12+ cos(θk) sin(θk)and Bk = 2.Since the lower frame bounds are uniformly bounded by 0 < 1 −√12+ cos(θ0) sin(θ0)and the subspaces generated by these frame sequences are mutually orthogonal, the claimfollows.Let T ∈ L(H) be the orthonormal projection onto K = span{e3k+1 , e3k+2 , k = 0, 1, . . .}.Since T (Ek) = T (Fk) = span{e3k+1 , e3k+2} for every k, we deduce that (T (Wi), wi)i∈N is afusion frame sequence for H. On the other side, it is not difficult to show thatγ(TPWi) ={sin2(θk), for i = 2k + 112, for i = 2(k + 1)and ‖TPWi‖ = 1 for every i ∈ N. Therefore, inf i∈N γ(TPWi )‖TPWi‖ = 0, so Eq.(6) is not a necessarycondition in order to have a fusion frame sequence which comes from another frame sequenceafter a closed range operator perturbation.References[1] J. Antezana, G. Corach, D. Stojanoff and M. Ruiz, Weighted projections andRiesz frames, Lin. Alg. Appl. 402 (2005), pp. 367-389[2] P.G. Casazza, M. Fickus, D. Mixon, Y.Wang, Z. Zhou, Constructing tight fusionframes, Appl. Comput. Harmon. Anal., 30 (2011), pp. 175-187[3] P.G. Casazza, G. Kutyniok, Frames of subspaces, Contemp. Math., 345 (2004),pp. 87-113[4] R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, A. Pezeshki, Sparsefusion frames: existence and construction, Adv. Comput. Math., 35 (2011), pp.1-31[5] G. Corach and A. Maestripieri, Redundant decompositions, angles between sub-spaces and oblique projections. Publ. Mat. 54 no. 2 (2010), pp. 461-4847[6] G. Corach, A. Maestripieri and D. Stojanoff, Schur complements and obliqueprojections. Acta Sci.Math 67 (2001), pp. 439-459[7] G. Corach, M. Pacheco and D. Stojanoff, Geometry of epimorphisms and frames,Proceedings of the AMS, 132 (2004), pp. 2039-2049[8] F. Deutsch, The angle between subspaces in Hilbert space, Approximation the-ory, wavelets and applications (S. P. Singh, editor), Kluwer, Netherlands, (1995)pp. 107-130[9] P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl., 333 (2007),pp. 871-879[10] M.S. Asgari, New characterizations of fusion frames (frames of subspaces), Proc.Indian Acad. Sci. Math. Sci., 119 (3) (2009), pp. 369-382[11] X.B. Li, S.Z. Yang and Y.C. Zhu Some results about operator perturbation offusion frames in Hilbert spaces, J. Math. Anal. Appl., 421 (2014), pp. 1417-1427[12] J. Wang, Y.Zhu, Some properties of Fusion Frames, J. Fuzhou Univ. Nat. Sci.Ed., 7 (1) (2009), pp. 1-5[13] P. G. Casazza, G. Kutyniok, S. Li and C. J. Rozell, Modeling Sensor Networkswith Fusion Frames, Wavelets XII (San Diego, CA, 2007), SPIE Proc.6701,SPIE, Belling-ham, WA, (2007)[14] C.J. Rozell and D.H. 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A note on perturbations of fusion frames

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A note on perturbations of fusion frames

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