Perturbations on K-fusion frames
Animesh Bhandari Saikat Mukherjee





A. Bhandari]A. Bhandari

Department of Mathematics
 National Institute of Technology

Meghalaya, India
animesh@nitm.ac.in

S. Mukherjee]S. Mukherjee

Department of Mathematics
 National Institute of Technology

Meghalaya, India
saikat.mukherjee@nitm.ac.in
Second author is partially supported by DST-SERB project MTR/2017/000797


[2010]42C15



 K-fusion frames are generalizations of fusion frames in frame theory. This article characterizes various kinds of property of K-fusion frames. Several perturbation results on K-fusion frames are formulated and analyzed.


Perturbations on K-Fusion Frames
    [
    December 30, 2023
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§ INTRODUCTION


The concept of Hilbert space frames was first introduced by Duffin and Schaeffer <cit.> in 1952. Later, in 1986, frame theory was reintroduced and popularized by Daubechies, Grossman and Meyer <cit.>. Since then frame theory has been widely used by mathematicians and engineers in various fields of mathematics and engineering, namely, operator theory <cit.>, harmonic analysis <cit.>, signal processing <cit.>, sensor network <cit.>, data analysis <cit.>, etc.

Frame theory literature became richer through several generalizations-fusion frame (frames of subspaces) <cit.> , G-frame (generalized frames) <cit.>, K-frame (atomic systems) <cit.>, K-fusion frame (atomic subspaces) <cit.>, etc. and these generalizations have been proved to be useful in various applications.


This article focuses on study, characterize and explore several properties of K-fusion frame. It is organized as follows. Section <ref> is devoted to the basic definitions and results related to frames and fusion frames. The characteristics of K-fusion frames are discussed in Section <ref>. Finally, results related to perturbation and erasure properties are established  in Section <ref>.

Throughout the paper, ℋ is a separable Hilbert space. We denote by ℒ(ℋ_1, ℋ_2) the space of all bounded linear operators from  ℋ_1 into ℋ_2, and ℒ(ℋ) for ℒ(ℋ, ℋ). For T ∈ℒ(ℋ), we denote D(T), N(T) and R(T) for domain, null space and range of T, respectively. For a collection of closed subspaces 𝒲_i of ℋ and scalars v_i, i∈ I, the weighted collection of closed subspaces { (𝒲_i , v_i) }_i ∈ I is denoted by 𝒲_v. We consider the index set I to be finite or countable.



§ PRELIMINARIES


In this section we recall basic definitions and results needed in this paper. We refer the books of Ole Christensen <cit.> and Casazza et.al. <cit.> for an introduction to frame theory.



 §.§ Frame
 A collection  { f_i }_i∈ I in ℋ is called a frame if there exist constants A,B >0 such that 
    Af^2 ≤ ∑_i∈ I |⟨ f,f_i⟩|^2  ≤ Bf^2,
for all f ∈ℋ. The numbers A, B are called frame bounds. The supremum over all A's and infimum over all B's satisfying above inequality are called the optimal frame bounds.

If a collection satisfies only the right inequality in (<ref>), it is called a Bessel sequence.

Given a frame {f_i}_i∈ I of ℋ. The pre-frame operator or synthesis operator is a bounded linear operator T: l^2(I)→ℋ and is defined by T{c_i} = ∑_i∈ I c_i f_i. The adjoint of T, T^*: ℋ→ l^2(I), given by T^*f = {⟨ f, f_i⟩}, is called the analysis operator. The frame operator, S=TT^*: ℋ→ℋ, is defined by 
    Sf=TT^*f = ∑_i∈ I⟨ f, f_i⟩ f_i.
 It is well-known that the frame operator is bounded, positive, self adjoint and invertible.











 §.§ Fusion Frame
 Consider a weighted collection of closed subspaces, 𝒲_v, of ℋ. Then 𝒲_v  is said to be a fusion frame for ℋ, if there exist constants 0<A ≤ B <∞ satisfying

    Af^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ Bf^2,

where P_𝒲_i is the orthogonal projection from ℋ onto 𝒲_i.

The constants A and B are called fusion frame bounds.

A collection of closed subspaces, satisfying only the right inequality in (<ref>), is called a fusion Bessel sequence.




Additionally, it is to be noted that for every fusion frame 𝒲_v, there are frames { f_ij}_j ∈ J_i for each 𝒲_i and these are called local frames for 𝒲_v. It is a well-known fact that 𝒲_v is a fusion frame if and only if { w_i f_ij}_j ∈ J_i, i ∈ I is a frame for ℋ, for details readers are referred to Theorem 3.2 in <cit.>.


For a family of closed subspaces, {𝒲_i }_i ∈ I, of ℋ, the associated l^2 space is defined by (∑_i ∈ I⊕𝒲_i)_l^2 = {{ f_i }_i ∈ I : f_i ∈𝒲_i, ∑_i ∈ If_i^2 < ∞} with the inner product ⟨{ f_i }, { g_i }⟩ = ∑_i ∈ I⟨ f_i , g_i ⟩_ℋ.

Let 𝒲_v be a fusion frame. Then the associated synthesis operator T_𝒲: (∑_i ∈ I⊕𝒲_i)_l^2→ℋ is defined as T_𝒲 (f)=∑_i ∈ I v_i f_i for all { f_i }_i ∈ I∈ (∑_i ∈ I⊕𝒲_i)_l^2 and the analysis operator T^*_𝒲: ℋ→  (∑_i ∈ I⊕𝒲_i)_l^2 is defined as  T^*_𝒲 (f)={ v_i P_𝒲_i (f) }_i ∈ I. It is well-known that (see <cit.>) the synthesis operator T_𝒲 of a fusion frame is bounded, linear and onto, whereas the corresponding analysis operator T_𝒲 ^*  is (possibly into) an isomorphism. Corresponding fusion frame operator is defined as S_𝒲 (f) = T_𝒲T_𝒲^*(f)=∑_i ∈ I v_i ^2 P_𝒲_i(f).  S_𝒲 is bounded, positive, self adjoint and invertible. Any signal f ∈ℋ can be expressed by its fusion frame measurements { v_i P_𝒲_i f }_i ∈ I as

    f=∑_i ∈ I v_i S_𝒲 ^-1 (v_i P_𝒲_i f) .




 §.§ K-fusion frame

In <cit.>, authors, introduced a generalization of fusion frame, K-fusion frame, and scrutinized the equivalence between atomic subspaces and K-fusion frames. K-fusion frame is used to reconstruct signals from range of a bounded linear operator K.

(K-fusion frame)
Let K∈ℒ(ℋ),  𝒲_v={(W_i,v_i) }_i ∈ I be a weighted collection of closed subspaces of ℋ. Then   𝒲_v is said to be a K-fusion frame for ℋ  if there exist positive constants A, B such that for all f ∈ℋ we have

    AK^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ Bf^2.





In the rest of this Section, we recall some fundamental results in Hilbert space that are necessary to present some outcomes of this article.

(Douglas' factorization theorem <cit.>) Let ℋ_1, ℋ_2, and ℋ be Hilbert spaces and S∈ℒ(ℋ_1, ℋ), T∈ℒ(ℋ_2, ℋ). Then the following are equivalent:


  * R(S) ⊆ R(T).

  * SS^* ≤α TT^* for some α >0.

  * S = TL for some L ∈ℒ(ℋ_1, ℋ_2).




(Moore-Penrose pseudo-inverse <cit.>) Let ℋ and 𝒦 be two Hilbert spaces and T ∈ℒ(ℋ,𝒦) be a closed range operator, then the followings hold:


  * TT^†=P_T(ℋ), T^†T=P_T^*(𝒦)

  * f/T^†≤T^*f for all f ∈ T(ℋ).

  * TT^†T=T, T^†TT^†=T^†, (TT^†)^*=TT^†, (T^†T)^*=T^†T.





(<cit.>)  Suppose  ℋ and 𝒦 be two Hilbert spaces and T ∈ℒ(ℋ,𝒦). consider 𝒲 be a closed subspace of ℋ and 𝒱 be a closed subspace of 𝒦. Then we have the followings:


  * P_𝒲T^*P_T 𝒲=P_𝒲T^*.

  * P_𝒲T^*P_𝒱=P_𝒲T^*. if and only if T 𝒲⊂𝒱.





(Drazin inverse <cit.>)  Let S, T ∈ℒ(ℋ), S is said to be the Drazin inverse of T if we have the following:


  * STS=S.

  * ST=TS.

  * TST^k=T^k, for some positive integer k.



It is to be noted that T ∈ℒ(ℋ) has the Drazin inverse in ℒ(ℋ) if and only if λ =0 is a pole of the resolvent operator (λ I - T)^-1. Moreover, the order of the pole is equal to the index of T. In particular 0 is not an accumulation point in the spectrum σ(T).




§ CHARACTERIZATION OF K-FUSION FRAMES

In this section, we characterize various properties of K-fusion frame. The following theorem provides a sufficient condition on a bounded, linear operator K under which the image of K-fusion frame remains a K-fusion frame.

 Let K ∈ℒ(ℋ) be an idempotent, closed range operator and  𝒲_v  be a K-fusion frame for ℋ with K^†K(𝒲_i)⊂𝒲_i. Then { (K 𝒲_i, v_i)}_i ∈ I constitutes a K-fusion frame for ℋ.


 First we prove for all i ∈ I, K(𝒲_i) is a closed subspace in  ℋ. Since K^†K(𝒲_i)⊂𝒲_i, K K^†K(𝒲_i)⊂ K(𝒲_i). Therefore using the Lemma 2.5.2 in <cit.> we have KK^* (KK^*)^-1K(𝒲_i)⊂ K(𝒲_i) and hence K(𝒲_i) is a closed subspace in  ℋ for all i ∈ I. Since { (𝒲_i, v_i)}_i ∈ I is a K-fusion frame in ℋ, there exist A, B>0 such that for all f ∈ℋ we have, 
    A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2.
 Again as K is idempotent, using the Lemma <ref> we obtain, 
    A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_i K^*f^2 ≤K^2 ∑_i ∈ I v_i ^2 P_K 𝒲_i f^2 .
 Therefore A/K^2K^*f^2 ≤∑_i ∈ I v_i ^2 P_K 𝒲_i f^2.

Again from the Lemma <ref> we have P_K 𝒲_i = P_K 𝒲_i K^† * P_𝒲_i K^*.  Therefore for all f ∈ℋ we obtain, 
    ∑_i ∈ I v_i ^2 P_K 𝒲_i f^2 ≤K^† *^2 ∑_i ∈ I v_i ^2 P_𝒲_i K^*f^2 ≤  B K^† *^2 K^2 f^2.
 Hence our assertion is tenable.



In the next result, we further characterize K-fusion frame by means of Drazin inverse.

 Let K∈ℒ(ℋ)  has non-zero Drazin inverse, S. Also suppose that  𝒲_v is a K-fusion frame for ℋ. Then the following hold:



  * 𝒲_v is a SKS-fusion frame for ℋ.


  * 𝒲_v is a SK-fusion frame in ℋ.



 Since 𝒲_v is a K-fusion frame for ℋ, there exist A, B>0 such that for all f ∈ℋ we have, A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2. Again as S is the non-zero Drazin inverse of K, for all f ∈ℋ we have


    A/S^ 4(SKS)^* f^2 ≤ A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2.
 and also 
    A/S^2(SK)^* f^2 ≤ A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2.

Hence the conclusions follow.


 It is to be noted that 𝒲_v is also a KS-fusion frame for ℋ but this result is obvious as for any (0≠) T ∈ℒ(ℋ),  
    A/T^2(KT)^*f^2 ≤ A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2,
 for all f ∈ℋ.



In the following theorem we scrutinize the robustness of K-fusion frames under erasure property.


 Let K ∈ℒ(ℋ) be a closed range operator and  𝒲_v be a K-fusion frame for ℋ with bounds A and B. Suppose J⊆ I such that ∑_i ∈ J v_i ^2 = C<∞ with (A- C K^†^2) >0. Then { (𝒲_i, v_i)}_i ∈I ∖ J forms a K-fusion frame for R(K) with bounds (A- C K^†^2) and B.


 Since K has closed range, for all f ∈  R(K) we have for all i∈ I,

    P_𝒲_i f ≤K^†K^*f
 and hence
 ∑_i∈ Jv_i^2P_𝒲_if^2 ≤ C K^†^2 K^*f^2 for all f ∈ R(K). Consequently for all f ∈ R(K),
 
    (A- C K^†^2) K^* f^2 = AK^* f^2 - C K^†^2 K^*f^2  ≤∑_i∈I ∖ Jv_i^2P_𝒲_if^2.

The upper bound follows directly from the assumption.





§ PERTURBATION PROPERTIES


In this section we analyze stability conditions of K-fusion frames under perturbations.


  Let K_1 ∈ℒ(ℋ), 𝒲_v be a K_1-fusion frame for ℋ. Suppose  K_2 ∈ℒ(ℋ) and a, b≥ 0 such that 
    (K_1 ^* - K_2 ^*)f≤ aK_1 ^* f + b K_2 ^* f,  for all f ∈ℋ.
 Then 𝒲_v is also a K_2-fusion frame for ℋ if b<1.



 Since 𝒲_v is K_1-fusion frame for ℋ, there exist A , B>0, for all f ∈ℋ we have A K_1 ^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2. Now for all f ∈ℋ we obtain

    K_2 ^*f≤(K_1 ^* - K_2 ^*)f + K_1 ^*f≤ (1+a)K_1 ^*f + bK_2 ^*f.

Therefore for all f ∈ℋ we have

    A(1-b/1+a)^2    K_2 ^*f^2 ≤ A K_1 ^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2.

Hence our assertion is tenable.



  Let K_1 ,K_2 ∈ℒ(ℋ) and 0≤ a,  b < 1 so that for all f ∈ℋ, 
    (K_1 ^* - K_2 ^*)f≤ aK_1 ^* f + b K_2 ^* f.
 Also suppose, 𝒲_v is a weighted collection of closed subspaces of ℋ. Then 𝒲_v is K_1-fusion frame for ℋ if and only if it is K_2-fusion frame for ℋ.



 The proof follows from above Lemma <ref> by interchanging the roles of K_1 and K_2.





Above results immediately provide the following result.
  Let K ∈ℒ(ℋ), 𝒲_v be  K-fusion frame for ℋ with 
    K^* f-f≤ a K^* f + b f,  for all f ∈ℋ,
 where 0≤ a,  b <1. Then 𝒲_v is fusion frame for ℋ.



Given a fusion Bessel sequence or K-fusion frame or fusion frame, can we construct a K-fusion frame? Following results address this using perturbations on projection operators.

 Let a, b, c≥ 0 and 𝒲_w, 𝒱_v be two weighted collections of closed subspaces of ℋ so that for all f∈ℋ, 
    (∑_i ∈ I(w_i P_𝒲_i - v_i P_𝒱_i)f ^2 )^1/2≤  a (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 + b  (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2 + c  Λ(f),
 for some Λ:ℋ→ℝ_+. Then the following results hold:


  * Let 𝒲_w be a fusion Bessel sequence in ℋ and b<1, c=0. Then 𝒱_v is a K-fusion frame for ℋ for any K ∈ℒ(ℋ) satisfying R(K) ⊆ R(T_𝒱), where T_𝒱 is the associated synthesis operator of   𝒱_v.


  *   Let K ∈ℒ(ℋ), 𝒲_w be a K-fusion frame for ℋ with bounds A,  B>0 and Λ(f)=K^*f. Then if a <1,  b < 1,  0≤c/1-a<√(A), 𝒱_v is also a  K-fusion frame for ℋ with bounds (√(A) (1-a) -c/(1+b))^2 and ((1+a) √(B) + c K/(1-b))^2.


  * Let 𝒲_w be a fusion frame for ℋ with bounds A,   B>0 and Λ(f) = f. Then 𝒱_v forms a fusion frame for ℋ with bounds (√(A) -c-a√(B)/1+b)^2 and ((1+a)√(B) +c/1-b)^2, if a√(B) + c <√(A),   b < 1.






  * Since 𝒲_w is a fusion Bessel sequence in ℋ, there exists B>0 such that for all f ∈ℋ we have ∑_i ∈ I w_i ^2 P_𝒲_if^2 ≤ B f^2. Using Minkowski's inequality, for all f ∈ℋ, we obtain


    (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2   ≤   (∑_i ∈ I(w_i P_𝒲_i - v_i P_𝒱_i)f ^2 )^1/2 +  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2
       ≤    (1+a)  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 + b  (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2,

Hence 
    ∑_i ∈ I v_i ^2 P_𝒱_if^2 ≤√(B)(1+a/1-b)^2 f^2,   ∀  f∈ℋ.

Therefore T_𝒱^* and hence T_𝒱 is well-defined.
The left hand inequality directly follows from Theorem <ref> .


  * Since 𝒲_w is a K-fusion frame for ℋ with bounds A , B >0, for all f ∈ℋ we have A K^*f^2 ≤∑_i ∈ I w_i ^2 P_𝒲_if^2 ≤ B f^2. Now

    (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2   ≤   (∑_i ∈ I(w_i P_𝒲_i - v_i P_𝒱_i)f ^2 )^1/2 +  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2
       ≤    (1+a)  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 + b  (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2 + c K^*f,

for all f ∈ℋ. Hence

    ∑_i ∈ I v_i ^2 P_𝒱_if^2 ≤((1+a) √(B) + c K/(1-b))^2 f^2,   ∀  f∈ℋ.


Similarly, for the lower bound we have,

    (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2≤ a (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 + (1+b) (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2  + c K^*f,
 for all f ∈ℋ. Hence we obtain

    (√(A) (1-a) -c/(1+b))^2 K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒱_if^2,   ∀ f∈ℋ.



  * Since 𝒲_w is a fusion frame for ℋ with bounds A , B >0, for all f ∈ℋ we have A f^2 ≤∑_i ∈ I w_i ^2 P_𝒲_if^2 ≤ B f^2. Also for all f ∈ℋ we obtain

    (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2   ≥   (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 - (∑_i ∈ I(w_i P_𝒲_i - v_i P_𝒱_i)f ^2 )^1/2
       ≥    (√(A) - c) f - a  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 - b  (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2,

Therefore  for all f ∈ℋ we obtain 
    (√(A) -c-a√(B)/1+b)^2 f^2 ≤∑_i ∈ I v_i ^2 P_𝒱_if^2.


Moreover, for all f ∈ℋ we have

    (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2   ≤   (∑_i ∈ I(w_i P_𝒲_i - v_i P_𝒱_i)f ^2 )^1/2 +  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2
       ≤    (1+a)  (∑_i ∈ I w_i ^2 P_𝒲_if^2)^1/2 + b  (∑_i ∈ I v_i ^2 P_𝒱_if^2)^1/2 + c f,

Hence for all f ∈ℋ we obtain 
    ∑_i ∈ I v_i ^2 P_𝒱_if^2 ≤ ((1+a)√(B) +c/1-b)^2 f^2.




We acknowledge that recently Li and Leng <cit.> proved a similar result as stated in the second statement of above theorem. We present the result here as this work has been done almost simultaneously with the work of Li and Leng.




In the following proposition we discuss another perturbation condition on the projection operators to obtain a K-fusion frame.

  Let K ∈ℒ(ℋ), 𝒲_w be a K-fusion frame for ℋ with bounds A,  B>0. Also suppose 𝒱_v  is any weighted collection of closed subspaces  of ℋ so that for all f ∈ℋ,

    ∑_i ∈ I |⟨ f, (w_i ^2 P_𝒲_i - v_i^2 P_𝒱_i)f ⟩| ≤ R K^* f^2,  where   0<R<A.
 Then 𝒱_v  forms a K-fusion frame for ℋ with bounds (A-R) and  (B+RK).



 Since 𝒲_w is a K-fusion frame for ℋ with bounds A, B>0, we have 
    A K^*f^2 ≤∑_i ∈ I v_i ^2 P_𝒲_if^2 ≤ B f^2,   ∀ f ∈ℋ.
 Now for all f ∈ℋ we obtain

    ∑_i ∈ I w_i^2P_𝒲_if^2    ≤   ∑_i ∈ I |⟨ f, (w_i ^2 P_𝒲_i - v_i^2 P_𝒱_i)f ⟩| + ∑_i ∈ I v_i^2P_𝒱_if^2  
       ≤    R K^* f^2 + ∑_i ∈ I v_i^2P_𝒱_if^2

Therefore for all f ∈ℋ we have, (A-R)K^* f^2 ≤∑_i ∈ I v_i ^2 P_𝒱_if^2.

Similarly, ∑_i ∈ I v_i ^2 P_𝒱_if^2 ≤ (B+RK) f^2, for  all f ∈ℋ.




In the following two results we analyze perturbation conditions under which a fusion Bessel sequence forms a K-fusion frame.

  Let 𝒲_w be a fusion Bessel sequence in ℋ with bound B>0 and J ⊊ I with T_𝒲 is the associated synthesis operator of { (𝒲_i, w_i)}_i ∈I ∖ J. Let a, b≥ 0 and K ∈ℒ(ℋ) satisfying (K^* - T_𝒲 T_𝒲^*)f≤ a K^*f + b T_𝒲^* f for all f ∈ℋ. Then { (𝒲_i, w_i)}_i ∈I ∖ J forms a K-fusion frame for ℋ with bounds (1-a/b+T_𝒲)^2 and B if a< 1.


 We have for  all f ∈ℋ, K^*f≤(K^* - T_𝒲T_𝒲^*)f + T_𝒲T_𝒲^*f≤  a K^*f + (b+T_𝒲)T_𝒲^* f. Therefore 
    (1-a/b+T_𝒲)^2 K^*f^2 ≤∑_i ∈I∖ J w_i^2 P_𝒲_if^2  ≤∑_i ∈ I w_i^2 P_𝒲_if^2 ≤ B f^2,
 for all f∈ℋ.





 Let 𝒲_w be a fusion Bessel sequence in ℋ with bound B>0 and let  J ⊊ I so that the associated  synthesis operator of { (𝒲_i, w_i)}_i ∈I ∖ J  is T_𝒲. Let a, b, c≥ 0 and K ∈ℒ(ℋ) be a closed range operator such that (K^* - T_𝒲T_𝒲^*)f≤ a K^*f + b T_𝒲^* f+c f for all f ∈ℋ. Then if a+cK^† <1, { (𝒲_i, w_i)}_i ∈I ∖ J is a K-fusion frame for R(K) with bounds (1-a-cK^†/b+T_𝒲) and B.


 For all f ∈ℋ we have, K^*f≤(K^* - T_𝒲T_𝒲^*)f + T_𝒲T_𝒲^*f≤  a K^*f + (b+T_𝒲)T_𝒲^* f+c f and hence for all f ∈ R(K),

    (1-a-cK^†)K^*f≤(b+T_𝒲)T_𝒲^* f.
 Therefore for all f ∈ R(K), we have the following: 
    (1-a-cK^†/b+T_𝒲) K^*f≤T_𝒲^* f = ∑_i ∈I∖ J w_i^2 P_𝒲_if^2 ≤∑_i ∈ I w_i^2 P_𝒲_if^2 ≤ B f^2.
 Consequently, our declaration is sustainable.






§ ACKNOWLEDGMENT


The first author acknowledges the financial support of MHRD, Government of India.


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