Some properties of K-frame in n-Hilbert space
aa r X i v : . [ m a t h . F A ] F e b Some properties of K -frame in n -Hilbert space Prasenjit Ghosh
Department of Pure Mathematics, University of Calcutta,35, Ballygunge Circular Road, Kolkata, 700019, West Bengal, Indiae-mail: [email protected]
T. K. Samanta
Department of Mathematics, Uluberia College,Uluberia, Howrah, 711315, West Bengal, Indiae-mail: mumpu − [email protected] Abstract
The notion of a K-frame in n-Hilbert space is presented and someof their characterizations are given. We verify that sum of two K-frames is also a K-frame in n-Hilbert space. Also, the concept oftight K-frame in n-Hilbert space is described and some properties ofits are going to be established.
Keywords:
Frame, K -frame, n-normed space, n-inner product space, Framein n-inner product space. In 1952, Duffin and Schaeffer introduced frames in Hilbert spaces in their funda-mental paper [5], they used frames as a tool in the study of nonharmonic Fourierseries. Later in 1986, the formal definition of frame in the abstract Hilbert spaceswere given by Daubechies, Grossman, Meyer [2]. A frame for a Hilbert space is ageneralization of an orthonormal basis and this is such a tool that also allows eachvector in this space can be written as a linear combination of elements from the framebut, linear independence among the frame elements is not required. Such frames playan important role in Gabor and wavelet analysis. Several generalizations of framesnamely, g -frame [16], K -frames [12] etc. have been introduced in recent times. K -frames for a separable Hilbert spaces were introduced by Lara Gavruta to study thebasic notions about atomic system for a bounded linear operator. In recent times, K -frame was presented to reconstruct elements from the range of a bounded linearoperator K in a separable Hilbert space. K -frames are more generalization than theordinary frames and many properties of ordinary frames may not holds for such gen-eralization of frames. Generalized atomic subspaces for operators in Hilbert spaceswere studied by P. Ghosh and T. K. Samanta [8] and they were also presented thestability of dual g -fusion frames in Hilbert spaces in [7].The concept of 2-inner product space was first introduced by Diminnie, Gahlerand White [3] in 1970’s. In 1989, A. Misiak [14] developed the generalization of a Prasenjit Ghosh & T. K. Samanta n ≥
2. Frame in n -Hilbert space was presented by P.Ghosh and T. K. Samanta [9]. They also discussed frame in tensor product of n -Hilbert spaces [10].In this paper, we shall present the notion of a K -frame relative to n -Hilbertspace and discuss some properties. Further it will be seen that the family of all K -frames is closed with respect to addition in n -Hilbert space. We also give the notionof a tight K -frame in n -Hilbert space.Throughout this paper, H will denote a separable Hilbert space with the innerproduct h · , · i and B ( H ) denote the space of all bounded linear operator on H . We also denote R ( T ) for range set of T where T ∈ B ( H ) and l ( N )denote the space of square summable scalar-valued sequences with index set N . Theorem 2.1. ( Douglas’ factorization theorem ) [4] Let
U, V ∈ B ( H ) . Thenthe following conditions are equivalent:( I ) R ( U ) ⊆ R ( V ) .( II ) U U ∗ ≤ λ V V ∗ for some λ > .( III ) U = V W for some W ∈ B ( H ) . Theorem 2.2. [6] Let
S, T, U ∈ B ( H ) . Then the following are equivalent:( I ) R ( S ) ⊆ R ( T ) + R ( U ) .( II ) S S ∗ ≤ λ ( T T ∗ + U U ∗ ) for some λ > .( III ) S = T A + U B for some
A , B ∈ B ( H ) . Theorem 2.3. ([1]) Let H , H be two Hilbert spaces and U : H → H bea bounded linear operator with closed range R U . Then there exists a bounded linearoperator U † : H → H such that U U † x = x ∀ x ∈ R U . Note 2.4.
The operator U † defined in Theorem (2.3), is called the pseudo-inverseof U . Definition 2.5. [1] A sequence { f i } ∞ i = 1 ⊆ H is said to be a frame for H ifthere exist constants < A ≤ B < ∞ such that A k f k ≤ ∞ X i = 1 | h f , f i i | ≤ B k f k ∀ f ∈ H The constants A and B are called frame bounds. If { f i } ∞ i = 1 satisfies the inequality ∞ X i = 1 | h f , f i i | ≤ B k f k ∀ f ∈ H then it is called a Bessel sequence with bound B . -Frame in n -Hilbert space Definition 2.6. [1] Let { f i } ∞ i = 1 be a frame for H . Then the bounded linear op-erator T : l ( N ) → H , defined by T { c i } = ∞ P i = 1 c i f i , is called pre-frame opera-tor and its adjoint operator T ∗ : H → l ( N ) , given by T ∗ ( f ) = { h f , f i i } ∞ i = 1 is called the analysis operator. The operator S : H → H defined by S ( f ) = T T ∗ ( f ) = ∞ P i = 1 h f , f i i f i ∀ f ∈ H is called the frame operator. Definition 2.7. [12] Let K ∈ B ( H ) . Then a sequence { f i } ∞ i = 1 in H is saidto be a K -frame for H if there exist constants < A ≤ B < ∞ such that A k K ∗ f k ≤ ∞ X i = 1 | h f , f i i | ≤ B k f k ∀ f ∈ H. Theorem 2.8. [17] Let K be a bounded linear operator on H . Then a Besselsequence { f i } ∞ i = 1 in H is a K -frame if and only if there exists λ > such that S ≥ λ K K ∗ , where S is the frame operator for { f i } ∞ i = 1 . Definition 2.9. [13] A real valued function k · , · · · , · k : X n → R is called an-norm on X if the following conditions hold:(I) k x , x , · · · , x n k = 0 if and only if x , · · · , x n are linearly dependent,(II) k x , x , · · · , x n k is invariant under any permutations of x , x , · · · , x n ,(III) k α x , x , · · · , x n k = | α | k x , x , · · · , x n k ∀ α ∈ K ,(IV) k x + y , x , · · · , x n k ≤ k x , x , · · · , x n k + k y , x , · · · , x n k .The pair ( X , k · , · · · , · k ) is then called a linear n-normed space. Definition 2.10. [14] Let n ∈ N and X be a linear space of dimension greaterthan or equal to n over the field K , where K is the real or complex numbers field. Afunction h · , · | · , · · · , · i : X n + 1 → K is satisfying the following five properties:(I) h x , x | x , · · · , x n i ≥ and h x , x | x , · · · , x n i = 0 if andonly if x , x , · · · , x n are linearly dependent,(II) h x , y | x , · · · , x n i = h x , y | x i , · · · , x i n i for every permutation ( i , · · · , i n ) of ( 2 , · · · , n ) ,(III) h x , y | x , · · · , x n i = h y , x | x , · · · , x n i ,(IV) h α x , y | x , · · · , x n i = α h x , y | x , · · · , x n i , for all α ∈ K ,(V) h x + y , z | x , · · · , x n i = h x , z | x , · · · , x n i + h y , z | x , · · · , x n i .is called an n-inner product on X and the pair ( X , h · , · | · , · · · , · i ) is called n-inner product space. Prasenjit Ghosh & T. K. Samanta
Theorem 2.11. [11] For n -inner product space ( X , h · , · | · , · · · , · i ) , | h x , y | x , · · · , x n i | ≤ k x , x , · · · , x n k k y , x , · · · , x n k hold for all x, y, x , · · · , x n ∈ X . Theorem 2.12. [14] For every n-inner product space ( X , h · , · | · , · · · , · i ) , k x , x , · · · , x n k = p h x , x | x , · · · , x n i defines a n-norm for which h x , y | x , · · · , x n i = 14 (cid:0) k x + y , x , · · · , x n k − k x − y , x , · · · , x n k (cid:1) , & k x + y , x , · · · , x n k + k x − y , x , · · · , x n k = 2 (cid:0) k x , x , · · · , x n k + k y , x , · · · , x n k (cid:1) hold for all x, y, x , x , · · · , x n ∈ X . Definition 2.13. [15] Let ( X , h · , · | · , · · · , · i ) be a n -inner product spaceand { e i } ni = 1 be linearly independent vectors in X . Then for a given set F = { a , · · · , a n } ⊆ X , if h e i , e j | a , · · · , a n i = δ i j i, j ∈ { , , · · · , n } where , δ i j = ( if i = j if i = j , the family { e i } ni = 1 is said to be F -orthonormal. If an F -orthonormal set is count-able, we can arrange it in the form of a sequence { e i } and call it F -orthonormalsequence. Definition 2.14. [13] A sequence { x k } in a linear n -normed space X is saidto be convergent to some x ∈ X if for every x , · · · , x n ∈ X lim k →∞ k x k − x , x , · · · , x n k = 0 and it is called a Cauchy sequence if lim l , k → ∞ k x l − x k , x , · · · , x n k = 0 for every x , · · · , x n ∈ X . The space X issaid to be complete if every Cauchy sequence in this space is convergent in X . An-inner product space is called n-Hilbert space if it is complete with respect to itsinduce norm. Note 2.15. [9] Let L F denote the linear subspace of X spanned by the non-empty finite set F = { a , a , · · · , a n } , where a , a , · · · , a n are fixed ele-ments in X . Then the quotient space X / L F is a normed linear space with re-spect to the norm, k x + L F k F = k x , a , · · · , a n k for every x ∈ X . Let M F be the algebraic complement of L F , then X = L F ⊕ M F . Define h x , y i F = -Frame in n -Hilbert space h x , y | a , · · · , a n i on X . Then h · , · i F is a semi-inner product on X and thissemi-inner product induces an inner product on X / L F which is given by h x + L F , y + L F i F = h x , y i F = h x , y | a , · · · , a n i ∀ x, y ∈ X. By identifying
X / L F with M F in an obvious way, we obtain an inner product on M F . Now for every x ∈ M F , we define k x k F = p h x , x i F and it can be easilyverify that ( M F , k · k F ) is a norm space. Let X F be the completion of the innerproduct space M F . For the remaining part of this paper, (
X , h · , · | · , · · · , · i ) is consider to be a n -Hilbert space and I will denote the identity operator on X F . Definition 2.16. [9] A sequence { f i } ∞ i = 1 in X is said to be a frame associatedto ( a , · · · , a n ) for X if there exist constants < A ≤ B < ∞ such that A k f , a , · · · , a n k ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k for all f ∈ X . The constants A and B are called the frame bounds. If the sequence { f i } ∞ i = 1 satisfies the inequality ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k ∀ f ∈ X is called a Bessel sequence associated to ( a , · · · , a n ) in X with bound B . Theorem 2.17. [9] Let { f i } ∞ i = 1 be a sequence in X . Then { f i } ∞ i = 1 is a frameassociated to ( a , · · · , a n ) with bounds A & B if and only if it is a frame forthe Hilbert space X F with bounds A & B . Definition 2.18. [9] Let { f i } ∞ i = 1 be a Bessel sequence associated to ( a , · · · , a n ) for X . Then the bounded linear operator T : l ( N ) → X F , defined by T { c i } = ∞ P i = 1 c i f i , is called pre-frame operator and its adjoint operator T ∗ : X F → l ( N ) ,given by T ∗ ( f ) = { h f , f i | a , · · · , a n i } ∞ i = 1 ∀ f ∈ X F is called the analy-sis operator . The operator S : X F → X F defined by S F ( f ) = T T ∗ ( f ) = ∞ P i = 1 h f , f i | a , · · · , a n i f i ∀ f ∈ X F is called the frame operator. Note 2.19. [9] If { f i } ∞ i = 1 is a frame associated to ( a , · · · , a n ) for X , thenthe frame operator S F is bounded, positive, self-adjoint and invertible. K -frame in n -Hilbert space Definition 3.1.
Let K be a bounded linear operator on X F . Then a sequence { f i } ∞ i = 1 ⊆ X is said to be a K -frame associated to ( a , · · · , a n ) for X if thereexist constants < A ≤ B < ∞ such that A k K ∗ f , a , · · · , a n k ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k , Prasenjit Ghosh & T. K. Samanta for all f ∈ X F . In particular, if K = I , then by Theorem (2.17), { f i } ∞ i = 1 is aframe associated to ( a , · · · , a n ) for X . Obviously, every K -frame associated to ( a , · · · , a n ) is a Bessel sequence associated to ( a , · · · , a n ) in X . Note 3.2.
In general, the frame operator of a K -frame associated to ( a , · · · , a n ) is not invertible. But, if K ∈ B ( X F ) has closed range, then S F : R ( K ) → S F ( R ( K ) ) is an invertible operator. For f ∈ R ( K ) , we have k f, a , · · · , a n k = (cid:13)(cid:13)(cid:13) ( K † ) ∗ K ∗ f, a , · · · , a n (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) K † (cid:13)(cid:13)(cid:13) k K ∗ f, a , · · · , a n k . Therefore, if { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ) for X , by defini-tion (3.1), we get A (cid:13)(cid:13)(cid:13) K † (cid:13)(cid:13)(cid:13) − k f, a , · · · , a n k ≤ h S F f, f | a , · · · , a n i ≤ B k f, a , · · · , a n k and furthermore for each f ∈ S F ( R ( K ) ) , we have B − k f, a , · · · , a n k ≤ (cid:10) S − F f, f | a , · · · , a n (cid:11) ≤ A − (cid:13)(cid:13)(cid:13) K † (cid:13)(cid:13)(cid:13) k f, a , · · · , a n k Theorem 3.3.
Let { f i } ∞ i = 1 be a K-frame associated to ( a , · · · , a n ) for X and T ∈ B ( X F ) with R ( T ) ⊂ R ( K ) . Then { f i } ∞ i = 1 is a T-frame associatedto ( a , · · · , a n ) for X .Proof. Suppose { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ) for X . Thenfor each f ∈ X F , there exist constants A, B > A k K ∗ f , a , · · · , a n k ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k . Since R ( T ) ⊂ R ( K ), by Theorem (2.1), ∃ λ > T T ∗ ≤ λ K K ∗ . Thus, Aλ k T ∗ f , a , · · · , a n k = Aλ h T T ∗ f , f | a , · · · , a n i = (cid:28) Aλ T T ∗ f , f | a , · · · , a n (cid:29) ≤ h A K K ∗ f , f | a , · · · , a n i = A k K ∗ f , a , · · · , a n k . Therefore, for each f ∈ X F , Aλ k T ∗ f , a , · · · , a n k ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k . Hence, { f i } ∞ i = 1 is a T -frame associated to ( a , · · · , a n ) for X . Theorem 3.4.
Let { f i } ∞ i = 1 be a K-frame associated to ( a , · · · , a n ) for X with bounds A, B and T ∈ B ( X F ) be an invertible with T K = K T , then { T f i } ∞ i = 1 is a K-frame associated to ( a , · · · , a n ) for X . -Frame in n -Hilbert space Proof.
Since T is invertible, for each f ∈ X F , k K ∗ f , a , · · · , a n k = (cid:13)(cid:13)(cid:13) (cid:0) T − (cid:1) ∗ T ∗ K ∗ f , a , · · · , a n (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) (cid:0) T − (cid:1) ∗ (cid:13)(cid:13)(cid:13) k T ∗ K ∗ f , a , · · · , a n k . ⇒ (cid:13)(cid:13) T − (cid:13)(cid:13) − k K ∗ f , a , · · · , a n k ≤ k T ∗ K ∗ f , a , · · · , a n k . (1)Also, since { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ), for f ∈ X F , ∞ X i = 1 | h f , T f i | a , · · · , a n i | = ∞ X i = 1 | h T ∗ f , f i | a , · · · , a n i | ≥ A k K ∗ T ∗ f , a , · · · , a n k = A k T ∗ K ∗ f , a , · · · , a n k [ since T K = K T ] ≥ A (cid:13)(cid:13) T − (cid:13)(cid:13) − k K ∗ f , a , · · · , a n k [ using (1)] . On the other hand, for all f ∈ X F , ∞ X i = 1 | h f , T f i | a , · · · , a n i | = ∞ X i = 1 | h T ∗ f , f i | a , · · · , a n i | ≤ B k T ∗ f , a , · · · , a n k ≤ B k T k k f , a , · · · , a n k . Hence, { T f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ) for X . Theorem 3.5.
Let { f i } ∞ i = 1 be a K-frame associated to ( a , · · · , a n ) for X with bounds A, B and T ∈ B ( X F ) such that T T ∗ = I with T K = K T . Then { T f i } ∞ i = 1 is a K-frame associated to ( a , · · · , a n ) for X .Proof. Since
T T ∗ = I , for f ∈ X F , k T ∗ f , a , · · · , a n k = k f , a , · · · , a n k and this implies that k T ∗ K ∗ f , a , · · · , a n k = k K ∗ f , a , · · · , a n k . Also,since { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ), for each f ∈ X F , ∞ X i = 1 | h f , T f i | a , · · · , a n i | = ∞ X i = 1 | h T ∗ f , f i | a , · · · , a n i | ≥ A k K ∗ T ∗ f , a , · · · , a n k = A k T ∗ K ∗ f , a , · · · , a n k = A k K ∗ f , a , · · · , a n k . Thus, we see that { T f i } ∞ i = 1 satisfies lower K -frame condition. Following the proofof the Theorem (3.4), it can be shown that it also satisfies upper K -frame conditionand therefore it is a K -frame associated to ( a , · · · , a n ) for X . Theorem 3.6.
Let { f i } ∞ i = 1 be a sequence in X . Then { f i } ∞ i = 1 is a K-frameassociated to ( a , · · · , a n ) for X if and only if there exists a bounded linear op-erator T : l ( N ) → X F such that f i = T e i and R ( K ) ⊂ R ( T ) , where { e i } ∞ i = 1 is an F -orthonormal basis for l ( N ) . Prasenjit Ghosh & T. K. Samanta
Proof.
First we suppose that { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ).Then, for each f ∈ X F , there exist constants A, B > A k K ∗ f , a , · · · , a n k ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k . Now, we consider the linear operator L : X F → l ( N ) defined by L ( f ) = ∞ X i = 1 h f , f i | a , · · · , a n i e i ∀ f ∈ X F . Since { e i } ∞ i = 1 is an F -orthonormal basis for l ( N ), we can write k L ( f ) k l = ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k . Thus, L is well-defined and bounded linear operator on X F . So, the adjoint operator L ∗ : l ( N ) → X F exists and then for each f ∈ X F , we get h L ∗ e i , f | a , · · · , a n i = h e i , L ( f ) | a , · · · , a n i = * e i , ∞ X i = 1 h f , f i | a , · · · , a n i e i | a , · · · , a n + = h f , f i | a , · · · , a n i = h f i , f | a , · · · , a n i . The above calculation shows that, L ∗ ( e i ) = f i . Also, from the definition (3.1),we get A k K ∗ f k F ≤ k L ( f ) k l and this implies that h A K K ∗ f , f | a , · · · , a n i ≤ h L ∗ L f , f | a , · · · , a n i⇒ A K K ∗ ≤ T T ∗ , where T = L ∗ and hence from the Theorem (2.1), R ( K ) ⊂ R ( T ).Conversely, suppose that T : l ( N ) → X F be a bounded linear operatorsuch that f i = T e i and R ( K ) ⊂ R ( T ). We have to show that { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ). Let g ∈ l ( N ) then g = ∞ P i = 1 c i e i , where c i = h g , e i | a , · · · , a n i . Now, for all g ∈ l ( N ), we have h T ∗ f, g | a , · · · , a n i = * T ∗ f, ∞ X i = 1 c i e i | a , · · · , a n + = ∞ X i = 1 c i h f, T e i | a , · · · , a n i = ∞ X i = 1 c i h f , f i | a , · · · , a n i -Frame in n -Hilbert space ∞ X i = 1 h g , e i | a , · · · , a n i h f , f i | a , · · · , a n i = ∞ X i = 1 h e i , g | a , · · · , a n i h f , f i | a , · · · , a n i = * ∞ X i = 1 h f , f i | a , · · · , a n i e i , g | a , · · · , a n + . ⇒ T ∗ ( f ) = ∞ X i = 1 h f , f i | a , · · · , a n i e i ∀ f ∈ X F . Thus, for all f ∈ X F , ∞ P i = 1 | h f , f i | a , · · · , a n i | = ∞ X i = 1 | h f , T e i | a , · · · , a n i | = ∞ X i = 1 | h T ∗ f , e i | a , · · · , a n i | = k T ∗ f , a , · · · , a n k [ since { e i } ∞ i = 1 is an F -orthonormal basis ] ≤ k T ∗ k k f , a , · · · , a n k = k T k k f , a , · · · , a n k ⇒ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ k T k k f , a , · · · , a n k ∀ f ∈ X F . Thus, { f i } ∞ i = 1 is a Bessel sequence associated to ( a , · · · , a n ). Since R ( K ) ⊂R ( T ), from Theorem (2.1), there exists A >
A K K ∗ ≤ T T ∗ . Hencefollowing the proof of the Theorem (3.3), for all f ∈ X F A k K ∗ f, a , · · · , a n k ≤ k T ∗ f, a , · · · , a n k = ∞ X i = 1 | h f , f i | a , · · · , a n i | . Hence, { f i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ) for X . Theorem 3.7.
Let { f i } ∞ i =1 and { g i } ∞ i = 1 be K-frames associated to ( a , · · · , a n ) for X with the corresponding pre frame operators T and L , respectively. If T L ∗ and L T ∗ are positive operators, then { f i + g i } ∞ i = 1 is also a K-frame associatedto ( a , · · · , a n ) for X .Proof. Let { f i } ∞ i = 1 and { g i } ∞ i = 1 be two K -frames associated to ( a , · · · , a n )for X . Then by Theorem (3.6), there exist bounded linear operators T and L suchthat T e i = f i , L e i = g i and R ( K ) ⊂ R ( T ) , R ( K ) ⊂ R ( L ), where { e i } ∞ i = 1 is an F -orthonormal basis for l ( N ). Now, we have R ( K ) ⊂ R ( T ) + R ( L ). By Theorem (2.2), K K ∗ ≤ λ ( T T ∗ + L L ∗ ), for some λ >
0. Now, for f ∈ X F , ∞ X i = 1 | h f , f i + g i | a , · · · , a n i | = ∞ X i = 1 | h f , T e i + L e i | a , · · · , a n i | Prasenjit Ghosh & T. K. Samanta = ∞ X i = 1 | h ( T + L ) ∗ f , e i | a , · · · , a n i | = k ( T + L ) ∗ f , a , · · · , a n k [ since { e i } is F-orthonormal ]= h ( T + L ) ∗ f , ( T + L ) ∗ f | a , · · · , a n i = h T ∗ f + L ∗ f , T ∗ f + L ∗ f | a , · · · , a n i = h T ∗ f , T ∗ f | a , · · · , a n i + h L ∗ f , T ∗ f | a , · · · , a n i + h T ∗ f , L ∗ f | a , · · · , a n i + h L ∗ f , L ∗ f | a , · · · , a n i = h T T ∗ f , f | a , · · · , a n i + h T L ∗ f , f | a , · · · , a n i + h L T ∗ f , f | a , · · · , a n i + h L L ∗ f , f | a , · · · , a n i≥ h ( T T ∗ + L L ∗ ) f , f | a , · · · , a n i [ since T L ∗ , L T ∗ are positive ] ≥ λ h K K ∗ f , f | a , · · · , a n i [ since K K ∗ ≤ λ ( T T ∗ + L L ∗ ) ]= 1 λ h K ∗ f , K ∗ f | a , · · · , a n i = 1 λ k K ∗ f , a , · · · , a n k . Therefore, for each f ∈ X F ,1 λ k K ∗ f , a , · · · , a n k ≤ ∞ X k = 1 | h f , f i + g i | a , · · · , a n i | . (2)On the other hand, using the Minkowski’s inequality, for each f ∈ X F , we have ∞ X i = 1 | h f , f i + g i | a , · · · , a n i | ! ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | !
12 + ∞ X i = 1 | h f , g i | a , · · · , a n i | ! ≤ √ A k f, a , · · · , a n k + √ B k f, a , · · · , a n k = (cid:16) √ A + √ B (cid:17) k f, a , · · · , a n k . This implies that ∞ X i = 1 | h f , f i + g i | a , · · · , a n i | ≤ (cid:16) √ A + √ B (cid:17) k f , a , · · · , a n k . (3)From (2) and (3), { f i + g i } ∞ i = 1 is a K -frame associated to ( a , · · · , a n ) for X . Theorem 3.8.
Let { f i } ∞ i = 1 be K-frame associated to ( a , · · · , a n ) for X and U : X F → X F be a positive operator. Then { f i + U f i } ∞ i = 1 is also a K-frameassociated to ( a , · · · , a n ) for X . -Frame in n -Hilbert space Proof.
Let { f i } ∞ i = 1 be K -frame associated to ( a , · · · , a n ) for X with frameoperator S F . Then for each f ∈ X F , there exist A, B > A k K ∗ f , a , · · · , a n k ≤ ∞ X i = 1 | h f , f i | a , · · · , a n i | ≤ B k f , a , · · · , a n k . It is easy to verify that h S F f , f | a , · · · , a n i = ∞ P i = 1 | h f , f i | a , · · · , a n i | . Thus, A k K ∗ f , a , · · · , a n k ≤ h S F f , f | a , · · · , a n i ≤ B k f , a , · · · , a n k . This implies that
A K K ∗ ≤ S F ≤ B I . Now, for each f ∈ X F , ∞ X i = 1 h f , f i + U f i | a , · · · , a n i ( f i + U f i )= ∞ X i = 1 h f , ( I + U ) f i | a , · · · , a n i ( I + A ) f i = ( I + U ) ∞ X i = 1 h ( I + U ) ∗ f , f i | a , · · · , a n i f i = ( I + U ) S F ( I + U ) ∗ f. This shows that the corresponding frame operator for { f i + U f i } ∞ i = 1 is ( I + U ) S F ( I + U ) ∗ . Since S F , U are positive operators, ( I + U ) S F ( I + U ) ∗ ≥ S F ≥ A K K ∗ , by Theorem (2.8), { f i + U f i } ∞ i = 1 is a K -frame associated to( a , · · · , a n ) for X . K -frame and its properties in n -Hilbert space Definition 4.1.
A sequence { f i } ∞ i = 1 in X is said to be a tight K-frame asso-ciated to ( a , · · · , a n ) for X if there exist constants < A ≤ B < ∞ suchthat ∞ X i = 1 | h f , f i | a , · · · , a n i | = A k K ∗ f , a , · · · , a n k ∀ f ∈ X F (4) If A = 1 , then { f i } ∞ i = 1 is called Parseval K-frame associated to ( a , · · · , a n ) for X . Remark 4.2.
From (4), we can write ∞ X i = 1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) √ A f , f i | a , · · · , a n (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) = k K ∗ f , a , · · · , a n k . Therefore, if { f i } ∞ i = 1 is a tight K-frame associated to ( a , · · · , a n ) with bound A then the family (cid:26) √ A f i (cid:27) ∞ i = 1 is a Parseval K-frame associated to ( a , · · · , a n ) for X . Prasenjit Ghosh & T. K. Samanta
Theorem 4.3.
Let { f i } ∞ i = 1 be a tight frame associated to ( a , · · · , a n ) for X with bound A and K ∈ B ( X F ) , then { K f i } ∞ i = 1 is a tight K-frame associatedto ( a , · · · , a n ) for X with bound A .Proof. Since { f i } ∞ i = 1 is a tight frame associated to ( a , · · · , a n ) for X withbound A , for any f ∈ X F , we have ∞ X i = 1 | h f , K f i | a , · · · , a n i | = ∞ X i = 1 | h K ∗ f , f i | a , · · · , a n i | = A k K ∗ f , a , · · · , a n k and hence { K f i } ∞ i = 1 is a tight K -frame associated to ( a , · · · , a n ) for X withbound A . Theorem 4.4.
Let
K, T ∈ B ( X F ) and { f i } ∞ i = 1 be a tight K-frame associ-ated to ( a , · · · , a n ) for X with bound A . Then { T f i } ∞ i = 1 is a tight T K-frameassociated to ( a , · · · , a n ) for X with bound A .Proof. Since { f i } ∞ i = 1 is a tight K -frame associated to ( a , · · · , a n ) for X withbound A , for any f ∈ X F , we have ∞ X i = 1 | h f , T f i | a , · · · , a n i | = ∞ X i = 1 | h T ∗ f , f i | a , · · · , a n i | = A k K ∗ ( T ∗ f ) , a , · · · , a n k = A k ( T K ) ∗ f, a , · · · , a n k and hence { T f i } ∞ i = 1 is a tight T K -frame associated to ( a , · · · , a n ) for X withbound A . Theorem 4.5.
Let { f i } ∞ i = 1 be a tight K-frame associated to ( a , · · · , a n ) for X with bound A . Then there exists a Bessel sequence { g i } ∞ i = 1 associated to ( a , · · · , a n ) with bound B such that for all f ∈ X F , K ( f ) = ∞ X i = 1 h f , g i | a , · · · , a n i f i and A B ≥ . Proof.
Let { f i } ∞ i = 1 be a tight K -frame associated to ( a , · · · , a n ) for X withbound A . Then by Theorem (2.4) of ([ ? ]), there exists a Bessel sequence { g i } ∞ i = 1 associated to ( a , · · · , a n ) with bound B such that K ( f ) = ∞ X i = 1 h f , g i | a , · · · , a n i f i and K ∗ ( f ) = ∞ X i = 1 h f , f i | a , · · · , a n i g i ∀ f ∈ X F . -Frame in n -Hilbert space { f i } ∞ i = 1 is a tight K -frame associated to ( a , · · · , a n ) for X , we have ∞ X i = 1 | h f , f i | a , · · · , a n i | = A k K ∗ f , a , · · · , a n k = A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i = 1 h f , f i | a , · · · , a n i g i , a , · · · , a n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = A sup k g , a , ··· , a n k = 1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i = 1 h h f , f i | a , · · · , a n i g i , g | a , · · · , a n i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = A sup k g , a , ··· , a n k = 1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i = 1 h f , f i | a , · · · , a n i h g i , g | a , · · · , a n i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ A sup k g , a , ··· , a n k = 1 ( ∞ X i = 1 | h f , f i | a , · · · , a n i | ∞ X i = 1 | h g i , g | a , · · · , a n i | ) ≤ A sup k g , a , ··· , a n k = 1 ( ∞ X i = 1 | h f , f i | a , · · · , a n i | B k g , a , · · · , a n k ) [ since { g i } ∞ i = 1 is a Bessel sequence associated to ( a , · · · , a n ) with bound B ]= A B ∞ X i = 1 | h f , f i | a , · · · , a n i | . The above calculation shows that
A B ≥ Theorem 4.6.
Let { f i } ∞ i = 1 and { g i } ∞ i = 1 be two Parseval K-frame associatedto ( a , · · · , a n ) for X with the corresponding synthesis operators T and L , re-spectively. If T L ∗ = θ , where θ is the null operator on X F then { f i + g i } ∞ i = 1 is a tight K-frame associated to ( a , · · · , a n ) with frame bound .Proof. Let { f i } ∞ i = 1 and { g i } ∞ i = 1 be Parseval K -frames associated to ( a , · · · , a n )for X . Then by Theorem (3.6), there exist synthesis operators T and L such that T e i = f i , L e i = g i with R ( K ) ⊂ R ( T ) , R ( K ) ⊂ R ( L ) respectively,where { e i } ∞ i = 1 is a F -orthonormal basis for l ( N ). Now, for f ∈ X F , T ∗ : X F → l ( N ) , T ∗ ( f ) = ∞ X i = 1 h f , f i | a , · · · , a n i e i , and L ∗ : X F → l ( N ) , L ∗ ( f ) = ∞ X i = 1 h f , g i | a , · · · , a n i e i . Now from the definition of Parseval K -frame associated to ( a , · · · , a n ), k K ∗ f, a , · · · , a n k = ∞ X i = 1 | h f, f i | a , · · · , a n i | = k T ∗ f, a , · · · , a n k , (5)4 Prasenjit Ghosh & T. K. Samanta k K ∗ f, a , · · · , a n k = ∞ X i = 1 | h f, f i | a , · · · , a n i | = k L ∗ f, a , · · · , a n k . (6)Following the proof of the Theorem (3.7), it can be shown that for each f ∈ X F , ∞ X i = 1 | h f , f i + g i | a , · · · , a n i | = ∞ X i = 1 | h f , ( T + L ) e i | a , · · · , a n i | = k ( T + L ) ∗ f , a , · · · , a n k = h T T ∗ f , f | a , · · · , a n i + h T L ∗ f , f | a , · · · , a n i + h L T ∗ f , f | a , · · · , a n i + h L L ∗ f , f | a , · · · , a n i = h T T ∗ f , f | a , · · · , a n i + h L L ∗ f , f | a , · · · , a n i [ since T L ∗ = θ = L T ∗ ]= h T ∗ f , T ∗ f | a , · · · , a n i + h L ∗ f , L ∗ f | a , · · · , a n i = k T ∗ f , a , · · · , a n k + k L ∗ f , a , · · · , a n k = k K ∗ f , a , · · · , a n k + k K ∗ f , a , · · · , a n k [ using (5) and (6) ]= 2 k K ∗ f , a , · · · , a n k . Hence, { f i + g i } ∞ i = 1 is a tight K -frame associated to ( a , · · · , a n ) for X withbound 2. References [1] O. Christensen,
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