Frame operator for K-frame in 2-inner product space

Prasenjit Ghosh Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road, Kolkata, 700019, West Bengal, India e-mail: [email protected]

Abstract In this paper we discuss a few properties of 2-frame in the aspect of 2-inner product spaces. We also give a relationship between 2-K- frames and quotient operators in 2-inner product spaces.

Keywords: Frame, K-frame, quotient operator, 2-inner product space, 2-normed space. 2010 Mathematics Subject Classification: 42C15, 46C50, 47B32.

1 Introduction

Duffin and Schaeffer were introduced frame in Hilbert space in their fundamental paper [10], they used frame as a tool in the study of nonharmonic Fourier series. After some decades, frame theory was popularized by Daubechies, Grossman, Meyer [6]. Every separable Hilbert space has a countable orthonormal basis i . e., every elements in this space can be represented by its Fourier series expansion with respect to the basis elements. A frame for a separable Hilbert space is a generalization of such an orthonormal basis and this is such a tool that also allows each vector in the space to be written as a linear combination of elements from the frame but, linear independence among the frame elements is not required. Several generalizations of frames namely, K-frame [12], Fusion frame [4], K-fusion frame [2], G-frame [13], etc. have been introduced in recent times. K-frames for a separable Hilbert space were introduced by Lara Gavruta to study the basic notions about atomic system for a bounded linear operator K. K-frame is more generalization than the ordinary frame and many properties of ordinary frame may not hold for such generalization of frame. After the introduction of 2-inner product space [5, 8, 9] 2-norm was introduced by S. Gahler [11]. The concepts of 2-inner product and 2-inner product spaces are closely related to the concepts of 2-norm and 2-normed space. The notion of a frame in a 2-inner product space has been introduced by A. Arefijamaal and G. Sadeghi [1] and they also established some fundamental proper- ties of 2-frames for 2-inner product space. The concept of 2-atomic systems which is a generalization of families of local 2-atoms in a 2-inner product spaces was in- troduced by B. Dastourian and M. Janfada [7] and they also defined 2-K-frame as the generalization of 2-frame. 2 Prasenjit Ghosh

In this paper, we shall discuss some properties of 2-frame and establish some relationship between 2-K-frame and quotient operators. Throughout this paper, X will denote a separable Hilbert space with the inner product h · , · i and B ( X ) denote the space of all bounded linear operator on X. We also denote R ( T ) for range set of T , N ( T ) for null space of T where T ∈ 2 B ( X ) and l ( N ) denote the space of square summable scalar-valued sequences with index set N.

2 Preliminaries

∞ Definition 2.1. [3] A sequence { f i }i = 1 of elements in X is said to a frame for X if there exist constants A, B > 0 such that ∞ 2 X 2 2 A k f k ≤ | h f , f i i | ≤ B k f k ∀ f ∈ X. i = 1 ∞ The constants A and B are called frame bounds.If the collection { f i }i = 1 satisfies ∞ X 2 2 | h f , f i i | ≤ B k f k ∀ f ∈ X i = 1 then it is called a Bessel sequence. ∞ Definition 2.2. [3] Let { f i }i = 1 be a frame for X then operator defined by ∞ 2 ∞ X T : l ( N ) → X,T ( { ci }i = 1 ) = c i f i i = 1 is called pre-frame operator and its adjoint operator given by

∗ 2 ∗ ∞ T : X → l ( N ) ,T ( f ) = { h f , fi i }i = 1 is called the analysis operator. The frame operator is given by ∞ ∗ X S : X → X , S f = TT f = h f , f i i f i. i = 1 Definition 2.3. [12] Let K : X → X be a bounded linear operator. Then ∞ a sequence { f i }i = 1 in X is said to be K-frame for X if there exist constants A, B > 0 such that ∞ ∗ 2 X 2 2 A k K f k ≤ | h f , f i i | ≤ B k f k ∀ f ∈ X. i = 1 Definition 2.4. [12] Let U, V : X → X be two bounded linear operator with N ( U ) ⊂ N ( V ). The quotient operator T = [ U/V ] is a linear operator which is defined as T = [ U/V ]: R ( V ) → R ( U ),T ( V x ) = U x. Some characterization of K-frame in 2-inner product space 3

Definition 2.5. [5, 8] Let X be a linear space of dimension greater than 1 over the field K, where K is the real or complex numbers field. A function h · , · | · i : X × X × X → K is said to be an 2-inner product on X if ( I 1 ) h x, x | z i ≥ 0 and h x, x | z i = 0 if and only if x, z are linearly dependent, ( I 2 ) h x , x | z i = h z , z | x i, ( I 3 ) h x , y | z i = h y , x | z i, ( I 4 ) h α x , y | z i = α h x , y | z i, for all α ∈ K, ( I 5 ) h x 1 + x 2 , y | z i = h x 1 , y | z i + h x 2 , y | z i. A linear space X equipped with an 2-inner product h · , · | · i on X is called an 2-inner product space [8]. Definition 2.6. [11] A 2-norm k · , · k on a linear space X is a real valued function defined on X × X satisfying the following conditions: ( N 1 ) k x , y k = 0 if and only if x, y are linearly dependent, ( N 2 ) k x , y k = k y , x k, ( N 3 ) k α x , y k = | α | k x , y k ∀ α ∈ R, ( N 4 ) k x , y + z k ≤ k x , y k + k x , z k. Then the pair ( X, k · , · k ) is called a linear 2-normed space. Theorem 2.7. [8] Let ( X, h · , · | · i ) be a 2-inner product space then

| h x , y | z i | ≤ k x , z k k y , z k hold for all x, y, z ∈ X. Theorem 2.8. [8] For every 2-inner product space ( X, h · , · | · i ),

k x , y k = ph x , x | y i defines a 2-norm for which 1 h x , y | z i = k x + y , z k 2 − k x − y , z k 2
, & 4 k x + y , z k 2 + k x − y , z k 2 = 2 k x , z k 2 + ky , zk 2
hold for all x , y , z ∈ X . Definition 2.9. [11] Let ( X, k · , · k ) be a linear 2-normed space. A sequence { x n } in X is said to be converges to some x ∈ X if

lim k x n − x, y k = 0 n→∞ for every y ∈ X and it is called Cauchy sequence if

lim k x n − x m, z k = 0 n , m→∞ for every z ∈ X. The space X is said to be complete if every Cauchy sequence in this space is convergent in X.A 2-inner product space is called 2-Hilbert space if it is complete with respect to its induce norm. 4 Prasenjit Ghosh

Definition 2.10. [1] Let ( X, h · , · | · i ) be a 2-Hilbert space and h ∈ X.A ∞ sequence { f i }i = 1 ⊆ X is called a 2-frame associated to h if there exist constants A, B > 0 such that

∞ 2 X 2 2 A k f , h k ≤ | h f , f i | h i | ≤ B k f , h k ∀ f ∈ X. i = 1 ∞ A sequence { f i }i = 1 which satisfies the inequality ∞ X 2 2 | h f , f i | h i | ≤ B k f , h k ∀ f ∈ X. i = 1 is called a 2-Bessel sequence associated to h.

Theorem 2.11. [1] Let ( X, h · , · | · i ) be a 2-Hilbert space and L h denote the 1-dimensional linear subspace generated by a fixed h ∈ X. Let M h be the algebraic complement of L h. Define h x , y i h = h x , y | h i on X. Then h · , · i h is a semi- inner product on X and this semi-inner product induces an inner product on the quotient space X/L h which is given by

h x + L h , y + L h i h = h x , y i h = h x , y | h i ∀ x, y ∈ X.

By identifying X/L h with M h in an obvious way, we obtain an inner product on p M h. Now, for x ∈ Mh, define k x k h = h x , x i h. Then ( M h, k · k h ) is a norm space.

Let X h be the completion of the inner product space M h. Theorem 2.12. [1] Let ( X, h · , · | · i ) be a 2-Hilbert space and h ∈ X. Then a ∞ sequence { f i }i = 1 in X is a 2-frame associated to h with bounds A & B if and only if it is a frame for the Hilbert space X h with bounds A & B. ∞ Theorem 2.13. [1] Let { f i }i = 1 be a 2-Bessel sequence associated to h then the 2-pre frame operator

∞ 2 ∞ X T h : l ( N ) → X h,T h ( { ci }i = 1 ) = c i f i i = 1 is well-defined and bounded and its adjoint operator given by,

∗ 2 ∗ ∞ T h : X h → l ( N ),T h ( f ) = { h f , fi | h i }i = 1 is also well-defined and bounded.

∞ Definition 2.14. [1] Let { f i }i = 1 be a 2-frame associated to h.The operator ∞ ∗ X S h : X h → X h,S h f = T h T h f = h f , f i | h i f i i = 1 ∞ is called the frame operator for { f i }i = 1. Some characterization of K-frame in 2-inner product space 5

Theorem 2.15. [1] The frame operator Sh is bounded, invertible, self-adjoint, and positive.

Definition 2.16. [7]) Let K h : X h → X h be a bounded linear operator. Then ∞ a sequence { f i }i = 1 ⊆ X is called a 2-K-frame associated to h if there exist constants A, B > 0 such that

∞ ∗ 2 X 2 2 A k K h f k h ≤ | h f , f i | h i | ≤ B k f k h ∀ f ∈ X h. i = 1

3 Some properties of 2-frame

Theorem 3.1. Let Y be a closed subspace of Xh and P Y be the orthogonal ∞ projection on Y . Then for a sequence { f i }i = 1 ⊆ Xh the following hold:

∞ (i) If { f i }i = 1 is a 2-frame associated to h for X with frame bounds A, B ∞ then { P Y f i }i = 1 is a frame for Y with the same bounds.

∞ (ii) If { f i }i = 1 is a frame for Y with frame operator Sh, then the orthogonal projection on Y is given by,

∞ X −1 P Y f = f , Sh f i | h f i ∀ f ∈ Xh. i = 1

Proof. Using the definition of an orthogonal projection of Xh onto Y , we get  f if f ∈ Y P f = (1) Y 0 if f ∈ Y ⊥.

∞ (i) Suppose { f i }i = 1 is a 2-frame associated to h for X with frame bounds ∞ A, B ⇒ { f i }i = 1 is a frame for Xh with frame bounds A, B. So we can write, ∞ 2 X 2 2 A k f kh ≤ | h f , f i ih | ≤ B k f kh ∀ f ∈ Xh i = 1 Using (1), the above can be write as

∞ 2 X 2 2 A k f kh ≤ | h f , P Y f i ih | ≤ B k f kh ∀ f ∈ Y. i = 1 ∞ This shows that { P Y f i }i = 1 is a frame for Y with the same bounds.

∞ (ii) Let { f i }i = 1 is a frame for Y with frame operator Sh. Then ∞ X −1 f = f , Sh f i | h f i ∀ f ∈ Y. i = 1 6 Prasenjit Ghosh

Therefore,

∞ X −1 P Y f = f , S h f i | h f i ∀ f ∈ Y [ using (1) ] i = 1

−1 ⊥ Since Sh is a bijection on Y , then S h f i ∈ Y . Now, if f ∈ Y then −1 ⊥ f , S h f i | h = 0 and P Y f = 0 if f ∈ Y . Therefore

∞ X −1 P Y f = f , S h f i | h f i ∀ f ∈ Xh. i = 1

∞ Note 3.2. Let { f i }i = 1 be a 2-frame associated to h for X. If for f ∈ Xh, f = ∞ P ∞ 2 c i f i for some { c i }i = 1 ∈ l (N), then i = 1

∞ ∞ ∞ X 2 X −1 2 X −1 2 | c i | = f , Sh f i | h + c i − f , S h f i | h . i = 1 i = 1 i = 1 ∞ Theorem 3.3. Let { f i }i = 1 be a 2-frame associated to h for X with pre frame operator T h. Then the pseudo-inverse of T h is given by

† 2 †  −1 ∞ Th : Xh → l ( N ),Th f = f , Sh f i | h i = 1 ,

where Sh be the corresponding frame operator. ∞ Proof. By the Theorem (2.15), { f i }i = 1 is a frame for Xh. Then for f ∈ Xh has ∞ P ∞ 2 a representation f = c i f i for some { c i }i = 1 ∈ l ( N ) and this can be writ- i = 1 ∞  −1 ∞ ten as Th { c i }i = 1 = f. By note (3.2), the frame coefficient f , Sh f i | h i = 1 have minimal l 2-norm among all sequences representing f. Hence, the above equa- †  −1 ∞ tion has a unique solution of minimal norm namely, T h f = f , Sh f i | h i = 1.

∞ Theorem 3.4. Let { f i }i = 1 be a 2-frame associated to h for X, then the optimal frame bounds A, B are given by

−2 −1 −1 † 2 A = Sh = Th ,B = k Sh k = k T h k ,

† where Th is the pre frame operator, Th is the pseudo inverse of Th and Sh is the frame operator.

Proof. By the definition, the optimal upper frame bound is given by

∞ X 2 B = sup | h f , f i | h i | = sup h Sh f , f | h i = k Sh k k f , h k = 1 i = 1 k f ,h k = 1 Some characterization of K-frame in 2-inner product space 7

∗ 2 Therefore, B = k Sh k = k Th Th k = k Th k . We know that the dual frame  −1 ∞ −1 −1 Sh f i i = 1 has frame operator Sh and the optimal upper bound is A . So −1 −1 −1 −1 by the above similar process A = Sh ⇒ A = Sh . Now, from the Theorem (3.3), we obtain

∞ 2 2 −1 X −1 2 † † Sh = sup f , Sh f k | h = sup T h f = T h . h k f , h k = 1 k = 1 k f k h = 1 −2 −1 −1 † Thus, A = Sh = Th . This completes the proof of the Theorem.

4 Frame operator for 2-K-frame

∞ Theorem 4.1. Let { f i }i = 1 be a 2-Bessel sequence in X with the frame op- ∞ erator Sh and Kh ∈ B ( Xh ). Then { f i }i = 1 is a 2-K-frame if and only if the  1  quotient operator  K ∗ /S 2  is bounded.  h h 

∞ Proof. Let { f i }i = 1 be a 2-K-frame for X. Then there exists positive constants A, B such that

∞ ∗ 2 X 2 2 A k K h f k h ≤ | h f , f i | h i | ≤ B k f k h ∀ f ∈ X h. (2) i = 1

Using the definition of frame operator Sh, we can write

∞ X 2 h S h f , f | h i = | h f , f i | h i | , ∀ f ∈ X h. (3) i = 1 Using ( 3 ), The inequality ( 2 ) can be written as

∗ 2 2 A k K h f k h ≤ h S h f , f | h i ≤ B k f k h ∀ f ∈ Xh

* 1 1 + ∗ 2 2 2 2 ⇒ A k Kh f kh ≤ S h f , S h f | h ≤ B k f kh ∀ f ∈ Xh

2 1

∗ 2 2 2 ⇒ A k K h f k h ≤ Sh f ≤ B k f k h ∀ f ∈ Xh (4)

h Let us now define the operator ,  1   1   1  T =  K ∗ /S 2  : R  S 2  → R ( K ∗ ) , by T  S 2 f  = K ∗ f ∀ f ∈ X .  h h   h  h  h  h h 8 Prasenjit Ghosh

  2 1 1 1

Now, let f ∈ N  S 2 . Then S 2 f = θ ⇒ S 2 f = 0, so by ( 4 ),  h  h h

h  1  A k K ∗ f k 2 = 0 ⇒ K ∗ f = θ ⇒ f ∈ N ( K ∗ ) ⇒ N  S 2  ⊆ N ( K ∗ ). h h h h  h  h

This shows that the quotient operator T is well-defined. Also for all f ∈ Xh,   1 1 1 T  S 2 f  = k K ∗ f k ≤ √ S 2 f  h  h h h A h h Hence, T is bounded.  1  Conversely, suppose that the quotient operator  K ∗ /S 2  is bounded. Then  h h  there exists B > 0 such that,

  2 2 1 1

T  S 2 f  ≤ B S 2 f ∀ f ∈ X  h  h h

h h 2 1 * 1 1 +

∗ 2 2 2 2 ⇒ k K h f k h ≤ B S h f = B S h f , S h f | h

h  1  = B h S f , f | h i  since S 2 is also self-adjoint  h  h 

∞ X 2 = B | h f , f i | h i | (5) i = 1 ∞ Also, { f i }i = 1 be a 2-Bessel sequence associated to h in X,so there exists C > 0 such that ∞ X 2 2 | h f , f i | h i | ≤ C k f k h , ∀ f ∈ X h (6) i = 1 ∞ Hence, from (5) and (6), { f i }i = 1 is a 2-K-frame associated to h for X. ∞ Theorem 4.2. Let { f i }i = 1 be a 2-K-frame sequence in X with the frame operator S h and T ∈ B ( X h ). Then the following are equivalent:

∞ (1) { T f i }i = 1 is a 2-TK-frame associated to h for X. Some characterization of K-frame in 2-inner product space 9

 1  (2)  ( TK ) ∗ /S 2 T ∗  is bounded.  h h 

 1  ∗ ∗ (3)  ( TK h ) / ( TS h T ) 2  is bounded.

∞ Proof. ( 1 ) ⇒ ( 2 ) Suppose that { T f i }i = 1 is a 2-TK-frame associated to h for X. Then there exists constants A, B > 0 such that ∞ ∗ 2 X 2 2 A k ( TK h ) f k h ≤ | h f , T f i | h i | ≤ B k f k h , ∀ f ∈ X h. (7) i = 1

Using the definition of frame operator S h, we can write ∞ X 2 h S h f , f | h i = | h f , f i | h i | , ∀ f ∈ X h. i = 1 Now, ∞ ∞ X 2 X ∗ 2 ∗ ∗ | h f , T f i | h i | = | h T f , f i | h i | = h S h ( T f ) ,T f | h i i = 1 i = 1 2 * 1 1 + 1

2 ∗ 2 ∗ 2 ∗ = S h ( T f ) ,S h ( T f ) | h = S h ( T f )

h Let us now consider the quotient operator,  1   1   ( TK ) ∗ /S 2 T ∗  : R  S 2 T ∗  → R (( TK ) ∗ ) by  h h   h  h

 1   S 2 T ∗  f 7→ ( TK ) ∗ f ∀ f ∈ X .  h  h h

From (7), we can write 2 1

∗ 2 2 ∗ A k ( TK h ) f k h ≤ S h ( T f ) ∀ f ∈ X h.

h 2 1 1 ∗ 2 2 ∗ ⇒ k ( TK h ) f k h ≤ S h ( T f ) ∀ f ∈ X h. A h 10 Prasenjit Ghosh

 1  This shows that the quotient operator  ( TK )∗ /S 2 T ∗  is bounded.  h h 

 1  ( 2 ) ⇒ ( 3 ) Suppose that the quotient operator  ( TK ) ∗ /S 2 T ∗  is bounded.  h h 

Then there exists constant B > 0 such that

2 1

∗ 2 2 ∗ k ( TK h ) f k h ≤ B S h ( T f ) ∀ f ∈ X h. (8)

h

Now, for each f ∈ X h, we have

2 1

2 ∗ ∗ ∗ ∗ S h ( T f ) = h S h ( T f ) ,T f | h i = h TS h ( T f ) , f | h i

h

2 * 1 1 + 1 ∗ ∗ ∗ = ( TS h T )2 f , ( TS h T )2 f | h = ( TS h T )2 f . (9)

h From (8) and (9), we get

2 1 ∗ 2 ∗ k ( TK h ) f k ≤ B ( TS h T )2 f ∀ f ∈ X h. h h  1  ∗ ∗ Hence, the quotient operator  ( TK h ) / ( TS h T ) 2  is bounded.

 1  ∗ ∗ ( 3 ) ⇒ ( 1 ) Suppose the quotient operator  ( TK h ) / ( TS h T ) 2  is bounded.

Then there exists constant B > 0 such that 2 1 ∗ 2 ∗ k ( TK h ) f k ≤ B ( TS h T )2 f ∀ f ∈ X h. (10) h h

∗ It is easy to verify that TS h T is self-adjoint and positive and hence the square ∗ root of TS h T exists. Now, for each f ∈ Xh, we have

∞ ∞ X 2 X ∗ 2 ∗ ∗ | h f , T f i | h i | = | h T f , f i | h i | = h S h ( T f ) ,T f | h i i = 1 i = 1 Some characterization of K-frame in 2-inner product space 11

  ∗ * 1 1 + * 1 1 + = S 2 T ∗ f , S 2 T ∗ f | h =  S 2 T ∗  S 2 T ∗ f , , f | h h h  h  h

* 1 1 + 2 2 ∗ ∗ = TS h S h T f , f | h = h TS h T f , f | h i

1 1 2 * + 1 2 2 ∗ ∗ ∗ = TS S T f , f | h = h TS h T f , f | h i = ( TS h T )2 f . (11) h h h From (10) and (11),

∞ 1 2 X k ( TK ) ∗ f k ≤ | h f , T f | h i | 2 ∀ f ∈ X . B h h i h i = 1

∞ On the other hand, since { f i }i = 1 is a 2-K-frame associated to h for X,

∞ ∞ X 2 X ∗ 2 2 2 | h f , T f i | h i | = | h T f , f i | h i | ≤ C k T k k f k h i = 1 i = 1

∞ Hence, { T f i }i = 1 is a 2-K-frame associated to h for X.

References

[1] Ali Akbar Arefijamaal, Gadhir Sadeghi, Frames in 2-inner Product Spaces, Ira- nian Journal of Mathematical Sciences and Informatics, Vol.8, No.2 ( 2013 ), pp 123-130.

[2] A. Bhandari, S. Mukherjee, Atomic subspaces for operators, Submitted, arXiv: 1705. 06042.

[3] O. Christensen, An introduction to frames and Riesz bases, Birkhauser (2008).

[4] P. Casazza, G. Kutyniok, Frames of subspaces, Contemporary Math, AMS 345 ( 2004 ) 87-114.

[5] Y. J. Cho, S. S Kim, A. Misiak, Theory of 2-inner product spaces, Nova Science Publishes New York ( 2001 ).

[6] I. Daubechies, A. Grossmann, Y. Mayer, Painless nonorthogonal expansions, Journal of Mathematical Physics 27 (5) ( 1986 ) 1271-1283.

[7] Bahram Dastourian, Mohammad Janfada, Atomic system in 2-inner product spaces, Iranian Journal of Mathematics Sciences and Informatics, vol .13, No. 1 (2018), pp 103-110. 12 Prasenjit Ghosh

[8] C. Diminnie, S. Gahler, A. White, 2-inner product spaces, Demonstratio Math. 6 (1973) 525-536.

[9] C. Diminnie, S. Gahler, A. White, 2-inner product spaces II, Demonstratio Math. 10(1977) 169-188.

[10] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc ., 72, ( 1952 ), 341-366.

[11] S. Gahler, Lineare 2-normierte Raume, Math. Nachr., 28, ( 1965 ), 1-43.

[12] Laura Gavruta, Frames for operator, Appl. Comput. Harmon. Anal. 32 (1), ( 2012 ) 139-144.

[13] W. Sun, G-frames and G-Riesz bases, Journal of Mathematical Analysis and Applications 322 (1) (2006), 437-452.

[14] X. Xiao, Y. Zhu, Gavruta, Some properties of K-frame in Hilbert spaces, Re- sults Math. 63 (3-4), ( 2013 ), 1243-1255.