On stability of retro Banach frame with respect to b-linear functional in n-Banach space

• 1.    Introduction and Preliminaries

  In 1946, D. Gabor [1] first initiated a technique for rebuilding signals using a family of elementary signals. In 1952, Duffin and Schaeffer [2] abstracted the fundamental notion of Gabor method for studying signal processing and they gave the formal definition of frame for Hilbert space. Later on, in 1986, it was reintroduced, developed and popularized by Daubechies et al. [3].

Frame for Hilbert space was defined as a sequence of basis-like elements in Hilbert space. A sequence { f i } i=i C H is called a frame for a separable Hilbert space (H, (• , •) ) , if there exist positive constants 0 < A ^ B <  to such that

∞

A Ilf 112 <Е 1^>1 2 <  B I f I 2 (V f ^E H).

i=1

• (0 2023 Ghosh, P. and Samanta, T. K.

But, in Banach space, due to the absence of inner product, frame was completely defined as a sequence of bounded linear functionals from the dual space of the Banach space. Before the notion of Banach frame was be formalized, it emerged in the fundamental work of Feichtinger and Groching [4, 5] related to the atomic decomposition for Banach spaces. Grochenig [6] introduced Banach frame in more general way in Banach space. Thereafter, further development of Banach frame was done by Casazza et al. [7]. P. K. Jain et al. [8] introduced and studied retro Banach frame and it was further developed by Vashisht [9]. Stability theorems for Banach frames were studied by Christensen and Heil [10] and P. K. Jain et al. [11]. S. Gahler [12] was the first to introduce the notion of linear 2 -normed space. A generalization of a linear 2 -normed space for n ^ 2 was developed by H. Gunawan and Mashadi [13]. P. Ghosh and T. K. Samanta [14–16] have studied the frames in n -Hilbert spaces and in their tensor products.

In this paper, we present the retro Banach frame relative to bounded b -linear functional in n -Banach space. A sufficient condition for the stability of retro Banach frame associated to (a 2 ,..., a n ) in n -Banach space under some perturbations is discussed. We establish that retro Banach frame associated to (a 2 ,..., a n ) is stable under perturbation of frame elements by positively confined sequence of scalars. Also, we consider the finite sum of retro Banach frame associated to (a 2 ,..., a n ) and establish a sufficient condition for the finite sum to be a retro Banach frame associated to (a 2 , ...,a _n ) in n -Banach space. Finally, retro Banach frame associated to (a 2 ,..., a n ) in Cartesian product of two n -Banach spaces is presented.

Throughout this paper, E is considered to be a separable Banach space and E ^∗ , it’s dual space. By B (E) we denotes the space of all bounded linear operators on E . Let E d be a sequence space, which is a Banach space and for which the co-ordinate functionals are continuous. Let { g i } i y C E* and S : E d ^ E be a bounded linear operator. Then the pair ( { g i } , S) is said to be a Banach frame for E with respect to E d if

• (i)    Mf)} e Ed (V f E E);

• (ii)    there exist B > A > 0 such that A^f ||e C l{gi(f)}^Ed C B|f ||e (Vf E E);

• (iii)    S({gi(f)}) = f (V f E E).

The constants A , B are called Banach frame bounds and S is called the reconstruction operator.

Let E _d ^∗ be a Banach space of scalar-valued sequences associated with E ^∗ indexed by N . Let { x k } C E and T : E d * ^ E * be given. The pair ( { x k } ,T) is called a retro Banach frame for E ^∗ with respect to E _d ^∗ if

• (i)    { f (x k ) } E E d for each f E E* ;

• (ii)    there exist positive constants A and B with 0 < A C B <  to such that A l f I e * C l{ f (x k )}| ^ C B l f I e * , f E E * ;

• (iii)    T is a bounded linear operator such that T( { f (x k ) } ) = f , f E E*.

The constants A and B are called frame bounds. The operator T is called the reconstruction operator or pre-frame operator.

A n -norm on a linear space X (over the field K of real or complex numbers) is a function

(xi,x2, ...,xn) I > |xi,x2, . . . ,xnl xi,x2, ...,xn E X, from Xn to the set R of all real numbers such that

• (i)    | x i , x 2 ,...,x n | = 0 if and only if x i ,...,x n are linearly dependent;

• (ii)    | x i , x 2 ,..., x _n | is invariant under permutations of x i , x 2 ,..., x n ;

• (iii)    |axi,x2,... ,xn| = |a||xi,x2,... ,xn|;

• (iv)    |x + y,x2, . . . ,xnl C |x,x2, ••• ,xn| + ly,x2, . . . , xn |,

• 2. Main Results

for every x i ,x 2 ,..., x n G X and a G K . A linear space X , together with a n -norm || • ,..., • ||, is called a linear n -normed space. A sequence { x k } in linear n -normed space X is said to be convergent in X if there exists x G X such that

lim |xk - x,e2, ...,en| =0 (V e2,...,6n G X), k^^

and it is called a Cauchy sequence if

lim |xi - xk ,e2, ...,en || = 0 (V e2,..., en G X). l,k^^

The space X is said to be complete or n -Banach space if every Cauchy sequence in this space is convergent in X .

In this section, the notion of retro Banach frame in n -Banach space X is introduced and some stability theorems for retro Banach frame relative to bounded b -linear functional in n -Banach space have been derived.

Now, we first define a bounded b -linear functional. Let (X, || • , ..., • | ) be a linear n -normed space and a 2 ,..., a n be fixed elements in X . Let W be a subspace of X and ( a i ) denote the subspaces of X generated by a i , for i = 2, 3,..., n . Then a map T : W x ( a ₂ ) x ... x ( a n ) ^ K is called a b -linear functional defined on W x ( a ₂ ) x ... x ( a n ) , if for every x, y G W and k G K , the following conditions hold:

• (i)    T(x + y, a2,..., an) = T(x, a2,..., an) + T(y, a2,..., an);

• (ii)    T(kx, a2,..., an) = k T(x, a2,..., an).

A b -linear functional is said to be bounded if there exists a real number M > 0 such that

|T(x, a2,..., an)| C M ||x, a2,..., an| (Vx G W).

The norm of the bounded b -linear functional T is defined by

|T| = inf {M > 0 : |T(x, a2,..., an)| C M |x, a2,..., an| (Vx G W)}.

The norm of T can be expressed by any one of the following equivalent formula:

• (i)    |T| = sup {|T (x,a2, ...,an)| |x,a2, ...,an| C 1};

• (ii)    |T| = sup {|T (x,a2,... ,an)| ||x,a2,„. ,an|| = 1};

I T(xM-A.) !

• ⁽ⁱⁱⁱ⁾    H T II =^sup^ || x,a ₂ ,...,a _n | ||x, ^a 2 , . . . , a n || = ⁰J .

For more details on bounded b -linear functional defined on X x ( a ₂ ) x ... x ( a n ) one can go through the paper [17]. For the remaining part of this paper, X denotes the n -Banach space with respect to the n -norm || • ,..., • | and X F denotes the Banach space of all bounded b -linear functional defined on X x ( a ₂ ) x ... x ( a n ) with respect to the norm given by above.

Definition 1. Let X be a n -Banach space and X ) be a Banach space of scalar-valued sequences associated to X F indexed by N . Let { x k } С X and S : X ) ^ X F be given. Then the pair ( { x k } , S) is said to be a retro Banach frame associated to (a ₂ , • • •, a n ) for X F with respect to X ) if

• (i)    { T(x k , a ₂ ,..., a n ) } G X ) for each T G X F ;

• (ii)    there exist constants 0 < A C B <  to such that

• (iii)    S is a bounded linear operator such that

S({T(xk,a2,...,an)})= T (VT E XF).

The constants A , B are called frame bounds. If A = B , then ( { x k } ,S) is called tight retro Banach frame associated to (a 2 ,..., a n ) and for A = B = 1 , it is called normalized tight retro Banach frame associated to (a 2 ,..., a n ) . The inequality (1) is called the frame inequality for the retro Banach frame associated to (a 2 ,..., a n ) . The operator S : X d ^ X F is called the reconstruction operator or the pre-frame operator.

Definition 2. A sequence { x k } С X is said to be a retro Banach Bessel sequence associated to (a 2 ,..., a n ) for X F with respect to X ^ if

• (i)    { T(x k , a 2 ,..., a n ) } E X ^ for each T E X F ;

• (ii)    there exists a constant B >  0 such that

\\{T (xk ,a2,...,an)}hxd < B \T ||xF (V T E XF)

The constant B is called a retro Banach Bessel bound for the retro Banach Bessel sequence { x k } associated to (a 2 ,..., a n ) . Let X b denotes the set of all retro Banach Bessel sequence associated to (a 2 ,..., a n ) for X F with respect to X ^ . For { x k } E X b , define

Ra : XF ^ Xd by Ra (T ) = {T (xk ,a2,***,an)} (V T E XF)*

Then it is easy to verify that R A is a bounded linear operator. The operator R A is called the analysis operator.

Next, we verify that scalar combinations of two retro Banach frames associated to ( a 2 , . . . , a _n ) becomes a retro Banach frame associated to ( a 2 , . . . , a _n ) .

Theorem 1. Let ( { x k } ,S) and ( { y k } ,S) be two retro Banach frames associated to (a 2 ,..., a n ) for X F with respect to X ^ having bounds A, B and C, D, respectively. Then for any scalars a, в, ({ ax k + ey k } , a++e S is a retro Banach frame associated to (a 2 ,..., a n ) for X _F ^∗ with respect to X _d ^∗ .

• < 1 For each T E X F , we have

||{T (axk + eyk ,a2,.. .,an)}^X * = ^{aT (xk ,a2, ...,an) + вТ (yk ,a2,*•., an H^X * Xd

• <    |a| |{T(xk,a2, * * * , an ) } ^ X * + И ||{T(yk ,a2,...,an )}|X * ^ (|a|B + |в |D)|T ||X * -ddF

On the other hand,

^{T (axk + eyk, a2, * * *, an)} ^X * > |a| ^{T (xk ,02, * * * ,an)}|X * Xd

• -    |в| |{T(yk, a2,***, an)}|X* > (|a|A - |в|С)\\T\x*, T E XF*

Also, for T E X F , we have

S ({T (xk ,a2,***,an)}) = T and S ({T (yk ,»2,***,an)}) = T*

Then for T E X F , we have

а + в S({T(axk + eyk, a2, * * *, an)}) = a+в [aS^{T(xk, a2, * * *, an)})

+ eS({T(yk, a2,***, an)})] = o+j (aT + eT) = T*

Hence, the family ({ ax k + вУ к } , a++e S) is a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X d having bounds ( | a | A — | в | С) and ( | a | B + | e | D) • »

In the next theorem, we will see that the sum of two retro Banach frames associated to (a 2 ,..., a n ) with different reconstructions operators is also a retro Banach frame associated to (a 2 ,..., a n ) •

Theorem 2. Let ({xk},S) and ({yk},P) be two retro Banach frames associated to (a2,..., an) for XF with respect to Xd having bounds A, B and C, D, respectively. Let R : Xd ^ Xd be a linear homeomorphism such that

R({T(xk, a2,..., an)}) = {T(yk, a2,..., an)}, T E XF.

Then there exists a reconstruction operator Q : Xd ^ XF such that the family ({xk + yk}, Q) is a retro Banach frame associated to (a2,..., an) for XF with respect to X,.

• <1 Let U , V be the corresponding coefficient mappings for the retro Banach Bessel sequences { x k } and { y k } , respectively and I denotes the identity mapping on X d, . Now, for each T E X F , we have

• 11 {T(xk + yk, a2, ... , an)} 11 x*     11{T(xk, a2,... , an) } + {T(yk, a2, . . . , an )} ||x*

dd

• 11 {T(xk, a2, . . . , an ) } + R({T(xk, a2, . . . , an ) }) 11 x*

X _d

• ^ ||I + R||{T(xk,a2,...,an)}|X* ^ B ||I + R^T||X*. dF

Similarly, for each T E X F , we have

||{T(xk + yk,a2,...,an)}|, ^ D ||I + R-1||T||, . dF

Thus, for each T E X F , we get

• ||{T (xk + yk ,a2,...,an)}| . ^ min {B ||I + R^,D ||I + R-1||}||T|| . . dF

On the other hand, for each T E X F , we have

• 11 {T (xk + yk, a2,..., an)} ||X ^.

  > || { T (x k ,a 2 , ...,a n ) } ^| X *

  d

  
  

  _—

  
  

|| { T (y k ,a 2 ,...,a n ) } |, >  A || I — R^T |_x * .

dF

Also, for each T E X F , we have

|| { T (x k + y k , a 2 ,..., a n ) } |_x * >  C 11 R ^{- 1} - 1 1| || T |_x * . dF

Therefore, for each T E X F , we get

• ||{T (xk + yk ,a2,...,an)}|X * > max {A |I — R|,C ||R-1 — I |}|T||X *.

dF

Now, for T E X F , we have

R({T (xk + yk, a2,..., an)}) = R({T (xk, a2,..., an)}) + R({T (yk, a2,..., an)}) = (I + R){T (yk, a2,..., an)} = (I + R)P-1T.

Therefore, if we take Q = ((I + R)P ¹ ) ¹ , then Q : X d ^ X F is a bounded linear operator such that

Q({T (xk + yk, a2,..., an)}) = T (V T E XF).

Hence, ( { x k + y k } , Q) is a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X d . ▻

Now, we start with a necessary and sufficient condition for the stability of a retro Banach frame associated to (a 2 , . . . , a _n ) .

Theorem 3. Let ( { x k } ,S) be a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X d having bounds A, B. Let { y k } be a sequence in X such that { T(y k ,a 2 ,...,a _n ) } G X ; T G X F . Suppose R : X d ; ^ X d ; be a bounded linear operator such that

R( { T(y k , a 2 ,..., a n ) } ) = { T(x k , a 2 ,..., a n ) } , T G X F .

Then there exists a bounded linear operator P : X ^ ^ X F such that ( { y k } ,P) is a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X ^ if and only if there exists a constant K >  1 such that

|| ^{{ T (} x k ^y k , a 2 , . . . , a n ^{) }} | | x *

d

< K min {|{ T (x k ,a 2 ,...,a n )}| x * , ||{ T (y k ,a 2 ,...,a n )}| x * }.

<1 First we suppose that ( { y k } , P ) is a retro Banach frame associated to (a 2 ,..., a n ) . Then there exist constants C,D >  0 such that

A I T | x _F ^ || { T (x k ,a 2 ,...,a n ) } | x * ^ B | T | x F   ( V T G X F ),             (3)

C I T | x f ^ || { T (y k ,a 2 ,...,a n ) } | x * <  D I T H x *    ( V T G X F ).             (4)

Therefore, for each T G X F , we have

^T (x k - y k ,a 2 ,... a)}^_d . <  ^T (x k ,a 2 ,..

(4)                                                         (3)

^ || { T (x k ,a 2 ,...,a n ) } | X * + D | T | x F ^

. , a n ^{) } 11} x * ^{+ 11 { T(y} k , ^a 2 , ... , a n ) ^{} 11} x *

dd

^^{1 +} A) ^{| { T(x} k , ^a 2 ,.

. ., a n ) ^} | x * .

X _d

Similarly, it can be shown that

^{| { T(x} k ^- y k ^,a 2 , . . . ^,a n ^{)} ||} х . < (^{1 +} B^ ^{| { T(}y k ^,a 2 , .

..,a n ) } | x * ( V T G X F ).

Thus, for each T G X F , we get

^ { T(x k - y k ,a 2 , ... ,a n ) } ^| _x * <  K min {|{ T(x k , a 2 ,... ,a n ) }| x * , |{ T(y k , a 2 ,... ,a n ) }| x * }, where K = max {( 1 + D ) , ( 1 + C )} .

Conversely, suppose that there exists K >  1 such that (2) holds. Now, for each T G X F we have

A ^{| T |} X * ^ ^{|| { T(x} k , ^a 2 , . . . , a n ^{) }} ^ x * ^ ^{|| { T(x} k    ^y k ,^a 2 ,...,^a n ^{) } |} x *

Fd                                         d

+ | { T(y k ,a 2 ,...,a n )} | _x * . ^ (K + 1) | { T(y k ,a 2 ,...,a n ) } | _x *, .

This implies that

A

( _{K +} 1) ^{| T |} x F <  ^{|| { T(}y k ^,a 2 ^,...^,a n ^{) } |} _x * .

On the other hand, for each T E X F , we have

||{T (yk, a2,...,an)}^X, ^ ||{T (xk - yk ,a2,...,an^^, + ||{T (xk, a2,..., an ^^, ^ (K + 1) ||{T(xk, a2,..., an)}^, ^ B(K + 1) |T|xF•

Now, take P = SR . Then P : X } ^ X F is a bounded linear operator such that

P ({T (yk, a2,..., an)}) = SR({T (yk, a2,..., an)}) = S{T (xk, a2,..., an)} = T, T E XF.

Thus, ( { y k } , P ) is a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X } . This completes the proof. >

The Theorem 3 shows that the stability of retro Banach frame associated to (a 2 , ..., a n ) depends on the value of K . For large value of K , the retro Banach frame inequality is lost. Therefore, to get optimal frame bounds, we still need to modify the stability conditions. In the following theorem, we give a sufficient conditions for the stability of a retro Banach frame associated to ( a 2 , . . . , a _n ) .

Theorem 4. Let ({xk},S) be a retro Banach frame associated to (a2,... ,an) for XF with respect to X}. Let {yk} С X be such that {T (yk ,a2,..., an)} E X}, T E XF and let U : XF ^ X}} be the coefficient mapping given by

U(T) = {T(xk,a2,...,an)},  T E XF.

If there exist positive constants а, в (< 1) and Ц such that

• (i)   max ('   2-               'Л < 1;

(1 - в)

• (ii)    ^{T (xk - yk,a2,... ,an)}|Xd, ^ а ^{T (xk, a2,..., an)} ^X,

+ в ||{T(yk, a2, . . . , an)}|X, + ^ llT hxF , T E XF, then there exists a reconstruction operator P : X}} ^ XF such that ({yk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect to X}.

• <1 Let V : X F ^ X } be an operator defined by

V(T) = {T(yk,a2,...,an)}, T EXF.

Using the operators U and V , condition ( ii ) can be written as

lUT - VT|xd ^ a lUT|xd + в lVT|xd + Ц |T|xF, T E XF.

Thus, for T E XF, we have l{T (yk ,a2,...,an)}lx* = llVT|x* ^ |UT - VT |xd. + |UT|x* ^ O+OlUlt^ ^ ^x^ .

d            d                   d            d          ¹ — ^{в             f}

Therefore, V is a bounded linear operator such that

lUT- VTlx* ^ Mo-^^^UH^ |TIX (VT E XF). d           1 - в             F

Now,

lip - SVII ^ ||SIIIIU - VII ^ ffl^+^-^MU^+^l < 1.

1-в

This shows that SV is an invertible operator with satisfying

II (SV) 1 ⁵₁   _s_ 2±a_U_+X_ •

¹            i - e

Now, take P = (SV) 1 S . Then PV = I f , where I f is the identity operator on X F . Thus, P : X ^ ^ X F is a bounded linear operator such that

P({T(yk,a2,...,an)})= T, T E XF

Now, for T E X F , we have

« T B x f = « ^PVT « x f 5 « P «« ^VT « Xd 5 —^^(^7 1^ « VT « Xd . ¹             i - e

This implies that

«S«-1 (1 - [(2 + a ^«U« + ^«S«)«T«XF 5 |{T(yk,a2,...,an)}Ix, (VT E XF). 1 — e                                           d

Hence, ( { y k } , P ) is a retro Banach frame associated to (a 2 , ..., a n ) for X F with respect to X ^ . This completes the proof. >

Next, we give a stability condition of a retro Banach frame associated to ( a 2 , . . . , a _n ) by using a given retro Banach Bessel sequence associated to ( a 2 , . . . , a _n ) .

Theorem 5. Let ({xk}, S) be a retro Banach frame associated to (a2,..., an) for XF with respect to X^ having bounds A,B. Let {yk} be a sequence in X such that ({T(yk, a2,..., an)} E X^), T E XF and for some constant K > 0

||{T(yk,a2,...,an)}|X, 5 K«T«xF (VT E XF).

IISII-1

Then for any non-zero constant A with |A| < ^K—, there exists a reconstruction operator P : Xd ^ XF such that ({xk + Ayk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect to X} having frame bounds («S«-1 — |A|K) and (B + |A|K).

• <1 Let V : X F ^ X } be a bounded linear operator defined by

V(T) = {T(yk,a2,...,an)}, TE XF, and U : XF ^ X^^ be a bounded linear operator given by

U(T) = {T(xk,a2,...,an)}, T E XF.

Then it is easy to verify that { T (x k + Ay k , a 2 ,... ,a _n ) } E X ^ , for all T E X F . Now, for each T E X F , we have

«UT + AVT «x* = ||{T (xk + Ayk,a2,...,an)}|X,

• 5 |{T(xk,a2,...,an)}|X* + |A| |{T(yk,a2,...,an)}|X* 5 (B + |A|K)«T«x*. d                                      dF

On the other hand, for each T E X F , we have

-1 — |A|K«T«xF 5 |{T(xk,a2,...,an)}|x*

^{| A | || { T(}y k ^,a 2 , . . .^,a n ^{) } |} X »

Define, L : X F ^ X ) by L(T ) = { T(x k + Ay k , a 2 ,..., a n ) } , T E X F . Then L is a bounded linear operator such that

^ UT — LT ||x d = || { T (x k ,a 2 , ... ,a n ) } - { T(x k + Ay k ,a 2 ,... ,a n ) } ^ X *

= | { AT (y k ,a 2 ,...,a n ) } | X , ^ | A | K | T | x F , T E X F .

This verifies that | U — L | ^ | A | K . Since SU = I f , I f is the identity operator on X F , we have

I I f — SL I = I SU — SL I ^ | S || U — L | < 1.

Thus SL is invertible. Take P = (SL) ^{- 1} S . Then P : X ) ^ X F is a bounded linear operator such that

P( { T(x k + Ay k , a 2 ,..., a n ) } ) = T, T E X F .

Hence, ( { x k + Ay k } , P ) is a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X ) having frame bounds ( | S | ^{- 1} — | A | K) and (B + | A | K) . >

Theorem 6. Let ( { x k } , S ) be a retro Banach frame associated to (a 2 , ..., a n ) for X F with respect to X ) Let { y k } С X and { a k } C R be any positively confined sequence such that { T (a k y k , a 2 ,...,a n ) }E X ) , T E X F .If V : X F ^ X d defined by

V(T) = {T(yk,a2,...,an)}, T EXF, such that IIV|| < ---—-----, then there exists a reconstruction operator P : X^ ^ XF such

• ¹¹ " sup k V ^a k ^,                                                    p                  ^{d        F}

that ({xk + akyk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect to Xd,-

• <1 Let U : X F ^ X ) be a bounded linear operator defined by

U(T ) = { T(x k ,a 2 ,...,a n ) } , T E X F .

It is easy to verify that { T(x k + a k y k ,a 2 ,...,a n ) } E X ) , for all T E X F . Now, for each T E X _F ⁾ , we have

¹1 ^{{ T (} x k ^{+ a} k ^y k , ^a 2 , ... , a n ^{) } 1}1 y* ^ ¹1 ^{{ T ( x} k , ^a 2 , ^{. . .} , a n ) ^{} 1}1 y* ^{+ |{ a} k ^{T ( y} k , ^a 2 , ^{. . .} , a n ) ^{} 1}1 j^*

dd   d

^ ^ { T (x k , a 2 ,... ,a n ) } ^| _X * + ( sup a k )^| { T (y k , a 2 ,..., a n ) } ^ X * ^d    \ 1 ^ k<^   /                          ^d

• < [^{| U 1 + | V}Ilf ^suP ^ak) ]^{| T} IIxf .

\ 1 ^ k<^ /

On the other hand, for each T E X _F ⁾ , we have

¹1 ^{{ T ( x} k ^{+ a} k ^y k , ^a 2 , ... , ^a n ^{) } 1} jy* ^ ¹1 ^{{ T ( x} k , ^a 2 ,^... , a n ) ^{} 1} jy*    ¹1 ^{{ a} k ^{T ( y} k , ^a 2 ,^... , a n ) ^{} 1}1 jy*

dd   d

• > f|S I ^{- 1} — I V Ilf sup ak) 1| T | x * .

L           \ Kk<~  /J

Define, L : X F ^ X d , by

Following the lines of proof of the Theorem 5, L is a bounded linear operator on X _F ^∗ such that \\ U — L \\ C sup i c k< ^ a k || V || and SL is invertible. Take P = (SL) - 1 S . Then P : X ^ ^ X F is a bounded linear operator such that

P( { T(x k + a k y k , a 2 ,..., a n ) } ) = T, T E X F .

Hence, ( { x k + a k y k } , P ) is a retro Banach frame associated to (a 2 , • • •, a n ) for X F with respect to X d . ▻

In the next theorem, we establish that retro Banach frame associated to (a 2 , • • •, a n ) is stable under perturbation of frame elements by positively confined sequence of scalars.

Theorem 7. Let ( { x k } ,S) be a retro Banach frame associated to (a 2 , • • •, a n ) for X F with respect to X^. Let { y k } С X be such that { T(y k ,a 2 , • • • ,a _n ) } E X ^ , T E X F . Let R : X d ^ X d be a bounded linear operator such that

R( { T (y k , a 2 ,..., a n ) } ) = { T(x k , a 2 , • • •, a n ) } ,    T E X F .

Suppose { a k } and { e k } are two positively confined sequences in R . If there exist constants A, p(0 C A, p <  1) and y such that

• ^{( i )}    Y< ⁽¹ - ^A )IISII^{- 1}f ^ ^ak);

1 C k< ^

• (ii)    ^ { a k T (x k ,a 2 ,.. •,a n ) } — { e k T (y k ,a 2 , •••, a n )} ^ _x * C A ^ { a k T (x k ,a 2 ,. .•,a n ) } ^| _X . dd

^{+ p | { e} k ^{T ( y} k , ^a 2 , • • • , a n ^{) } |} X * ^{+ Y} ||^T\ X F , ^{T E X} F • Then there exists a reconstruction operator P : X ^ ^ X F such that ( { y k } ,P ) is a retro Banach frame associated to (a 2 , • ••, a n ) for X F with respect to X^.

• <1 Let U : X F ^ X ^ be a bounded linear operator defined by

U(T) = { T (x k ,a 2 ,...,a n ) } , T E X F .

Since the operator SU : X F ^ X F is an identity operator, for T E X F ,

\ T | x f = \ SU(T )| x f C | S | | { T (x k ,a 2 ,...,a n )} ^ x * .

Now, for each T E X F , we have

• ¹ 1 ^{{ e} k ^{T ( y} k , ^a 2 , • • • , a n ^{) } 1 |} x *

d

C ^ { a k T (x k ,a 2 , • • .,a n ) } ^ X * + || { « k T (x k ,a 2 , • • .,a n ) } — { e k T (y k ,a 2 , • • .,a n )} ^ _x *

X _d                                                 X _d

C (1 + A) ^ { a k T (x k ,a 2 ,...,a n ) } ^ X * + p ^ { e k T (y k ,a 2 ,...,an. ) } ^ X * + Y l|T | x * • d                                       d^F

Thus,

^{(1 —} pF ^{l { e} k T ⁽ y k ,a 2 ,•••^,a n ^{) } |} _X * ^C [^{(1 + a )} \ U||f sup ^a k) ⁺ y] \ ^t \ x f • ^d L                \ 1 Ck<^ / _l

This implies that

⁽¹ - P)( inf ^ek) || ^{{ T (}y k ^,a 2 ,. у 1 C k< ^ у

.

.^,a n ^{)} |} _x * ^C [^{(1 + A}) \ U Ilf sup op) ⁺ y"| || T \ xf . ^d L                \ 1 C k<^   /     _|

On the other hand, by condition (ii) , we get

(1 + д) ||{вкT(yk,a2,.. .,an)}|X* > (1 - A) ||{akT(xk,a2,...,an)^L* - y ||T||x*

dd

> [(1 - A) \S\-1f inf a?) - y1 \T|x*, T G XF 1^k<^

Therefore, for each T G X F , we have

(1 + д)( sup вк ) ^{T(Ук, a2, • • •, an)}HX* ^ (1 + ^) WkT(Ук, a2, • • • >an)}|X« \ 1^k<^ /                          dd

> [(1 - A) \S\-1f inf a?) - y1 \\T|x*. 1^k<^

Thus, for each T G X F , we have

^{(1 - a )| s} \l 1( inf ^ak) ^-

γ

- \ T \ x _F <  | { T(y k ,a 2 ,...,a n )} | x * .

x 1^k<^/

(1 + X sup ek )

\ 1^k<^/

(1 + A) \ U \| ( sup аЛ

+y

---\TIX -

<

\ 1^k<^/

⁽¹ - ^)С1^^П^ ^ek) у 1 ^ k< ^ у

Now, take P = SV . Then P : X d ^ X F is a bounded linear operator such that

P({T(yk,a2,...,an)}) = T, T G XF.

Hence, ( { y k } ,P ) is a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X d | . ▻

Definition 3. A sequence { x k } in X is said to be total over X F if

{T G XF : T(xk,a2,...,an) = 0 (Vk)} = {0}, where 0 G XF is the null operator.

In the following two theorems, some sufficient condition will be describe under which the finite sum of retro Banach frame associated to (a 2 ,..., a n ) is again a retro Banach frame associated to (a 2 , . . . , a _n ) .

Theorem 8. For i G E m = { 1, 2,..., m } , let ( { x k,i } , S i ) be retro Banach frames associated to (a 2 ,..., a n ) for X F with respect to X d ^ . Then there exists a reconstruction operator P : X d ^ X F such that ( { ^2^ =1 X k,i } , P) is a tight retro Banach frame associated to (a 2 ,..., a n ) for X _F ^∗ with respect to X _d ^∗ , provided

||{T(xk,j ,a2, ...

an)}|Xd ^

T i=1

xk,i, a2, . . . , an

)}

X

∗

d

for T G XF, for some j G Em.

<1 For T E X F , we have

H T « X F = ^ S i ( { T Ek,, ,« 2 ,...

a n ) } ) H, V « « S iHK T(x k,j ,a 2 ,...,a n ) } ^ _x.

Fd

^ INI

{ / m

T ■

x k_,i , a 2 , . . . , a _n

)}

X ^∗

d

Thus, { T (E m= , 1 X k,i , « 2 , ..., a n ) } is total over X F . Therefore, by Remark 7.1 in [18], there exists an associated Banach space

Xdl   I V (^ Xk’i’°2’---’°n) ^ T E Xf} equipped with the norm

{ / m

Tls

x k,i , a 2 , . . . , a n

X d 1

= « T « X F ,

T E X F ,

and a bounded linear operator P : X d ^ X F defined by

P

^T i=1

x k_,i , a 2 , . . . , a _n

)l)

= T,

T E X F ,

such that ( {Em i X k,i } ,P ) is a tight retro Banach frame associated to (« 2 , ..., a n ) for X F with respect to X d } . >

Theorem 9. Let ( { x k,i } , S i ), i E E m — { 1, 2,... , m } be retro Banach frames associated to (a 2 ,..., a n ) for X F with respect to X d . Let { y k,i } С X be such that { T (y k,i , a 2 ,... ,a _n ) } E X d } , T E X F . Suppose R : X d } ^ X d } be a bounded linear operator such that

r({t (f} y k,i ,a 2 ...

• , an^ ।^ — {T(xk,p, a2, • • • j an) }, T E XF, for some p E Em and for each i E Em, let Ui : XF ^ Xd} be an operator defined by

U i (T ) — { T (X k,i ,a 2 ,...,a n ) } , T E X F .

If there exist constants а, в >  0 such that

• (i)    a ^ « U i || + тв <  « S j « ^{- 1} - E « U i II, for some j E E m ; i G E m                             i G E m ,

i = j

• ⁽ⁱⁱ⁾    || ^{{ T(x} k,i ^{— y} k,i ^{, a} 2 , . . . , a n ^)} ^ x * ^ ^a || ^{{ T(x} k,i , ^a 2 , . . . , a n ) } ^ X * ^+в « ^T « X * , ^{T E X} F , i ^E E m , d                                      d^F

then there exists a bounded linear operator P : X d } ^ X F such that the family ( { _{i G E} y k_,i } , P ) is a retro Banach frame associated to ( a 2 , . . . , a _n ) for X _F ^} with respect to X _d ^} . ^m

• <    For each i E E _m , S i U i is an identity operator on X _F ^} . Therefore, for each T E X _F ^} , we have

« T « x F — « S i U i T « x _F <  « S i « | { T (x k,i ,a 2 ,...,a n ) } | _x * .                  (5)

Also, for T E X F , we have

{T( E

L   \ i^E m

xk,i, a2, . . . , an

)}

X ^∗

d

E IT(xk,i,a2, • • • ,an)} iGEm

^ E IUHHTIX•

X ;    i G E m

dm

Now, for each T E X F , we have

I t( E yk,i,a2,...,an)|

L X iGEm                 / J

>

T ( x k,i , a 2 , i G E m

X ^∗

d

E {T (xk,i, a2,---. an) - T (xk,i — yki. a2,---. an)} iGEm

. , a _n

-

X ^∗ d

X ^∗ d

^ {T(xkj. a2. ... . an)} + } {T(xk,i. a2. ... , an )}

i G E m , i = j

X ^∗ d

X ^∗ d

^ } { ^{T ( x} k,i   ^y k,i ^,a 2 ,* ** ^,a n )}

i G E m

X ^∗ d

'' IS I- - a E bUib + E ^Ui^ + me \\TIX. L iGEm         iGEm, i=j

On the other hand, using (6), for each T E X F , we get

T         y k,i ,a 2 ,..

i G E m

..an)}    < ((1 + a) E ISII + в) ITIX.

X d            i G E m

Now, we take P = SpR, where p is fixed. Then P : Xd ^ XF is a bounded linear operator such that

р(Ы E yk,i,a2,...,an)| V T, iGEm

T E X F .

Hence, ({^2 i _G E_m y k,i } ,P ) is a retro Banach frame associated to (a 2 . ..., a n ) for X F with respect to X d . >

We end this section by discussing retro Banach frame associated to (a 2 , . . . , a _n ) in Cartesian product of two n -Banach spaces.

^{Let (X}. II • .... . • \ x ) ^{and (Y}. II • ...., • II Y ) be two n -Banach spaces. Then the Cartesian product of X and Y is denoted by X ® Y and defined to be an n -Banach space with respect to the n -norm

\xi ® У1.Х2 ® У2, . . . ,Xn ® yn\ = \xi,X2, . . . ,Xn\x + ||У1,У2, - - - .yn^Y, for all xi®yi,X2®У2,... ,Xn®yn E X®Y, and xi,X2,...,Xn E X; yi,y2,... ,yn E Y. Consider YG as the Banach space of all bounded b-linear functional defined on Y x (b2) x ... x (bn) and Zf®G as the Banach space of all bounded b-linear functional defined on X ® Y x (a2 ® b2) x ... x (an ® bn), where b2,.... bn E Y and a2 ® b2,.... an ® bn E X ® Y are fixed elements. Now, if T E XF and U E YG, for all x ® y E X ® Y, we define T ® U E ZF®g by

(T ® U)(x ® y.a2 ® b2.....an ® bn) = T(x. a2..... an) ® U(y. b2.....bn) (Vx E X. Vy E Y).

Let us consider Y d and Z d as the Banach spaces of scalar-valued sequences associated with Y G and Z F ^ g , respectively.

Theorem 10. Let ( { x k } , S x ) be a retro Banach frame associated to (a 2 ,..., a n ) for X F with respect to X ) having bounds A,B and ( { y k } ,S y ) be a retro Banach frame associated to (b 2 , ..., b n ) for Y Q with respect to Y ) having bounds C, D. Then ( { x k ф У к } , S x ф S y ) is a retro Banach frame associated to (a 2 ф b 2 ,..., a n ф b n ) for Z F ^ q with respect to Z )

<1 Since ({xk}, Sx) is a retro Banach frame associated to (a2,..., an) for XF with respect to X)) and ({yk}, Sy) is a retro Banach frame associated to (b2,..., bn) for YQ with respect to Yd∗ , we have

A W T ll x _F <  ^ { T (x k ,a 2 ,...,a n ) } ^ x * <  B || T | x _F   ( V T E X F ),             (7)

C \l R \l y G ^ ^ { R(y k ,b 2 ,...,b n ) } || Y, ^ D | R | x F   ( V R E Y Q ).              (8)

Adding (7) and (8), we get

A|T|x * + C^R^y* ^ ||{T (xk, a2,..., a„)}| ^, + ^{R(yk, b2,..., bn )}|| y, ^B |T|x * +D|R|y*.

F   G          d          d    FG

This implies that

min(A, C ){|T|xF + llRllYG*} ^ ||{T (xk, a2,..., an)} ф {R(yk, 62,..., bn)} ^Z*,

^ max(B,D) { W T Ц х * + №• } .

Thus,

min(A, C )|T ф R^z *   < ||{(T Ф R)(xk Ф yk, a2 Ф b2,---,an Ф bn)}Wz,■

F⊕G                                                   d

^ max(B, D)|T ф R^zf^g (VT ф R E ZF^q).

Also, we have

Sx ({T (xk ,a2 ,...,an)}) = T, T E XF, and Sy ({R(yk ,b2,...,bn)}) = R, R E Y)

Now,

(Sx ф Sy)({(Tф R)(xk ф yk,a2 ф b2,... ,an ф bn)}) = (Sx ф Sy) ({T(xk, a2,..., an)} ф {R(yk, b2,..., bn)})

= Sx ({T(xk, a2,..., an)}) ф Sy ({R(yk, b2,..., bn)}) = T ф R (VT ф R E ZF^q).

Hence, the family ( { x k ф y k } , S x ф S y ) is a retro Banach frame associated to (a 2 ф b 2 ,..., a n ф b n ) for Z F ® q with respect to Z) ) . >