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1
Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 1, P. 48-63
YAK 517.982.22
DOI 10.46698/o3961-3328-9819-i
ON STABILITY OF RETRO BANACH FRAME WITH RESPECT TO b-LINEAR FUNCTIONAL IN n-BANACH SPACE
P. Ghosh1 and T. K. Samanta2
1 Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata 700019, West Bengal, India; 2 Department of Mathematics, Uluberia College, Uluberia, Howrah 711315, West Bengal, India E-mail: prasenjitpuremath@gmail.com, mumpu_tapas5@yahoo.co.in
Abstract. We introduce the notion of a retro Banach frame relative to a bounded b-linear functional in n-Banach space and see that the sum of two retro Banach frames in n-Banach space with different reconstructions operators is also a retro Banach frame in n-Banach space. Also, we define retro Banach Bessel sequence with respect to a bounded b-linear functional in n-Banach space. A necessary and sufficient condition for the stability of retro Banach frame with respect to bounded b-linear functional in n-Banach space is being obtained. Further, we prove that retro Banach frame with respect to bounded b-linear functional in n-Banach space is stable under perturbation of frame elements by positively confined sequence of scalars. In n-Banach space, some perturbation results of retro Banach frame with the help of bounded b-linear functional in n-Banach space have been studied. Finally, we give a sufficient condition for finite sum of retro Banach frames to be a retro Banach frame in n-Banach space. At the end, we discuss retro Banach frame with respect to a bounded b-linear functional in Cartesian product of two n-Banach spaces.
Keywords: frame, Banach frame, retro Banach frame, stability, n-Banach space, b-linear functional. AMS Subject Classification: 42C15, 46C07, 46M05, 47A80.
For citation: Ghosh, P. and Samanta, T. K. On Stability of Retro Banach Frame with Respect to b-Linear Functional in n-Banach Space, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 48-63. DOI: 10.46698/o3961-3328-9819-i.
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 |/i}|=1 C H is called a frame for a separable Hilbert space (H, (■, ■)), if there exist positive constants 0 < A ^ B < to such that
<x
A ||/1|2 |(/,/i)| 2 < B ||/1|2 (V / € H).
i=1
© 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 (a2,... , an) in n-Banach space under some perturbations is discussed. We establish that retro Banach frame associated to (a2,..., an) 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 (a2,... , an) and establish a sufficient condition for the finite sum to be a retro Banach frame associated to (a2,...,an) in n-Banach space. Finally, retro Banach frame associated to (a2,..., an) 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* be a sequence space, which is a Banach space and for which the co-ordinate functionals are continuous. Let {ft}^/ C E* and S : E* — E be a bounded linear operator. Then the pair ({gj}, S) is said to be a Banach frame for E with respect to E* if
(i) {gi(/)}€ Ed (V / € E);
(ii) there exist B > A > 0 such that A||/ ||E < ||{gi(/)}||e < B||f ||e (V/ € E);
(iii) S({gi(/)}) = / (V/ € E).
The constants A, B are called Banach frame bounds and S is called the reconstruction operator.
Let Ed be a Banach space of scalar-valued sequences associated with E* indexed by N. Let {xk} C E and T : E** — E* be given. The pair ({xk},T) is called a retro Banach frame for E* with respect to Ed* if
(i) {/(xk)} € E* for each / € E*;
(ii) there exist positive constants A and B with 0 < A ^ B < to such that A||/||e < ||{/(xk)}||e* < B||/||e, / € E*;
(iii) T is a bounded linear operator such that T({/(xk)}) = /, / € 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
(X1,X2, ...,Xn) I-> ||X1,X2, . . . ,Xn||, Xi,X2, ...,Xn € X,
from Xn to the set R of all real numbers such that
(i) ||x1,x2,..., xn|| =0 if and only if x1,..., xn are linearly dependent;
(ii) ||x1,x2,... ,Xn|| is invariant under permutations of x1, x2, . . . , xn;
(iii) ||ax1, x2,..., xn|| = |a| ||x1, x2,..., xn||;
(iv) ||x + y,X2, . . . ,Xn| ^ ||x,X2, . . . ,Xn|| + ||y,X2, . . . ,Xn|,
for every xi, x2,..., xn € X and a € K. A linear space X, together with a n-norm || ■,..., ■ ||, is called a linear n-normed space. A sequence {xk} in linear n-normed space X is said to be convergent in X if there exists x € X such that
lim ||xfc - x,e2, ...,en|| =0 (V e2,...,en € X),
fc^TO
and it is called a Cauchy sequence if
lim ||xi - xfc ,e2, ...,en || =0 (V e2, ...,en € X).
The space X is said to be complete or n-Banach space if every Cauchy sequence in this space is convergent in X.
2. Main Results
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 a2,..., an be fixed elements in X. Let W be a subspace of X and (a} denote the subspaces of X generated by a», for i = 2,3,..., n. Then a map T : W x (a2} x ... x (an} ^ K is called a b-linear functional defined on W x (a2} x ... x (an}, if for every x, y € W and k € K, the following conditions hold:
(i) T(x + y,a2,... ,an) = T(x,a2,... ,an) + T(y,a2,... ,an);
(ii) T(kx, a2,..., an) = kT(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)| ^ M ||x,a2,... ,an|| (Vx € W). The norm of the bounded b-linear functional T is defined by
||T || = inf {M> 0: |T (x,a2,...,an )| < M ||x, a2,..., aj (V x € 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|| ^ 1};
(ii) ||T|| = sup {|T(x,a2,... ,an)| ||x,a2,... ,an|| = 1};
(Hi) ||T|| = SUP til' II®. «2, • • •, aj / o}.
For more details on bounded b-linear functional defined on X x (a2} x ... x (an} 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 XF denotes the Banach space of all bounded b-linear functional defined on X x (a2} x ... x (an} 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 XF indexed by N. Let {xk} C X and S : X* ^ XF be given. Then the pair ({xk}, S) is said to be a retro Banach frame associated to (a2,..., an) for XF with respect to X* if
(i) {T(xk, a2,..., an)} € X* for each T € X*F;
(ii) there exist constants 0 < A ^ B < to such that
A ||T||xf < ||{T(xk, a2,..., an)}||x* < B ||T||XF (VT € X*F); (1)
(iii) S is a bounded linear operator such that
S({T(xfc,02,..., a„)})= T (VT € XF).
The constants A, B are called frame bounds. If A = B, then ({xk},S) is called tight retro Banach frame associated to (a2,..., 0n) and for A = B = 1, it is called normalized tight retro Banach frame associated to (a2,..., 0n). The inequality (1) is called the frame inequality for the retro Banach frame associated to (a2,..., 0n). The operator S : X^ ^ XF is called the reconstruction operator or the pre-frame operator.
Definition 2. A sequence } С X is said to be a retro Banach Bessel sequence associated to (a2,..., 0n) for XF with respect to X^ if
(i) {T(xk, a2,..., an)} € X^ for each T € XF;
(ii) there exists a constant B > 0 such that
||{T (xfc ,02 ,...,0„)}||х* < B ||T Их» (V T € XF)
The constant B is called a retro Banach Bessel bound for the retro Banach Bessel sequence
} associated to (a2,... ,an). Let XB denotes the set of all retro Banach Bessel sequence associated to (a2,..., 0n) for XF with respect to X^. For } € XB, define
Ra : X*F ^ Xd by Ra (T) = {T (xfc ,a2,...,a„)} (V T € XF).
Then it is easy to verify that Ra is a bounded linear operator. The operator Ra is called the analysis operator.
Next, we verify that scalar combinations of two retro Banach frames associated to (a2,..., 0n) becomes a retro Banach frame associated to (a2,..., 0n).
Theorem 1. Let ({xk},S) and ({yk},S) be two retro Banach frames associated to (a2,... ,an) for XF with respect to X^ having bounds A, B and C, D, respectively. Then for any scalars а, /3, ({axfc + is a retro Banach frame associated to (a2,..., an)
for XF with respect to Xd.
< For each T € XF, we have
||{T(axfc + вУк,«2,... )}||X» = ||{aT(xfc,02,... ,0«) + ^T(yfc,02,... ,0«)}|„»
d d
< |a| ||{T (xfc ,02 ,...,a„)}|Xd + |в| ||{T (yfc ,02,...,^ )}||X, < (|a|B + |в |D) ||T ||X.. On the other hand,
¡{T(axfc + вУк,«2,... )}!X» ^ |a| ¡{T(xfc,02,... ,0«)}^X»
d d
- |e|||{T (yfc ,02,...,0n )}||X. ^ (|a|A -|в1С IIT ||xF, T € XF. Also, for T € XF, we have
S({T(xfc, 02,..., 0«)}) = T and S({T(yfc, 02,..., 0«)}) = T. Then for T € XF, we have
a l+ p S ({T(axk + /Зук, а2,..., ага)}) = ^ ^ aS" ({T(xfc, a2,...,an)})
+ /?5({T(yfc, a2,..., ara)})] = (аГ + /?Т) = Т.
Hence, the family ({axk + ftyk}, is a retro Banach frame associated to (02,... ,an)
for X*F with respect to X*d having bounds (|a|A - |^|C) and (|a|B + >
In the next theorem, we will see that the sum of two retro Banach frames associated to (a2,... , an) with different reconstructions operators is also a retro Banach frame associated to (a2,... ,a„).
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(yfc, a2,..., a„)}, T € X*F.
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 Xd.
< Let U, V be the corresponding coefficient mappings for the retro Banach Bessel sequences {xk} and {yk}, respectively and I denotes the identity mapping on Xd. Now, for each T € XF, we have
|{T(xk + yk,«2, • • • = ||{T(xk,a2,... ,a„)} + {T(yfc,a2,... ,a„)},,X
= || {T (xk ,a2,... ,an)} + R({T (xk ,a2,.. .,a„ )})|X*
d
< ||1 + R||||{T(xk,a2,...,an)}||X* < B ||1 + R||||T||x* 11 1111 IIXd M ......
Similarly, for each T € XF, we have
|{T (Xk + yk, «2, )}||X* < D ||1 + R-1||||T || 1 IIXd M ........
Thus, for each T € X*F, we get
Fi
|{T(Xk + yk, «2,..., an)}||X < min {B ||1 + R||, D ||l + R-1||} ||T|
On the other hand, for each T € XF, we have
H{T(Xk + yk,«2,... ,an)}NX*
d
> ||{T(xk ,«2 ,...,a„)}|L, - ||{T(yk ,a2,...,an)}|X* ^ A ||l - R||||T | .
11 "Xd " Xd M ......
Also, for each T € XF, we have
|{T (xk + yk,a2,...,a„)}|Xd ^ C ||R-1 - I ||||T ||x.
Therefore, for each T € XF, we get
||{T(xk + yk,a2,... ,an)}||X* ^ max {A ||1 - R||,C ||R-1 - IN}NT||x*.
II ll^d 11 11 11
Now, for T € XF, 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 : Xd ^ XF is a bounded linear operator such that
Q({T(xk + yk, a2,..., an)}) = T (VT € X*F).
Hence, ({xk + }, Q) is a retro Banach frame associated to (a2,..., an) for X^ with respect to X*. >
Now, we start with a necessary and sufficient condition for the stability of a retro Banach frame associated to (a2,..., an).
Theorem 3. Let ({xk},S) be a retro Banach frame associated to (a2,...,an) for X^ with respect to X* having bounds A, B. Let {yk} be a sequence in X such that {T(yk,a2,...,an)} € X*, T € X^. Suppose R : X* ^ X* be a bounded linear operator such that
R({T(yfc, a2,..., a„)}) = {T(xfc, a2,..., a„)}, T € .
Then there exists a bounded linear operator P : X* ^ X^ such that ({yk},P) is a retro Banach frame associated to (a2,..., an) for X^ with respect to X* if and only if there exists a constant K > 1 such that
II {T (xfc - yfc ,a2 ,...,a„ )}| _
IIXd (2) < Kmin {||{T(xfc, a2,..., a„)}||, ||{T(yfc, a2,..., a„)}||}.
< First we suppose that ({yk}, P) is a retro Banach frame associated to (a2,..., an). Then there exist constants C, D > 0 such that
A ||T ||x* < || {T (xfc ,a2,...,ara)}| xd < B ||T ||x* (V T € X>), (3)
C ||T ||x* < |{T(yfc,a2,...,a„)}| < D ||T ||x* (V T € X». (4)
d
Therefore, for each T € X^, we have
|{T(xfc - yfc,a2,... ,a„)}! x ^ ¡{T(xfc,a2,... ,an)}^Xd + ¡{T(yfc,a2,... ,a„)}^ x
d d
(4) Mr , ______ (3) / D
^ \\{T(xk,a2,...,an)}
Similarly, it can be shown that
< \\{T(xk,a2,...,arl)}\\x.d+D\\T\\x* < f 1 + j J ||{T(xfc,a2,... ,ara)}||x*
||{T(xfc - yk, a2,..., ara)}||x* ^ + f) ||№fe> a2,..., an)}\\x, (VT e X*F).
Thus, for each T € , we get 11 {T (xfc - yfc ,a2,... ,a„)}! ^ K min { ||{T (xfc ,a2,... ,a„)}|| x*d, ||{T (yfc ,a2,... ,a„)}|| xdd} ,
where K = max{(l + §), (l + §)}.
Conversely, suppose that there exists K > 1 such that (2) holds. Now, for each T € X^, we have
A ||T |X ^ ¡{T (xfc ,a2,... ,a„)}! Xd ^ ¡{T (xfc - yfc ,a2,...,a„ x
d d
|, |, (2) |. |, + ||{T (yfc ,a2,...,a„ )}|| Xd ^ (K + 1)||{T (yfc, a2,..., a„)}|| Xd.
This implies that
A
-\\T\\x*F < ||{T(yfc,a2,...,ara)}||x*.
(K + 1)
On the other hand, for each T € XF, we have
|{T(yk,a2,... ,a„)}yXd ^ ^{T(xk - yk,a2,... ,a„+ ^{T(xk,a2,... ,a„
< (K + 1) ||{T(xk, a2,..., an)}|X* < B(K + 1) ||T||xF.
Now, take P = SR. Then P : Xd ^ XF is a bounded linear operator such that
p({T(yk, a2,..., an)}) = SR({T(yk, a2,..., an)}) = S{T(xk, a2,..., an)} = T, T € x*f.
Thus, ({yk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect to Xd. This completes the proof. >
The Theorem 3 shows that the stability of retro Banach frame associated to (a2,..., an) 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 (a2, . . . , an).
Theorem 4. Let ({xk},S) be a retro Banach frame associated to (a2,...,an) for XF with respect to Let {yk} C X be such that {T(yk,a2,... ,an)} € Xd, T € XF and let U : XF ^ Xd be the coefficient mapping given by
U (T) = {T (xk ,a2,...,an)}, T € x*f.
If there exist positive constants a, P (< 1) and m such that
(,) max{l'S'l"2 + °:g'|[,|l+"',/;}<!;
(ii) ¡{T(xk - yk, a2,..., an)}11 X* ^ a ¡{T(xk, a2,..., an)}!X*
d d
+ P ||{T(yk, a2,..., an)}|X* + M ||T||xF, T € XF, ii ii^d F
then there exists a reconstruction operator P : Xd ^ XF such that ({yk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect to Xd.
< Let V : XF ^ Xd be an operator defined by
V(T) = {T(yk,a2,...,an)}, T € XF. Using the operators U and V, condition (ii) can be written as
||UT - VT||Xd* < a ||UT||x* + P ||VT||x* + M ||T||x>, T € x*f. Thus, for T € XF, we have
||{T(yk,a2,...,ara)}|U* = \\VT\\X. < ||f/T-FT|U* + ||[/T|U* < il±^M±£|№|l. Therefore, V is a bounded linear operator such that
(2 + a -mu\\ + »
1-/3
\\ut — vt\\x* < , T" (yrex*F).
Now,
||/F - < ||5||r " < + < i.
1 - p
This shows that SV is an invertible operator with satisfying
II CSV)-1 N 1
-, ||g||[(2+a-/3)||£/||+/i] • 1
Now, take P = (SV) Then PV = 1F, where 1F is the identity operator on XF. Thus, P : Xd ^ XF is a bounded linear operator such that
P({T(yfc,02,...,fln)})= T, T € XF.
Now, for T € XF, we have
imu* = \\PVT\\X.F < ||P||||FT|U* < iisiipyj-^iiE/H+.r
1 Ï-/3
This implies that
||s||-, ^ _ [(2 + a-ff)||f/||+Mll|S||^ ||r|u, ( ||{T(№a2i (VT £ xp)
Hence, ({yk}, P) is a retro Banach frame associated to (o2,..., on) for XF with respect to X^. This completes the proof. >
Next, we give a stability condition of a retro Banach frame associated to (o2,..., on) by using a given retro Banach Bessel sequence associated to (a2,..., on).
Theorem 5. Let ({xk},S) be a retro Banach frame associated to (o2,...,on) for XF with respect to XF having bounds A, B. Let {yk} be a sequence in X such that ({T(yk, a2,..., on)} € XF), T € XF and for some constant K > 0
||{T(yfc,a2,...,a„)}||x* < K||T||x. (VT € XF).
Tiien for any non-zero constant A with |A| < ^jj, , tiiere exists a reconstruction operator P : XF ^ XF such that ({xk + Ayk}, P) is a retro Banach frame associated to (a2,..., on) for XF with respect to XF having frame bounds (||S||-1 — |A|K) and (B + |A|K). < Let V : XF ^ XF be a bounded linear operator defined by
V(T) = {T(yfc,02,...,0„)}, T € XF,
and U : XF ^ XF be a bounded linear operator given by
U(T) = {T(xfc,02,...,0„)}, T € XF.
Then it is easy to verify that {T(xk + Ayk, o2,..., on)} € XF, for all T € XF. Now, for each T € XF, we have
||UT + AVT ||xt = ||{T (xk + Ayfc,02 ,...,o„)}|
xt
< ||{T(Xk,02,...,ora)}||Xt + |A| ||{T(yk,02,...,o„)}|Xt < (B + |A|K)||T||xt,.
d d
On the other hand, for each T € XF, we have
1 — |A|K)||T||xt < ||{T(Xk,02,...,0n)}|xd — |A| ||{T(yk,02,...,0n)}|xd ^ H{T(Xk + Ayk,02,... ,0n)}|
lxr
Define, L : X*F ^ X*d by L(T) = {T(xfc + Ayfc, ,..., an)}, T e XF. Then L is a bounded linear operator such that
||UT - LT||x = ||{T(xfc,a2,...,a„)} - {T(xfc + Ayfc,a2,...,ara)}||X*
= ||{AT(yfc,a2,...,a„)}|X, < |A|K||T||xF, T e X*F.
This verifies that ||U — L|| ^ |A|K. Since SU = 1F, 1F is the identity operator on XF, we have
||1f — SL|| = ||SU — SL|| < ||S||||U — L|| < 1.
Thus SL is invertible. Take P = (SL)-1S. Then P : Xd ^ XF is a bounded linear operator such that
P({T(xfc + Ayfc, a2,..., a„)}) = T, T e XF.
Hence, ({xk + Ayk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect to Xd having frame bounds (||S||-1 — |A|K) and (B + |A|K). >
Theorem 6. Let ({xk}, S) be a retro Banach frame associated to (a2,..., an) for XF with respect to Xd. Let {yk} C X and {ak} C R be any positively confined sequence such that {T (a yfc, a2,..., an)} e Xd, T e XF .If V : XF ^ Xd defined by
V(T) = {T(yfc,a2,...,an)}, T e XF,
such that IIFII < ——-, then there exists a reconstruction operator P : X*, —> Xt such
11 11 suPi^fc<TC afc' 1 d F
that ({xk + yk}, P) is a retro Banach frame associated to (a2,..., an) for XF with respect d.
< Let U : XF ^ Xd be a bounded linear operator defined by
to X *
U (T) = {T (xfc ,a2,...,a„)}, T e X*.
It is easy to verify that {T(xk + ak,a2,...,an)} e X*, for all T e X*. Now, for each T e X*, we have
||{T(xfc + afcyfc,a2,... ,a«)}|X» ^ ||{T(xfc,a2,... ,ara)}||X» + ||{afcT(yfc,a2,... ,ara)}||X»
d d d
^ H{T(xfc,a2,... ,ara)}¡X* + 1 sup afc I ¡{T(yfc, «2,..., a«)}^*
d V / d
< [||Uy + ||V||( sup afc)]||TIxf.
On the other hand, for each T e X*, we have
H{T(xfc + afcyfc,a2,... ,a„)}!X* ^ ^T(xfc,a2,... , a„* - ||{afcT(yfc,a2,... , a„*
Xd Xd Xd
i
sup afc
1<fc<oo
l|T yxF.
Define, L : X* ^ X* by
L(T) = {T(xfc + afcyfc, o>2,..., «n)}, T e X*.
Following the lines of proof of the Theorem 5, L is a bounded linear operator on X^ such that ||U - L|| < sup1<k<^ ak||V|| and SL is invertible. Take P = (SL)-1S. Then P : Xd ^ XjF is a bounded linear operator such that
P({T(xfc + afcyk, a2,..., a„)}) = T, T € X>.
Hence, ({xk + akyk}, P) is a retro Banach frame associated to (a2,..., an) for X^ with respect
to Xd. >
In the next theorem, we establish that retro Banach frame associated to (a2,... , an) is stable under perturbation of frame elements by positively confined sequence of scalars.
Theorem 7. Let ({xk},S) be a retro Banach frame associated to (a2,...,an) for X|, with respect to X^. Let {yk} C X be such that {T(yk, a2,...,an)} € Xd, T € X|,. Let R : Xd ^ Xd be a bounded linear operator such that
R({T(yk, a2,..., a„)}) = {T(xfc, a2,..., an)}, T € .
Suppose {ak} and {^k} are two positively confined sequences in R. If there exist constants X, ^(0 ^ X, ^ < 1) and y such that
(i) Y < (1 - X)||S||-1( inf a^ ;
(ii) ||{akT(xk,a2,... ,a„)} - {^kT(yk,a2,... ,a„)}|X» ^ X ||{akT(xk,a2,... ,an)}|X*
d d
+ ^ ||{AT(yk, a2,..., ara)} IX + Y ||T||x*, T € X>.
Then there exists a reconstruction operator P : Xd ^ X^ such that ({yk},P) is a retro Banach frame associated to (a2,..., an) for X^ with respect to Xd.
< Let U : X^ ^ Xd be a bounded linear operator defined by
U(T) = {T(xk,a2,...,a„)}, T € X^. Since the operator SU : X^ ^ X^ is an identity operator, for T € X|>,
||T||x* = ||SU(T)||x* < ||S||||{T(xk,a2,...,an)}|
lxr
Now, for each T € X|,, we have
|{^kT(yk,a2,... ,a„)}|
^ ¡{akT(xk,a2,... ,a„)}^Xt + ¡{akT(xk,a2,... ,a„)} - T(yk,a2,... ,a„)}|
Ixt
^ (1 + X) y{ak T (xk ,a2,...,a„ )}^X « + ^ ^k T (yk ,a2,...,a„ « + y ||T ||x*p .
d
Thus,
(1 - T (yk ,a2 ,...,a„)}|X» ^
(1 + X)
sup ak + Y
1<k<oo
||T ||x*
This implies that
(1 - inf ^k y{T (yk ,a2 ,...,a„ )H|X « ^ / 11 llXd
(1 + X)
sup ak J + Y
1<k<oo
||T ||x*
On the other hand, by condition (ii), we get
(1+ feT(yfc
>o„)}||X» ^ (1 - A) ||{afc T (xfc ,02,...,0„ )}||X » - 7 ||T ||xf
d d
(1 - A)
-1
inf afc - y
1<fc<oo '
||T ||
xí
Te X
F •
Therefore, for each T € XF, we have
(1+ SUp Pfc ) H {T(yfc,fl2,
. 1<fc<oo
Thus, for each T € XF, we have (1 - A)||S||-1( afc) - Y
^Lj ^ (1 + H{^fcT(yfc,«2,... ,o„)}
ni III V*
(1 - A)
1
inf afc - y
1<fc<oo '
|T ||XF
(1 + ( sup pfc
||T||xF < H{T(yfc,02,...,a„)}|
<
(1 + A)||U||( sup afc ) + y
(1 - i<inf &
||T ||xF
Now, take P = SV. Then P : Xd ^ XF is a bounded linear operator such that
P({T(yfc,a2,...,a„)}) = T, T € XF.
Hence, ({yk},P) is a retro Banach frame associated to (a2,...,an) for XF with respect to XdF. >
Definition 3. A sequence {xk} in X is said to be total over XF if
{T € XF : T(xfc,a2,...,fln) = 0 (Vk)} = {0},
where 0 € 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 (a2,...,an) is again a retro Banach frame associated to (a2,..., an).
Theorem 8. For i € = {1, 2,..., m}, let ({xk;i}, Sj) be retro Banach frames associated to (a2,..., an) for XF with respect to X^. Then there exists a reconstruction operator P : Xd ^ XF such that ({^xk)i},P) is a tight retro Banach frame associated to (a2,...,an) for XF with respect to Xd, provided
||{T (xk j, 02, . . . , 0n)}||X* ^ for T € XF, for some j €
T( X^Xfc,¿,02, . . .
i=1
0
n
< For T € XF, we have
||THx* = ||Si({T(xk,j,a2,...,ara)})||X* < ||Sj||||{T(xk,j,a2,...,a„)}
nn IIX*
< ||Si
T X^xk,i,a2,...
j=1
Xd
Thus, {T(^m=1 xk)i, a2,..., an)} is total over XF. Therefore, by Remark 7.1 in [18], there exists an associated Banach space
Xdx = | |t( ^xk,i,a2,... ,i equipped with the norm
: T € X *
F
j=1
T ^xk,i,a2, . . .
j=1
= ||T||x*, T € X*,
Xdi
and a bounded linear operator P : Xd — X* defined by
pNT( ¿xk,i,a2,...,aU = T, T € X
F,
i=1
such that ({^m=1 xk)i},P) is a tight retro Banach frame associated to (a2,..., an) for X* with respect to Xd. >
Theorem 9. Let ({xk;j}, Sj), i € = {1, 2,... , m} be retro Banach frames associated to (a2,..., an) for X* with respect to Xd. Let {yk)j} C X be such that {T(yk;j, a2,..., an)} € Xd, T € X*. Suppose R : Xd —^ Xd be a bounded linear operator such that
R T^Ty^^,...
= {T(xk,p,a2,...,an)}, T € X*,
j=1
for some p € and for each i € let U : X* — Xd* be an operator defined by
Uj(T) = {T(xk,j,a2,...,an)}, T € X*.
If there exist constants a, ^ > 0 such that
(i) a £ ||Uj|| + m£ < ||Sj||-1 - £ ||Uj||, for some j € Em;
ieEm
ieEm , j=j
(ii) ||{T(xk,j-yk,j, a2,..., an)} ||X* < a ||{T(xk,j, a2,..., an)} ||X* +£ ||T||x*, T € X*, i € Em,
Xd Xd
then there exists a bounded linear operator P : Xd* — X* such that the family ({£ i£Em yk)j},P) is a retro Banach frame associated to (a2,...,an) for X* with respect
to Xd.
< For each i € SjUj is an identity operator on X*. Therefore, for each T € XF*, we
have
||T||x* = ||SjUjT||x* < ||Sj|| ||{T(xk,j,a2,...,an)}
n Xd
(5)
a
n
n
a
n
a
n
Also, for T € XF, we have
T e Xfc,i,a2, • • •,an >
ViSEm / J
Now, for each T € XF, we have
^ {Ta2, • • •, a„)}
i€E„
<
E iiu
X* íeE„
(6)
. íeEm
X*
y] {T(Xk,i,a2, • • • ,an) - T(xfc)i - yk,i,a2, • • • ,a„)}
íeE„
X*
y] {T(xk,i,a2, • • • ,a„)}
i€E„
X*
y] {T(xk,i - Vk,i, a2, • • •, an)}
i€E„
X*
{T(xk,j,a2, • • • ,an)} +E {T(Xk,i,a2, • • •,an)}
ieEm
X*
E{T(xk,i - yk,i,a2, • • • ,a„)}
ieEm
X*
(5),(6)
^ llS
i-i
a E llUil + E llUil + mP
. íeEm ieEm,
i|t |xf •
On the other hand, using (6), for each T € XF, we get
T( e yk,i,a2, • • • ,i
i€E„
< ((1 + a) E lUill + p) ||T|xf•
^ ieEm '
= T, T € X
F •
Now, we take P = SpR, where p is fixed. Then P : X^ ^ XF is a bounded linear operator such that
p({T( e yfc,i'a2' • • • 'an
Hence, ({^ ieEm yfc,i},P) is a retro Banach frame associated to (a2,...,an) for XF with respect to X^. >
We end this section by discussing retro Banach frame associated to (a2,...,an) in Cartesian product of two n-Banach spaces.
Let (X, || ■,..., ■ ||X) and (Y, || ■,..., ■ ||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
||Xl ® yi,X2 ® y2, . . . ,X„ ® = ||X1,X2, . . . ,Xn||x + ||yi,^2, . . . ,yn||r,
for all Xi®yi,X2®y2,... ®y„ € X®Y, and xi,x2,... € X; yi,y2,... ,yn € 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 € Y and a2 ® b2,..., an ® bn € X ® Y are fixed elements. Now, if T € XF and U € Y£, for all x ® y € X ® Y, we define T ® U € ZFeG by
(T ® U )(x ® y,a2 ® b2,...,a
n ® bn) — T(x, a.2,..., &n) ® U(y, b2,..., bn) (V x € X, V y € Y).
Let us consider Yd* and Z^ as the Banach spaces of scalar-valued sequences associated with Yq and ZFe G, respectively.
XF
X*
n
X*
Theorem 10. Let ({xk}, SX) be a retro Banach frame associated to (a2,..an) for XF with respect to X^ having bounds A, B and ({yk},SY) be a retro Banach frame associated to (62, • • •, 6n) for YG with respect to Yd* having bounds C, D. Then ({xk ® }, SX ® SY) is a retro Banach frame associated to (a2 ® 62,..., an ® ) for ZFeG with respect to
< 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 (62, • • •, ) for YG with respect
to Yd*, we have
Ayr||x* < ||{T(xfc,a2,...,a„)}||x» < B||T||x» (VT € X*), (7)
d
C||R||yG < ||{R(yfc,62< D||R||x* (VR € Yg)• (8)
Adding (7) and (8), we get
A||T||x* + C||R||yG < ||{T(xfc, a2, • • •, ara)}||X, + ||{R(yfc, 62, • • •, 6n)}||Y; <B||T||x^+D||R||yG. This implies that
min(A, C ){||T + ||R||yG } < ||{T (xfc ,a2,...,ara)} ® {R(yfc, 62, • • •, 6n)}||Z.
< max(B, D) {||T+ ||R||yg }•
Thus,
min(A, C)||T ® R||z*ffiG < ||{(T ® R)(xfc ® yfc, a2 ® 62, • • •, a„ ® 6„)}||z*
< max(B, D) ||T ® R||^ffiG (VT ® R € ZFeG).
Also, we have
Sx ({T(xfc,a2,...,a„)}) = T, T € XF, and Sy ({R(yfc, 62, • • •, 6n)}) = R, R € YG Now,
(Sx ® Sy) ({(T ® R)(xfc ® yfc, a2 ® 62, • • •, ara ® )}) = (Sx ® Sy)( {T (x& ,a2,...,an)} ® {R(yfc ,62 ,...,6n)}) = Sx({T(xfc, a2, • • •, a„)}) ® Sy({R(yfc, 62, • • •, 6n)}) = T ® R (VT ® R € ZFeG).
Hence, the family ({xk ® }, SX ® SY) is a retro Banach frame associated to (a2 ® 62,... an ® ) for ZFeG with respect to Z^. >
References
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Received November 9, 2021 Prasenjit Ghosh
Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata 700019, West Bengal, India, Research Scholar
E-mail: prasenj itpuremath@gmail. com
https://orcid.org/0000-0001-6450-5436
Tapas Kumar Samanta
Department of Mathematics, Uluberia College, Uluberia 711315, Howrah, West Bengal, India, Associate Professor E-mail: mumpu_tapas5@yahoo.co.in
Владикавказский математический журнал 2023, Том 25, Выпуск 1, С. 48-63
ОБ УСТОЙЧИВОСТИ РЕТРО БАНАХОВА ФРЕЙМА ОТНОСИТЕЛЬНО b-ЛИНЕЙНОГО ФУНКЦИОНАЛА В n-БАНАХОВОМ ПРОСТРАНСТВЕ
Гхош П.1, Саманта Т. К.2
1 Калькуттский университет, Индия, Западная Бенгалия, 700019, Калькутта;
2 Колледж Улуберия, Индия, Западная Бенгалия, Ховрах, 711315, Улуберия
E-mail: prasenjitpuremath@gmail.com, mumpu_tapas5@yahoo.co.in
Аннотация. Вводится понятие ретро банахова фрейма относительно ограниченного 6-линейного функционала в n-банаховом пространстве и устанавливается, что сумма двух ретро банаховых фреймов в n-банаховом пространстве с разными операторами реконструкции также является ретро банаховым фреймом в n-банаховом пространстве. Также определяется ретро банахова последовательность Бесселя относительно ограниченного 6-линейного функционала в n-банаховом пространстве. Получено необходимое и достаточное условие устойчивости ретро банахова фрейма относительно ограниченного 6-линейного функционала в n-банаховом пространстве. Далее, доказано, что ретро банахов фрейм относительно ограниченного 6-линейного функционала в n-банаховом пространстве устойчив по отношению к возмущению элементов фрейма положительно ограниченной последовательностью скаляров. Изучены некоторые свойства возмущении ретро банахова фрейма в n-банаховом пространстве с помощью ограниченного 6-линейного функционала. Наконец, даются достаточное условие того, чтобы конечная сумма ретро банаховых фреймов была ретро банаховым фреймом в n-банаховом пространстве. В заключении рассматривается ретро банахов фрейм относительно ограниченного 6-линейного функционала в декартовом произведении двух n-банаховых пространств.
Ключевые слова: фрейм, банахов фрейм, ретро банахов фрейм, устойчивость, n-банахово пространство, 6-линейный функционал.
AMS Subject Classification: 42C15, 46C07, 46M05, 47A80.
Образец цитирования: Ghosh P. and Samanta T. K. On Stability of Retro Banach Frame with Respect to 6-Linear Functional in n-Banach Space // Владикавк. мат. журн.—2023.—Т. 25, № 1.—C. 4863 (in English). DOI: 10.46698/o3961-3328-9819-i.
