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Frames in 2-Inner Product Spaces Iranian Journal of Mathematical Sciences and Informatics Vol. 8, No. 2 (2013), pp 123-130 Frames in 2-inner Product Spaces Ali Akbar Arefijamaal∗ and Ghadir Sadeghi Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran E-mail: [email protected] E-mail: [email protected] Abstract. In this paper, we introduce the notion of a frame in a 2- inner product space and give some characterizations. These frames can be considered as a usual frame in a Hilbert space, so they share many useful properties with frames. Keywords: 2-inner product space, 2-norm space, Frame, Frame operator. 2010 Mathematics subject classification: Primary 46C50; Secondary 42C15. 1. Introduction and preliminaries The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer [12] in 1952 to study some deep problems in nonharmonic Fourier se- ries. Various generalizations of frames have been proposed; frame of subspaces [2, 6], pseudo-frames [18], oblique frames [10], continuous frames [1, 4, 14] and so on. The concept of frames in Banach spaces have been introduced by Grochenig [16], Casazza, Han and Larson [5] and Christensen and Stoeva [11]. The concept of linear 2-normed spaces has been investigated by S. Gahler in 1965 [15] and has been developed extensively in different subjects by many authors [3, 7, 8, 13, 14, 17]. A concept which is related to a 2-normed space is 2-inner product space which have been intensively studied by many math- ematicians in the last three decades. A systematic presentation of the recent results related to the theory of 2-inner product spaces as well as an extensive ∗ Corresponding author Received 16 November 2011; Accepted 29 April 2012 c 2013 Academic Center for Education, Culture and Research TMU 123 124 Arefijamaal, Sadeghi list of the related references can be found in the book [7]. Here we give the basic definitions and the elementary properties of 2-inner product spaces. Let be a linear space of dimension greater than 1 over the field K, where K X is the real or complex numbers field. Suppose that (., . .) is a K-valued function | defined on satisfying the following conditions: X×X×X (I1)(x, x z) 0 and (x, x z) = 0 if and only if x and z are linearly dependent, | ≥ | (I2)(x, x z) = (z,z x), | | (I3)(y, x z)= (x, y z), | | (I4)(αx, y z)= α(x, y z) for all α K, | | ∈ (I5)(x1 + x2,y z) = (x1,y z) + (x2,y z). | | | (., . .) is called a 2-inner product on and ( , (., . .)) is called a 2-inner product | X X | space (or 2-pre Hilbert space). Some basic properties of 2-inner product (., . .) | can be immediately obtained as follows ([8, 13]): (0,y z) = (x, 0 z) = (x, y 0) = 0, • | | | (x, αy z)= α(x, y z), • | | (x, y αz)= α 2(x, y z), for all x,y,z and α K. • | | | | ∈ X ∈ Using the above properties, we can prove the Cauchy-Schwarz inequality (1.1) (x, y z) 2 (x, x z)(y,y z). | | | ≤ | | Example 1.1. If ( , ., . ) is an inner product space, then the standard 2-inner X h i product (., . .) is defined on by | X x, y x, z (1.2) (x, y z)= h i h i = x, y z,z x, z z,y , | z,y z,z h ih i − h ih i h i h i for all x,y,z . ∈ X In any given 2-inner product space ( , (., . .)), we can define a function ., . X | k k on by X × X 1 (1.3) x, z = (x, x z) 2 , k k | for all x, z . ∈ X It is easy to see that, this function satisfies the following conditions: (N1) x, z 0 and x, z = 0 if and only if x and z are linearly dependent, k k≥ k k (N2) x, z = z, x , k k k k (N3) αx, z = α x, z for all α K, k k | |k k ∈ (N4) x1 + x2,z x1,z + x2,z . k k≤k k k k Any function ., . defined on and satisfying the conditions (N1)- k k X × X (N4) is called a 2-norm on X and ( , ., . ) is called a linear 2-normed space. X k k Whenever a 2-inner product space ( , (., . .)) is given, we consider it as a linear X | 2-normed space ( , ., . ) with the 2-norm defined by (1.3). X k k Framesin2-innerProductSpaces 125 In the present paper, we shall introduce 2-frames for a 2-inner product space and describe some fundamental properties of them. This implies that each el- ement in the underlying 2-inner product space can be written as an uncondi- tionally convergent infinite linear combination of the frame elements. 2. Frames in the standard 2-inner product spaces Throughout this paper, we assume that is a separable Hilbert space, with H the inner product ., . chosen to be linear in the first entry. We first review h i some basic facts about frames in , then try to define them in a standard H 2-inner product space. ∞ Definition 2.1. A sequence fi is called a frame for if there exist { }i=1 ⊆ H H A,B > 0 such that ∞ 2 2 2 (2.1) A f f,fi B f , (f ). k k ≤ X |h i| ≤ k k ∈ H i=1 The numbers A, B are called frame bounds. The frame is called tight if ∞ A = B. Given a frame fi , the frame operator is defined by { }i=1 ∞ Sf = f,fi fi. Xh i i=1 The series defining Sf converges unconditionally for all f and S is a ∈ H bounded, invertible, and self-adjoint operator. This leads to the frame decom- position: ∞ −1 −1 f = S Sf = f,S fi fi, (f ). Xh i ∈ H i=1 The possibility of representing every f in this way is the main feature of a −1 ∞ ∈ H frame. The coefficients f,S fi are called frame coefficients. A sequence {h i}i=1 satisfying the upper frame condition is called a Bessel sequence. A sequence ∞ ∞ fi is Bessel sequence if and only if the operator T : ci cifi { }i=1 { } 7→ Pi=1 is a well-defined operator from l2 into . In that case T, which is called the H ∞ pre-frame operator, is automatically bounded. When fi is a frame, the { }i=1 pre-frame operator T is well-defined and S = TT ∗. For more details see [9, Section 5.1]. Also see [19] for a class of finite frames. ∞ Let be a 2-inner product space. A sequence an of is said to be X { }n=1 X convergent if there exists an element a such that limn an a, x = 0, ∈ X →∞ k − k for all x . Similarly, we can define a Cauchy sequence in . A 2-inner ∈ X X product space is called a 2-Hilbert space if it is complete. That is, every X Cauchy sequence in is convergent in this space [17]. Clearly, the limit of any X convergent sequence is unique and if ( , (., . .)) is the standard 2-inner product, X | then this topology is the original topology on . X Now we are ready to define 2-frames on a 2-Hilbert space. 126 Arefijamaal, Sadeghi Definition 2.2. Let ( , (., . .)) be a 2-Hilbert space and ξ . A sequence ∞ X | ∈ X xi of elements in is called a 2-frame (associated to ξ) if there exist { }i=1 X A,B > 0 such that ∞ 2 2 2 (2.2) A x, ξ (x, xi ξ) B x, ξ , (x ). k k ≤ X | | | ≤ k k ∈ X i=1 A sequence satisfying the upper 2-frame condition is called a 2-Bessel sequence. In (2.2) we may assume that every xi is linearly independent to ξ , by (1.1) and the property (I1). The following proposition shows that in the standard 2-inner product spaces every frame is a 2-frame. ∞ Proposition 2.3. Let ( , ., . ) be a Hilbert space and xi is a frame for H h i { }i=1 . Then it is a 2-frame with the standard 2-inner product on . H H ∞ Proof. Suppose that xi is a frame with the bounds A, B and ξ such { }i=1 ∈ H that ξ = 1. Then by using (2.1) and (1.2) for every x we have k k ∈ H ∞ ∞ 2 2 (x, xi ξ) = x x, ξ ξ, xi X | | | X |h − h i i| i=1 i=1 B x x, ξ ξ 2 ≤ k − h i k B( x 2 x, ξ 2) ≤ k k − |h i| = B(x, x ξ). | The argument for lower bound is similar. The converse of the above proposition is not true. In fact, by the following proposition, every 2-frame is a frame for a closed subspace of with codi- H mension 1. For each ξ we denote by Lξ the subspace generated with ∈ H ξ. Proposition 2.4. Let ( , ., . ) be a Hilbert space and ξ . Every 2-frame H h i⊥ ∈ H associated to ξ is a frame for Lξ . ∞ Proof. If xi is a 2-frame with the bounds A, B then (2.2) implies that { }i=1 there exist A,B > 0 such that ∞ 2 2 2 2 2 A( x x, ξ ) x x, ξ ξ, xi B( x x, ξ ), (x ). k k − |h i| ≤ X |h − h i i| ≤ k k − |h i| ∈ H i=1 ∞ ⊥ Therefore, xi is a frame for the Hilbert space L . { }i=1 ξ ∞ Remark 2.5.
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