Weighted and Controlled Continuous g-Frames and their Multipliers in Hilbert Spaces

Çankaya University Journal of Science and Engineering Volume 13, No. 1 (2016) 031–039 Weighted and Controlled Continuous g-Frames and their Multipliers in Hilbert Spaces Sayyed Mehrab Ramezani, Akbar Nazari Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran. e-mail: mr , Abstract: A generalization of weighted, multiplier, controlled from frame and Bessel sequences to continuous g-frames and continuous g-Bessel sequences in Hilbert spaces is presented in this study. Moreover, we find a dual of a continuous g-frame in the case that the multiplier operator is invertible. Finally, it is demonstrated that a controlled continuous g-frame is equivalent to a continuous g-frame. Keywords: Continuous g-frame, continuous g-multiplier, weighted continuous g-frame, controlled continuous g-frame. 1. Introduction Frames for Hilbert space were formally defined by Duffin and Schaeffer [11] in 1952 for studying some problems in non-harmonic Fourier series. Wenchang Sun [21] introduced a generalization of frames and showed that this includes more other cases of generalizations of the frame concept and proved that many basic properties can be derived within this more general context. Continuous frames were proposed by G. Kaiser [15] and independently by Ali, Antoine and Gazeau [2] to a family indexed by some locally compact space endowed with a Radon measure. Gabardo and Han in [13] denotes these frames as frames associated with measurable spaces. Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces [4]. However, they have been used earlier in [6] for spherical wavelets. Gabor multipliers [10, 12], Gabor filters [16] and some other applications of frames led Peter Balazs to introduce Bessel and frame multipliers for abstract Hilbert spaces H1 and H2 . These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. A. Rahimi and A. Fereydooni [20] defined the concept of controlled g-frames and they showed that any controlled g-frame is equivalent a g-frame. P. Balazs, D. Bayer and A. Rahimi [5] defined continuous ISSN 1309 - 6788 c 2016 Çankaya University 32 S. M. Ramezani, A. Nazari Bessel and continuous frame multipliers as generalizations of discrete Bessel and frame multipliers. They further developed their theory and proved a number of statements on the compactness of multipliers as well as on mapping properties with respect to Schatten classes. A generalization of the concept of controlled from frame and Bessel sequences, which is proved in [4, 5, 18, 19, 20], to continuous g-frames and continuous g-Bessel sequences in Hilbert spaces is presented in this study. In this paper we extend the concepts of weighted and multiplier from continuous frames to continuous g-Bessel sequences and continuous g-frames. It is shown that the dual of a continuous g-frame in the case that the continuous g-multiplier operator is invertible, is {m(ω )Θω M−1 }ω ∈Ω (see Theorem 1). Moreover, in Theorem 2, we show that a controlled continuous g-frame is equivalent to a continuous g-frame. Finally, we define the multiplier for C2 -controlled g-frames in Hilbert spaces. 2. Preliminaries In the following, we briefly recall some definitions and basic properties of continuous g-frames in Hilbert spaces. We first give some notations which are needed later. Throughout this paper, (Ω, µ ) is a measure space. H and K are two Hilbert spaces and {Kω }ω ∈Ω is a sequence of closed Hilbert subspaces of K . For each ω ∈ Ω, B (H , Kω ) is the collection of all bounded linear operators from H to Kω . We also write M ω ∈Ω Kω = {g = {gω } : gω ∈ Kω and Z Ω kgω k2 d µ (ω ) < ∞}. A bounded operator T : H −→ K is called positive (respectively non-negative), if hT f , f i > 0 for all f 6= 0 (respectively hT f , f i ≥ 0 for all f ∈ H ). Let G L (H ) be the set of all bounded operators with a bounded inverse and G L + (H ) be the set of positive operators in G L (H ). Definition 1. We call a sequence {Λω ∈ B(H , Kω ) : ω ∈ Ω} a continuous g-frame for H with respect to {Kω }ω ∈Ω , if 1. for each f ∈ H , {Λω f }ω ∈Ω is strongly measurable, 2. there are two constants 0 < A ≤ B < ∞ such that, 2 Ak f k ≤ Z Ω kΛω f k2 d µ (ω ) ≤ Bk f k2 , ( f ∈ H ). (1) We call A and B the lower and upper continuous g-frame bounds, respectively. If only the righthand inequality of (1) is satisfied, we call {Λω }ω ∈Ω the continuous g-Bessel sequence for H with respect to {Kω }ω ∈Ω with continuous g-Bessel bound B. If A = B = λ , we call {Λω }ω ∈Ω the λ tight continuous g-frame. Moreover, if λ = 1, {Λω }ω ∈Ω is called the parseval continuous g-frame. CUJSE 13, No. 1 (2016) Weighted and Controlled Continuous g-Frames and their Multipliers in Hilbert Spaces For any { fω }ω ∈Ω , {gω }ω ∈Ω ∈ L 33 ω ∈Ω Kω , if the inner product is defined by h f , gi = Z Ω h fω , gω id µ (ω ), and the norm is defined by 1 k f k = |h f , f i| 2 , then L ω ∈Ω Kω is a Hilbert space. We define the synthesis operator for a continuous g-Bessel sequence {Λω }ω ∈Ω as follows: ! Z hTΛ { fω }ω ∈Ω , gi = Ω h fω , Λω gid µ (ω ), { fω }ω ∈Ω ∈ M ω ∈Ω Kω , g ∈ H . Operator TΛ is well-defined and bounded, therefore, operator TΛ∗ defined for map TΛ∗ : H −→ M TΛ∗ ( f ) = {Λω f }ω ∈Ω , Kω , ω ∈Ω is the adjoint of TΛ and is called the analysis operator. The bounded linear operator SΛ defined by SΛ : H −→ H, hSΛ f , gi = Z Ω h f , Λ∗ω Λω gid µ (ω ), is called the continuous g-frame operator of {Λω }ω ∈Ω . Remark 2.1. A continuous frame is equivalent to a continuous g-frame, whenever Kω = C, for all ω ∈ Ω. Definition 2. Let {Λω }ω ∈Ω and {Γω }ω ∈Ω be two continuous g-frames for H with respect to {Kω }ω ∈Ω such that h f , gi = Z Ω h f , Γ∗ω Λω gid µ (ω ). (2) Then, {Γω }ω ∈Ω is called an alternate dual continuous g-frame of {Λω }ω ∈Ω . Definition 3. Let {Λ j ∈ B(H , K j ) : j ∈ J} and {Θ j ∈ B(H , K j ) : j ∈ J} be g-Bessel sequences. If for m = {m j } j∈J ∈ l ∞ , the operator Mm,Λ,Θ : H −→ H , (3) Mm,Λ,Θ ( f ) = ∑ m j Λ∗j Θ j f , (4) j∈J is well-defined, then Mm,Λ,Θ is called the g-multiplier of Λ, Θ and m. Definition 4. Let C,C′ ∈ G L + (H ). The family {Λ j } j∈J is called a (C,C′ )-controlled g-frame for H with respect to {K j } j∈J , if {Λ j } j∈J is a g-Bessel sequence and there exist constants A > 0 and B < ∞ such that Ak f k2 ≤ ∑ hΛω C f , Λω C′ f i ≤ Bk f k2 , j∈J ( f ∈ H ). (5) 34 S. M. Ramezani, A. Nazari A and B will be called (C,C′ )-controlled g-frame bounds. If C′ = I (or, C′ = C ), we call {Λ j } j∈J a C-controlled g-frame (respectively, C2 -controlled g-frame ) for H with bounds A and B. If the second part of the above inequality holds, it will be called (C,C′ )-controlled g-Bessel sequence with bound B. 3. Main results In this section we define the concept of weighted c (...truncated)