Annals of Mathematics and Computer Science ISSN: 2789-7206 Vol 22 (2024) 45-53 https://doi.org/10.56947/amcs.v22.281 EQUAL-NORM PARSEVAL CONTINUOUS K-FRAMES IN HILBERT SPACES HAFIDA MASSIT 1 AND MOHAMED ROSSAFI 2* This paper is dedicated to Professor Samir Kabbaj Abstract. In this paper, we present some fundamental results for the Parse- val continuous K−frames in Hilbert space and we study some properties of an equal-norm of vectors. Furthermore, we show that a set of K−norm vectors can be extended to become a K−norm of K−frame. 1. Introduction and preliminaries The concept of frame in Hilbert spaces has been introduced by Duffin and Schaffer [4] in 1952 to study some deep problems in nonharmonic Fourier series, after the fundamental paper [3] by Daubechies, Grossman and Meyer, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames. The majority of these applications requires frames in finite-dimensional spaces. For example, Jamali et al. [9] and Javanshiri et al. [10], were obtained results that are interesting in applications of frames. The continuous frames has been defined by Ali, Antoine and Gazeau [1], called these kinds of frames, frames associated with measurable space. The concept of continuous K −frame in Hilbert space have been introduced by G˘ avruta, in [6] for Hilbert spaces to study atomic systems with respect to a bounded linear operator. Due to the structure of K −frames, there are many differences between K −frames and standard frames. K −frames, which are a generalization of frames, allow us in a stable way, to reconstruct elements from the range of a bounded linear operator in a Hilbert space. For more, see [7, 8]. In this paper we try to give a generalization of the results given in [15] moving from the discrete case to the continuous case, we get some new results on the Parseval continuous K −frames. Throughout this paper, assume that (A,μ) be a measure space with positive measure μ, H, H 0 , H 1 and H 2 are Hilbert spaces. (A,μ) is a σ-finite measures pace, L(H 0 ; H) is the set of all linear mappings of H 0 to H and B(H 0 ; H) is the set of all bounded linear mappings. Instead of B(H; H), we simply write B(H). Date : Received: Feb 20, 2024; Accepted: Mar 7, 2024. * Corresponding author. 2010 Mathematics Subject Classification. 41A58, 42C15, 46L05. Key words and phrases. continuous K-frame; Parseval frame; K- dual frames. 45