K-block frames in Hilbert spaces

This paper introduces and develops the theory of [Formula: see text]-block frames in Hilbert spaces, presenting an extension of frame theory and operator theory. The concept of atomic systems for operators have been generalized to the framework of block-atomic systems, focusing on bounded linear operators in separable Hilbert spaces. The notion of [Formula: see text]-block frames is defined and thoroughly analyzed, demonstrating connections to block-atomic decompositions. The paper presents necessary and sufficient conditions for a sequence to qualify a [Formula: see text]-block frame or block-atomic systems. These ideas are illustrated with concrete examples and establish essential connections.