Some Results on Analysis, Synthesis and Frame Operators Associated with a Fixed Element in 2-Hilbert Spaces

Abstract

This paper develops the operator-theoretic framework of ξ-frames in 2-Hilbert spaces through the analysis, synthesis, and ξ-frame operators associated with a fixed element. The ξ-analysis operator is introduced for ξ-Bessel sequences, and its linearity and boundedness are established. Since ξ-frame coefficients depend only on equivalence classes determined by the fixed element, the construction is formulated on the corresponding quotient Hilbert space. The ξ-synthesis operator is then obtained as the adjoint of the quotient analysis operator and lifted to the original 2-Hilbert space by means of a complementary subspace. Furthermore, the ξ-frame operator is defined as the composition of the synthesis and analysis operators, and its boundedness, self-adjointness, positivity, and operator form of the ξ-frame inequalities are proved. The invertibility of the induced ξ-frame operator leads naturally to the canonical dual ξ-frame and the associated reconstruction formula. A finite-dimensional example is presented to illustrate the theoretical results. These findings provide a rigorous foundation for reconstruction theory and further investigations of generalized frame structures in 2-Hilbert spaces.