ON HILBERT-SCHMIDT OPERATORS BETWEEN NON-SEPARABLE HILBERT SPACES

Lucien Tebua Limalo Koto (Preparatory section, Higher Institute of Applied Technics of Kinshasa, Democratic Republic of Congo)
Nestor Anzola Kibamba (Department of Mathematics, statistics and Computer Science, Faculty of Sciences and Technology, University of Kinshasa, Democratic Republic of Congo)
Fidele Balibuno Luganda (Department of Mathematics, statistics and Computer Science, Faculty of Sciences and Technology, University of Kinshasa, Democratic Republic of Congo)
Joseph-Desiré Bukweli Kyemba (Department of Mathematics, statistics and Computer Science, Faculty of Sciences and Technology, University of Kinshasa, Democratic Republic of Congo)
Alain Musesa Landa (Department of Mathematics, statistics and Computer Science, Faculty of Sciences and Technology, University of Kinshasa, Democratic Republic of Congo)

AL-ULUM: JURNAL SAINS DAN TEKNOLOGI
Vol 12 No 2 (2026)

We first study Hilbert–Schmidt operators between non-separable Hilbert spaces. We employ concepts such as summable families in normed spaces, the packet summation theorem, and Hilbert bases. We define Hilbert–Schmidt operators and investigate the structure of the Hilbert space (HS(X,Y)), establishing its relationship with the spaces of compact operators (CO(X,Y)), finite-rank operators (F_i(X,Y)), and continuous linear operators (LC(X,Y)). We then consider particular classes of Hilbert–Schmidt operators on the space of almost periodic functions  where. Finally, we establish a spectral representation theorem for compact Hermitian operators on non-separable Hilbert spaces, extending the well-known result for separable Hilbert spaces.