NA-Isometric Operators on Hilbert Spaces

A great deal of excellent work has been done for partial isometrics. Thanks to early work of I. Erdelyi & P. R. Halmos, among others; they have played a fundamental role in structural study of Hilbert space operators, especially, in the theory of the polar decomposition of arbitrary operators and in the dimension theory of von Neumann algebras. They have also arisen in quantum physics (Bock et al. in Lett. Math. Phys. 112(2):1-11, 2022; Bracci and Picasso in Bull. Lond. Math. Soc. 39(5):792-802, 2007; Lai et al. in Quantum Inf. Process. 21(3):1-17, 2022). Based on the study of partial isometrics (Erdelyi in J. Math. Anal. Appl. 22:546-551, 1968; Ezzahraoui et al. in Arch. Math. 110(3):251-259, 2018; Halmos and McLaughlin in Pac. J. Math. 13:585-596, 1963; Halmos and Wallen in J. Math. Mech. 19:657-663, 1970; Mostafa and Skhiri in Integral Equ. Oper. Theory 38:334-349, 2000; Wallen in Bull. Am. Math. Soc. 75:763-764, 1969) and semi-generalized partial isometries (Garbouj and Skhiri in Results Math. 75(1):15, 2020), for a given linear bounded non-zero operator A, we introduce a new class of operators called N-A-isometries. We present its basic properties, and show a variety of results which improve and extend some works related to classical partial isometrics.