polynomially partial isometry. Our investigation is inspired by previous work in the field [5,
30, 31]. By extending the definition of partial-$ A $-isometry, we provide new insights into the
properties and applications of these mathematical objects. In particular, we define the notion
of left $ p $-partial-$ A $-isometry as a broader class of operators, including partial-$ A $-
isometry and left polynomially partial isometry. Some basic properties of a left $ p $-partial …