2023, Vol. 8, Issue 4, Part A

In this research paper, we explained and established the results for the space of operators in Hilbert space. Some of them are if the space of all operators on a Hilbert space H, B(H) is a Banach Space as well as here we construct a set of operators R on a Hilbert space H in the following way R = {Ti: Ti² = 0, TiTj = 0} and we have proved that R is a Banach Space. Also If R and R’ be linear spaces and T: R → R’ be a linear transformation then Kernel T is a linear subspace of R. We discussed on isomorphism concept also here if T be a linear transformation of R onto R’ then T is an isomorphism if and only if Ker T = {0}. We have also discussed algebraic structure, we have seen that the set of all non-singular transformations on a linear space R forms a group with respect to multiplication.