Study of the Special type of operator called (42) – jection

In this book I have generalized projection operator by constructing a new operator (42)-jection defined as E4 = E2.It is shown that a (42)-jection E on R2 or C2 is either a projection or E2 = 0 E2 = I or E2 = -E.(42)-Jection operator can be represented in the form of matrix and since computer is based on programming and programming is based on matrix so one can utilize (42)-jection operator in computer. (42)-jection operator can also be utilized in Hilbert space. The relation between defined ranges and null space of square of two given (42)-jection operators under different conditions has been discussed in this book. Different subspace of range associated with a (42)-jection is also discussed here. Theorem based on idea of pell’s equations from theory of numbers has been proved.A spectral decomposition of a (42)-jection operator has also been discussed. Resolvent set and spectral radius of E2 E being a (42)-jection.If a researcher wants to do research on operator theory this book will prove to be very helpful for the researcher.