Positive Operators–Isometries–Real Inner Product Spaces

In this chapter, we proceed with special normal operators such as the nonnegative operators. The analogical comparison of normal operators with the complex numbers continues in that we now introduce nonnegative operators which correspond to nonnegative real numbers. Subsequently, we discuss isometries. These are closely related to symmetries in physics, in particular to symmetries in quantum mechanics. We then use properties of operators in complex vector spaces to derive properties of operators in real vector spaces. An instrument for this method is complexification which we explain in detail. In this way, we obtain the spectral theorem for real normal operators which have not been accessible so far.