I have the following question and since I am not an expert on C*-algebras, I thought I ask it here:
I know that in general the sum and product of normal elements need not be normal. It is even true that every element in a C*-algebra is the sum of two normal elements. What I do not know is if that is true for multiplication, too, i.e. if every element in a C*-algebra is a product of normal elements.
I think I know that every invertible element can be written as a product of a unitary and a positive element but this decomposition does not work for non-invertible elements.
My standard example is the right shift operator on the l^2(N)-space. This is a non-normal operator, which is not invertible and I have no idea how to write it as a product of two normal operators.
I would be very grateful if anyone could help.