Let X and Y be Banach spaces
and T, respectively S, be a bounded linear transformation mapping X into Y ,
respectively Y into X. It is well-known that a nonzero complex number λ belongs to
the spectrum of ST precisely when λ belongs to the spectrum of TS. The main result
of §2 shows that for λ≠0 the states of the operators ST − λIX,TS − λIY
agree.
Sufficient conditions are obtained for this same result to hold when T and S are
unbounded closed linear transformations from X into Y and Y into X respectively.
Section 4 compares spectral decompositions of ST and TS when these sufficient
conditions are satisfied.
