Comparison of the states of closed linear transformations
J. Douglas Faires · Pacific Journal of Mathematics · 1974
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 λΦO the states of the operators ST -λl x , TS -λl γ 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. Throughout this paper D(A) and R(A) will denote the domain and range of A.The resolvent of A will be denoted p(A), the spectrum σ(A), the point spectrum p(A) and the approximate point spectrum a(A).[X, Y] will denote the set of all bounded linear transformations, defined on the Banach space X into the Banach space Y. Any other notation used will agree with that of [3].When no confusion will arise the identity operator will be denoted by I regardless of the space.The following preliminary result can be easily variίied.PROPOSITION I.I.If T: D(T) c X-> Y, S: D(S) c Y-+Xand λ Φ 0, then Xep(TS) if and only if Xep(ST). 2. Continuous transformations• PROPOSITION 2.1.// λ Φ 0 then R(ST -λl) = X precisely when E(TS-XI) = Y.