Vol. 54, No. 1, 1974

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Comparison of the states of closed linear transformations

John Douglas Faires

Vol. 54 (1974), No. 1, 123–133

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 λ0 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.

Mathematical Subject Classification 2000
Primary: 47A10
Received: 16 March 1973
Revised: 21 September 1973
Published: 1 September 1974
John Douglas Faires