On operators with rational resolvent

Abstract

It is shown that a bounded linear operator T on a complex Banach space into itself has a rational resolvent if and only if every bounded linear operator which commutes with every bounded linear operator that commutes with T can be expressed as a polynomial in T. In this note, we present a characterization of the Caradus class a of bounded linear operators on a complex Banach space X into itself [1], that is the class of bounded linear operators with rational resolvent [5, p. 314]. This confirms a suggestion of Caradus in [2]. We shall appeal to a version of the minima] equation theorem in which the following terminology is used. DEFINITION. Let T be a bounded linear operator on a complex Banach space X into itself We define the spectral order VT(i) of the point A of the spectrum of T to be the order of A as a pole of the resolvent of T; if A is not a pole of the resolvent of T, we give VT(O) the conventional value oo. We could extend this definition by putting VT(Q)=O when A is in the resolvent set of T. THEOREM 1 (MINIMAL EQUATION THEOREM). Let T be a bounded linear operator on a complex Banach space X into itself, and let f and g be functions holomorphic on open sets which contain the spectrum of T. Then f(T) =g(T) if and only if, for every point A of the spectrum of T, f(r) ()= g(r)(;) when r is an integer with O<r<VT(A), where VT(i) is the spectral order of A. This follows immediately from [3, Theorem VII.3.16, p. 571]. THEOREM 2. Let X be a complex Banach space. The bounded linear operator T on X into itself belongs to the Caradus class a if and only if it has the property that a bounded linear operator S (on X into itself) commutes with every bounded linear operator which commutes with T only if there is a polynomial p such that S=p(T). Received by the editors July 6, 1971. AMS 1970 subject classifications. Primary 47A10, 47B10; Secondary 46B99, 47A60.