Abstract
LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=∫fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that σ(T) is precisely the closedP-essential range of the functionf or equivalently, that σ(T) is equal to the support of the (unique) equicontinuous spectral measureQ * defined on the Borel sets of the extended complex plane ℂ* such thatQ *({∞})=0 andT=∫zdQ *(z). This result is then used to prove a spectral mapping theorem; namely, thatg(σ(T))=σ(g(T)) for anyQ *-integrable functiong: ℂ* → ℂ* which is continuous on σ(T). This is an improvement on previous results of this type since it covers the case wheng(σ(T))/{∞} is an unbounded set inℂ a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.
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Ricker, W. A spectral mapping theorem for scalar-type spectral operators in locally convex spaces. Integr equ oper theory 8, 276–288 (1985). https://doi.org/10.1007/BF01202816
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DOI: https://doi.org/10.1007/BF01202816