Abstract
Use the spectral theorem to represent A as a multiplication by, say, φ. If λ ∈ spec A and if N is an arbitrary neighborhood of F(λ), then F-1(N) is a neighborhood of λ, and therefore φ-1(F-1(N)) has positive measure. Since φ-1(F-1(N)) = (F ∘ φ)-1(N), it follows that F(λ) is in the essential range of F ∘ φ, so that F(λ) ∈ spec F(A). This proves that F(spec A) ⊂ spec F(A).
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© 1982 Springer-Verlag New York Inc.
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Halmos, P.R. (1982). Partial Isometries. In: A Hilbert Space Problem Book. Graduate Texts in Mathematics, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9330-6_65
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DOI: https://doi.org/10.1007/978-1-4684-9330-6_65
Publisher Name: Springer, New York, NY
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