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Spectral Theory

Example 1.1 Define, for h ∈ R, the operator τh on L2(R) by τhf(x) = f(x − h). Show that τh is bounded. Obviously, τh is linear, and it follows from τhf 22 = +∞ −∞ |f(x − h)|2dx = +∞ −∞ |f(x)|2dx = f 22 , that Tf 2 = f 2 for all f ∈ L2(R), hence T = 1. Remark 1.1 Here we add that τh is also regular. In fact, if τhf = 0, then f(x−h) = 0 for all x ∈ R, thus f ≡ 0. This shows that τh is injective, hence the inverse operator exists. Then we get by the change of variable y = x − h, i.e | bookboon.com Spectral Theory Functional Analysis Examples c-4 Leif Mejlbro Download free books at bookboon.com Leif Mejlbro Spectral Theory Download free ebooks at bookboon.com 2 Spectral Theory Leif Mejlbro Ventus Publishing ApS ISBN 978-87-7681-530-1 Disclaimer The texts of the advertisements are the sole responsibility of Ventus Publishing no endorsement of them by the author is either stated or implied. Download free ebooks at bookboon.com