3. Let H be a Hilbert space and T & B(H). 0 T* TO (b) A continuous function f: R→ C is called even if V ER: f(-x) = f(x)

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3 Let H Be A Hilbert Space And T B H 0 T To B A Continuous Function F R C Is Called Even If V Er F X F X 1 (23.32 KiB) Viewed 8 times

3. Let H be a Hilbert space and T & B(H). 0 T* TO (b) A continuous function f: R→ C is called even if V ER: f(-x) = f(x) and odd R (8 %) S (a) Show that == is a self-adjoint operator in B(HH). if Vr = R: f(-x) = -f(x). Show that f(T) is of the form f(T) := R ·(). S (c) Show that there exists SE B(H) such that T = SS S. even and f(1) = if f is odd, where R, S = B(H). if f is