4. Let T = B(X) be a continuous operator on a Banach space X and let f(2)==0 anz" be a power series with radius of conve

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4 Let T B X Be A Continuous Operator On A Banach Space X And Let F 2 0 Anz Be A Power Series With Radius Of Conve 1 (35.45 KiB) Viewed 12 times

4. Let T = B(X) be a continuous operator on a Banach space X and let f(2)==0 anz" be a power series with radius of convergence bigger than r(T). (a) Prove that f(T): EanT" converges in B(X). € n=0 (b) If g(2) bnz" is another power series with radius of convergence bigger than r(T), then fg(T) = f(T)g(T). (c) Show that f(o(T)) Co(f(T)). Remark: The other inclusion "" holds as well, but our current tools do not seem sufficient to prove this. (d) Assume now that f is a self-adjoint operator on a Hilbert space X = H. Show that the f(T) defined in (a) agrees with the f(A) defined by the continuous functional calculus in the lecture.