Problem 1. Let H be a Hilbert space and p: HH be a projection, i.e. it is a linear application such that pop = p. 1. Show that Imp = ker (Idи - p) and H=kerp Imp. 2. Suppose that p is a nonzero continuous operator. (a) Show that ||p|| > 1. (b) Show that the adjoint operator p is also a projection. 3. Suppose that p is a nonzero continuous operator such that p is Hermitian (i.e. p*= p).

Problem 1. Let H be a Hilbert space and p: HH be a projection, i.e. it is a linear application such that pop = p. 1. Show that Imp = ker (Idи - p) and H=kerp Imp. 2. Suppose that p is a nonzero continuous operator. (a) Show that ||p|| > 1. (b) Show that the adjoint operator p is also a projection. 3. Suppose that p is a nonzero continuous operator such that p is Hermitian (i.e. p*= p).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 30E

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