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Problem 1. Let H be a Hilbert space and p: H→ H 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: H→ H 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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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operayor on hilbert space part 3
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 (Idup) 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). (a) Show that ||p|| = 1. (b) Show that p is the orthogonal projection on Imp. 4. Suppose that p is a nonzero continuous operator such that ||p|| = 1. (a) Expand |x-p*x||2 and deduce that ker(Id - p) = ker (Idи - p*). (b) Show that p is Hermitian.
Transcribed Image Text: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 (Idup) 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). (a) Show that ||p|| = 1. (b) Show that p is the orthogonal projection on Imp. 4. Suppose that p is a nonzero continuous operator such that ||p|| = 1. (a) Expand |x-p*x||2 and deduce that ker(Id - p) = ker (Idи - p*). (b) Show that p is Hermitian.
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