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.
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.
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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Please problem 3
![Problem 1. Let H be a Hilbert space and p: HH be a projection, i.e. it is a
linear application such that po p = p.
1. Show that Imp = ker(IdHp) 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 Р is a nonzero continuous operator such that ||p|| = 1.
(a) Expand ||x - p*x||² and deduce that ker(Idµ − p) = ker(Idµ – p*).
(b) Show that Р is Hermitian.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2F9d08cbcb-5b44-48df-85ae-a664247f9e0b%2Ftf5yg2f_processed.jpeg&w=3840&q=75)
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 po p = p.
1. Show that Imp = ker(IdHp) 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 Р is a nonzero continuous operator such that ||p|| = 1.
(a) Expand ||x - p*x||² and deduce that ker(Idµ − p) = ker(Idµ – p*).
(b) Show that Р is Hermitian.
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