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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.

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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 22E
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Please solve number 1-2
M, MEGA 2.pdf Assignment 2 Open with Google Docs 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(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 p is a nonzero continuous operator such that ||p|| = 1. (a) Expand ||x - p*x||2 and deduce that ker(IdHp) = ker(IdH - p*). (b) Show that P is Hermitian. Problem 2. Let H be an Hilbert space. For all operator T on H, we define v(T) = sup{(Tx, x); ||*|| = 1}.
Transcribed Image Text:M, MEGA 2.pdf Assignment 2 Open with Google Docs 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(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 p is a nonzero continuous operator such that ||p|| = 1. (a) Expand ||x - p*x||2 and deduce that ker(IdHp) = ker(IdH - p*). (b) Show that P is Hermitian. Problem 2. Let H be an Hilbert space. For all operator T on H, we define v(T) = sup{(Tx, x); ||*|| = 1}.
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