Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

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
See similar textbooks
icon
Related questions
Question
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}.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS