8. Suppose T :V V is a linear operator on an inner product spaceT has an adjoint (hence, a unique adjoint) T*, prove that Range(T")- =null(T).V. If

Answered: 8. Suppose T: V V is a linear operator…

8. Suppose T: V V is a linear operator on an inner product space V. If T has an adjoint (hence, a unique adjoint) T", prove that Range(T")- = null(T). %3D

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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8. Suppose T :V V is a linear operator on an inner product space
T has an adjoint (hence, a unique adjoint) T*, prove that Range(T")- =
null(T).
V. If

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Let A E BL(H). Then A is called normal if A*A = AA*, uni- tary if A*A = I = AA*, that is, A* = A-¹, and self-adjoint if A* = A. Clearly, if A is unitary or self-adjoint, then A is normal. How- ever, a normal operator need be neither unitary nor self-adjoint. For example, let H = K² and for x = (x(1), x(2)) in H, A(x) = (x(1) − x(2), x(1) + x(2)). Then it can be easily seen that for x = H, A*(x) = (x(1) + x(2), −x(1) + x(2)), Request A*A(z) = 2(x(1),z(2)) = AA*(x). Hence A*A = AA*, but A*A# I and A* ‡ A. explain these steps in détail.
Let A E BL(H). Then A is called normal if A*A = AA*, uni- tary if A*A = I = AA*, that is, A* = A−¹, and self-adjoint if A* = A. Clearly, if A is unitary or self-adjoint, then A is normal. How- ever, a normal operator need be neither unitary nor self-adjoint. For example, let H = K² and for x = (x(1), x(2)) in H, A(x) = (x(1) − x(2), x(1) + x(2)). Then it can be easily seen that for x = H, A*(x) = (x(1) + x(2), −x(1) + x(2)), A*A(x) = 2(x(1), x(2)) = AA*(x). Hence A*A = AA*, but A*A ‡ I and A* # A. It is easy to see that if B is a normal operator and a C is a bounded operator such that C*C = I, then the operator A = CBC* is normal. Note that a bounded operator C may satisfy C*C = I without being a unitary operator. For example, let H = ² and for x = (x(1), x(2),...) in H, let C(x) = (0, x(1), x(2),...). 26 Normal, Unitary and Self-Adjoint Operators Then C*(x) = (x(2), x(3),…) for x € H, Hence Request explain Iin detail for better understan- 461 -ding C*C(x) = C*((0, x(1), x(2),...)) =…
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