Let M be an anti-self-adjoint operator and L(s) be a family of operators satisfyingthe equation0,L(s) = [M, L]%3Dwhere [M, L] = ML - LM. Show that:1. If the operator Lo is self-adjoint then L(s) is self-adjoint for any s.2. The operators L(s) and Lo have the same spectrum.

Use the given two equations to prove the following. If one operator is self-adjoint, then use the…

Answered: Let M be an anti-self-adjoint operator…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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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.
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Let M be an anti-self-adjoint operator and L(s) be a family of operators satisfying the equation 8,L(s) = [M, L] where [M, L] = ML - LM. Show that: 1. If the operator Lo is self-adjoint then L(s) is self-adjoint for any s. 2. The operators L(s) and Lo have the same spectrum.