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(b) Since (AB)*AB = B*A*AB = B*B = I and AB(AB)* = ABB* A* = AA* = I, we see that AB is unitary. Also, for all x € H, ((A + B)(x), (A + B)(x)) (A(x), A(x)) + (B(x), B(x)) + (A(x), B(x)) + (B(x), A(x)) = 2(x, x) + 2Re (A(1), B(1)). Request explain Hence 26.2(b) implies that A+B is unitary if and only if it is surjective step and (x,x) + 2Re (A(x), B(x)) = 0 for all x € H. =

(b) Since (AB)*AB = B*A*AB = B*B = I and AB(AB)* = ABB* A* = AA* = I, we see that AB is unitary. Also, for all x € H, ((A + B)(x), (A + B)(x)) (A(x), A(x)) + (B(x), B(x)) + (A(x), B(x)) + (B(x), A(x)) = 2(x, x) + 2Re (A(1), B(1)). Request explain Hence 26.2(b) implies that A+B is unitary if and only if it is surjective step and (x,x) + 2Re (A(x), B(x)) = 0 for all x € H. =

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
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(b) Let A and B be unitary. Then AB is unitary. Also, A + B is unitary if and only if it is surjective and Re (A(r), B(x)) = -1/2 for every x H with ||x|| = 1. (b) Since (AB)*AB = B*A*AB = B*B = I and AB(AB)* = ABB* A* = AA* = I, we see that AB is unitary. Also, for all z € H, ((A + B)(x), (A + B)(x)) (A(x), A(x)) + (B(x), B(x)) + (A(x), B(x)) + (B(x), A(x)) = 2(x, x) + 2Re (A(1), B(x)). Request explain Hence 26.2(b) implies that A+B is unitary if and only if it is surjective the step and (x,x) + 2Re (A(x), B(x)) = 0 for all ¤ ¤ H.
Transcribed Image Text:(b) Let A and B be unitary. Then AB is unitary. Also, A + B is unitary if and only if it is surjective and Re (A(r), B(x)) = -1/2 for every x H with ||x|| = 1. (b) Since (AB)*AB = B*A*AB = B*B = I and AB(AB)* = ABB* A* = AA* = I, we see that AB is unitary. Also, for all z € H, ((A + B)(x), (A + B)(x)) (A(x), A(x)) + (B(x), B(x)) + (A(x), B(x)) + (B(x), A(x)) = 2(x, x) + 2Re (A(1), B(x)). Request explain Hence 26.2(b) implies that A+B is unitary if and only if it is surjective the step and (x,x) + 2Re (A(x), B(x)) = 0 for all ¤ ¤ H.
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