Assume that T is a linear operator on a complex (not necessarily finite-dimensional)inner product space V with an adjoint T∗. Prove the following results. (a)If T is self-adjoint, then <t(x), x>is real for all x ∈V. (b) If T satisfies <t(x), x>= 0 for all x ∈V, then T = T0. Hint: Replace x by x + y and then by x+ iy, and expand the resulting inner products. (c) If <T(x), x>is real for all x ∈V, then T = T∗.

Assume that T is a linear operator on a complex (not necessarily finite-dimensional)inner product space V with an adjoint T∗. Prove the following results. (a)If T is self-adjoint, then <t(x), x>is real for all x ∈V. (b) If T satisfies <t(x), x>= 0 for all x ∈V, then T = T0. Hint: Replace x by x + y and then by x+ iy, and expand the resulting inner products. (c) If <T(x), x>is real for all x ∈V, then T = T∗.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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