16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.5.7

Operators on Inner Product Spaces

Mathematics — Linear Algebra · operators-on-matrix-space

Problem

Let

V

be the space of complex

n \times n

matrices equipped with the inner product

(A|B) = \operatorname{tr}(AB^*)

. If

B

is an element of

V

, let

L_B

R_B

, and

T_B

denote the linear operators on

V

defined by (a)

L_B(A) = BA

. (b)

R_B(A) = AB

. (c)

T_B(A) = BA - AB

. Consider the three families of operators obtained by letting

B

vary over all diagonal matrices. Show that each of these families is a commutative self-adjoint algebra and find their spectral resolutions.