Prove that for the usual matrices [math]A,B\in M_N(\mathbb C)[/math] we have

where [math]\sigma^+[/math] denotes the set of eigenvalues, taken with multiplicities.

Clarify, with examples and counterexamples, the relation between the eigenvalues of an operator [math]T\in B(H)[/math], and its spectrum [math]\sigma(T)\subset\mathbb C[/math].

Draw the picture of the following function, and of its inverse,

with [math]r\in\mathbb R[/math], and prove that for [math]r \gt \gt 0[/math] and [math]T=T^*[/math], the element [math]f(T)[/math] is well-defined.

Comment on the spectral radius theorem, stating that for a normal operator, [math]TT^*=T^*T[/math], the spectral radius is equal to the norm,

with examples and counterexamples, and simpler proofs of well, in various particular cases of interest, such as the finite dimensional one.

Develop a theory of [math]*[/math]-algebras [math]A[/math] for which the quantity

defines a norm, for the elements [math]a\in A[/math].

Find and write down a proof for the spectral theorem for normal operators in the spirit of the proof for normal matrices from chapter 1, and vice versa.

Find and write down an enhancement of the proof given above for the spectral theorem, as for [math]\bar{z}\to T^*[/math] to appear way before the end of the proof.