This is a concept I had not encountered before, and I am worried by it - and slightly ashamed to have to seek help.

Start here. (I am working from a text by Paul Halmos on Hilbert spaces).

I am reminded that the spectrum of an arbitrary operator

(

is in the first instance a finite-dimensional Hilbert space) is the the set of all complex numbers

for which

. This comes from the classical definition:

. (Actually, it is common to pop the identity operator as factor in the first equality, but that, I believe, is merely to simplify subsequent calculations).

First question: Halmos now writes this for the

*general* case as

by insisting that

be a

*unit* vector. Sorry but I cannot see how using a unit vector allows me move from the determinant to the norm.

Halmos then introduces the concept of an approximate eigenvalue by

for some non-negative number

and

still a unit vector.

Second question: Why on earth might one be interested in "approximate" eigenvalues?

Finally this: I am told that the last inequality is equivalent to

for any

*arbitrary* vector

. Insult to injury is added by his remark that this is "easy to verify".

I guess I need some help here too.