Matrix-valued analytic function?

[AxiomOfChoice]

Consider the matrix-valued function T(\beta): \mathbb C \to \mathscr L(\mathbb C^2), the bounded linear operators on \mathbb C^2, given by

<br /> T(\beta) = \begin{bmatrix} 1 &amp; \beta \\ \beta &amp; -1 \end{bmatrix}.<br />

According to Reed-Simon, Volume 4, this is a "matrix-valued analytic function" with singularities at \beta = \pm i. I'm confused as to how...

1. ...we are supposed to show that T(\beta) is analytic. The claim made in the book is that it is easier (in general) to show that a vector-valued analytic function is weakly analytic than strongly analytic, but I don't see how that is the case here.
2. ...we are supposed to see that this function has singularities at \pm i.

Can anyone help with either of the above? Thanks!

 

[AxiomOfChoice]

Update

It seems T(\beta) is actually an entire matrix-valued analytic function of \beta; it is the eigenvalues \lambda_\pm (\beta) = \pm \sqrt{\beta^2 + 1} that have singularities at \pm i. My question still stands, though...why are \pm i singularities of this function? What's wrong with taking the square root of zero, which is what one winds up doing at those values?

 

[pasmith]

What's wrong with taking the square root of zero, which is what one winds up doing at those values?

Nothing. But there is a problem with evaluating the derivative of the square root function at zero, and that's what causes \sqrt{\beta^2 + 1} to fail to be analytic at \beta = \pm i.