Consider the matrix-valued function [itex]T(\beta): \mathbb C \to \mathscr L(\mathbb C^2)[/itex], the bounded linear operators on [itex]\mathbb C^2[/itex], given by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

T(\beta) = \begin{bmatrix} 1 & \beta \\ \beta & -1 \end{bmatrix}.

[/tex]

According to Reed-Simon, Volume 4, this is a "matrix-valuedanalyticfunction" with singularities at [itex]\beta = \pm i[/itex]. I'm confused as to how...

Can anyone help with either of the above? Thanks!

- ...we are supposed to show that [itex]T(\beta)[/itex] 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.
- ...we are supposed to see that this function has singularities at [itex]\pm i[/itex].

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# Matrix-valued analytic function?

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