From: jbaez@zermelo.mit.edu (John C. Baez)
Subject: Re: Help - non-integral power of a matrix?
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Fri, 14 Aug 92 20:44:04 GMT

In article <KOSOWSKY.92Aug14122403@schottky.harvard.edu> kosowsky@schottky.harvard.edu (Jeffrey J. Kosowsky) writes:
[I wrote:]
>   To clarify, perhaps, let me add that this is not only "obvious", it's
>   true, at least if A is diagonalizable.  If A is a
>   not-necessarily-diagonalizable matrix log A is defined if ||A - 1|| < 1;
>>   I don't think the eigenvalue condition above is sufficient.
>
>
>Aside:
>
>The theory of taking analytic functions of a matrix (or more generally
>any bounded linear operator, T, on a Banach space, X) is called the
>Riesz (or functional) analytic calculus. Given a holomorphice
>(ie: analytic) funtion, f, defined on an open neighborhood of the
>spectrum, \sigma(T) of a bounded linear operator, T, we can define
>f(T) by an extension of the Cauchy integral formula. If the Taylor
>expansion of the function about some point contains the spectrum in
>its region of convergence, then f(T) can equivalently be written as
>the sum of the Taylor series terms where powers of T replace the
>independent variable in the Taylor series. Much more could of course
>be said about the properties of the functional analytic calculus. 
>On normal elements of C* algebras (eg:  L(H) is a C* algebra where H
>is a Hilbert space), there is a more general extension called the
>Borel functional calculus. Suppose T is a normal element of a C*
>algebra, A, and f is a continuous function on the spectrum of T, then
>we can define f(T) via the inverse of the Gelfand transform.

I'm particularly worried about the non-normal case, of course.  You are
right about the holomorphic functional calculus so we are free to say
that my series 

ln A = (A - 1) - (A - 1)^2/2 + ....

converges when the spectral radius of A - 1 is < 1. If our space is
finite-dimensional this just means that all eigenvalues of A have
|lambda - 1| < 1.   If A is normal the spectral radius equals the norm
so we may equivalently demand ||A - 1|| less than 1.   (In the
finite-dim case "normal" just means "diagonalizable".)

>In any case, the original poster was presumably talking about finite
>dimensional matrices so that the spectrum \sigma(T) is equal to the
>set of eigenvalues. Since the region of convergence of the logarithm
>about 1 is 1, Arthur Rubin's characterization is obviously correct.
>Assuming that John Baez is using the standard (2-norm) on linear
>operators, then his criterion is equivalent since ||A -1|| equals the
>largest magnitude eigenvalue of (A-1), so ||A-1|| < 1 iff all
>eigenvalues lie within the unit circle around 1.

Only if A is normal need the spectral radius equal the norm.  We can
have 

A  = 1   1000
     0    1

so 

A - 1 =  0  1000
         0   0

Then the spectral radius is less than 1 but the norm is not, since the
only eigenvalue of A - 1 is zero even though A - 1 is really big.

In any event, I guess it was me who brought norms into the picture so
I'm to blame for introducing these subtleties.  I agree that the
original condition for convergence in terms of spectral radius was fine.
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