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I just found out that there is this interesting book about fractional powers of operators.

Maybe it can be useful for our purposes. For example on can consider the infinite Carlemann-matrix of to be an operator (which is the infinite version of our matrix power method). However I didnt have a look into the book yet, so I dont know what they for example say about uniqueness etc.
Cannot be found at elsevier.
Here is a table of contents: (6'2010)
http://elib.tu-darmstadt.de/tocs/95952543.pdf

Gottfried
Here is an open access paper with a similar title:

http://projecteuclid.org/DPubS/Repositor...1102993838