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book: the theory of fractional powers of operators
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)

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

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