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Given a pseudo-differential operator $P$ of order zero, Seeley showed that the holomorphic family of operators $\lbrace P^{z} : z\in \mathbb{C} \rbrace$ of all complex powers is contained in the class of pseudo-differential operators.

Apart from knowing that we can take powers of these operators, is there any application of this theory? I understand the utility of raising operators to fractional powers. But, irrational and complex powers are not clear.

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Math Sci Net lists 181 citations for that paper. Have you looked at any of those? –  Zack Wolske Aug 19 '12 at 5:34
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You might be interested in an older question mathoverflow.net/questions/71429/… where I also gave an answer for a specific example of the exponential function –  Gottfried Helms Aug 19 '12 at 6:28
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up vote 16 down vote accepted

There are many reasons to do this. A principal one (that was Seeley's motivation) is that the meromorphic operator family $P^{-s}$ is trace-class when the real part of $s$ is greater than $-n/m$ where $n$ is the dimension and $m$ is the order of $P$. This gives a meromorphic continuation of the spectral zeta function $\zeta(s) = \sum_{j=1}^\infty \lambda_j^{-s}$. This is an interesting generalization of the Riemann zeta function. Furthermore, its behaviour sheds light on the distribution of eigenvalues (the Weyl law). This zeta function is equivalent to the heat trace $\sum_{j=1}^\infty e^{-\lambda_j t}$ via the Mellin transform. There are lots of other things that come out of this, however -- take a look at the recent book by Scott ``Traces and Determinants of Pseudodifferential Operators''.

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Thanks, Rafe. So, 'Spectral theory of pseudo-differential operators' is the answer to my question. –  Uday Aug 21 '12 at 5:27
    
@Rafe Mazzeo: I am very interrested by this book : "Traces and Determinants of Pseudodifferential Operators". –  user36539 Sep 9 '13 at 19:43
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Bounded imaginary powers of differential operators may give informations about maximal regularity of evolution equations. see Buzano &Nicola : Complex powers of hypoelliptic pseudodifferential operators.

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