Assume that $B$ is a self-adjoint operator and $\alpha\in(0,1)$. I need a reference for the following equality $$B^{-\alpha}=\frac{\sin\alpha \pi}{\pi}\int_0^\infty \lambda^{-\alpha}(\lambda+B)^{-1}\mathrm d\lambda.$$

  • 4
    $\begingroup$ Functional calculus plus an integral equality, I'd guess. So you don't need a reference. Is there an additional condition on the spectrum of $B$ for convergence issues? $\endgroup$ – Marc Palm Feb 20 '14 at 9:19
  • $\begingroup$ As suggested by Marc one needs some condition on the operator---positivity. Again as in the above comment one can reduce it to the scalar case by using the spectral theorem to assume that the operator is a mutiplication operator on an $L^2$ space. If $B$ is bounded and bounded away from zero, then this is routine. Otherwise (and this is, of course, the interesting case), the result and the interpretation of the integral are more delicate and require splitting the Hilbert space into a direct sum of subspaces on each of which these conditions hold. $\endgroup$ – alpha Feb 20 '14 at 12:42

See pp. 231-232 in Birman M.S., Solomyak M.Z. Spectral Theory of Self-Adjoint Operators in Hilbert Space (Reidel, 1987).


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