Fractional powers as spectral packet transfer

Proof of Theorem 10.1

All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example,

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Proof of Theorem 10.2

By the spectral theorem it suffices to prove the scalar identity

for \(\lambda\ge0\). This is the classical Gaussian integral after the substitution \(u=t/(2\sqrt s)\). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.

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Proof of Theorem 10.3

For \(0<\alpha<1\), the Balakrishnan formula gives

and likewise for \(B\). Using

and the rectangular resolvent identity

the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.

For small \(t\),

because the two positive resolvent factors are contractions. For large \(t\), the resolvent identity gives the bound \({\left\lVert D\right\rVert}/t\). Splitting the integral at

when \(C\) and \(D\) are nonzero yields

The zero cases are immediate.

If \(A,B\ge mI\), then both resolvents have norm at most \((t+m)^{-1}\), so

After \(t=mu\), the beta integral equals

which gives the linear estimate.

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Proof of Corollary 10.4

Differentiate

and integrate from \(0\) to \(s\). Positive fractional semigroups are contractions; under the gap, the product of their two norm bounds is \(e^{-sm^\alpha}\).

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