Paper VI: Fractional functional calculus

E11 Fractional functional calculus

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Let \(H\) be a finite dimensional Hilbert space and let \(L=L^*\geq0\). Write its spectral resolution as

\[ L=\sum_j\lambda_jP_j. \]

For \(\alpha>0\), define

\[ L^\alpha = \sum_j\lambda_j^\alpha P_j \]

and

\[ T_\alpha(t) = {\mathrm e}^{-tL^\alpha} = \sum_j{\mathrm e}^{-t\lambda_j^\alpha}P_j. \]

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Theorem E11.1: Fractional packet calculus

For \(\alpha,\beta>0\),

\[ L^\alpha L^\beta=L^{\alpha+\beta}. \]

For \(s,t\geq0\),

\[ T_\alpha(s)T_\alpha(t)=T_\alpha(s+t). \]

Every spectral packet \(P_jH\) reduces \(L^\alpha\) and \(T_\alpha(t)\).

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Proof of Theorem E11.1

The claims hold on each spectral packet and the projectors are mutually orthogonal.

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Theorem E11.2: Balakrishnan formula in finite dimension

For \(0<\alpha<1\),

\[ L^\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}L(sI+L)^{-1} {\,\mathrm d} s. \]

The integral converges in operator norm.

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Proof of Theorem E11.2

It is enough to prove the formula on an eigenvector with eigenvalue \(x\geq0\). For \(x>0\), the substitution \(s=xu\) gives

\[ \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}\frac{x}{s+x}{\,\mathrm d} s =x^\alpha \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty\frac{u^{\alpha-1}}{1+u}{\,\mathrm d} u =x^\alpha. \]

The value at \(x=0\) is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in [Balakrishnan1960].

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Theorem E11.3: Exact fractional naturality

Let \(L_f\) and \(L_c\) be finite dimensional positive self adjoint operators and let \(C\) satisfy

\[ CL_f=L_cC. \]

Then, for every \(\alpha>0\) and \(t\geq0\),

\[ CL_f^\alpha=L_c^\alpha C \]

and

\[ C{\mathrm e}^{-tL_f^\alpha} = {\mathrm e}^{-tL_c^\alpha}C. \]

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Proof of Theorem E11.3

The union of the two finite spectra is a finite set. There is a polynomial that agrees with \(x^\alpha\) on that set, and another polynomial that agrees with \({\mathrm e}^{-tx^\alpha}\). Exact intertwining passes to every polynomial by induction.

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Theorem E11.4: Fractional defect bound

Let \(0<\alpha<1\). Suppose

\[ L_f\geq mI, \qquad L_c\geq mI \]

for some \(m>0\). Define

\[ \Delta_L=CL_f-L_cC. \]

Then

\[ {\left\lVert CL_f^\alpha-L_c^\alpha C\right\rVert} \leq \alpha m^{\alpha-1}{\left\lVert\Delta_L\right\rVert}. \]

Consequently,

\[ {\left\lVert C{\mathrm e}^{-tL_f^\alpha} - {\mathrm e}^{-tL_c^\alpha}C \right\rVert} \leq \alpha t m^{\alpha-1}{\left\lVert\Delta_L\right\rVert}. \]

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Proof of Theorem E11.4

Set

\[ R_f(s)=(sI+L_f)^{-1}, \qquad R_c(s)=(sI+L_c)^{-1}. \]

The resolvent identity gives

\[ CR_f(s)-R_c(s)C = -R_c(s)\Delta_LR_f(s). \]

Since \(LR=I-sR\),

\[ CL_fR_f(s)-L_cR_c(s)C = sR_c(s)\Delta_LR_f(s). \]

Insert this into Theorem E11.2. The spectral lower bound gives

\[ {\left\lVert R_f(s)\right\rVert},{\left\lVert R_c(s)\right\rVert} \leq \frac{1}{s+m}. \]

Therefore

\[ \begin{aligned} {\left\lVert CL_f^\alpha-L_c^\alpha C\right\rVert} &\leq \frac{\sin(\pi\alpha)}{\pi} {\left\lVert\Delta_L\right\rVert} \int_0^\infty \frac{s^\alpha}{(s+m)^2}{\,\mathrm d} s\\ &= \alpha m^{\alpha-1}{\left\lVert\Delta_L\right\rVert}. \end{aligned} \]

The semigroup estimate follows from Theorem E3.1, applied to the contraction generators \(-L_f^\alpha\) and \(-L_c^\alpha\).

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Remark E11.5

The factor \(m^{\alpha-1}\) records the singular behavior of \(x^\alpha\) near zero. Without a common positive lower bound, a uniform Lipschitz estimate of this form cannot be expected.

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Standard fact E11.6: Stable subordination

Let \(0<\alpha<1\) and \(t>0\). There is a probability measure \(\eta_{\alpha,t}\) on \([0,\infty)\) such that

\[ \int_0^\infty {\mathrm e}^{-s\lambda}\eta_{\alpha,t}({\,\mathrm d} s) = {\mathrm e}^{-t\lambda^\alpha} \]

for every \(\lambda\geq0\). Consequently, for a finite dimensional positive self adjoint operator,

\[ {\mathrm e}^{-tL^\alpha} = \int_0^\infty {\mathrm e}^{-sL}\eta_{\alpha,t}({\,\mathrm d} s) \]

in operator norm. The measures satisfy the convolution law

\[ \eta_{\alpha,s}*\eta_{\alpha,t} = \eta_{\alpha,s+t}. \]

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The existence of \(\eta_{\alpha,t}\) follows from complete monotonicity of \({\mathrm e}^{-t\lambda^\alpha}\). The operator identity is obtained by applying the scalar Laplace transform on every spectral packet. This is the finite spectral form of Bochner subordination. See [SchillingSongVondracek2012].

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Theorem E11.7: Poisson subordination

For \(t>0\),

\[ {\mathrm e}^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2} {\mathrm e}^{-t^2/(4s)} {\mathrm e}^{-sL} {\,\mathrm d} s. \]

The integral converges in operator norm.

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Proof of Theorem E11.7

On a spectral packet with eigenvalue \(\lambda\), the formula reduces to

\[ {\mathrm e}^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}{\mathrm e}^{-t^2/(4s)-\lambda s}{\,\mathrm d} s. \]

This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.

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Fractional semigroup multipliers at time one. Smaller exponents preserve more of the high spectral range.

Remark E11.8

For sectorial and nonnormal operators, fractional powers require a resolvent or contour calculus together with domain control. The finite self adjoint theory proved here does not remove those analytic requirements. A broader account appears in [SchillingSongVondracek2012].

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