Linear clutching splits into spectral subbundles

For $0\le t<1$, the fractional-linear map $z\mapsto (z+t)\mathrm{/}(1+tz)$ carries $S^1$ to itself, and $1+tz\mathrm{\ne }0$ there. Therefore $H_t(z)=(1+tz)(a(z+t)\mathrm{/}(1+tz)+b)=(a+tb)z+ta+b$ is a homotopy through linear clutching automorphisms from $az+b$. At $t=1$, the coefficient $a+b$ is the original automorphism evaluated at $z=1$. Openness of bundle automorphisms and compactness of $X$ give $t_0<1$ for which $C=a+t_{0}b$ is invertible on every fiber.

Right multiplication of $H_{t_0}$ by the constant automorphism $C^{-1}$ does not change the glued bundle by [F1]. Since scalar $z$ commutes with $C$, it gives $zI+B$, where $B=(t_{0}a+b)C^{-1}$. Put $A=-B$. Then $zI-A$ is invertible on $S^1$, so $A_x$ has no eigenvalue of modulus one.

For one fiber $V$, factor the characteristic polynomial of $A$ as $q=q_{>}{q}_{<}$, with the roots of the monic factors respectively outside and inside $S^1$. Bézout polynomials for the relatively prime factors and Cayley–Hamilton give $V_{>}=\ker q_{>}(A)=\mathrm{im}{q}_{<}(A)$, $V_{<}=\ker q_{<}(A)=\mathrm{im}{q}_{>}(A)$, and $V=V_{>}\oplus V_{<}$. Both spaces are $A$-invariant and have precisely the indicated generalized eigenvalues. This also proves uniqueness: any invariant splitting with the same spectral locations is annihilated by the corresponding factor and therefore equals these kernels.

These fiber splittings vary continuously. Around each root cluster choose a small circle disjoint from all roots. For a sufficiently small change of the polynomial, the straight-line change stays nonzero on each circle, so [F2] preserves the winding number of $q\mathrm{/}\mathrm{\mid }q\mathrm{\mid }$. Factoring $q$ into linear factors shows this winding is exactly the number of enclosed roots counted with multiplicity. Thus the inside and outside monic factors vary continuously in their coefficients. In a local frame choose vectors whose images under $q_{<}(A)$ and $q_{>}(A)$ form the bases in step 3.1; the same determinant remains nonzero nearby. Their images therefore give local frames for $E_{>}$ and $E_{<}$, proving that the fiberwise spaces are complementary subbundles.

On $E_{>}$, the family $tzI-A{\mathrm{\mid }}_{E_{>}}$ is invertible for $0\le t\le 1$, since every eigenvalue of $A{\mathrm{\mid }}_{E_{>}}$ has modulus greater than one; it deforms $zI-A$ to the constant $-A$, which [F1] identifies with $I$. On $E_{<}$, $zI-tA{\mathrm{\mid }}_{E_{<}}$ stays invertible because all eigenvalues have modulus less than one; it deforms $zI-A$ to $zI$. Hence [F1] gives $[E,zI-A]\cong [E_{>},I]\oplus [E_{<},z]$.

Applying [F3] and the convention [F4] to step 5.1 gives the stated $K$-class, hence a combination of $1$ and $\beta$. For a block direct sum, the characteristic polynomial factors and the unique inside/outside invariant splitting in step 3.1 is the direct sum of the individual splittings, so the construction is additive. Rank zero gives two zero subbundles and the same formula.