Factorization of Operator Functions in a Hilbert Space

Let $H$ be a Hilbert space, $L=L(H)$ the algebra of bounded linear operators in $H$, $I$ the identity operator, and $H_\alpha^{+}(\Gamma,L)$ the algebra of operator functions defined on the circle $\Gamma=\{|\zeta|=1\}$, satisfying the Hölder condition with exponent $\alpha\in (0,1)$, ranging in $L$, and admitting holomorphic continuation to the disk $|\lambda|<1$. We show that if $A(\zeta)\in H_\alpha^{+}(\Gamma,L)$ and if, for any $\zeta\in\Gamma$, the point $z=0$ does not belong to the convex hull of the spectrum of $A(\zeta)$, then the factorization
\begin{gather*} A(\lambda)=A_{1,+}(\lambda)(\lambda^k I+\sum_{n=0}^{k-1}\lambda^n B_n) A_{2,+}(\lambda),\qquad|\lambda|\le1,\\ A_{j,+}(\lambda)\in H^{+}_\alpha(\Gamma, L),\quad j=1,2, \quad B_n\in L, \quad k=\operatorname{ind}_\Gamma\!A(\zeta), \end{gather*}
holds, where the operators $A_{j,+}(\lambda)$ are invertible for $|\lambda|\le1$.