Factorization of Operator Functions in a Hilbert Space

A. M. Gomilko

Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 1, pp. 19-24.

Résumé

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 $|\lambda|\le1$.

Détail

Keywords: Hilbert space, convex hull of the spectrum of operator, index of operator function, factorization of operator functions.

Bibliographie

[1] Markus A. S., Matsaev V. I., “K spektralnoi teorii golomorfnykh operator-funktsii v gilbertovom prostranstve”, Funkts. analiz i ego pril., 9:1 (1975), 76–77 | MR | Zbl

[2] Markus A. S., Matsaev V. I., “O spektralnykh svoistvakh golomorfnykh operator-funktsii v gilbertovom prostranstve”, Matem. issledovaniya, 9:4 (1974), 79–90 | MR

[3] Khalmosh P., Gilbertovo prostranstvo v zadachakh, Mir, M., 1970 | MR | Zbl

[4] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970 | MR

[5] Gomilko A. M., Radzievskii G. V., “O chislovykh obrazakh semeistva kommutiruyuschikh operatorov”, Matem. zametki, 62:5 (1997), 787–791 | DOI | MR | Zbl

[6] Krasnoselskii M. A., Vektornye polya na ploskosti, GIFML, M., 1963

[7] Borisovich Yu. G., Gelman B. D., Myshkis A. D., Obukhovskii V. V., “Topologicheskie metody v teorii nepodvizhnykh tochek mnogoznachnykh otobrazhenii”, UMN, 35:1 (1980), 59–126 | MR | Zbl

[8] Gohberg I., Laiterer Yu., “The factorization of operator-functions relative to a contour. II: The canonical factorization of operator-functions that are close to the identity operator”, Math. Nachr., 54 (1972), 41–74 | DOI | MR