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Zbl 0665.46058
Vasilescu, F.-H.
Spectral theory in quotient Fréchet spaces. I. (English)
[J] Rev. Roum. Math. Pures Appl. 32, 561-579 (1987). ISSN 0035-3965

Let E be a Fréchet space. A linear subspace $E\sb 0\subset E$ is said to be a Fréchet subspace of E if $E\sb 0$ has a Fréchet space structure which makes the inclusion $E\sb 0\subset E$ continuous. A quotient Fréchet space is a linear space of the form $E/E\sb 0$ where E is Fréchet space and $E\sb 0$ is a Fréchet subspace of E. A morphism of $E/E\sb 0$ into $F/F\sb 0$ is any linear map $u:E/E\sb 0\to F/F\sb 0$ whose graph is itself a quotient Fréchet space. The one-point compactification of the complex plane ${\bbfC}$, is denoted by ${\bbfC}\sb{\infty}$ and for any open subset $V\subset {\bbfC}\sb{\infty}$, Op(V) stands for the family of all open subsets of V. \par In this paper, the author considers some aspects of the spectral theory of morphisms in the category of quotient Fréchet spaces. In particular, he shows that if $E\ne E\sb 0$, then for every analytic sheaf morphism $u:\sb 0E/E\sb 0\to E/E\sb 0$, the spectrum of u is not empty. Here ${\cal O}\sb E$ denotes the analytic sheaf of germs of holomorphic E- valued function over ${\bbfC}\sb{\infty}$, and ${}\sb 0{\cal O}\sb E$ denotes the analytic subsheaf of ${\cal O}\sb E$ consisting of the stalks (${\cal O}\sb E)\sb z$ for $z\ne \infty$, and (${\cal O}\sb E)\sb{\infty}=(\sb 0{\cal O}\sb{\infty})({\cal O}\sb E)\sb{\infty}$. Moreover, if $u\in {\cal L}(E/E\sb 0)$ and $V\in Op({\bbfC}\sb{\infty})$ is a neighbourhood of the spectrum of u, then there is a unital algebra morphism from ${\cal O}(V)$ into ${\cal L}(E/E\sb 0)$ denoted by: $\phi$ $\to \phi (u)$ such that if $\phi\sb 0\in\sb 0{\cal O}(V)$, one has $\phi\sb 1(u)=u$ $\phi\sb 0(u)$, where $\phi\sb 1=\zeta \phi\sb 0$.
[J.Zafarani]

MSC 2000:: *46M20 Methods of algebraic topology in functional analysis
46A04 Locally convex Frechet spaces, etc.
47A10 Spectrum and resolvent of linear operators

Keywords: Fréchet space; Fréchet subspace; quotient Fréchet space; spectral theory of morphisms in the category of quotient Fréchet spaces; analytic sheaf morphism; analytic sheaf of germs of holomorphic E-valued function; stalks

Cited in: Zbl 0782.46005

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