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Analyticity preserving properties of resolvents for degenerate diffusion operators in one dimension

Author: Masaaki Tsuchiya
Journal: Proc. Amer. Math. Soc. 90 (1984), 91-94
MSC: Primary 47D05; Secondary 26E05, 34A25, 35A99, 60J60
MathSciNet review: 722422
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Abstract: Let $ L = a(x)({d^2}/d{x^2}) + b(x)(d/dx) + c(x)$ be a diffusion operator on a compact interval $ I = [{r_0},{r_1}]$ investigated by S. N. Ethier. Here, assume that $ a$, $ b$ and $ c$ are real analytic functions on $ I$, $ a(x) > 0$ for $ x \in ({r_0},{r_1})$, $ a({r_i}) = 0 \leqslant {( - 1)^i}b({r_i})(i = 0,1)$, and both $ {r_i}(i = 0,1)$ are simple zeros of $ a(x)$. It is shown that the resolvent $ \left\{ {{G_\lambda }} \right\}$ for $ L$ has the analyticity preserving property for sufficiently large $ \lambda $, so that the equation $ (L - \lambda )u = f$ is solvable in the space of real analytic functions on $ I$. Some examples are given to show that the condition on $ L$ is best possible.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1984 American Mathematical Society

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