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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-1984-0722422-2
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