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

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- by Masaaki Tsuchiya
- Proc. Amer. Math. Soc.
**90**(1984), 91-94 - DOI: https://doi.org/10.1090/S0002-9939-1984-0722422-2
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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

- S. N. Ethier,
*Differentiability-preserving properties of Markov semigroups associated with one-dimensional diffusions*, Z. Wahrsch. Verw. Gebiete**45**(1978), no. 3, 225–238. MR**510027**, DOI 10.1007/BF00535304
M. Hukuhara, - Hikosaburo Komatsu,
*On the regularity of hyperfunction solutions of linear ordinary differential equations with real analytic coefficients*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**20**(1973), 107–119. MR**328584**
L. M. Milne-Thomson, - Oskar Perron,
*Über Summengleichungen und Poincarésche Differenzengleichungen*, Math. Ann.**84**(1921), no. 1-2, 1–15 (German). MR**1512016**, DOI 10.1007/BF01458689

*Ordinary differential equations*, 2nd ed., Iwanami, Tokyo, 1980. (Japanese)

*The calculus of finite differences*, 2nd. ed., Chelsea, New York, 1981.

## Bibliographic Information

- © Copyright 1984 American Mathematical Society
- 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