# Article

Full entry | PDF   (0.2 MB)
Keywords:
reflexive Banach space; $L_1$-space of vector-valued functions; closed operator; resolvent set; range and domain; linear contraction; $C_0$-semigroup; strongly continuous cosine family of operators
Summary:
Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R(A)$ of $A$ as follows. Let $0\neq \lambda_{n}\in \rho(A)$ for all $1\leq n < \infty$, where $\rho(A)$ denotes the resolvent set of $A$, and assume that $\lim_{n\rightarrow \infty} \lambda_{n}=0$ and $\sup_{n\geq 1} \|\lambda_{n}(\lambda_{n}-A)^{-1}\| < \infty$. Furthermore, assume that there exists $\lambda_{\infty}\in \rho(A)$ such that $\|\lambda_{\infty}(\lambda_{\infty}-A)^{-1}\|\leq 1$. Then $f\in R(A)$ is equivalent to $\sup_{n\geq 1} \|(\lambda_{n}-A)^{-1}f\|_{1}<\infty$. This generalizes Shaw's result for scalar-valued functions.
References:
[1] Assani I.: A note on the equation $y=(I-T)x$ in $L^{1}$. Illinois J. Math. 43 (1999), 540-541. MR 1700608
[2] Browder F.E.: On the iteration of transformations in non-compact minimal dynamical systems. Proc. Amer. Math. Soc. 9 (1958), 773-780. MR 0096975 | Zbl 0092.12602
[3] Diestel J., Uhr J.J., Jr.: Vector Measures. Amer. Math. Soc., Providence, 1977. MR 0453964
[4] Fonf V., Lin M., Rubinov A.: On the uniform ergodic theorem in Banach spaces that do not contain duals. Studia Math. 121 (1996), 67-85. MR 1414895 | Zbl 0861.47006
[5] Gottschalk W.H., Hedlund G.A.: Topological Dynamics. Amer. Math. Soc. Colloq. Publ. 36, Amer. Math. Soc., Providence, 1955. MR 0074810 | Zbl 0067.15204
[6] Krengel U., Lin M.: On the range of the generator of a Markovian semigroup. Math. Z. 185 (1984), 553-565. MR 0733775 | Zbl 0525.60080
[7] Li Y.-C., Sato R., Shaw S.-Y.: Boundedness and growth orders of means of discrete and continuous semigroups of operators. preprint. MR 2410881 | Zbl 1151.47048
[8] Lin M., Sine R.: Ergodic theory and the functional equation $(I-T)x=y$. J. Operator Theory 10 (1983), 153-166. MR 0715565 | Zbl 0553.47006
[9] Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983. MR 0710486 | Zbl 0516.47023
[10] Sato R.: Solvability of the functional equation $f=(T-I)h$ for vector-valued functions. Colloq. Math. 99 (2004), 253-265. MR 2079330 | Zbl 1072.47010
[11] Shaw S.-Y.: On the range of a closed operator. J. Operator Theory 22 (1989), 157-163. MR 1026079 | Zbl 0703.47003
[12] Shaw S.-Y., Li Y.-C.: On solvability of $Ax=y$, approximate solutions, and uniform ergodicity. Rend. Circ. Mat. Palermo (2) Suppl. 2002, no. 68, part II, 805-819. MR 1975488 | Zbl 1050.47013
[13] Sova M.: Cosine operator functions. Rozprawy Mat. 49 (1966), 1-47. MR 0193525 | Zbl 0156.15404
[14] Travis C.C., Webb G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar. 32 (1978), 75-96. MR 0499581 | Zbl 0388.34039
[15] Zygmund A.: Trigonometric Series. Vol. I, Cambridge University Press, Cambridge, 1959. MR 0107776 | Zbl 1084.42003

Partner of