C-admissibility and analytic \(C\)-semigroups. (English) Zbl 1236.47041
This paper gives a uniqueness result for the solution of the operator equation \(AX-XB=CD\) in the case of \(A\) being the generator of an analytic \(C\)-regularized semigroup in a Banach space \(F\) and \(B\) being a closed linear operator in \(F\) with some further properties. When \(C=I\), the result was proved by Q. P. Vu and E. Schüler in [J. Differ. Equations 145, No. 2, 394–419 (1998; Zbl 0918.34059)].
Reviewer: Jin Liang (Shanghai)
MSC:
| 47D60 | \(C\)-semigroups, regularized semigroups |
Citations:
Zbl 0918.34059References:
| [1] | Vu, Quoc-Phong; Schuler, E., The operator equation \(A X - X B = C\), admissibility, and asymptotic behavior of differential equations, J. Differential Equations, 145, 394-419 (1998) · Zbl 0918.34059 |
| [2] | Vu, Quoc-Phong, The operator equation \(A X - X B = C\) with unbounded operators \(A\) and \(B\) and related abstract Cauchy problems, Math. Z., 208, 567-588 (1991) · Zbl 0726.47029 |
| [3] | Miyadera, I.; Tanaka, N., A remark on exponentially bounded C-semigroups, Proc. Japan Acad. Ser. A, 66, 31-34 (1990) · Zbl 0731.47040 |
| [4] | DeLaubenfels, R., C-semigroups and strongly continuous semigroups, Israel J. Math., 81, 227-255 (1993) · Zbl 0803.47034 |
| [5] | Tanaka, N., Holomorphic C-semigroups and holomorphic semigroups, Semigroup Forum, 38, 253-261 (1989) · Zbl 0651.47028 |
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