Geodesic
Functional models of non-selfadjoint operators, strongly continuous semigroups, and matrix Muckenhoupt weights
Izvestiya. Mathematics , Tome 75 (2011) no. 2, pp. 287-346

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We consider unbounded continuously invertible operators $A$, $A_0$ on a Hilbert space $\mathfrak{H}$ such that the operator $A^{-1}-A^{-1}_0$ has finite rank. Assuming that $\sigma(A_0)=\varnothing$ and the semigroup $V_+(t):=\exp\{iA_0t\}$, $t\geqslant0$, is of class $C_0$, we state criteria under which the semigroups $U_\pm(t):=\exp\{\pm iAt\}$, $t\geqslant0$, are also of class $C_0$. We give applications to the theory of mean-periodic functions. The investigation is based on functional models of non-selfadjoint operators and on the technique of matrix Muckenhoupt weights.
Keywords: $C_0$-semigroups, functional models of non-selfadjoint operators, matrix Muckenhoupt weights, Hilbert spaces of entire functions. 
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     author = {G. M. Gubreev and Yu. D. Latushkin},
     title = {Functional models of non-selfadjoint operators, strongly continuous semigroups, and matrix {Muckenhoupt} weights},
     journal = {Izvestiya. Mathematics },
     pages = {287--346},
     publisher = {mathdoc},
     volume = {75},
     number = {2},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_2_a3/}
}
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G. M. Gubreev; Yu. D. Latushkin. Functional models of non-selfadjoint operators, strongly continuous semigroups, and matrix Muckenhoupt weights. Izvestiya. Mathematics , Tome 75 (2011) no. 2, pp. 287-346. http://geodesic.mathdoc.fr/item/IM2_2011_75_2_a3/