相关论文: A remark on the growth rate of operator semigroups…
We study rates of growth of $\|AT(t)\|$ as $t \downarrow 0$ for an immediately differentiable $C_0$-semigroup $(T(t))_{t \geq 0}$ with generator $A$. We assume that the resolvent of the semigroup generator decays on the imaginary axis at…
We investigate rates of decay for $C_0$-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to…
We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the…
We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm…
We study decay rates for bounded $C_0$-semigroups from the perspective of $L^p$-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the…
We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show…
We prove lower bounds on $||T_t||$, where $T_t$ is a one-parameter semigroup, starting from information on the resolvent norms, i.e. the pseudospectra. We provide a physically important example in which the growth of the semigroup norm…
For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators,…
The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator $T$ on a Banach space we have $||T^n(I-T)\|\to0$ if and only if…
The purpose of this paper is to revisit the proof of the Gearhart-Pr\"uss-Huang-Greiner theorem for a semigroup $S(t)$, following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the…
We study rates of decay for $C_0$-semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with $\vert s\vert^{\beta}\log(\vert s\vert)^b$, $\beta, b \geq 0$, as $\vert…
The purpose of this note is to revisit the proof of the Gearhardt-Pr\"uss-Hwang-Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the norm…
Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some…
The conversion of resolvent conditions into semigroup estimates is crucial in the stability analysis of hyperbolic partial differential equations. For two families of multiple Toeplitz operators, we relate the power bound with a resolvent…
The purpose of this paper is to revisit the proof of the Gearhardt-Pr\"uss-Hwang-Greiner theorem for a semigroup $S(t)$, following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on $\Vert…
We give an elementary proof of weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit…
A regular operator T on a Hilbert C^*-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I+T^*T) is dense. Semiregular operators are a slightly larger class of operators that may…
We stretch the spectral bound equal growth bound condition along with a generalized Lyapunov stability theorem, known to hold for $C_0$-semigroups of normal operators on complex Hilbert spaces, to $C_0$-semigroups of scalar type spectral…
We give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the…
In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a…