Abstract
We prove that a Kreiss bounded \(C_0\) semigroup \((T_t)_{t \ge 0}\) on a Hilbert space has asymptotics \(\left\| T_t\right\| = {\mathcal{O}}\big (t/\sqrt{\log (t)}\big ).\) Then, we give an application to perturbed wave equation.
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1 Introduction
Let \(-A\) be the generator of a \(C_0\)-semigroup \((T_t)_{t\ge 0}\) on a complex Banach space X. For \(\alpha > 0,\) we say that \((T_t)_{t\ge 0}\) is \(\alpha\)-Kreiss bounded (or simply Kreiss bounded when \(\alpha = 1\)) if \(\sigma (A) \subset \overline{{\mathbb{C}}_+} = \{ \lambda \in {\mathbb{C}}: \, {\text{Re}}(\lambda ) \ge 0\}\) and
According to [7, (2.6)] (see also [12, Theorem 1.1.(1)] and [8]) , if \((T_t)_{t\ge 0}\) is an \(\alpha\)-Kreiss bounded \(C_0\)-semigroup on a Hilbert space for \(\alpha >1,\) then \(\left\| T_t\right\| = O(t^{\alpha }).\) In the following, we do not need to consider the case \(0< \alpha < 1.\) Indeed when \(0< \alpha < 1\) and \((T_t)_{t\ge 0}\) is \(\alpha\)-Kreiss bounded, then it is exponentially stable (see [4, Remark 1.22 page 90]). Let us fix \(\gamma \in (0,1).\) In [5], the authors give an example (see [5, Example 4.4.]) of a Kreiss bounded \(C_0\)-semigroup for which there exists a constant \(C>0\) such that \(\left\| T_t\right\| \ge Ct^{\gamma }\) for \(t\ge 1.\) A natural question (listed as an open question in [11, Appendix B]) is whether the estimate \(\left\| T_t\right\| = O(t^{\alpha }),\) for an \(\alpha\)-Kreiss bounded \(C_0\)-semigroup \((T_t)_{t\ge 0}\) on a Hilbert space, is sharp. In the next section we answer this question for \(\alpha =1.\)
2 Main results
Theorem 2.1
Let \(-A\) be the generator of a Kreiss bounded \(C_0\)-semigroup \((T_t)_{t\ge 0 }\) on a Hilbert space. Then
The proof of Theorem 2.1 is based on techniques given in [3]. We first state and prove two lemmas. It is known (see [6] or [13]) that a \(C_0\)-semigroup \((T_t)_{t\ge 0}\) on a Hilbert space, with negative generator A, is bounded if and only if
and
The following lemma gives an analogous necessary condition for a \(C_0\)-semigroup to be \(\alpha\)-Kreiss bounded, for \(\alpha >0.\)
Lemma 2.2
Let \(\alpha >0\) and \((T_t)_{t\ge 0 }\) be a \(C_0\)-semigroup on a Hilbert space with negative generator A. Assume \((T_t)_{t\ge 0}\) is \(\alpha\)-Kreiss bounded then
and
Proof
Let \(x \in H.\) First, by [9, formula (7.1) section 1.7], we have
This means that for \(r>0,\)
where \({\mathcal{F}} : L^2({\mathbb{R}},H) \mapsto L^2({\mathbb{R}},H)\) is the Fourier–Plancherel operator. By the Fourier–Plancherel Theorem
The Gearhart–Prüss Theorem (see [1, Theorem 5.2.1.]) says that for each \(r>0,\) \(\underset{t>0}{\sup }\left\| {\mathrm{e}}^{-rt}T_t\right\| < \infty .\) This implies that there exists \(K>0\) such that
and then
Now, let \(0<r<1\) and \(a = r+1>1.\) By the resolvent identity, we have, for \(\beta \in {\mathbb{R}},\)
In the same way we obtain (2.3). \(\square\)
The next lemma says that if \((T_t)_{t\ge 0}\) and \((T^*_t)_{t\ge 0}\) satisfy a kind of Cesàro condition then \(\left\| T_t\right\|\) satisfies (2.1).
Lemma 2.3
Let \((T_t)_{t\ge 0}\) be a \(C_0\)-semigroup on a Banach space X. Assume that there exists a \(C>0\) such that for each \(t>1,\)
and
Then, \(\left\| T_t\right\| = {\mathcal{O}}(t/\sqrt{\log (t)}).\)
Proof
Let \(t >2\) and \(1<P<Q<t.\) We have
Finally taking the supremum over \(\{\left\| x^*\right\| =1 \}\) one obtains,
Now we set \(L:= \left\lfloor \log (t)/\log (2) \right\rfloor .\) For \(0 \le l \le L-1,\) \(Q=2^{l+1}\) and \(P= 2^{l}\) we have
Therefore
Hence for \(t>2,\)
\(\square\)
We are now ready to prove Theorem 2.1.
Proof of Theorem 2.1
Let \(x\in H\) and \(r>0.\) By (2.4), (2.5) and Lemma 2.2,
Furthermore
Taking \(t = \frac{1}{r}\) we obtain
Hence, for \(t>1,\)
where \(C'=4{\mathrm{e}}^2C.\) Similarly, since \((T_t^*)_{t\ge 0}\) is a Kreiss bounded \(C_0\)-semigroup on a Hilbert space, for \(t>1\) and \(x^*\in H,\) we have
Finally, Lemma 2.3 allows us to conclude that \(\left\| T_t\right\| = {\mathcal{O}}(t/\sqrt{\log (t)}).\) \(\square\)
Remark 2.4
Let \(\alpha >1\) and \((T_t)_{t\ge 0}\) be an \(\alpha\)-Kreiss bounded \(C_0\)-semigroup. The computations which lead to (2.7) with \(Q=2\) and \(P=1\) give
Then using Lemma 2.2 and following the proof of Theorem 2.1 we can state \(\left\| T_t\right\| = O(t^{\alpha }).\) This gives another proof of [12, Theorem 1.1.(1) with \(g(s) = s^{\alpha }\) for \(\alpha >1\)].
Question 2.5
In view of [2], we can naturally ask whether the estimate (1.1) can be improved. In the aforementioned paper, the authors show that if \((T_t)_{t\ge 0}\) is a positive Kreiss bounded \(C_0\)-semigroup on a \(L^1\)-space, then there exists \(0<\epsilon <1\) such that \(\left\| T_t\right\| = {\mathcal{O}}(t^{1-\epsilon }).\) Do we have same estimate on Hilbert spaces?
Let us give an application of the Theorem 2.1. Let \(W^{2,s}(\mathbb {T}^2)\) be the second order Sobolev space equipped with the standard norm. Let A be the operator with domain \(D(A) = W^{2,2}(\mathbb {T}^2) \times W^{2,1}(\mathbb {T}^2)\) defined by
where \(\Delta\) is the Laplacian with \(D(\Delta ) = W^{2,2}(\mathbb {T}^2)\) and \(M :L^2(\mathbb {T}^2) \rightarrow L^2(\mathbb {T}^2)\) is the multiplication operator given by \(M(h)(x,y) = {\mathrm{e}}^{iy}h(x,y).\) Then, it is known (see [12, section 4] or [10] for details) that \(-A\) generates a \(C_0\)-group \((T_t)_{t\in {\mathbb{R}}}\) on the Hilbert space \(H = W^{1,2}(\mathbb {T}^2)\times L^2(\mathbb {T}^2),\) satisfying \(\left\| T_t\right\| = \underset{t \rightarrow \pm \infty }{{\mathcal{O}}}(|t|{\mathrm{e}}^{|t|/2}).\) With Theorem 2.1, we are able to improve this estimate:
Proposition 2.6
Let \((T_t)_{t \in {\mathbb{R}}}\) be the \(C_0\)-group on H generated by \(-A.\) Then
Proof
According to [12, Lemma 4.4.], for \(\lambda \in {\mathbb{C}}_+,\) with \({\text{Re}}(\lambda ) \in (0,1),\)
Then, according to Remark 2.4(2), \(({\mathrm{e}}^{-\frac{t}{2}}T_t)_{t\ge 0}\) (resp. \(({\mathrm{e}}^{-\frac{t}{2}}T_{-t})_{t\ge 0}\) ) which is generated by \(-(A+\frac{1}{2})\) (resp. \(A+\frac{1}{2}\)), satisfies \(\left\| {\mathrm{e}}^{-1/2t}T_t\right\| = {\mathcal{O}}(t/\sqrt{\log (t)})\) (resp. \(\left\| {\mathrm{e}}^{-1/2t}T_{-t}\right\| = {\mathcal{O}}(t/\sqrt{\log (t)})\)). This yields the estimate (2.8). \(\square\)
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Acknowledgements
This work was supported by the ERC grant Rigidity of groups and higher index theory under the European Union’s Horizon 2020 research and innovation program (Grant agreement no. 677120-INDEX.
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Communicated by Mark Veraar.
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Arnold, L. Behavior of Kreiss bounded \(C_0\)-semigroups on a Hilbert space. Adv. Oper. Theory 7, 62 (2022). https://doi.org/10.1007/s43036-022-00223-z
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DOI: https://doi.org/10.1007/s43036-022-00223-z