论文标题
希尔伯特空间上有界的$ C_0 $ -Semigroup的功能演算
Functional calculus for a bounded $C_0$-semigroup on Hilbert space
论文作者
论文摘要
我们在$ {\ Mathbb c} _+= \ = \ {z \ in {\ Mathbb c} \，： $ {\ mathbb r} $上的figa-talamenca-herz代数。然后，我们证明，Hilbert Space上的任何有限的$ C_0 $ -Semigroup的负发电机$ a $ a $ h $ a $ h $允许一个有限的（天然）功能性计算$ρ_a\ colon {\ mathcal a}（{\ Mathbb c} _+）\ to b（h）$。我们证明，这是对Batty-gomilko-Tomilov最近在某些BATTY-GOMILKO-TOMILOV上设计的有限功能性计算$ {\ MATHCAL B}（{\ MATHBB C} _+）\ to B（H）$的$ C} _+$，通过显示$ {\ Mathcal B}（{\ Mathbb C} _+）\ subset {\ Mathcal a}（{\ Mathbb C} _+）$和$ {\ Mathcal B}（\ Mathcal B}（ C} _+）$。在Banach空间设置中，我们为负面的$ C_0 $ -Semigroups提供了类似的结果。 $ {\ Mathcal a}（{\ Mathbb c} _+）$的研究需要与Hardy Space上的傅立叶乘数$ h^1（{\ Mathbb r}）\ subset l^1（{\ subset l^1（{\ \ Mathbb r}）$的分析功能的$。
We introduce a new Banach algebra ${\mathcal A}({\mathbb C}_+)$ of bounded analytic functions on ${\mathbb C}_+=\{z\in{\mathbb C}\, :\, {\rm Re}(z)>0\}$ which is an analytic version of the Figa-Talamenca-Herz algebras on ${\mathbb R}$. Then we prove that the negative generator $A$ of any bounded $C_0$-semigroup on Hilbert space $H$ admits a bounded (natural) functional calculus $ρ_A\colon {\mathcal A}({\mathbb C}_+)\to B(H)$. We prove that this is an improvement of the bounded functional calculus ${\mathcal B}({\mathbb C}_+)\to B(H)$ recently devised by Batty-Gomilko-Tomilov on a certain Besov algebra ${\mathcal B}({\mathbb C}_+)$ of analytic functions on ${\mathbb C}_+$, by showing that ${\mathcal B}({\mathbb C}_+)\subset {\mathcal A}({\mathbb C}_+)$ and ${\mathcal B}({\mathbb C}_+)\not= {\mathcal A}({\mathbb C}_+)$. In the Banach space setting, we give similar results for negative generators of $γ$-bounded $C_0$-semigroups. The study of ${\mathcal A}({\mathbb C}_+)$ requires to deal with Fourier multipliers on the Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ of analytic functions.
