An operator-valued transference principle and maximal regularity on vector-valued \(L_p\)-spaces (Q2708554)

Consider the space NEWLINE\[NEWLINEM(\mathbb{R})= \biggl\{m\in C^1\bigl( \mathbb{R} \setminus \{0\}:B (X)\bigr): M(\rho)(i \lambda-A)^{-1}= (i\lambda-A)^{- 1}M (\rho)\lambda, \rho\in \mathbb{R}\biggr\}NEWLINE\]NEWLINE and set NEWLINE\[NEWLINERM(\mathbb{R})= \biggl\{m\in M: \bigl\{m(\rho): \rho\neq 0\bigr\} \quad\text{and}\quad \bigl\{\rho m'(\rho): \rho\neq 0\bigr\} \text{ are }R\text{-bounded}\biggr\}.NEWLINE\]NEWLINE Then \(RM(\mathbb{R})\) forms a Banach algebra equipped with the norm NEWLINE\[NEWLINE\|m\|_{RM}= R\biggl(\bigl\{ M( \rho): \rho\neq 0\bigr\} \biggr)+R \biggl(\bigl\{ \rho m'(\rho):\rho\neq 0\bigr\} \biggr),NEWLINE\]NEWLINE where \(R(T)\) is \(R\)-bound.NEWLINENEWLINENEWLINETheorem. Let \(X\) be a \(UMD\) space. Then Phillips calculus extends to the Banach algebra \(RM(\mathbb{R})\) such that the defined homomorphism \(\psi:R M(\mathbb{R})\to B(X)\). There is a constant \(C>0\) such that \(\|M(A)\|_{B(X)}\leq C\|m\|_{RM}\), \(m\in RM(\mathbb{R})\), where \(M(A)_x= \int_Re^{-At} L(t)(1+A)^4 A^{-2}xdt\), and \(A\) is the generator of the \(C\)-semigroup.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].