CRITERIA FOR ANALYTICITY OF

MULTIDIMENSIONALLY

SUBORDINATE SEMIGROUPS

A. R. Mirotin<EMAIL_ADDRESS>
Dedicated to the memory of Yu. I. Lyubich

Key words and phrases: strongly continuous semigroup, holomorphic semigroup,
subordination, Bochner-Phillips functional calculus, Bernstein function.

Let $\psi$ be a Bernstein function in one variable. A. Carasso and T. Kato
obtained necessary and sufficient conditions for $\psi$ to have a property
that $\psi(A)$ generates a quasibounded holomorphic semigroup for every
generator $A$ of a bounded $C_{0}$-semigroup in a Banach space and deduced
necessary conditions as well. We generalize their results to the
multidimensional case and also give sufficient conditions for the property
mentioned above.

##  1\. Introduction

The celebrated theorem of Yosida [12] states that a fractional power
$-(-A)^{\alpha},0<\alpha<1$ of a generator $A$ of a bounded $C_{0}$-semigroup
on a Banach space $X$ is a genetrator of a holomorphic semigroup on $X$. A.
Carasso and T. Kato [10, Theorem 4] obtained a description of such (non-
negative) Bernstein functions $\psi$ in one variable that for every generator
$A$ of a bounded $C_{0}$-semigroup on a Banach space $X$ the operator
$-\psi(-A)$ is a generator of a quasibounded holomorphic semigroup on $X$. The
alternative criteria and sufficient conditions were obtained in [17]. Another
sufficient conditions were obtained in [13].

The present paper is devoted to some generalizations and analogs of A. Carasso
and T. Kato’s results in terms of multidimensional Bochner-Phillips calculus.
111The multidimensional Bochner-Phillips calculus was developed in [20], [21],
[2], [19], [24], [15], [16]. These results provide a new method of
constructing a large class of holomorphic semigroups.

##  2\. Preliminaries

We shall denote o by ${\mathscr{M}}(\mathbb{R}_{+}^{n})$
(${\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$,
${\mathscr{M}}^{1}(\mathbb{R}_{+}^{n})$) the convolution algebra of all
positive (respectively bounded positive, probability) measures on
$\mathbb{R}_{+}^{n}$. Throughout, $X$ stands for a complex Banach space,
$B(X)$ denotes the algebra of linear bounded operators on $X$, and $I$, the
identity operator on $X$.

######  Definition 2.1.

A family of operators $T(t))_{t\in\mathbb{R}_{+}}\subset B(X)$ is called a
$C_{0}$-semigroup if

(i) $T(0)=I$ and $T(t_{1}+t_{2})=T(t_{1})T(t_{2})$ for all
$t_{1},t_{2}\in\mathbb{R}_{+}$.

(ii) The map $t\mapsto T(t)x$ is continuous on $\mathbb{R}_{+}\forall x\in X$.

If, in addition,

(iii) $T(t)$ is bounded on $\mathbb{R}_{+}$,

we call $T(t))_{t\in\mathbb{R}_{+}}$ a bounded $C_{0}$\- semigroup.

In the following we denote by $C_{0}(X)$ the set of bounded strongly
continuous semigroups in a complex Banach space $X$. Further,
$T_{1},\dots,T_{n}$ will denote pairwise commuting semigroups from $C_{0}(X)$
with generators $A_{1},\dots,A_{n}$ respectively satisfying the condition
$\|T_{j}(t)\|_{B(X)}\leqslant M_{j}\quad(t\in\mathbb{R}_{+}),M_{j}={\rm
const}$. We put also $M:=\prod_{j=1}^{n}M_{j}$. We denote the domain of
$A_{j}$ by $D(A_{j})$ and set $A=(A_{1},\dots,A_{n})$. By the commutation of
operators $A_{1},\dots,A_{n}$ we mean the commutation of the corresponding
semigroups. By ${\rm Gen}(X)$ we denote the set of all generators of uniformly
bounded $C_{0}$-semigroups on $X$ and by ${\rm Gen}(X)^{n}$, the set of all
$n$-tuples $(A_{1},\dots,A_{n})$ where $A_{j}\in{\rm Gen}(X)$ commute. An
$B(X)$-valued function $T(u)=T_{A}(u):=T_{1}(u_{1})\dots T_{n}(u_{n})$
$(u\in\mathbb{R}_{+}^{n})$ is a bounded $n$-parameter $C_{0}$ semigroup with
$\|T(u)\|_{B(X)}\leqslant M$ ($u\in\mathbb{R}_{+}^{n}$). The linear manifold
$D(A):=\cap_{j=1}^{n}D(A_{j})$ is dense in $X$ [14, Sec. 10.10].

We denote by $\mathscr{L}\nu$ the $n$-dimensional Laplace transform

$\mathscr{L}\nu(z)=\int\limits_{\mathbb{R}_{+}^{n}}e^{-z\cdot u}d\nu(u)$ (2.1)

of a measure $\nu\in{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$. In this case we
put also for bounded $n$-parameter $C_{0}$-semigroup $T=T_{A}$ and a function
$g(s)=\mathscr{L}\nu(-s)$

$g(A):=\langle\nu,T_{A}\rangle:=\int\limits_{\mathbb{R}_{+}^{n}}T_{A}(u)d\nu(u)$
(2.2)

(the Bochner integral). Then $\|g(A)\|_{B(X)}\leqslant
M\|\nu\|_{{\mathscr{M}}^{b}}$ if $\|T_{A}(u)\|\leqslant M$.

######  Definition 2.2.

[5] We say that a function $\psi\in C^{\infty}((-\infty;0)^{n})$ is a
nonpositive Bernstein function of $n$ variables and write
$\psi\in{\mathscr{T}}_{n}$ if it is nonpositive and all of its first partial
derivatives are absolutely monotone (a function in
$C^{\infty}((-\infty;0)^{n})$ is said to be absolutely monotone if it is
nonnegative together with its partial derivatives of all orders).

Obviously, $\psi\in{\mathscr{T}}_{n}$ if and only if $-\psi(-s)$ is a
nonnegative Bernstein function of $n$ variables on $(0,\infty)^{n}$, and
${\mathscr{T}}_{n}$ is a cone under the pointwise addition of functions and
multiplication by scalars. Moreover ${\mathscr{T}}_{n}$ is an operada [2, p.
321]. As is known [5] (see also [22], [25]), each function
$\psi\in{\mathscr{T}}_{n}$ admits an integral representation of the form (here
the dot denotes inner product in $\mathbb{R}^{n}$ and the expression $s\to-0$
means that $s_{1}\to-0,\ldots,s_{n}\to-0$)

$\psi(s)=c_{0}+c_{1}\cdot
s+\int\limits_{\mathbb{R}_{+}^{n}\setminus\\{0\\}}(e^{s\cdot
v}-1)d\mu(v)\quad(s\in(-\infty;0)^{n}),$ (2.3)

where $c_{0}=\psi(-0):=\lim\limits_{s\to-0}\psi(s)$,
$c_{1}=(c_{1}^{j})_{j=1}^{n}\in\mathbb{R}_{+}^{n}$,
$c_{1}^{j}=\lim\limits_{s_{j}\to-\infty}\psi(s)/s_{j}$, and $\mu$ is a
positive measure on $\mathbb{R}_{+}^{n}\setminus\\{0\\}$; $\mu$ are determined
by $\psi$.

A lot of examples of (positive) Bernstein function of one variable one can
found in [6] (see also [17], [25]). Now we shall describe a family of
functions from ${\mathscr{T}}_{2}$ (for the multidimensional generalization
see [16, Lemma 1]).

######  Example 2.1.

[2, Theorem 1.11] Let a function $\psi_{1}\in{\mathscr{T}}_{1}$ have the
integral representation

$\psi_{1}(s)=c_{0}+\int\limits_{0}^{\infty}(e^{sv}-1)d\mu_{1}(v),$

where $\omega:=\psi_{1}^{\prime}(0-)\neq\infty$. Then the function

$\psi(s_{1},s_{2}):=\frac{\psi_{1}(s_{1})-\psi_{1}(s_{2})}{s_{1}-s_{2}}-\omega$

(under the value of the function $\psi$ for $s_{1}=s_{2}$ we, as usually
understand its limit at $s_{2}\to s_{1}$) belongs to ${\mathscr{T}}_{2}$ and
has the integral representation

$\psi(s_{1},s_{2})=\int\limits_{\mathbb{R}_{+}^{2}\setminus\\{0\\}}(e^{s_{1}u_{1}+s_{2}u_{2}}-1)d\mu(u_{1},u_{2}),$

where $d\mu(u_{1},u_{2})$ is the image of the measure $d\mu_{1}(v)dw$ under
the map

$u_{1}=\frac{v+w}{2},\quad u_{2}=\frac{v-w}{2}$

(in particular if $d\mu_{1}(v)=p(v)dv$ then
$d\mu(u_{1},u_{2})=p(u_{1}+u_{2})du_{1}du_{2}$).

Each function $\psi\in{\mathscr{T}}_{n}$ according to the formula

$\psi(z)=c_{0}+c_{1}\cdot
z+\int\limits_{\mathbb{R}_{+}^{n}\setminus\\{0\\}}(e^{z\cdot u}-1)d\mu(u)$

extends uniquely to a function that is holomorphic in the domain

$\\{{\rm Re}z<0\\}:=\\{{\rm Re}z_{j}<0,j=1,\ldots,n\\}\subset\mathbb{C}^{n}$

and continuous on its closure [2].

######  Definition 2.3.

[22] The value of a function $\psi\in{\mathscr{T}}_{n}$ of the form (2.3) at
$A=(A_{1},\ldots,A_{n})$ applied to $x\in D(A)$ is defined by

$\psi(A)x=c_{0}x+c_{1}\cdot
Ax+\int\limits_{\mathbb{R}_{+}^{n}\setminus\\{0\\}}(T_{A}(u)-I)xd\mu(u),$

where the integral is taken in the sense of Bochner, and $c_{1}\cdot
Ax:=\sum_{j=1}^{n}c_{1}^{j}A_{j}x$.

Given $\psi\in{\mathscr{T}}_{n}$ and $t\geqslant 0$, the function
$g_{t}(z):=e^{t\psi(z)}$ is absolutely monotone on $(-\infty;0)^{n}$. It is
also obvious that $g_{t}(z)\leqslant 1$ on $(-\infty;0)^{n}$. By virtue of the
multidimensional version of the Bernstein-Widder Theorem (see, e.g., [5],
[9]), there exists a unique sub-probability measure $\nu_{t}$ on
$\mathbb{R}_{+}^{n}$, such that, for $z\in(-\infty;0)^{n}$, we have

$g_{t}(z)=\int\limits_{\mathbb{R}_{+}^{n}}e^{z\cdot
u}d\nu_{t}(u)=\mathscr{L}(\nu_{t})(-z).$ (2.4)

If $\psi(0)=0$, the family $(\nu_{t})_{t\geqslant 0}$, where
$\nu_{0}:=\delta_{0}$ (the Dirac measure) is convolution semigroup of
probability measures.

The following theorem describes the family of measures in (2.4).

Recall that a net $(\mu_{n})\subset{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$
converges weakly to $\mu\in{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ (we write
$\mu=w\lim_{n}\mu_{n}$) if for each $f\in C^{b}(\mathbb{R}_{+}^{n})$

$\lim_{n}\int_{\mathbb{R}_{+}^{n}}fd\mu_{n}=\int_{\mathbb{R}_{+}^{n}}fd\mu.$

######  Theorem 2.1.

For the family $(\nu_{t})_{t\geqslant
0}\subset\mathscr{M}^{1}(\mathbb{R}_{+}^{n})$ the following statements are
equivalent:

(i) the mapping $t\mapsto\nu_{t}$,
$\mathbb{R}_{+}\to\mathscr{M}^{b}(\mathbb{R}_{+}^{n})$ is weakly continuous;

(ii) $\mathscr{L}\nu_{t}(-z)=e^{t\psi(z)}$ for some function
$\psi\in\mathscr{T}_{n}$, $\psi(0)=0$ ($t\in\mathbb{R}_{+}$, ${\rm
Re}z\leqslant 0$).

###### Proof.

(i)$\Rightarrow$ (ii). Let the mapping $t\mapsto\nu_{t}$ be weakly continuous.
Let $g_{t}(s):=(\mathscr{L}\nu_{t})(-s)$. Then for any $a>0$ the equality
$g_{at}(s)=(g_{t}(s))^{a}$ is true. In fact, for every $m\in\mathbb{N}$

$g_{mt}(s)=\int\limits_{\mathbb{R}^{n}_{+}}e^{s\cdot
u}d\nu_{mt}(u)=\int\limits_{\mathbb{R}^{n}_{+}}e^{s\cdot u}d\nu_{t}^{\ast
m}(u)=(g_{t}(s))^{m},$

where $\nu_{t}^{\ast m}$ denotes the $m$th convolutional degree. Replacing $t$
in the last equality with $(k/m)t$, we get, that for any positive rational
$r=k/m,k,m\in\mathbb{N}$, we have $g_{rt}(s)=(g_{t}(s))^{r}$. If now the
sequence of positive rational numbers $r_{k}$ converges to $a$, then due to of
our continuity condition
$g_{at}(s)=\lim_{k\to\infty}g_{r_{k}t}(s)=(g_{t}(s))^{a}$ (see, for example,
[8, Chapter IX, §5, Proposition 14]). In particular,
$g_{t}(s)=(g_{1}(s))^{t}$, and therefore the function $(1/t){\rm
log}g_{t}(s)={\rm log}g_{1}(s)$ does not depend on $t$. Further, a completely
monotonic function $g_{1}(-r)=({\mathscr{L}}\nu_{1})(r)$, as was shown above,
satisfies the equality
$g_{1}(-r)^{1/m}=g_{1/m}(-r),m\in\mathbf{N},r\in(0;\infty)^{n}$, and therefore
it is infinitely divisible. Consequently (see, e. g., [3])
$g_{1}(s)=e^{\psi(s)}$, where $\psi\in\mathscr{T}_{n}$, $\psi(0)=0$. Thus,
$g_{t}(s)=e^{t\psi(s)}$.

(ii)$\Rightarrow$ (i). If $\mathscr{L}\nu_{t}(-z)=e^{t\psi(z)}$, and
$t_{0}\geqslant 0$, then $\lim_{t\to
t_{0}}\mathscr{L}\nu_{t}(-z)=\mathscr{L}\nu_{t_{0}}(-z)$ for all $z$ with
${\rm Re}z\leqslant 0$. Therefore by the continuity theorem for the Laplace
transform we have $w\lim_{t\to t_{0}}\nu_{t}=\nu_{t_{0}}$. This completes the
proof. ∎

######  Definition 2.4.

In the notation introduced above, we set

$g_{t}(A)x:=\langle\nu_{t},T_{A}x\rangle=\int\limits_{\mathbb{R}_{+}^{n}}T_{A}(u)xd\nu_{t}(u)\
(x\in X).$ (2.5)

Obviously, $\|g_{t}(A)\|_{B(X)}\leqslant M$. The map $g(A):t\mapsto g_{t}(A)$
is a $C_{0}$-semigroup. In the one-dimensional case, it is called the
semigroup subordinate to $T_{A}$ (in a sense of Bochner). In the general case,
we call this semigroup multidimansionally subordinate to $T_{A}$.

In [26] it was proved that the closure of the operator $\psi(A)$ exists and is
the generator of the $C_{0}$-semigroup $g(A)$ (cf. [22].) Since it is no easy
to find this statement in the literature, we sketch its proof below.

######  Theorem 2.2.

[26] The closure of the operator $\psi(A)$ exists and is the generator of the
semigroup $g(A)$ of class $C_{0}$ defined by formula (2.5).

###### Proof.

It was proved in [2] that the operator $\psi(A)$ is closable, and its
extension is the generator $G$ of the $C_{0}$-semigroup $g(A)$ . Therefore,
the operator $G$ is an extension of the closure $\overline{\psi(A)}$ of
$\psi(A)$. We show that an equality holds here. Since the operators $g_{t}(A)$
commute with $T_{j}(s)$ for all $j$, it is easy to check that
$g_{t}(A):D(A_{j})\to D(A_{j})$ ) for all $j$, and therefore $g_{t}(A):D(A)\to
D(A)$. It follows that $D(A)$ is an operator core for the generator $G$ (see,
e. g., [7, Corollary 3.1.7]). On the other hand, $D(A)$ is an operator core
for the operator $\psi(A)$, and the restriction of the operators $G$ to $D(A)$
coincide with $\psi(A)$. Therefore $\overline{\psi(A)}=G$, which completes the
proof. ∎

The previous result suggests the following final version of the definition of
the operator $\psi(A)$.

######  Definition 2.5.

[22] By the value of a function $\psi\in{\mathscr{T}}_{n}$ at an $n$-tuple
$A=(A_{1},\dots,A_{n})$ of commuting operators in ${\rm Gen}(X)$ we understand
the generator of the semigroup $g(A)$, i.e., the closure of the operator
defined in the Definition 2.3. This value is denoted by $\psi(A)$.

The functional calculus thus arising is called multidimensional Bochner-
Phillips calculus, or $\mathscr{T}_{n}$-calculus.

Without loss of generality we shall assume that $c_{0}=0$ in (2.3). We assume
also that $c_{1}=0$.The corresponding subclass of ${\mathscr{T}}_{n}$ will be
denoted by ${\mathscr{T}}^{0}_{n}$.

The notation and constraints introduced above are used in what follows without
additional explanations.

######  Definition 2.6.

Let

$\Sigma(\theta):=\\{z\in\mathbb{C}:\mathrm{Re}(z)>0,|\arg(z)|<\theta<\pi/2\\}.$

A family of operators $(V(z))_{z\in\Sigma(\theta)\cup\\{0\\}}\subset B(X)$ is
called a holomorphic (analytic) semigroup (of angle $\theta\in(0,\pi/2]$) if

(i) $V(0)=I$ and $V(z_{1}+z_{2})=V(z_{1})V(z_{2})$ for all
$z_{1},z_{2}\in\Sigma(\theta)$.

(ii) The map $z\mapsto V(z)$ is analytic in $\Sigma(\theta)$.

(iii) $\underset{z\in\Sigma(\theta^{\prime})\ni z\to 0}{\lim}V(z)x=x$ $\forall
x\in X$ and $0<\theta^{\prime}<\theta$.

If, in addition,

(iv) $V(z)$ is bounded in $\Sigma(\theta^{\prime})$ for every
$0<\theta^{\prime}<\theta$,

we call $(V(z))_{z\in\Sigma(\theta)\cup\\{0\\}}$ a bounded holomorphic
semigroup.

A result by Yosida [11] asserts that if the bounded $C_{0}$-semigroup $U$ with
generator $G$ on $X$ satisfies

$U(t)X\subset D(G),t>0,\quad{\rm and}\quad\lim\sup\limits_{t\to
0+}(t\|GU(t)\|)<\infty,$ $None$

then for any $\beta>0$, $e^{-\beta t}U(t)$ can be extended to a bounded
holomorphic semigroup on $X$. In this case, the semigroup $U$ will be called a
quasibounded holomorphic semigroup.

We shall denote by ${\mathscr{T}}_{n}^{Y}$ the set of all
$\psi\in{\mathscr{T}}_{n}^{0}$ such that $\psi(G)$ generates a bounded
$C_{0}$-semigroup with property (Y) for every generator $G$ of a bounded
$C_{0}$-semigroup in a Banach space.

##  3\. Criteria for analyticity

In the sequel, for any measure $\mu\in{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$
we put

$(Z_{\mu}f)(x):=(\mu\ast f)(x)=\int_{\mathbb{R}_{+}^{n}}f(x-u)d\mu(u),\quad
f\in L^{1}(\mathbb{R}^{n}_{+}).$

Note, that the semigroup $Z_{\nu_{t}}$ is of the form
$\langle\nu_{t},T\rangle$ where $T(u)f(x)=f(x-u)$ ($u\in\mathbb{R}^{n}_{+}$)
is the left translation $n$-parameter semigroup in
$L^{1}(\mathbb{R}^{n}_{+})$.

For the convolution semigroup $(\nu_{t})_{t\geqslant
0}\subset{\mathscr{M}}^{1}(\mathbb{R}_{+}^{n})$ we define the weak derivative
as

$w\nu^{\prime}_{t}:=w\lim_{\Delta t\to 0}\frac{1}{\Delta t}(\nu_{t+\Delta
t}-\nu_{t}).$

######  Theorem 3.1.

(cf. [10]). Let $\psi\in{\mathscr{T}}_{n}^{0}$. The following conditions are
equivalent:

(a) for all $T_{j}\in C_{0}(X)$ ($j=1,\dots,n$) the semigroup
$g_{t}(A)=\langle\nu_{t},T\rangle$ satisfies the Yosida condition (Y) for
every Banach space $X$ (i.e. $\psi(A)$ is a generator of a quasibounded
holomorphic semigroup $g(A)$ in $X$);

(b) the semigroup $Z_{\nu_{t}}$ satisfies the Yosida condition (Y) in the
space $L^{1}(\mathbb{R}^{n}_{+})$;

(c) the map $t\mapsto\nu_{t}$,
$\mathbb{R}_{+}\to{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ is continuously
differentiable for $t>0$, with
$\|\nu_{t}^{\prime}\|_{{\mathscr{M}}^{b}}=O(t^{-1})$ as $t\to 0+$.

(d) the map $t\mapsto\nu_{t}$,
$\mathbb{R}_{+}\to{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ is weakly
differentiable for $t>0$, with
$\|w\nu^{\prime}_{t}\|_{{\mathscr{M}}^{b}}=O(t^{-1})$ as $t\to 0+$.

###### Proof.

$(a)\Rightarrow(b)$ This is obvious, since
$Z_{\nu_{t}}=\langle\nu_{t},T\rangle$ where $T$ stands for the left
translation semigroup in $L^{1}(\mathbb{R}^{n}_{+})$.

$(b)\Rightarrow(c)$ Since $Z_{\nu_{t}}$ satisfies the condition (Y) in
$L^{1}(\mathbb{R}^{n}_{+})$, the semigroup $U(t)=e^{-\beta t}Z_{\nu_{t}}$ is
holomorphic n $L^{1}(\mathbb{R}^{n}_{+})$ by the Yosida theorem. Then the
semigroup $Z_{\nu_{t}}$ is norm differentiable in $L^{1}(\mathbb{R}^{n}_{+})$
for $t>0$, as well. Due to the isometric isomorphism $\nu\mapsto Z_{\nu}$,
${\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})\to B(L^{1}(\mathbb{R}^{n}_{+}))$ we
conclude that

$\nu^{\prime}_{t}=\lim\limits_{\Delta t\to 0+}\frac{\nu_{t+\Delta
t}-\nu_{t}}{\Delta t}$ (3.1)

in the ${\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ norm and

$\|\nu^{\prime}_{t}\|_{{\mathscr{M}}^{b}}=\|Z_{\nu^{\prime}_{t}}\|_{B(L^{1})}.$

Moreover, since the semigroup $Z_{\nu_{t}}$ satisfies the Yosida condition
(Y), we have $\limsup\limits_{t\to
0+}(t\|\nu^{\prime}_{t}\|_{{\mathscr{M}}^{b}})<\infty$, and thus
$\|\nu_{t}^{\prime}\|_{{\mathscr{M}}^{b}}=O(t^{-1})$ as $t\to 0+$.

Now since $\nu_{t}^{\prime}\in{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ for all
$t>0$, we have
$\nu_{t/2}^{\prime}\ast\nu_{t/2}^{\prime}\in{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$,
too. Taking into account that $({\mathscr{L}}\nu_{t})(-z)=e^{t\psi(z)}$ we
deduce that $\nu_{t/2}^{\prime}\ast\nu_{t/2}^{\prime}=\nu_{t}^{\prime\prime}$
(the Laplace transform of the both sides of the last equality is equal to
$\psi(z)^{2}e^{t\psi(z)}$). So
$\|\nu_{t}^{\prime\prime}\|_{{\mathscr{M}}^{b}}\leqslant\|\nu_{t/2}^{\prime}\|_{{\mathscr{M}}^{b}}^{2}$.
In particular, the function $t\mapsto\|\nu_{t}^{\prime}\|_{{\mathscr{M}}^{b}}$
is bounded and continuous on each interval
$(t_{1},t_{2})\subset\mathbb{R}_{+}$.

$(c)\Rightarrow(d)$ This is obvious.

$(d)\Rightarrow(a)$ Note that for every $x\in X$ and every continuous linear
functional $l\in X^{*}$ the function $u\mapsto l(T(u)x)$ belongs to
$C^{b}(\mathbb{R}_{+}^{n})$. Then

$\displaystyle l\left(\frac{d}{dt}g_{t}(A)x\right)$
$\displaystyle=\frac{d}{dt}l\left(g_{t}(A)x\right)=\lim_{\Delta t\to
0}\frac{1}{\Delta t}(l\left(g_{t+\Delta
t}(A)x\right)-l\left(g_{t}(A)x\right))$ $\displaystyle=w\lim_{\Delta t\to
0}\frac{1}{\Delta t}\int\limits_{\mathbb{R}_{+}^{n}}l(T(u)x)d(\nu_{t+\Delta
t}-\nu_{t})(u)$ $\displaystyle=l\left(w\lim_{\Delta t\to 0}\frac{1}{\Delta
t}\int\limits_{\mathbb{R}_{+}^{n}}T(u)xd(\nu_{t+\Delta t}-\nu_{t})(u)\right)$
$\displaystyle=l\left(\int\limits_{\mathbb{R}_{+}^{n}}T(u)xd\left(w\nu^{\prime}_{t}\right)(u)\right).$

Therefore

$\left\|\frac{d}{dt}g_{t}(A)x\right\|_{B(X)}=\left\|\int\limits_{\mathbb{R}_{+}^{n}}T(u)xd\left(w\nu^{\prime}_{t}\right)(u)\right\|\leqslant
M\left\|w\nu^{\prime}_{t}\right\|_{{\mathscr{M}}^{b}}\|x\|,\quad t>0.$

Since $\left\|w\nu^{\prime}_{t}\right\|_{{\mathscr{M}}^{b}}=O(t^{-1})$ as
$t\to 0+,(a)$ follows. ∎

To compute the weak derivative $w\nu^{\prime}_{t}$ we introduce the following
notion. Consider the vector space of exponential polynomials on
$\mathbb{R}^{n}_{+}$

$E(\mathbb{R}^{n}_{+}):=\left\\{p(r)=\sum_{j}c_{j}e^{s_{j}\cdot
r}:c_{j}\in\mathbb{C},s_{j}\in(-\infty,0)^{n}\right\\}$

equipped with the ${\rm sup}$ norm.

######  Definition 3.1.

For $p\in E(\mathbb{R}^{n}_{+})$, $t>0$ let

$b_{t}(p):=\int\limits_{\mathbb{R}_{+}^{n}}\int\limits_{\mathbb{R}_{+}^{n}}p(r)d_{r}(\nu_{t}(r-u)-\nu_{t}(r))d\mu(u).$

Below we identify a finite measure on $\mathbb{R}^{n}_{+}$ with the
corresponding integral, i. e. with the corresponding bounded linear functional
on $C_{0}(\mathbb{R}^{n}_{+})$.

######  Theorem 3.2.

If the map $t\mapsto\nu_{t}$,
$\mathbb{R}_{+}\to{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ is weakly
differentiable for $t>0$ then the linear functional $b_{t}$ extends uniquely
from $E(\mathbb{R}^{n}_{+})$ to some bounded Radon measure on
$\mathbb{R}_{+}^{n}$ and $w\nu^{\prime}_{t}=b_{t}$.

###### Proof.

For $p\in E(\mathbb{R}^{n}_{+})$, $p(r)=\sum_{j}c_{j}e^{s_{j}\cdot r}$, $t>0$
we have, since $\nu_{t}$ is concentrated on $\mathbb{R}^{n}_{+}$,

$\displaystyle\int\limits_{\mathbb{R}_{+}^{n}}p(r)d_{r}(\nu_{t}(r-u)-\nu_{t}(r))$
$\displaystyle=\int\limits_{\mathbb{R}_{+}^{n}}p(r)d_{r}\nu_{t}(r-u)-\int\limits_{\mathbb{R}_{+}^{n}}p(r)d_{r}\nu_{t}(r)$
$\displaystyle=\int\limits_{-u+\mathbb{R}_{+}^{n}}p(r)d_{r}\nu_{t}(r)-\int\limits_{\mathbb{R}_{+}^{n}}p(r)d_{r}\nu_{t}(r)$
$\displaystyle=\int\limits_{\mathbb{R}_{+}^{n}}(p(r+u)-p(r))d_{r}\nu_{t}(r)$
$\displaystyle=\int\limits_{\mathbb{R}_{+}^{n}}\sum_{j}c_{j}e^{s_{j}\cdot
r}(e^{s_{j}\cdot u}-1)d_{r}\nu_{t}(r)$
$\displaystyle=\sum_{j}c_{j}\left(e^{s_{j}\cdot u}-1\right)e^{t\psi(s_{j})}.$

Therefore

$\displaystyle b_{t}(p)$
$\displaystyle=\int\limits_{\mathbb{R}_{+}^{n}}\left(\sum_{j}c_{j}\left(e^{s_{j}\cdot
u}-1\right)e^{t\psi(s_{j})}\right)d\mu(u)$
$\displaystyle=\sum_{j}c_{j}e^{t\psi(s_{j})}\int\limits_{\mathbb{R}_{+}^{n}}\left(e^{s_{j}\cdot
u}-1\right)d\mu(u)$ $\displaystyle=\sum_{j}c_{j}e^{t\psi(s_{j})}\psi(s_{j}).$

On the other hand,

$\displaystyle\nu_{t}(p)\equiv\int\limits_{\mathbb{R}_{+}^{n}}p(r)d\nu_{t}(r)=\sum_{j}c_{j}e^{t\psi(s_{j})}.$

It follows that

$\displaystyle\frac{d}{dt}\nu_{t}(p)=\sum_{j}c_{j}e^{t\psi(s_{j})}\psi(s_{j})=b_{t}(p).$
(3.2)

Note that $E(\mathbb{R}^{n}_{+})$ is a dense sub-algebra of
$C_{0}(\mathbb{R}^{n}_{+})$ by the Stone-Weierstrass theorem. So, if
$w\nu^{\prime}_{t}$ exists, then by (3.2) the linear functional $b_{t}$
extends uniquely from $E(\mathbb{R}^{n}_{+})$ and its extension equals to
$w\nu^{\prime}_{t}$.

∎

##  4\. The necessary conditions

For $z\in\mathbb{C}^{n}$ we put $\|z\|:=\max\limits_{1\leqslant j\leqslant
n}|z_{j}|$, $\mathbb{D}^{n}:=\\{z:\|z\|<1\\}$,
$\Pi^{n}_{-}:=\\{z\in\mathbb{C}^{n}:\mathrm{Re}z_{j}<0,j=1,\dots,n\\}$, and
$\overline{\Pi^{n}_{-}}$ the closure of $\Pi^{n}_{-}$.

######  Theorem 4.1.

(cf. [10]). If the conditions $(a)-(c)$ of Theorem 3.1 hold then $\psi$ maps
the region $\Pi^{n}_{-}$ into a truncated sector

$S(\theta,\beta):=(\beta+\\{|\arg(-z)|<\theta\\})\cap\Pi^{1}_{-}.$

of opening $2\theta<\pi$ and there exist constants $K>0$ and $\gamma$,
$\gamma\in(0,1)$, such that

$|\psi(z)|\leqslant K\|z\|^{\gamma}\quad\mbox{ {\rm for all}
}z\in\overline{\Pi^{n}_{-}},\|z\|\geqslant 1.$

###### Proof.

Step 1. This step use arguments that are similar to the arguments from the
corresponding part of the proof of Theorem 4 in [10]. Fix a number $\beta>0$.
Due to the condition $(b)$ of Theorem 3.1, the semigroup $e^{-\beta
t}Z_{\nu_{t}}$ on $L^{1}(\mathbb{R}^{n}_{+})$ can be extended to a bounded
holomorphic semigroup in a sector $\Sigma(\theta)$ ($0<\theta<\pi/2$). We
denote this analytic continuation by $e^{-\beta z}Z_{\nu}(z)$. Thus,
$e^{-\beta t}Z_{\nu}(t)=e^{-\beta t}Z_{\nu_{t}}$.

In [10] was noticed that the condition (Y) together with
$\|Z_{\nu_{t}}\|_{B(L^{1})}\leqslant M$ imply

$C_{\beta}:=\sup\limits_{t>0}{t(\|(e^{-\beta
t}Z_{\nu}(t))^{\prime}\|_{B(L^{1})}<\infty}.$

Moreover, since $U^{(n)}(t)=(U^{\prime}(t/n))^{n}$ for every holomorphic
semigroup $U$ (see, e.g., [27, p. 98]), we have

$\displaystyle\|(e^{-\beta t}Z_{\nu}(t))^{(n)}\|_{B(L^{1})}$
$\displaystyle\leqslant$ $\displaystyle\|(e^{-\beta
t/n}Z_{\nu}(t/n))^{\prime}\|_{B(L^{1})}^{n}$ $\displaystyle\leqslant$
$\displaystyle(nt^{-1}C_{\beta})^{n}\leqslant n!(et^{-1}C_{\beta})^{n},\quad
t>0,n\geqslant 1.$

For $n=0$ the last estimate is true, too, because $\nu_{t}$ is a sub-
probability measure.

Now fix $q\in(0,1)$, $\theta\in(0,\arctan(q/(eC_{\beta}))$ and consider the
Taylor series

$\displaystyle e^{-\beta
z}Z_{\nu}(z)=\sum_{n=0}^{\infty}\frac{1}{n!}\left(e^{-\beta\mathrm{Re}z}Z_{\nu}(\mathrm{Re}z)\right)^{(n)}(\imath\mathrm{Im}z)^{n}.$

This series converges uniformly in $B(L^{1}(\mathbb{R}^{n}_{+}))$ for
$\mathrm{Re}z>0$, $|\arg(z)|<\theta$, since

$\displaystyle\left\|\frac{1}{n!}\left(e^{-\beta\mathrm{Re}z}Z_{\nu}(\mathrm{Re}z)\right)^{(n)}(\imath\mathrm{Im}z)^{n}\right\|_{B(L^{1})}\leqslant(eC_{\beta})^{n}\left(\frac{|\mathrm{Im}z|}{\mathrm{Re}z}\right)^{n}\leqslant
q^{n}.$

This estimate shows also that for $z\in\Sigma(\theta/2)$

$\displaystyle\left\|e^{-\beta
z}Z_{\nu}(z)\right\|_{B(L^{1})}\leqslant\frac{1}{1-q}.$

Now the isometric isomorphism $\nu\mapsto Z_{\nu}$,
${\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})\to B(L^{1}(\mathbb{R}^{n}_{+}))$
implies that the map $t\mapsto\nu_{t}$,
$(0,\infty)\to{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ has the analytic
continuation $\zeta\mapsto\nu(\zeta)$,
$\Sigma(\theta)\to{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ and

$\displaystyle\|e^{-\beta\zeta}\nu(\zeta)\|_{{\mathscr{M}}^{b}}\leqslant\frac{1}{1-q},\quad\zeta\in\Sigma(\theta).$

Note that by analyticity (2.4) is valid for $t\in\Sigma(\theta)$. The
application of the Laplace transform to the measure $e^{-\beta t}\nu(t)$
yields

$\displaystyle|e^{-t(\beta-\psi(w))}|\leqslant\|e^{-\beta
t}\nu(t)\|_{{\mathscr{M}}^{b}}\leqslant\frac{1}{1-q},\quad
w\in\overline{\Pi^{n}_{-}},\ t\in\Sigma(\theta).$ (4.1)

Let $\overline{\Pi^{n}}:=-\overline{\Pi^{n}_{-}}$, $\Pi^{n}:=-\Pi^{n}_{-}$,
and $\xi(z):=\beta-\psi(-z)$, where $z\in\overline{\Pi^{n}}$. The inequality
(4.1) implies that $\xi$ maps $\Pi^{n}$ into $\Sigma((\pi/2-\theta)$ (the
boundedness of $t\mapsto e^{-t\zeta}$ on the sector $\Sigma(\theta)$ implies
$\zeta\in\Sigma((\pi/2-\theta)$). It follows that $\psi$ maps the region
$\Pi^{n}_{-}$ into a truncated sector
$(\beta-\Sigma((\pi/2-\theta))\cap\overline{\Pi_{1}^{-}}$ of opening $<\pi$.

Step 2. Now let

$\displaystyle f(z):=\left(\prod_{j=1}^{n}z_{j}^{\frac{2\theta}{\pi
n}}\right)\xi(z)/\xi(1,\dots,1),$

and

$\displaystyle
z=(z_{j})_{j=1}^{n}:=\left(\frac{1+w_{j}}{1-w_{j}}\right)_{j=1}^{n},\
h(w):=f(z),\ g(w)=\frac{h(w)-1}{h(w)+1},\quad
w=(w_{j})_{j=1}^{n}\in\mathbb{D}^{n}.$

Since $f:\Pi^{n}\to\Pi^{1}$, $f(1,\dots,1)=1$, we get
$h:\mathbb{D}^{n}\to\Pi^{1}$, $h(0)=1$, and thus
$g:\mathbb{D}^{n}\to\mathbb{D}$, $g(0)=0$. By the Scwatrz Lemma

$\displaystyle|g(w)|\leqslant\|w\|,\quad w\in\mathbb{D}^{n}.$

But we have $h(w)=(1+g(w))/(1-g(w))$, and so

$\displaystyle|f(z)|=|h(w)|\leqslant\frac{1+|g(w)|}{1-|g(w)|}\leqslant\frac{1+\max_{j}|w_{j}|}{1-\max_{j}|w_{j}|},\quad
w\in\mathbb{D}^{n}.$

In other words,

$\displaystyle|f(z)|\leqslant\frac{1+\max_{j}\left|\frac{z_{j}-1}{z_{j}+1}\right|}{1-\max_{j}\left|\frac{z_{j}-1}{z_{j}+1}\right|},\quad
z\in\Pi^{n}.$

If we consider $z_{j}=x_{j}\geqslant 1$ for all $j=1,\dots,n$, then $f(x)>0$
and

$\displaystyle
f(x)\leqslant\frac{1+\max_{j}\frac{x_{j}-1}{x_{j}+1}}{1-\max_{j}\frac{x_{j}-1}{x_{j}+1}}=\frac{1+\frac{x_{k}-1}{x_{k}+1}}{1-\frac{x_{k}-1}{x_{k}+1}}=x_{k}$

for some $k$, $1\leqslant k\leqslant n$. Hence,
$0<f(x)\leqslant\max_{k}x_{k}$. This inequality means that for all $\beta>0$

$\displaystyle\prod_{j=1}^{n}x_{j}^{\frac{2\theta}{\pi
n}}(\beta-\psi(-x))\leqslant(\beta-\psi(-1,\dots,-1))\max_{k}x_{k}$

Putting here $\beta\to 0$, we get

$\displaystyle-\psi(-x)\leqslant
A(\max_{k}x_{k})\prod_{j=1}^{n}x_{j}^{-\frac{2\theta}{\pi n}}\leqslant
A(\max_{k}x_{k})(\max_{k}x_{k})^{-\frac{2\theta}{\pi}}=A\|x\|^{\gamma},$ (4.2)

where $A=-\psi(-1,\dots,-1)$, $\gamma=1-\frac{2\theta}{\pi}\in(0,1)$, and all
$x_{j}$ are real, $\|x\|\geqslant 1$.

Step 3. To finish the proof, we shall use the representation (2.3) with
$c_{0}=c_{1}=0$ to obtain the similar estimate for
$z\in\overline{\Pi^{n}_{-}}$.

First note that for $a,u\in\mathbb{R}_{+}^{n}$, $\sigma=1-e^{-1}$

$1-e^{-a\cdot u}\geqslant\sigma,\ \mbox{ as }a\cdot u\geqslant 1,\quad
1-e^{-a\cdot u}\geqslant\sigma a\cdot u\mbox{ as }a\cdot u\leqslant 1.$

Therefore, by (4.1) we have for $a\in\mathbb{R}_{+}^{n}$ with $\|a\|\geqslant
1$

$\displaystyle A\|a\|^{\gamma}$
$\displaystyle\geqslant-\psi(-a)=\int_{\mathbb{R}_{+}^{n}}(1-e^{-a\cdot
u})d\mu(u)$ $\displaystyle=\left(\int_{a\cdot u\leqslant 1}+\int_{a\cdot
u>1}\right)(1-e^{-a\cdot u})d\mu(u)$
$\displaystyle\geqslant\sigma\int_{\\{a\cdot u\leqslant 1\\}}a\cdot
ud\mu(u)+\sigma\int_{\\{a\cdot u>1\\}}d\mu(u).$

In particular, if we take $\forall ja_{j}=c$, $c\geqslant 1$, then

$Ac^{\gamma}\geqslant\sigma c\int_{\\{c\sum_{j}u_{j}\leqslant
1\\}}\sum_{j=1}^{n}u_{j}d\mu(u)+\sigma\int_{\\{c\sum_{j}u_{j}>1\\}}d\mu(u).$

It follows that for all $c\geqslant 1$

$\int_{\\{c\sum_{j}u_{j}\leqslant
1\\}}\sum_{j=1}^{n}u_{j}d\mu(u)\leqslant\frac{A}{\sigma}c^{\gamma-1},\quad\int_{\\{c\sum_{j}u_{j}>1\\}}d\mu(u)\leqslant\frac{A}{\sigma}c^{\gamma}.$
(4.3)

Next, since for all $z\in\overline{\Pi^{n}_{-}}$, $u\in\mathbb{R}_{+}^{n}$ we
have

$\left|1-e^{z\cdot u}\right|\leqslant\min\\{z\cdot
u,2\\}\leqslant\min\\{\|z\|\sum_{j=1}^{n}u_{j},2\\}.$

This estimate and (4.3) imply for all $z\in\overline{\Pi^{n}_{-}}$,
$c\geqslant 1$

$\displaystyle|\psi(z)|$
$\displaystyle\leqslant\int_{\mathbb{R}_{+}^{n}}|1-e^{z\cdot u}|d\mu(u)$
$\displaystyle=\left(\int_{\\{c\sum_{j}u_{j}\leqslant
1\\}}+\int_{\\{c\sum_{j}u_{j}>1}\right)|1-e^{z\cdot u}|d\mu(u)$
$\displaystyle\leqslant\|z\|\int_{\\{c\sum_{j}u_{j}\leqslant
1\\}}\sum_{j=1}^{n}u_{j}d\mu(u)+2\int_{\\{c\sum_{j}u_{j}>1\\}}d\mu(u)$
$\displaystyle\leqslant\|z\|\frac{A}{\sigma}c^{\gamma-1}+2\frac{A}{\sigma}c^{\gamma}.$

Putting here $c=\|z\|$, we get $|\psi(z)\leqslant
3\frac{A}{\sigma}\|z\|^{\gamma}$ for $\|z\|\geqslant 1$. This completes the
proof. ∎

##  5\. Sufficient conditions

It is known that the condition

$\limsup\limits_{t\to 0+}\|I-T(t)\|<2$

is sufficient for the analyticity of a one parametric $C_{0}$-semigroup $T$ in
a Banach space $X$. Although in general the converse fails, this condition is
necessary for analyticity of $T$ if $X$ is locally convex.

The next theorem from [23] generalizes this result.

######  Theorem 5.1.

Assume that $C_{0}$-semigroups $T_{j}$ satisfy the conditions
$\|T_{j}(t)\|\leqslant M_{j}\ (j=1,\dots,n)$ and

$\sum\limits_{j=1}^{n}C_{j}\limsup\limits_{t\to 0+}\|I-T_{j}(t)\|<2,$

where $C_{j}=\prod_{k=1}^{j-1}M_{k}$ as $j>1$, $C_{1}=1$. Then for each
function $\psi\in\mathscr{T}_{n}$ the operator $\psi(A)$ is a generator of a
holomorphic semigroup.

We are going to deduce several sufficient conditions for
$\psi\in\mathscr{T}^{Y}_{n}$ from Theorem 3.1.

In the following we shall denote by $\mathscr{F}$ the Fourier transform on
$\mathbb{R}^{n}$ in a sense of distributions, and by ${\mathscr{F}}^{-1}$ the
inverse of ${\mathscr{F}}$. Let

$F_{t}(\lambda)=e^{t\psi(\imath\lambda)}\psi(\imath\lambda)\quad({\rm
Im}\lambda_{j}\geqslant 0,t>0).$

The restriction $F_{t}|\mathbb{R}^{n}$ will be so denoted by $F_{t}$, too. We
put also

$J_{t}(\psi)=\int\limits_{\mathbb{R}^{n}_{+}}\left|(\mathscr{F}F_{t})(u)\right|du$

(if this expression makes sense).

######  Theorem 5.2.

Let the Fourier transform $p_{t}=\mathscr{F}(F_{t})$) belongs to
$L^{1}(\mathbb{R}^{n})$ and is concentrated on $\mathbb{R}^{n}_{+}$. If

$\displaystyle\limsup\limits_{t\to 0+}tJ_{t}(\psi)<\infty,$

then $\psi\in\mathscr{T}^{Y}_{n}$ (i.e. $\psi(A)$ is a generator of a
quasibounded holomorphic semigroup $g(A)$ in $X$).

###### Proof.

Since $p_{t}\in L^{1}(\mathbb{R}^{n}_{+})$ we have by the inverse formula for
the Fourier transform

$F_{t}(y)=\frac{1}{(2\pi)^{n/2}}\int\limits_{\mathbb{R}^{n}_{+}}p_{t}(u)e^{\imath
y\cdot u}du$

for a. e. $y\in\mathbb{R}^{n}$. By the continuity the last equality holds for
all $y\in\mathbb{R}^{n}$. Therefore we have for the Laplace transform

$\mathscr{L}p_{t}(-z)=\int\limits_{\mathbb{R}^{n}_{+}}e^{z\cdot
r}p_{t}(r)dr=(2\pi)^{n/2}e^{t\psi(z)}\psi(z),\quad{\rm Re}z_{j}\leqslant 0,$
(5.1)

by the Pinchuk boundary uniqueness theorem [28] because both sides here are
analytic on the domain $\Pi^{n}_{-}$, continuous on its closure, and coincide
on $\imath\mathbb{R}^{n}$.

On the other hand,

$\mathscr{L}\nu_{t}(-z)=\int\limits_{\mathbb{R}^{n}_{+}}e^{z\cdot
r}d\nu_{t}(r)=e^{t\psi(z)},\quad{\rm Re}z_{j}\leqslant 0.$

It follows that

$e^{t\psi(z)}\psi(z)=\frac{d}{dt}\int\limits_{\mathbb{R}^{n}_{+}}e^{z\cdot
r}d\nu_{t}(r)=\int\limits_{\mathbb{R}^{n}_{+}}e^{z\cdot
r}d(w\nu^{\prime}_{t})(r).$ (5.2)

Comparing (5.1) and (5.2) we conclude that

$d(w\nu^{\prime}_{t})(r)=(2\pi)^{n/2}p_{t}(r)dr.$

Consequently,

$\|w\nu^{\prime}_{t}\|_{\mathscr{M}^{b}}=(2\pi)^{n/2}\|p_{t}\|_{L^{1}}=J_{t}(\psi),$

and the result follows from Theorem 3.1. ∎

Let

$K(\nu_{t},\mu):=\sup\limits_{\|p\|_{E(\mathbb{R}_{+}^{n})}=1}\left|\int\limits_{\mathbb{R}_{+}^{n}}\int\limits_{\mathbb{R}_{+}^{n}}p(r)d_{r}(\nu_{t}(r-u)-\nu_{t}(r))d\mu(u)\right|.$

######  Theorem 5.3.

Let

$\displaystyle\limsup\limits_{t\to 0+}tK(\nu_{t},\mu)<\infty.$

If the map $t\mapsto\nu_{t}$,
$\mathbb{R}_{+}\to{\mathscr{M}}^{b}(\mathbb{R}_{+}^{n})$ is weakly
differentiable for $t>0$, then $\psi\in\mathscr{T}^{Y}_{n}$ (i.e. $\psi(A)$ is
a generator of a quasibounded holomorphic semigroup $g(A)$ in $X$).

###### Proof.

Since $E(\mathbb{R}_{+}^{n})$ is dense in $C_{0}(\mathbb{R}_{+}^{n})$ by the
Stone-Weierstrass theorem, we have $\|b_{t}\|=K(\nu_{t},\mu)$ (see Definition
3.1). On the other hand, $w\nu^{\prime}_{t}=b_{t}$ by Theorem 3.2, and the
result follows from Theorem 3.1.

∎

We shall deduce one more sufficient condition for $\psi\in\mathscr{T}^{Y}_{n}$
from Theorem 3.1 using the following lemma.

######  Lemma 5.1.

Let $f\in L^{1}(\mathbb{R}_{+}^{n})$, $g(s)=\mathscr{L}f(-s)$, and
$\mu\in{\mathscr{M}}(\mathbb{R}_{+}^{n})$. Assume that

$k(f,\mu):=\int_{\mathbb{R}_{+}^{n}}\int_{\mathbb{R}_{+}^{n}}|f(r-u)-f(r)|drd\mu(u)<\infty.$

Then

1) for every $\psi\in\mathscr{T}^{0}_{n}$ there is such $b\in
L^{1}(\mathbb{R}_{+}^{n})$ that

$h(s):=\psi(s)g(s)=\mathscr{L}b(-s),\mbox{ and }\|b\|_{L^{1}}\leqslant
k(f,\mu);$

2) for every $A\in\mathrm{Gen}(X)^{n}$ with $\|T_{A}\|_{B(X)}\leqslant M\mbox{
we have }$

$h(A)=\psi(A)g(A)\mbox{ and }\|h(A)\|_{B(X)}\leqslant Mk(f,\mu).$

###### Proof.

1) Since $f=0$ on $\mathbb{R}^{n}\setminus\mathbb{R}_{+}^{n}$, we have for
$u\in\mathbb{R}_{+}^{n}$, $s\in(-\infty,0)^{n}$

$\displaystyle(e^{s\cdot u}-1)g(s)$
$\displaystyle=\int_{\mathbb{R}_{+}^{n}}e^{s\cdot(u+r)}f(r)dr-\int_{\mathbb{R}_{+}^{n}}e^{s\cdot
r}f(r)dr$ $\displaystyle=\int_{\mathbb{R}_{+}^{n}}e^{s\cdot
r}(f(r-u)-f(r))dr.$

Then

$\displaystyle h(s)$
$\displaystyle=\psi(s)g(s)=\int_{\mathbb{R}_{+}^{n}}(e^{s\cdot
u}-1)g(s)d\mu(u)$
$\displaystyle=\int_{\mathbb{R}_{+}^{n}}\int_{\mathbb{R}_{+}^{n}}e^{s\cdot
r}(f(r-u)-f(r))dr\mu(u)$ $\displaystyle=\int_{\mathbb{R}_{+}^{n}}e^{s\cdot
r}\int_{\mathbb{R}_{+}^{n}}e^{s\cdot r}(f(r-u)-f(r))\mu(u)dr$
$\displaystyle=\mathscr{L}b(-s),$

where $b(r):=\int_{\mathbb{R}_{+}^{n}}(f(r-u)-f(r))\mu(u)$,
$\|b\|_{L^{1}}\leqslant k(f,\mu)$. Above (and below) the application of Fubini
theorem is justified, since $k(f,\mu)<\infty$.

2) For $A\in\mathrm{Gen}(X)^{n}$ consider the Bochner integral

$\displaystyle h(A)$ $\displaystyle=\int_{\mathbb{R}_{+}^{n}}T_{A}(r)b(r)dr$
$\displaystyle=\int_{\mathbb{R}_{+}^{n}}T_{A}(r)\int_{\mathbb{R}_{+}^{n}}(f(r-u)-f(r))d\mu(u)dr$
$\displaystyle=\int_{\mathbb{R}_{+}^{n}}\left(\int_{\mathbb{R}_{+}^{n}}(f(r-u)-f(r))T_{A}(r)dr\right)d\mu(u)$
$\displaystyle=\int_{\mathbb{R}_{+}^{n}}\int_{\mathbb{R}_{+}^{n}}f(r-u)T_{A}(r)drd\mu(u)-\int_{\mathbb{R}_{+}^{n}}\int_{\mathbb{R}_{+}^{n}}f(r)T_{A}(r)drd\mu(u).$

Furthermore, since $f$ is concentrated on $\mathbb{R}_{+}^{n}$,

$\displaystyle\int_{\mathbb{R}_{+}^{n}}f(r-u)T_{A}(r)dr$
$\displaystyle=\int_{\mathbb{R}_{+}^{n}}f(r)T_{A}(r+u)dr$
$\displaystyle=T_{A}(u)\int_{\mathbb{R}_{+}^{n}}f(r)T_{A}(r)dr.$

Then

$\displaystyle
h(A)=\int_{\mathbb{R}_{+}^{n}}(T_{A}(u)-I)\int_{\mathbb{R}_{+}^{n}}f(r)T_{A}(r)drd\mu(u)=\psi(A)g(A).$

Finally,

$\displaystyle\|h(A)\|_{B(X)}\leqslant\int_{\mathbb{R}_{+}^{n}}\|T_{A}(r)\|_{B(X)}|b(r)|dr\leqslant
M\|b\|_{L^{1}}\leqslant Mk(f,\mu).$

∎

######  Theorem 5.4.

Let $\psi\in\mathscr{T}^{0}_{n}$ with a representing measure $\mu$, and
$g_{t}(s):=e^{t\psi(s)}=\mathscr{L}f_{t}(-s)$ with $f_{t}\in
L^{1}(\mathbb{R}_{+}^{n}),$ $t>0$. If

$\displaystyle\limsup\limits_{t\to 0+}tk(f_{t},\mu)<\infty,$

then $\psi\in\mathscr{T}^{Y}_{n}$ (i.e. $\psi(A)$ is a generator of a
quasibounded holomorphic semigroup $g(A)$ in $X$).

###### Proof.

Indeed, by Lemma 5.1

$\displaystyle\limsup\limits_{t\to
0+}t\left\|\frac{d}{dt}g_{t}(A)\right\|_{B(X)}=\limsup\limits_{t\to
0+}t\|\psi(A)g_{t}(A)\|_{B(X)}\leqslant M\limsup\limits_{t\to
0+}tk(f_{t},\mu)<\infty.$

Now the result follows from Theorem 3.1. ∎

######  Corollary 5.1.

Let in addition $f_{t}\geqslant 0$ and $f_{t}(r-u)\geqslant f_{t}(r)$ for all
$r,u,r-u\in\mathbb{R}_{+}^{n},t>0$. Let
$\Gamma_{u}:=\mathbb{R}_{+}^{n}\setminus(u+\mathbb{R}_{+}^{n})$. If

$\displaystyle\limsup\limits_{t\to
0+}t\int_{\mathbb{R}_{+}^{n}}\int_{\Gamma_{u}}f(r)drd\mu(u)<\infty,$

then $\psi\in\mathscr{T}^{Y}_{n}$ (i.e. $\psi(A)$ is a generator of a
quasibounded holomorphic semigroup $g(A)$ in $X$).

###### Proof.

It is sufficient to prove that

$\displaystyle
k(f_{t},\mu)=2\int_{\mathbb{R}_{+}^{n}}\int_{\Gamma_{u}}f_{t}(r)drd\mu(u).$

To this end, note that

$\displaystyle\int_{\mathbb{R}_{+}^{n}}|f_{t}(r-u)-f_{t}(r)|dr$
$\displaystyle=\int_{\Gamma_{u}}f_{t}(r)dr+\int_{\mathbb{R}_{+}^{n}\setminus\Gamma_{u}}(f_{t}(r-u)-f_{t}(r))dr$
$\displaystyle=\int_{\Gamma_{u}}f_{t}(r)dr+\int_{u+\mathbb{R}_{+}^{n}}f_{t}(r-u)dr-\int_{\mathbb{R}_{+}^{n}\setminus\Gamma_{u}}f_{t}(r)dr$
$\displaystyle=\int_{\Gamma_{u}}f_{t}(r)dr+\int_{\mathbb{R}_{+}^{n}}f_{t}(r)dr-\int_{\mathbb{R}_{+}^{n}\setminus\Gamma_{u}}f_{t}(r)dr$
$\displaystyle=2\int_{\Gamma_{u}}f_{t}(r)dr.$

The proof is complete. ∎

##  References

  * 1\.
  * 2\.  A. R. Mirotin, Multivariable ${\mathscr{T}}$-calculus in generators of $C_{0}$-semigroups, St. Petersburg Math. J., 11 (2000), 315-335.
  * 3\.  A. R. Mirotin, Completely monotone functions on lie semigroups, Ukr Math J., 52, 964–968 (2000). https://doi.org/10.1007/BF02591791
  * 4\.  H. Heyer, Probability measures on locally compact groups, Springer-Verlag, Berlin-Heidelberg-New York (1977).
  * 5\.  S. Bochner, Harmonic analysis and the theory of probabylity, University of California Press, Berkeley and Los Angeles (1955).
  * 6\.  R. Shilling, R. Song, Z. Vondracek, Bernstein functions. Theory and applications, de Greyter, Berlin-New York (2010).
  * 7\.  O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, $C^{*}$ and $W^{*}$-Algebras. Symmetry Groups. Decomposition of States, Springer-Verlag, Berlin – Heidelberg – New York (1979).
  * 8\.  N. Bourbaki, Elements de mathematique. Livre VI. Integration. 2nd ed., Ch. 1 – 9, Hermann, Paris (1965 – 1969).
  * 9\.  Ch. Berg, J.P.R. Christensen, P. Ressel, Harmonic analysis on semigroups, Grad. Texts in Math., vol. 100, Springer-Verlag, New York-Berlin (1984).
  * 10\.  A. S. Carasso and T. Kato, On subordinated holomorphic semigroups, Trans. Am. Math. Soc. 327, No. 2, 867–878 (1991).
  * 11\.  K. Yosida, Abstract analyticity in time for solutions of a diffusion equation, Proc. Japan. Acad. 35,109–113 (1959).
  * 12\.  K. Yosida,  Fractional powers of infinitesimal generators and the analyticity of the semi-groups generated by them, Proc. Japan Acad. 36, 86–89 (1960).
  * 13\.  J. Fujita, A sufficient condition for the Carasso-Kato theorem, Math. Ann. 297 (1993), 335–341 ; ’Erratum to: A sufficient condition for the Carasso-Kato theorem’,Math. Ann. 299, 390 (1994).
  * 14\.  E. Hille and R. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, R.I. (1957).
  * 15\.  A. R. Mirotin, Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, Operators and Matrices, 11, No 1, 199–217 (2017), doi:10.7153/oam-11-14No.
  * 16\.  A. R. Mirotin, Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, II, Operators and Matrices, 12, No. 2, 445–463 (2018).
  * 17\.  A. R. Mirotin, Criteria for Analyticity of Subordinate Semigroups, Semigroup Forum, 78, No. 2, 262–275 (2009).
  * 18\.  A. R. Mirotin, On some functional calculus of closed operators on Banach space. III. Some topics in perturbation theory, Russian Math., 12, 61, No. 12, 19–28 (2017).
  * 19\.  A.R. Mirotin, On the $\mathscr{T}$-calculus of generators for $C_{0}$-semigroups, Sib. Math. J., 39, No. 3, 493–503 (1998).
  * 20\.  A.R. Mirotin, Functions from the Schoenberg class $\mathscr{T}$ on the cone of dissipative elements of a Banach algebra, Math. Notes, 61, No. 3–4, 524–527 (1997).
  * 21\.  A.R. Mirotin, Functions from the Schoenberg class $\mathscr{T}$ act in the cone of dissipative elements of a Banach algebra, II, Math. Notes, 64, No. 3–4, 364–370 (1998).
  * 22\.  A.R. Mirotin, Multidimensional $\mathscr{T}$-calculus for generators of $C_{0}$-semigroups, St. Petersburg Math. J., 11, No. 2, 315–335 (1999).
  * 23\.  A.R. Mirotin, On some properties of the multidimensional Bochner-Phillips functional calculus, Siberian Mathematical Journal 52, 6, 1032–1041 (2011).
  * 24\.  A.R. Mirotin, On joint spectra of families of unbounded operators, Izvestiya: Mathematics, 79, No. 6, 1235–1259 (2015).
  * 25\.  A.R. Mirotin, Properties of Bernstein functions of several complex variables, Math. Notes, 93, No. 2 (2013).
  * 26\.  A.R. Mirotin, On multidimensional Bochner-Phillips functional calculus, Probl. Fiz. Mat. Tekh., 1, No. 1, 63–66 (2009) (Russian).
  * 27\.  K.-J. Engel, R. Nagel, A Short Course on Operator Semigroups, Springer (2006)
  * 28\.  S. I. Pinchuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Mathematical Notes of the Academy of Sciences of the USSR, 15, 116–120 (1974). https://doi.org/10.1007/BF02102390
  * 29\.