J. Austral. Math. Soc. (Series A) 64 (1998), 368–396
SUBORDINATION IN THE SENSE OF BOCHNER
AND A RELATED FUNCTIONAL CALCULUS
RENÉ L. SCHILLING
(Received 13 January 1997)
Communicated by P. G. Dodds
Abstract
We prove a new representation of the generator of a subordinate semigroup as limit of bounded operators.
Our construction yields, in particular, a characterization of the domain of the generator. The generator
of a subordinate semigroup can be viewed as a function of the generator of the original semigroup. For
a large class these functions we show that operations at the level of functions has its counterpart at the
level of operators.
1991 Mathematics subject classification (Amer. Math. Soc.): primary 47D03, 47A60, 60J35.
Keywords and phrases: strongly continuous semigroup, subordination, functional calculus, functions of
an operator.
0. Introduction
Let fTt gt ½0 be a contraction semigroup of type .C0 / (that is, strongly continuous
semigroup) on a Banach space .X; k Ð k/ with infinitesimal generator .A; D.A//.
Subordination (in the sense of Bochner) is a method of getting new .C0 /-semigroups
from the original one fTt gt ½0 by integrating fTt gt ½0 (as function of t) with respect
to a subordinator, that is a vaguely continuous semigroup f¼t gt ½0 of sub-probability
measures on [0; 1/. Thus, the subordinate semigroup fTt f gt ½0 is given by the Bochner
integral
(0.1)
Tt f u D
Z
Ts u¼t .ds/;
t ½ 0; u 2 X:
[0;1/
The superscript f in Tt f refers to the Bernstein function f : .0; 1/ ! [0; 1/ which
is the logarithm of the (one-sided) Laplace transform of ¼1 ,
c 1998 Australian Mathematical Society 0263-6115/98 $A2.00 + 0.00
368
[2]
(0.2)
Subordination in the sense of Bochner and a functional calculus
¼
bt .x/ D
Z
e
sx
¼t .ds/ D e
t f .x /
;
369
t; x ½ 0;
[0;1/
and is given by the Lévy-Khinchine-type formula
Z
f .x/ D bx C
(0.3)
.1 e t x / ¼.dt/;
.0;1/
x > 0;
R
where b ½ 0 and .0;1/ t=.1 C t/ ¼.dt/ < 1. The generator of fTt f gt ½0 will consequently be denoted by .A f ; D.A f //. In [15] Phillips obtained the following representation formula for A f :
(0.4)
A f u D b Au C
Z
.0;1/
.Tt u
u/ ¼.dt/;
u 2 D.A/:
If we write et A for Tt , this result shows—at least at a formal level—that A f D f . A/.
In fact, it gave rise to a functional calculus which is sometimes referred to as BochnerPhillips calculus, see [2]. It is well-known for contractive semigroups fTt gt ½0 and
one-sided Þ-stable subordinators f .x/ D x Þ , Þ 2 .0; 1/, that A f is indeed the
fractional power . A/Þ (in the sense of Balakrishnan), see [23, 13]. In [11, 2, 19]
the complete Bernstein functions, a sufficiently rich subclass of the Bernstein functions
(containing, for example, the above fractional powers), was considered and the relation
A f D f . A/ established whenever f is defined on the spectrum of A, see
Proposition 1.3. Here, f . A/ is characterized via its resolvent in terms of the
Dunford-Taylor integral, see Section 4 and [2, 19].
In [11, 2, 19] the representation formula
Z
f
A u D b Au C
(0.5)
A.A ½/ 1 ².d½/; u 2 D.A/;
.0;1/
R
for A f was used, where f .x/ D bx C .0;1/ x.½ C x/ 1 ².d½/ is a complete Bernstein
function with a representation which is particular to this class of functions. Note
that (0.5) generalizes Balakrishnan’s formula for fractional powers, [23, Chapter
IX.11]. Both formulae (0.4) and (0.5) are only defined for u 2 D.A/ and fail, in
general, to describe A f as a whole. There is little information on D.A f /, only that
D.An / ² D.A f /, n 2 N, is an operator core for A f and that D.A f / D D.A/ if
and only if lim x !1 x 1 f .x/ 6D 0, see [19]. In [18, 19] we used for self-adjoint
Hilbert space operators A; B and complete Bernstein functions f a Heinz-Kato type
theorem so as to compare D.A f / and D.B f /: if the graph norms of A and B are
comparable, so are those of A f and B f and, in particular, D.A f / D D.B f /. This
370
René L. Schilling
[3]
technique can be fruitful if one works in concrete situations, see [12] for an application
to pseudo-differential operators.
In this paper we will give an alternative description of .A f ; D.A f // by approximating A f through a sequence of bounded infinitesimal generators. The idea itself
does not seem to be new and was employed by Westphal [22] in order to get a similar
approximation of Balakrishnan’s fractional powers . A/Þ of an infinitesimal generator A. Westphal showed that there is a sequence gn .x/ ! x Þ as n ! 1 such
that Q n .x/ gn .x/x Þ is a Laplace transform of a measure qÞ;n .x/1.0;1/ dx. Putting
formally x D A, we can interpret Q n . A/ as bounded operator and approximate
. A/Þ . Since the densities qÞ;n .x/ can be explicitly calculated, the reasoning in [22]
was straightforward and could easily be made rigorous.
We will go along similar lines, although we cannot use explicit formulae when
approximating the complete Bernstein function f . Here we will construct an approximating sequence f f n gn2N for f consisting of bounded complete Bernstein functions.
Now the operators A fn are well-defined, for example, by Phillips’ formula (0.4), and
the following definition makes sense:
8
< AQ f
D weak- limn!1 A fn
(0.6)
:
D. AQ f / D fu 2 X : the above weak limit existsg :
We show that . AQ f ; D. AQ f // is a closed dissipative extension of .A f ; D.A f //, hence
AQ f D A f .
In particular, the above characterization of D.A f / yields the following asymptotic
result
1 kTž u uk D O
as ž ! 0 for u 2 D.A f /;
(0.7)
f .ž 1 /
which allows for a comparison of the domains of A f and Ag , where g is some other
Bernstein function.
In our final section we make some first steps towards a functional calculus for
generators of subordinate semigroups. We show that operations at the level of complete
Bernstein functions—scalar multiplication, addition, composition, multiplication, and
convergence—have their counterparts at the level of operators. In particular, we can
show for complete Bernstein functions f; g,
(0.8) if f Ð g is a complete Bernstein function, then A f g D
A f Ž Ag D
Ag Ž A f
where equality is understood in the sense of operators. Putting f . A/ D A f we
can extend (0.8) to the case where f Ð g is no longer a Bernstein function:
(0.9)
. f g/. A/ D f . A/ Ž g. A/ D g. A/ Ž f . A/:
[4]
Subordination in the sense of Bochner and a functional calculus
371
Hirsch derived in [11] a representation of .A f ; D.A f // for the class of complete
Bernstein functions that is basically identical with (0.6) but uses a different approximation of the function f . In fact, Corollary 2.10 and Theorem 4.1 (5) can already
be found there. However, the method used in [11] is quite different from ours. It is
motivated by potential theoretic considerations and uses an approach via the resolvent
rather than the semigroup. Since our results in sections 3 and 4 strongly depend on
our approach we will, nevertheless, include our proofs in full.
1. Notations and auxiliary results
Let us recall some results from semigroup theory, see, for example, [8] and [14].
A .C0 /-semigroup of type !0 2 [ 1; 1/ on a Banach space .X; k Ð k/ is a oneparameter family of operators fTt gt ½0 on X satisfying Tt Cs u D Tt Ts u for all t; s ½ 0,
and limt !0 kTt u uk D 0 for all u 2 X, and
kTt k M! e !t ;
(1.1)
t ½ 0;
for all ! > !0 with a suitable constant M! > 0. The infinitesimal generator .A; D.A//
of fTt gt ½0 is the operator
(1.2)
Au D lim
t !0
Tt u
u
t
on
D.A/;
where
(1.3)
ý
D.A/ D u 2 X : the limit (1.2) exists strongly :
The resolvent set ².A/ of A contains the complex half-plane fz 2 C : Re z > !0g,
and the following estimate holds for z 2 ².A/, Re z > !, ! > !0 :
(1.4)
k.z
A/ 1 uk M!
kuk;
Re z !
for all u 2 X:
If A is the generator of a contraction semigroup, that is, a semigroup where (1.1)
becomes kTt k 1 for all t ½ 0, it is a dissipative operator:
(1.5)
kAu
zuk ½ Re zkuk;
z 2 C; Re z > 0;
holds for all u 2 D.A/. In the general situation, we can always turn a semigroup
fTt gt ½0 into a contraction semigroup with generator AQ D A ! on .X; jjj Ð jjj/) where
TQt D e !t Tt is defined on X equipped with the new norm jjjujjj D M! 1 sups½0 kTQs uk.
Hence, (1.4) and (1.5) are, in this sense, the same properties. Let us recall from [21,
pp. 22–24] the following result for dissipative operators.
372
René L. Schilling
[5]
THEOREM 1.1. For any [closed] operator B on a Banach space .X; k Ð k/ the
following assertions are equivalent:
.1/ B is dissipative, that is, k.B ½/uk ½ Re ½kuk for all u 2 D.B/ and some [all]
½ 2 C such that Re ½ > 0.
.2/ k.B ½/uk ½ ½kuk for all u 2 D.B/ and some [all] ½ > 0.
.3/ For all u 2 D.B/ there is a 2 XŁ such that kuk2 D kk2Ł D X hu; i X Ł and
Re X hBu; i X Ł 0.
If B is closed, dissipative, and densely defined with .½ B/.D.B// D X for some
½ > 0, .3/ holds for all 2 X Ł with kuk2 D kk2Ł D X hu; i X Ł .
By the above remark, the operator B D A !, where ! > !0 and A is the generator
of a .C0 /-semigroup of type !0, satisfies the conditions of the above theorem.
A subordinator is a vaguely continuous convolution semigroup f¼t gt ½0 of subprobability measures ¼t on [0; 1/. By Bochner’s theorem the Laplace transform of
¼t has the form
(1.6)
¼t .x/ D
b
Z
1
e
sx
¼t .ds/ D e
t f .x /
;
t; x ½ 0;
0
where f is a Bernstein function (BF ), that is a C 1 -function on .0; 1/ with representation
Z
f .x/ D a C bx C
(1.7)
.1 e t x / ¼.dt/
.0;1/
R
with a; b ½ 0 and a measure ¼ on .0; 1/ such that .0;1/ t=.1 C t/ ¼.dt/ < 1. We
will also need the related class of completely monotone functions C M which consists
of the Laplace transforms of measures in [0; 1/. For an exhaustive account on BF
and C M we refer to [3, §9].
If the semigroup fTt gt ½0 is not equi-bounded
R but of type !0 , not every subordinator
is eligible for subordination. If, however, [0;1/ e s! ¼t .ds/ < 1 for some ! > !0
from (1.1), the Bochner integral
(1.8)
f
Z
Tt u D
Ts u ¼t .ds/;
u 2 X; t ½ 0;
[0;1/
makes sense and defines a new .C0 /-semigroup fTt f gt ½0 .
DEFINITION 1.2. Let fTt gt ½0 be a .C0 /-semigroup of type !0 and f¼t gt ½0 be an
admissible subordinator with f 2 BF . Then the semigroup fTt f gt ½0 of (1.8) is called
subordinate (in the sense of Bochner) with respect to fTt gt ½0.
[6]
Subordination in the sense of Bochner and a functional calculus
373
The idea of subordination is probably due to Bochner [4], also [5, Chapter 4.4]. The
following result characterizes those subordinators which can be used for subordination
if the semigroup is not equi-bounded, see [18, Satz 2.13] or [19]. This, in particular,
allows us later on to restrict ourselves to contractive semigroups.
PROPOSITION 1.3. Let f¼t gt ½0 be a subordinator with f 2 BF as in (1.7). For all
þ ½ 0 the following assertions are equivalent:
R
.1/ [0;1/ e sþ ¼t .ds/ < 1 for all t ½ 0;
.2/ f extends analytically onto fz 2 C : Re z > þg and continuously up to the
boundary;
R
.3/ .1;1/ e sþ ¼.ds/ < 1.
Obviously, the type !0f of fTt f gt ½0 is less or equal to
Later on, we will only consider a subset of BF .
f . !0 / D inf !>!0 . f . !//.
DEFINITION 1.4. A function f : .0; 1/ ! R is a complete Bernstein function
(C BF ), if
Z
2
f .x/ D x
(1.9)
e sx .s/ ds; x > 0;
.0;1/
holds with some 2 BF .
Complete Bernstein functions are sometimes also called operator monotone functions, see, for example, [9]. Our notation follows closely [16].
Examples for complete Bernstein functions are x 7! x=.x C c/, c ½ 0, x 7! x Þ ,
.0 Þ 1/, or x 7! log.1 C x/, whereas x 7! 1 e cx , .c > 0/, is contained in
BF n C BF . The following theorem gives a precise characterization of the class
C BF :
THEOREM 1.5. (see [18]) Each of the following five properties of f : .0; 1/ ! R
implies the other four:
.1/ f 2 CBF ;
.2/ x !
7
f .x/=x is a Stieltjes transform, that is
a
f .x/
D CbC
x
x
Z
.0;1/
1
¦ .dt/;
t Cx
x > 0;
with a; b ½ 0 and a measure ¦ on .0; 1/.
.3/ f : .0; 1/ ! [0; 1/ extends analytically onto C n . 1; 0] such that f .z/ D
f .z/ and Im z Im f .z/ ½ 0. .In other words: f preserves upper and lower half-planes
in C/.
374
René L. Schilling
[7]
.4/ f 2 BF with representation
f .x/ D a C bx C
where a; b ½ 0, m.s/ D
².dt/ D t¦ .dt/ of .2/:/
.5/ x= f .x/ 2 C BF .
R
e
.0;1/
ts
Z
.0;1/
.1
².dt/, and
sx
e
R
/m.s/ ds;
.0;1/
x > 0;
1=.t .1 C t// ².dt/ < 1. .In fact,
Some of the above implications can be found in [2] and [9], a detailed proof for .1/–
.4/ is given in [18]. Only .5/ seems to be new, but it is straightforward that .1/ and .3/
imply .5/, whereas .1/ follows from .5/ because of the identity f .x/ D x=.x= f .x//.
ASSUMPTION 1.6. Throughout this paper, fTt gt ½0 will always be a .C0 / contraction
semigroup on the Banach space .X; k Ð k/, its generator will be denoted by .A; D.A//,
and f¼t gt ½0 will be a subordinator with Bernstein function f . The subordinate semigroup is denoted by fTt f gt ½0 and its generator by .A f ; D.A f //.
2. On the domain of A f
In his 1952 paper on the generation of semigroups of linear operators [15] Phillips
showed that D.A/ ² D.A f / is a core of A f and gave the following representation
formula for the generator of the subordinate semigroup
Z
f
A u D au C b Au C
(2.1)
.Ts u u/ ¼.ds/; u 2 D.A/I
.0;1/
the spectrum of A f satisfies ¦ .A f / ¦ f . ¦ .A//. Little seems to be known about
the domain D.A f / of A f . It was shown in [19, Theorem 5.1] that D.A f / D D.A/ if
and only if either A is bounded or if limx !1 x 1 f .x/ D b > 0.
LEMMA 2.1. Let f be a Bernstein function with representation (1.7) where a D b D
0. Then f is bounded if and only if ¼..0; 1// < 1. The operator A f is bounded if
A is bounded or if f is bounded.
PROOF. If ¼..0; 1// < 1, we have for all x ½ 0
Z
Z
tx
f .x/ D
.1 e / ¼.dt/ .0;1/
.0;1/
¼.dt/ < 1:
Conversely, if f is bounded, we find from Fatou’s lemma
Z
Z
¼.dt/ lim inf
.1 e t x / ¼.dt/ D sup f .x/ < 1:
.0;1/
x !1
.0;1/
x >0
[8]
Subordination in the sense of Bochner and a functional calculus
375
Suppose now that kAk < 1. Then for u 2 D.A/ D X
kA f uk Z
uk ¼.dt/
kTt u
.0;1/
Z
uk ¼.dt/ C
kTt u
.0;1/
Z
¼.dt/kuk
[1;1/
Z
kAk
.0;1/
t ¼.dt/ C
Z
¼.dt/ kuk;
[1;1/
hence, kA f k < 1 (use Proposition 1.3 (3)). If f is bounded, we have by much the
same calculation
Z
kA f uk ¼.dt/kuk;
.0;1/
hence, kA f k < 1.
In what follows we will always assume that f is a complete Bernstein function
C BF . As a consequence of the above Lemma we may furthermore assume that f is
not bounded, that is, that ¼..0; 1// D 1 and has no linear part. Thus,
f .x/ D
Z
.0;1/
.1
Z
(2.2)
m.t/ D
tx
e
st
e
.0;1/
/m.t/ dt;
Z
².ds/;
with
.0;1/
².ds/
< 1:
s.1 C s/
The following approximations of f will be important
f k .x/ D
(2.3)
m k .t/ D
Z
.0;1/
.1
Z
e
.0;k/
st
e
tx
/m k .t/ dt;
².ds/
k2N
with ² as in (2.2):
Note that f k 2 C BF and is bounded, since
f k .x/ Z
.0;1/
m k .t/ dt D
Z
Z
e
.0;k/
.0;1/
st
dt ².ds/ D
Z
.0;k/
².ds/
< 1;
s
and that f .x/ D limk!1 f k .x/ for all x > 0. We use fk in order to give an alternative
definition of the operator A f .
376
René L. Schilling
[9]
DEFINITION 2.2. Let fTt gt ½0 and A be as in Assumption 1.6, f¼t gt ½0 be a subordinator with f 2 BF as in (2.2), and the sequence f f k gk2N given by (2.3). Then the
Bochner integral
Z
f
Q
(2.4)
.Tt u u/mk .t/ dt
A u D weak- lim
k!1
.0;1/
defines an operator on X with domain
(2.5)
D. AQ f / D fu 2 X : the limit (2.4) exists weaklyg :
Obviously, D.A/ ² D. AQ f / and AQ f extends A f jD.A/. We shall show that, in fact,
A D A f jD.A/ D AQ f . In order to do so, we need some preparations.
f
LEMMA 2.3. Let f; f k be as in (2.2), (2.3). Then there exist .possibly signed/
measures k .depending on a constant b ½ 0/ such that
.b C f /b
k D b C f k ;
(2.6)
k 2 N;
holds for the one-sided Laplace transform bk of k .
PROOF. Since f 2 BF , f 6 0, implies 1=.b C f / 2 C M for any b ½ 0, see [3,
example 9.9], we find
Z
1
D
(2.7)
e t x ¹.dt/ D b
¹.x/
b C f .x/
[0;1/
with a suitable measure ¹ on [0; 1/. By the convolution theorem for Laplace transforms we have
Z
1
1
b C f k .x/
ck .x/
D bC
m
m k .t/ dt
b C f .x/
b C f .x/
b C f .x/
.0;1/
Z
ck .x/b
m k .t/ dt b
¹ .x/ m
¹ .x/
D bC
.0;1/
D
Z
bC
.0;1/
m k .t/ dt ¹
m k ? ¹ b.x/;
and so
(2.8)
Z
k .ds/ D b C
.0;1/
m k .t/ dt ¹.ds/
mk ? ¹.s/ ds:
[10]
Subordination in the sense of Bochner and a functional calculus
377
In fact, under the above assumptions on f and f k , the measures k are positive
measures as the following Lemma shows.
LEMMA 2.4. Let f; f k ; k , and b ½ 0 be as in Lemma 2.3. Then the function
x 7! .b C f k .x//=.b C f .x// is completely monotone and its representing measure
k is positive.
PROOF. Assume that b > 0. Since f k 2 C BF , we know from Theorem 1.5 (3)
that f k is analytic on C n . 1; 0] and maps fIm z > 0g onto itself. Thus, z 7!
. f .z/ fk .z//=.b C fk .z// is an analytic function on C n . 1; 0], maps .0; 1/ into
.0; 1/, satisfies
lim
x !0C
f .0/ f k .0/
f .x/ f k .x/
D
D 0;
b C f k .x/
b C f k .0/
and
f .z/ f k .z/
f .z/ f k .z/
D
D
b C f k .z/
b C f k .z/
f .z/ f k .z/
;
b C f k .z/
and preserves the upper complex half-plane:
Im
f .z/ f k .z/
b C f k .z/
1
D
Re. f .z/
jb C f k .z/j2
Ð
fk .z// Im.b C fk .z//
Ð
C Im. f .z/ fk .z// Re.b C fk .z//
Ð
1
D
Im. f .z/ fk .z// Re.b C fk .z//
2
jb C f k .z/j
Ð
Re. f .z/ fk .z// Im fk .z// :
Using the Stieltjes representation (see Theorem 1.5 (2)) for f and f k we get
Z
z
².dt/
t Cz
and
z
ty
D
t Cz
.t C x/2 C y 2
and
f k .z/ D
.0;k/
f .z/ D
Z
.0;1/
z
².dt/
t Cz
and since for z D x C i y
Im
Re
t x C x 2 C y2
z
D
t Cz
.t C x/2 C y 2
378
René L. Schilling
[11]
is valid, we have
Im
1
f .z/ f k .z/
D
b C f k .z/
jb C f k .z/j2
Z
t 2.k;1/
Z
s2.0;k/
h
ty
sx C x 2 C y 2 b
C
.t C x/2 C y 2
.s C x/2 C y 2
i
t x C x 2 C y2
sy
².ds/ ².dt/
.t C x/2 C y 2 .s C x/2 C y 2
"
Z
Z
.t s/y.x 2 C y 2 /
1
D
jb C f k .z/j2 t 2.k;1/ s2.0;k/ ..t C x/2 C y 2/..s C x/2 C y 2 /
#
bt y
².ds/ ².dt/
C
.t C x/2 C y 2
which is positive whenever y D Im z ½ 0. Hence, Theorem 1.5 shows that . f
f k /=.b C fk / 2 C BF , thus .b C f /=.b C fk / D . f
f k /=.b C fk / C 1 2 BF and,
by [3, example 9.9], its reciprocal value is completely monotone. Now Bernstein’s
theorem, see [3, Theorem 9.3] shows that the representing measure k (from Lemma
2.3) of this completely monotone function is positive.
We still have to treat the case where b D 0. In this situation choose a sequence fbn gn2N such that bn > 0 and bn ! 0 and observe that the pointwise limit
f k .x/= f .x/ D limn!1 .bn C f k .x//=.bn C f .x// is again a completely monotone
function. In particular, k is positive.
The measures k are even (sub-)probability measures,
Z
b C f k .x/
k .dt/ D lim bk .x/ D lim
D1
x
!0
x
!0
b C f .x/
[0;1/
(if b D 0, a limiting argument as above still gives k .[0; 1// 1), whose Laplace
transforms approach the function x 7! 1 as k ! 1. By Lévy’s continuity theorem
we have
k ! Ž0
as k ! 1
vaguely, and by the uniform boundedness of the sequence fk gk2N also weakly. We
can use this fact to construct approximations for every u 2 X.
LEMMA 2.5. Let fTt gt ½0, A, f¼t gt ½0 be as in Assumption 1.6, f 2 C BF with
representation (2.2) and fk gk2N as in Lemma 2.3. Then the Bochner integrals
Z
(2.9)
Tt u k .dt/; u 2 X; k 2 N;
[0;1/
[12]
Subordination in the sense of Bochner and a functional calculus
379
define a sequence of bounded operators on X which strongly converges to the identity
operator on X.
PROOF. Write Þk D 1=
R
[0;1/
k .dt/ and observe that limk!1 Þk D 1. Since
Z
Þk
[0;1/
Tt u k .dt/ kuk;
(2.9) defines indeed a bounded operator. For some small Ž > 0 we find
Z
Z
Þk
Tt u k .dt/ u
.Tt u u/ k .dt/
D Þk
[0;1/
[0;1/
Þk
Z
kTt u
uk k .dt/ C 2Þk
[0;Ž/
k .dt/ kuk
uk C 2Þk
t Ž
[Ž;1/
Letting first k ! 1 and then Ž ! 0 we get both
Z
Z
Þk
Tt u k .dt/ ! u and
Tt u k .dt/ ! u
.0;1/
k .dt/ kuk
[Ž;1/
Z
sup kTt u
Z
(strongly)
.0;1/
as k ! 1 since Þk ! 1.
Lemma 2.5 enables us to mimic the usual proof of the closedness of an infinitesimal
generator of a .C0 /-semigroup.
PROPOSITION 2.6. Let fTt gt ½0 , A, f¼t gt ½0 be as in Assumption 1.6, f 2 C BF with
representation (2.2), and fk gk2N as in Lemma 2.3. We then have the identity
Z
Z
.Tt 1/m k .t/ dt b Ž
Ts u l .ds/
.0;1/
[0;1/
Z
(2.10)
Z
D
.Tt 1/ml .t/ dt b Ž
Ts u k .ds/
.0;1/
[0;1/
for all k; l 2 N, b ½ 0, and u 2 X.
PROOF. Using the Stieltjes representation of f ,
Z
x
².dt/;
f .x/ D
.0;1/ t .x C t/
380
René L. Schilling
it is seen that for v 2 X
Z
.Tt
.0;1/
Z
1/v mk .t/ dt [13]
1k m k .t/ dt kvk
kTt
.0;1/
Z
Z
e
.0;k/
Z
D
.0;k/
st
.0;1/
dt ².ds/ kvk
².ds/
kvk;
s
that is, both sides of the identity (2.10) are well-defined. Fixing k; l 2 N, k 6D l, we
observe
Z
Z
.Tt 1/m k .t/ dt b Ž
Ts u l .ds/
.0;1/
[0;1/
Z
Z
Z
.Tt 1/Ts um k .t/l .ds/ dt b
Ts u l .ds/
D
.0;1/ [0;1/
[0;1/
Z
Z
Tt Cs um k .t/l .ds/ dt
D
.0;1/ [0;1/
Z
Z
Z
Ts um k .t/l .ds/ dt b
Ts u l .ds/
.0;1/ [0;1/
[0;1/
Z
Z
Z
D
Tr u.mk ? l /.r / dr
bC
m k .t/ dt
Tr u l .dr /
[0;1/
.0;1/
[0;1/
Z
Z
D
Tr u .m k ? l /.r /½[0;1/
bC
m k .t/ dt l .dr /
.0;1/
[0;1/
where ½[0;1/ denotes the one-dimensional Lebesgue measure on the half-line. The
Laplace transforms of
Z
b C f l .x/
.m k ? l /½[0;1/
bC
m k .t/ dt l b.x/ D
. b f k .x//
b C f .x/
.0;1/
and of
.ml ? k /½[0;1/
Z
bC
b C f k .x/
. b
m l .t/ dt k b.x/ D
b C f .x/
.0;1/
fl .x//
coincide, therefore the measures coincide and the same calculation as above with k
and l interchanged proves (2.10).
REMARK 2.7. The key in the proof of Proposition 2.6 was to show that the representation measures of both sides of (2.10) are the same. This was done via their
Laplace transforms. This, however, amounts to checking the identity (2.10) just for
the semigroup fe t x gt ½0 , x > 0 fixed.
[14]
Subordination in the sense of Bochner and a functional calculus
381
We shall combine (2.10) with the approximation result of Lemma 2.5 to show the
closedness of . AQ f ; D. AQ f //.
THEOREM 2.8. Let fTt gt ½0, A, f¼t gt ½0 be as in Assumption
1.6, f 2 C BF of the
R
form (2.2) and fk gk2N as in Lemma 2.3. Then we have [0;1/ Tt u k .dt/ 2 D. AQ f / for
all k 2 N and all u 2 X,
Z
Z
f
Q
(2.11) . A
b/
Tt u k .dt/ D bu C
.Tt 1/u mk .t/ dt; u 2 X;
.0;1/
[0;1/
and if u 2 D. AQ f /,
Z
(2.12)
Tt . AQ f
b/u k .dt/
Z
bu C
.Tt 1/u mk .t/ dt;
[0;1/
D
.0;1/
u 2 D. AQ f /:
Moreover, D. AQ f / is dense in X and . AQ f ; D. AQ f // is a closed operator.
PROOF. Throughout the proof we denote by hÐ; Ði the dual pairing of X; X Ł and always denotes an (arbitrary) vector of X Ł .
R
In the proof of Proposition 2.6 we saw that u 7! .0;1/ .Tt u u/ mk .t/ dt is a
(strongly) bounded operator. Lemma 2.5 therefore allows us, for fixed k 2 N, to pass
to the weak limit l ! 1 in the identity (2.10). This yields on the left-hand side
Z
×
−Z
lim
.Tt 1/m k .t/ dt b Ž
Ts u l .ds/; l!1
.0;1/
[0;1/
×
−Z
.Tt 1/m k .t/ dt b u; D
.0;1/
for all u 2 X and 2 X Ł . Since the limit on the right-hand side exists, the very
definition of AQ f gives
Z
×
−Z
lim
.Tt 1/ml .t/ dt b Ž
Ts u k .ds/; l!1
.0;1/
[0;1/
−
×
Z
f
Q
D .A
b/
Ts u k .ds/; :
[0;1/
The definition
of D. AQ / now implies—note that the limit on the left side exists—that
R
we have [0;1/ Ts u k .ds/ 2 D. AQ f /. Since there were no restrictions on 2 X Ł ,
(2.11) follows; (2.12) is derived similarly. Just observe that
Z
Z
.Tt 1/ml .t/ dt b Ž
Ts u k .ds/
.0;1/
[0;1/
Z
Z
Ts
.Tt 1/ml .t/ dt b u k .ds/
D
f
[0;1/
.0;1/
382
René L. Schilling
holds for all u 2 X and that for u 2 D. AQ f /
Z
weak- lim
.Tt 1/u ml .t/ dt
l!1
.0;1/
[15]
bu D . AQ f
b/u:
That D. AQ f / ² X is (strongly) dense follows at once from Lemma 2.5. In order
to check the closedness of AQ f on D. AQ f /, we choose any sequence fu n gn2N in D. AQ f /
satisfying
un ! u 2 X
AQ f u n ! v 2 X
strongly and
strongly as n ! 1:
We have to show that u 2 D. AQ f / and AQ f u D v. By (2.12) we have for any 2 X Ł
DZ
E DZ
E
f
Q
Tt . A
b/u n k .dt/; D
.Tt 1/u n m k .t/ dt bun ; ;
.0;1/
[0;1/
and as n ! 1
DZ
Tt .v
E
bu/ k .dt/; D
DZ
.0;1/
[0;1/
.Tt
1/u mk .t/ dt
E
bu; :
By Lemma 2.5 the strong limit
Z
strong- lim
k!1
Tt .v
bu/ k .dt/ D v
bu
[0;1/
exists, and according to the definition of D. AQ f /,
u 2 D. AQ f /
and
AQ f u D v:
We can now identify the generator of the subordinate semigroup, A f , and the
operator AQ f given by (2.4), (2.5).
COROLLARY 2.9. With the assumptions of Theorem 2.8 we have A f D AQ f .
PROOF. Clearly, the operators .A fk ; X/, k 2 N, generate .C0 / contraction semigroups. An application of Theorem 1.1 yields
Þ
(2.13)
Re A fk u; 0
for all u 2 X and all 2 X Ł (note: for all k we have D.A fk / D X as kA fk k < 1)
satisfying kuk2 D kk2Ł D hu; i. Passing to the limit k ! 1 in (2.13) we get
Þ
(2.14)
Re AQ f u; 0
[16]
Subordination in the sense of Bochner and a functional calculus
383
for all u 2 D. AQ f / and all 2 X Ł such that kuk2 D kk2Ł D hu; i. This is but the
dissipativity of AQ f .
Since .A f ; D.A f // generates a .C0 /-semigroup it is maximal dissipative (see [17,
p. 237, Lemma 4.17], [8, p. 21, Proposition 4.1]). But AQ f is a closed, dissipative
extension of .A f ; D.A f // and this is impossible unless AQ f D A f as operators.
COROLLARY 2.10. Let fTt gt ½0 , A, f¼t gt ½0 be as in Assumption 1.6 and f 2 C BF
as in (2.2). Then for the domain of A f .as generator of the subordinate semigroup
fTt f gt ½0 / we have
²
Z
D.A / D u 2 X : lim
.Tt u
k!1 .0;1/
²
Z
D u 2 X : lim
.Tt u
f
(2.15)
k!1
.0;1/
¦
u/ mk .t/ dt exists strongly
¦
u/ mk .t/ dt exists weakly :
PROOF. Write (only in this proof) AQ f;w D AQ f and AQ f;s for the strong version of
Definition 2.2. We then have
A f ² AQ f;s ² AQ f;w
where ‘²’ means the extension in the sense of operators. By Corollary 2.9, however,
we also have
AQ f;w ² A f ;
which completes the proof.
REMARK 2.11. A closer look at our method reveals that every approximation
f f n gn2N of some f 2 BF (not necessarily in C BF ) satisfying
lim f n .x/ D f .x/
n!1
for all x > 0
and fn 2 BF ;
and, writing b
n D f n = f (n is a signed measure, the proof of Lemma 2.3 remains
valid),
n ! Ž0
weakly in the sense of signed measures
leads to the same operator, namely limn!1 A fn D A f .
384
René L. Schilling
[17]
3. An asymptotic result
We will now study a converse of (2.6). Consider the equality
(3.1)
1
bl .x/ D 1
.b C f .x//þ
f .l/
e
x =l
;
x > 0; l 2 N;
bl is the (one-sided) Laplace transform of the (signed) measure
where b > 0 and þ
þl D f .l/.¹
(3.2)
Ž1=l ? ¹/I
recall that ¹ is the representing measure of
Z
1
D
(3.3)
e
b C f .x/
[0;1/
tx
¹.dt/:
Assume moreover that ¹ has a decreasing density n with respect to Lebesgue measure
on [0; 1/—this is always the case when f 2 C BF since then 1=.b C f .x// is a
Stieltjes transform with representing density n 2 C M . Then þl has a density
(3.4)
dþl
.x/ D f .l/ n.x//
dx
1[l
1 ;1/
.x/n.x
Ð
l 1/ ;
x > 0:
Since n is decreasing, the total variation of þl satisfies
þ
þ
Z
þ dþl þ
þ
.x/þþ dx
kþl k D
þ
[0;1/ dx
Z
Z
Ð
1
D f .l/
n.x/ dx C
n.x l / n.x/ dx
[0;l 1 /
[l 1 ;1/
Z
Z
f .l/
2e:
D 2 f .l/
n.x/ dx 2 f .l/e
e xl n.x/ dx 2e
b C f .l/
[0;l 1 /
[0;l 1 /
Thus, the family of measures fþl gl2N has uniformly bounded total mass. This immediately gives the following result.
LEMMA 3.1. Let fTt gt ½0 , A, f¼t gt ½0 be as in Assumption 1.6, f 2 BF and ¹.dt/ D
n.t/1.0;1/ .t/ dt as above, and þl as in (3.1). Then
Z
sup (3.5)
Tt u þl .dt/
2e kuk
l2N
[0;1/
holds for all u 2 X.
Exactly as in the proof of Theorem 2.8, we obtain the following result.
[18]
Subordination in the sense of Bochner and a functional calculus
385
PROPOSITION 3.2. Let fTt gt ½0 , A, f¼t gt ½0 be as in Assumption 1.6, f 2
Lemma 3.1, and þl given by (3.1). Then
Z
f .l/.T1=l 1/
Tt u k .dt/
[0;1/
Z
Z
(3.6)
D
.Tt 1/m k .t/ dt b Ž
Ts u þl .ds/
.0;1/
BF as in
[0;1/
holds for all k; l 2 N, and u 2 X.
REMARK 3.3. One should observe that in equality (3.6) both k and þl will depend
on b > 0.
PROOF. As in the proof of Proposition 2.6, see Remark 2.7, it suffices to check (3.6)
for the semigroup e t x with x > 0 fixed:
f .l/.1
e
x =l
/b
k .x/ D
f .l/.1
e
x =l
/
f k .x/ C b
f .x/ C b
1 e x =l
f .x/ C b
D
. f k .x/ C b/ f .l/
D
bl .x/:
. f k .x/ C b/ þ
The combination of Lemma 3.1 and Proposition 3.2 yields the analogue of Theorem 2.8.
THEOREM 3.4. Let fTt gt ½0, A, f¼t gt ½0 be Ras in Assumption 1.6, f 2 C BF as in
(2.2), and fþl gl2N ; fk gk2N as before. Then [0;1/ Tt u þl .dt/ 2 D.A f / for all u 2 X
and l 2 N. Moreover
Z
f
f .l/.T1=l 1/u D .A
(3.7)
b/
Ts u þl .ds/; u 2 X;
[0;1/
and, if u 2 D.A f /, also
(3.8)
f .l/.T1=l
Z
Ts .A f
1/u D
b/u þl .ds/;
u 2 D.A f /:
[0;1/
Note that the proof of (3.8) uses the closedness of the operator A f
COROLLARY 3.5. With the assumptions of Theorem 3.4 we have
1 k.Tž 1/uk D O
(3.9)
as ž ! 0
f .ž 1 /
for all u 2 D.A f /.
b.
386
René L. Schilling
[19]
PROOF. If u 2 D.A f /, (3.8) gives for l 1 D ž,
Z
Ð
1
f
jþ
j.ds/
bkuk
C
kA
uk
kTž u uk (3.10)
;
1=ž
f .ž 1 / [0;1/
and the assertion follows from kþl k 2e, see Lemma 3.1.
The following corollary contains the converse of Lemma 2.1.
COROLLARY 3.6. Under the assumptions of Theorem 3.4, the operator A f is
bounded if and only if either A is bounded or f is bounded or both are bounded.
PROOF. The sufficiency has already been established (under less restrictive assumptions) in Lemma 2.1. In order to see the necessity, let A f be a bounded operator, that
is, kA f uk ckuk for some c > 0 and all u 2 X, and A be an unbounded operator.
Suppose lim x !1 f .x/ D 1. In this case, (3.10) gives
kTž u
uk Ð
2e
2e
bkuk C kA f uk .b C c/kuk
1
f .ž /
f .ž 1 /
for all u 2 X. Thus,
sup
06Du2X
kTž u uk
D kTž
kuk
1k C
;
f .ž 1 /
ž 2 .0; 1/;
with a constant C not depending on ž. Letting ž ! 0 we find
lim kTž
ž!0
1k lim
ž!0
C
C
D lim
D 0;
1
f .ž / l!1 f .l/
that is, fTt gt ½0 is continuous in the uniform operator norm. Therefore, see [14, p. 2,
Theorem 1.2], its generator A is bounded which contradicts our assumption.
Our next corollary is a generalization of the well-known relation D.Aþ / ² D.AÞ /
if 0 < Þ þ 1.
COROLLARY 3.7. Let fTt gt ½0, A, f¼t gt ½0 be as in Assumption 1.6, f 2 C BF given
by (2.2) and g be another, not necessarily complete, Bernstein function of the form
(1.7) with representing measure ¼g and a D b D 0. If
Z
¼g .dt/
(3.11)
<1
for some ž > 0;
1/
.0;ž/ f .t
then we have D.Ag / ¦ D.A f /.
[20]
Subordination in the sense of Bochner and a functional calculus
PROOF. If u 2 D.A f /, we get for any ž > 0
Z
Z
Z
g
g
.Tt u u/ ¼ .dt/ kTt u uk ¼ .dt/ C
.0;1/
.0;ž/
387
¼g .dt/ kuk
[ž;1/
Z
C
.0;ž/
Ð
¼g .dt/
kA f uk C kuk
1
f .t /
with a constant C depending on g; ž, and b. Assumption (3.11) allows us to choose
ž 2 .0; 1/ such that the above integral converges. Therefore, u 2 D.Ag /, and the
proof is complete.
REMARK 3.8. It is easy to see that
Z
1
g
.1
¼ .[Ž; ž// .1 e 1/ [Ž;1/
holds for any 0 < Ž < ž. Since
Z
lim
Ž!0
[Ž;ž/
¼g .dt/
D
f .t 1 /
s=Ž
e
/¼g .ds/ Z
.0;ž/
e
e
1
g.Ž 1/
¼g .dt/
;
f .t 1 /
whenever the integral on the right hand side exists, and, since by integration by parts
Z
Z
¼g .[Ž; ž//
¼g .dt/
t 2 f 0 .t 1 / g
D
C
¼ .[t; ž// dt
1
f .Ž 1 /
f 2 .t 1 /
[Ž;ž/ f .t /
[Ž;ž/
Z
t 2 f 0 .t 1 /g.t 1 / e g.Ž 1/
C
dt
e 1 f .Ž 1 /
f 2 .t 1 /
[Ž;ž/
for some fixed ž > 0 and Ž ! 0, it is a sufficient condition for (3.11) to hold that
− 7!
f 0 .− /g.− /
f 2 .− /
is integrable in some neighborhood of C1 and that
g.− /
D O.1/ as − ! 1:
f .− /
The above remark contains an interesting special case. Since for Bernstein functions
f .x/= f .x/ 1=x holds (see, for example, [12]), both conditions in Remark 3.8 are
met if g.x/= f .x/ decays like some power x ½ . Let us state this criterion in the
following corollary.
0
COROLLARY 3.9. Let fTt gt ½0 , A, f¼t gt ½0, f 2 C BF , and g 2 BF be as in
Corollary 3.7. If for some ½ > 0 the ratio g.x/= f .x/ D O.x ½ / as x ! 1, then
D.Ag / ¦ D.A f /.
388
René L. Schilling
[21]
4. Towards a functional calculus
In our last section we will show that subordination in the sense of Bochner gives
rise to a reasonable functional calculus for the generators of subordinate semigroups.
In particular, we will prove some multiplication rule for these generators. First of
all, however, let us discuss the relation to other functional calculi. In [2] and [19] it
was shown that the above construction of A f extends the Dunford-Taylor functional
calculus [7, Chapter VII.3, VII.9] in the following sense: the resolvent of .A ž/ f ,
ž > 0, is given by
(4.1)
.½
.A
ž/ f / 1 u D .½ C f .ž
A// 1 u;
½>
f . !/;
where the right hand side is to be understood in the sense of the Dunford-Taylor
functional calculus. Both sides of (4.1) converge as ž ! 0, see [19, Theorem 4.3] or
[2, Proposition 5.17; Theorem 4.2]. It is therefore justified to write f . A/ D A f
for f 2 C BF .
From now on we will, however, follow a somewhat different route which is closer
to Dunford and Schwartz’s definition of functions of an infinitesimal generator, see
[7, Chapter VIII.2] and [10, Chapter XV.1,2]. In particular, our definition of A f for
bounded f , for example, for the approximations f k of (2.3), matches the definition
given in Hille and Phillips [10, equation (15.4.1)].
Here are some rules for generators of subordinate semigroups.
THEOREM 4.1. Let fTt gt ½0 , A, f¼t gt ½0 be as in Assumption 1.6 and f; g 2
with representations of type (2.2). Then we have
.1/ Þ f 2 C BF .Þ > 0/ and AÞ f D Þ A f
.2/ f C g 2 C BF and A f Cg D A fÐC Ag
f
.3/ f Ž g 2 C BF and A f Žg D Ag
.4/ if a ½ 0 then AaCx C f D a C A C A f
.5/ if f g 2 C BF then A f g D A f Ž Ag D
(5-bis) g.x/ D x= f .x/; g 2 CBF and A D
Ag Ž A f
A f Ž Ag D
C BF
Ag Ž A f
where the right-hand sides of (1)–(5-bis) are equalities in the sense of closed operators
with the usual domains for sums, compositions, and so on, of operators. Note, that
we can always identify A f , Ag , and A f g with f . A/, g. A/, and . f g/. A/,
respectively.
PROOF. Clearly, f; g 2 C BF implies that Þ f; f C g; f Ž g 2 C BF , see, for
example, [2, 18]. Since D.A/ ² D.A f / \ D.Ag /, all of the above operators are
densely defined. By the linearity of the definition of AQ f , that is, of A f (Definition
2.2), (1), (2), and (3) are immediate, (3) being a consequence of the transitivity of
[22]
Subordination in the sense of Bochner and a functional calculus
389
Ðf
subordination, Tt g D Tt f Žg , see [5, Theorem 4.4.3]. In order to prove (4), observe
that A f is an A–bounded operator in the sense that kA f uk žkAuk C cž kuk, see, for
example, the proof of Lemma 2.1 or [19, Theorem 5.1]. Thus, .b C A C A f ; D.A//
is a closed operator and the assertion follows from formula (2.1).
For (5) we proceed as in the proof of Theorem 2.8 (with b D 0) and note the
following identity (k; l; m 2 N)
f k .x/gl .x/b
mf g .x/ D . f g/m .x/b
lg .x/b
kf .x/
(4.2)
where we used the notation of Lemma 2.3, that is,
bkh .x/ D
h k .x/
2 CM;
h.x/
x > 0; k 2 N;
with h D f; g; f g and h k means the truncation as done in (2.3). Now (4.2) shows for
all u 2 X
Z
Z
Z
f k gl
fg
. f g/m
(4.3) A A
Tt u m .dt/ D A
Tt Cs u lg .dt/ kf .ds/:
[0;1/
[0;1/
[0;1/
If u 2 D.A f Ž Ag / D fu 2 X : u 2 D.Ag / and Ag u 2 D.A f /g, we find by letting
l ! 1; k ! 1, and then m ! 1 in (4.3) that u 2 D.A f g /, defined as in Definition
2.2, hence A f Ž Ag ² A f g and, by symmetry, Ag Ž A f ² A f g .
If, conversely, u 2 D.A f g / (given by Definition 2.2 and Corollary 2.10), we first let
m ! 1 in (4.3) and afterwardsl ! 1 and k ! 1. This shows that u 2 D.A f Ž Ag /
and therefore A f g ² A f Ž Ag and A f g ² Ag Ž A f , respectively.
Assertion (5-bis) follows directly from (5) and Theorem 1.5. Note, that with our
assumptions on f one has limx !1 g.x/=x D 0 (that is, g has no linear part) but it
might occur that limx !0 g.x/ D 1= f 0 .0/ > 0 (if f 0 .0/ < 1). Thus, g is up to a trivial
part of the form (2.2) and (5) is indeed applicable (see also (4)).
REMARK 4.2. .a/ Theorem 4.1 shows, in particular, that for f; g 2 C BF such
that f g 2 CBF always D.A f Ž Ag / D D.Ag Ž A f / D D.A f g / holds true. This is
but to say
u 2 D.A f / and A f u 2 D.Ag / if and only if
u 2 D.Ag / and Ag u 2 D.A f /:
Therefore, it is possible to speak of A f and Ag as commuting operators whenever
f g 2 C BF .
.b/ Another consequence of 4.1 (5) is that for f 2 C BF
p
p Ð2
p
Af D A f
hence
Af D A f
390
René L. Schilling
[23]
.c/ The representing measure of f g can be explicitly calculated. If f has the
representation
Z
Z
f k .x/
¦ f .dt/
¦ fk .dt/
f .x/
D
;
D
I
x
x
.0;1/ t C x
.0;k/ t C x
with ¦ fk D 1.0;k/ .Ð/¦ f —see Theorem 1.5 (2), (2.2), (2.3)—and similarly x 1 g.x/ and
x 1 gk .x/, we can use a convolution-type theorem for the Stieltjes transform, see [20,
(2),(3)], that implies
Z
Z
¦ g .d y/ f
¦ f .d y/ g
fg
¦ C
¦ :
¦ D
t
t
.0;1/ y
.0;1/ y
Here, the integrals are to be understood as Cauchy principal values.
.d/ It might be instructive to note that Theorem 4.1 covers the case of the Yosida
approximation of an infinitesimal generator A. The Yosida approximation is the family
A½ D ½A.½
A/ 1 ;
½>0
of bounded linear operators that strongly approach A on D.A/ as ½ ! 1.
If we put y.½I x/ D ½x.½ C x/ 1 , we find on X
A y.½IÐ/ D
y.½I
A/ D ½A.½
A/
1
D A½
and similarly
Af
Ðy.½IÐ/
D A y.½I f .Ð// D A f
for any f 2 CBF .
Ð
½
The requirement that f g 2 C BF for f; g 2 C BF in (5) of Theorem 4.1 is
quite restrictive. We can overcome this difficulty if we use Berg’s result [1] that the
cone of Stieltjes functions is logarithmically convex, that is to say that for all Stieltjes
transforms 8; 0 and any 0 ½ 1; 8½ 0 1 ½ is again a Stieltjes transform. Combining
this with Theorem 1.5 (1), (2) we get
(4.4)
f ½ g1
½
2 C BF
for all
f; g 2 C BF ; 0 ½ 1:
This observation together with Theorem 4.1 enables us to write down a consistent
formula for the operator . f g/. A/ and all f; g 2 C BF —even if A f g has no longer
any meaning since it cannot be the generator of a subordinate semigroup if f g 62 BF .
DEFINITION 4.3. For f 1 ; f 2; : : : ; f N 2 C BF , N 2 N, we set
ð
1=N
1=N
1=N Ł N
(4.5)
. f 1 Ð f 2 Ð Ð Ð f N /. A/ D . A f1 / Ž . A f2 / Ž Ð Ð Ð Ž . A f N /
on its natural domain.
[24]
Subordination in the sense of Bochner and a functional calculus
391
Let A D f f 1 f 2 Ð Ð Ð f N : N 2 N; f j 2 CBF ; 0 j N g denote the set of
all finite products of C BF -functions. We want to show that the above definition
extends Theorem 4.1 (5) to the set A . First, however, we have to check that (4.5)
gives a well-defined operator. Let F 2 A with representations F D f 1 f 2 Ð Ð Ð f N and
F D g1 g2 Ð Ð Ð g M , M ½ N , f j ; g j 2 C BF . Extending (4.4) to N -fold products we
see, by Theorem 4.1 (5)
h
iN
h 1=N i N h
iN
1=N
1=N
1=N
2
1=N
. A f1 / Ž . A f2 / Ž Ð Ð Ð Ž . A f N / D . 1/N A F
D
AF
(note that . 1/N D . 1/N ) and an analogous formula for the other representation
of F. In particular, we find F 1=N ; F 1=M 2 CBF and it remains to show that
1=N
1=M
[ A F ] N D [ A F ] M . However,
2
h
A
F 1=N
i N ½1=M
D
h
AF
1=N
iN=M
D
Af
1=M
;
where the first equality is well-known for (arbitrary) powers of closed operators, while
the second identity follows from Theorem 4.1 (3). (All equalities are to be understood
in the sense of closed operators.) Thus, (4.5) is well-defined for A and the following
corollary shows that it indeed extends Theorem 4.1 (5).
COROLLARY 4.4. Let fTt gt ½0 , A, f¼t gt ½0 be as in Assumption 1.6, f; g 2
with representations of type (2.2) and put H D f g 2 A . Then
C BF
H . A/ D A f Ž Ag D Ag Ž A f
(4.6)
.x f /. A/ D A Ž A f D A f Ž A:
(4.7)
More generally, if H D F G with F; G 2 C BF , then
(4.8)
H . A/ D F. A/ Ž G. A/ D G. A/ Ž F. A/:
The above equalities are to be understood in the sense of closed operators.
p
PROOF.p Put Bp D A f g . By definition, H . A/ D B2 , and B D A
A g D A g Ž A f by Theorem 4.1 (5). Since
p
p
n
p p
D.B 2 / D u 2 X : u 2 D.A f g / and
A
p p
f g
p p
f g
u 2 D.A
fg
p
D A
o
/
p
p
ý
p
p
p
p
D u 2 X : u 2 D.A g /; A g u 2 D.A f /; A f A g u 2 D.A g /
p
p p
p
and A g A f A g u 2 D.A f /
f
Ž
392
René L. Schilling
[25]
we get from Remark 4.2 (1) that D.B 2 / D D.A f Ž Ag / D D.Ag Ž A f / and, again by
Remark 4.2 (1), (2), that
p p p p p p p
p Ð
B 2 D A f A g Ž A f A g D A f A f Ž A g A g D A f Ž Ag D Ag Ž A f :
p p
This proves (4.6). Now (4.7) follows from (4.6) since we have x f .x/ D x x f .x/
p
and x 2 C BF which has also a representation of type (2.2).
Finally, (4.6), (4.7), and Theorem 4.1 (5) prove (4.8) if F; G are in CBF , but do not
necessarily have the form (2.2). Assume now that F 2 C BF and G D g1 g2 Ð Ð Ð g M
with g j 2 C BF . Then H D F Ð g1 g2 Ð Ð Ð g M 2 A and
H . A/ D F. A/ Ž g1. A/ Ž Ð Ð Ð Ž gM . A/ D F. A/ Ž G. A/
which proves (4.8) first for F 2 C BF and G 2 A and, by iteration, for F; G 2 A .
Let us point out the connection of Corollary 4.4 with results obtained by deLaubenfels [6] where automatic extensions of (bounded) functional calculi to wider classes of
functions are considered. Our extension resembles Method II introduced there, see in
particular [6, Examples 4.1, 4.2]. However, in our situation these constructions seem
not directly applicable since neither C BF nor A are algebras. It is obvious that we
cannot transfer our results to the algebra generated by C BF or A without losing all
the nice properties, for example, information on closedness and so on.
We finally prove some convergence results for sequences of subordinate generators.
THEOREM 4.5. Let fTt gt ½0, A, f¼t gt ½0 be as in Assumption 1.6 and f f .n/ gn2N be a
sequence of complete Bernstein functions of the form (2.2) with pointwise limit f . If
f has no linear part, that is, if limx !1 x 1 f .x/ D 0, then
.n/
strong- lim A f u D A f u
(4.9)
for all u2 D.A f /\
n!1
T1
nD1 D.A
f .n/
/. .Note that always D.A/ ² D.A f /\
T1
nD1 D.A
f .n/
/:/
PROOF. As pointwise limit of complete Bernstein functions f is itself in C BF .
Define as in (2.3) for k 2 N the truncation . f .n/ /k of f .n/ with n 2 N being
R fixed. Since
.n/
f .n/ is of the form (2.2), some elementary
calculations
show
f
.x/
D
x=.t .x C
.0;1/
R
t//² .n/ .dt/ and . f .n/ /k .x/ D .0;k/ x=.t .x C t//².n/ .dt/, see also Theorem 1.5 (2)
and (4).
We claim that
(4.10)
lim . f .n/ /k .x/ D lim lim . f .n/ /k .x/ D lim lim . f .n/ /k .x/ D f .x/;
n;k!1
k!1 n!1
n!1 k!1
x > 0;
[26]
Subordination in the sense of Bochner and a functional calculus
393
holds true. In order to prove (4.10) we note
þ
þ þ
þ þ
þ
þ f .x/ . f .n/ /k .x/þ þ f .x/ f .n/ .x/þ C þ f .n/ .x/ . f .n/ /k .x/þ
þ Z
þ
x
D þ f .x/ f .n/ .x/þ C
² .n/.dt/:
t
.x
C
t/
[k;1/
Moreover, for every l > 1
Z
Z
x C k=l
x
xl
.n/
² .dt/ ² .n/ .dt/
x C k [k;1/ t .xl C t/
[k;1/ t .x C t/
x C k=l .n/
f .xl/
x Ck
which implies
lim lim
k!1 n!1
f .xl/
x C k=l .n/
x C k=l .n/
f .xl/ D lim lim
f .xl/ D
:
n!1 k!1 x C k
x Ck
l
Since f has no linear part, liml!1 f .xl/l
Using the results of Section 2 we get
. f .n/ /k
.n/
D d
k
f .n/
1
D 0, and (4.10) follows.
f .n/
.n/
D d
f
and
k; n 2 N
for suitable sub-probability measures k.n/ and .n/ . Hence,
Ð
. f .n/ /k
. f .n/ /k f .n/
D
D k.n/ ? .n/ b
.n/
f
f
f
with another sub-probability measure k.n/ ? .n/ . Therefore, Remark 2.11 applies and
shows that the double sequence
lim A. f
k;n!1
.n/
/k
u D Afu
converges strongly for all u 2 D.A f /.
For fixed u 2 D.A f / and any ž > 0 there is a number N D N .ž; u/ such that
f
. f .n/ /k u ž for all n; k ½ N :
A u A
By the triangle inequality we get for u 2 D.A f / and n; k ½ N .ž; u/
.n/
f
. f /k
f
f .n/ . f .n/ /k f .n/ u
A
u
A
u
A
u
C
A
u
A
u
A
.n/
ž C A. f /k u
.n/
.n/ A f u
If also u 2 D.A f / for all n ½ N .u; ž/, we find that limn!1 kA f u
k ! 1 and the assertion follows as ž ! 0.
.n/
A f uk ž for
394
René L. Schilling
[27]
The assumption lim x !1 x 1 f .x/ D 0 in the statement of Theorem 4.5 is essential.
For example, put f .n/ .x/ D nx.n C x/ 1 which is a complete Bernstein function
with limiting function f .x/ D x as n ! 1 (note that f .n/ . A/ is just the Yosida
approximation of A). Clearly, ² .n/ D nŽn and therefore
lim lim . f .n/ /k .x/ D x 6D lim lim . f .n/ /k .x/ D 0;
n!1 k!1
k!1 n!1
x > 0;
.n/
though still limn!1 A f D A strongly on D.A/. The reason that the proof of
Theorem 4.5 fails is that ² .n/ as n ! 1 pushes too much mass too fast to 1—this is
the only way to produce linear growth in the limiting function if for the approximations
lim x !1 x 1 f .n/ .x/ D 0, n 2 N, holds. Therefore we have to adapt our truncation
procedure, that is, we have to choose in . f .n/ /k the truncation k 2 N in dependence of
n 2 N.
COROLLARY 4.6. In the situation of Theorem 4.5 assume that the sequence f .n/
has a limit f such that lim x !1 x 1 f .x/ D b > 0. Then D.A f / D D.A/ and for
.n/
u 2 D.A/ we have limn!1 A f u D A f u in the strong topology.
PROOF. That RD.A f / D D.A/ if (and only if) b > 0 was, for example, shown
in [19]. Since [0;1/ .t .1R C t// 1 ² .n/ .dt/ < 1 for every n 2 N, we can choose a
k D k.n/ 2 N such that [k.n/;1/ t 2 ² .n/ .dt/ n 1 . Thus,
þ
þ þ
þ þ
þ
þ f .x/ . f .n/ /k.n/ .x/þ þ f .x/ f .n/ .x/þ C þ f .n/ .x/ . f .n/ /k.n/ .x/þ
þ
þ 1
þ f .x/ f .n/ .x/þ C ;
n
that is, limn!1. f .n/ /k.n/ .x/ D f .x/ and with the same reasoning as in the proof of
Theorem 4.5 we find for every ž > 0 and u 2 D.A/ a number N .u; ž/ such that
.n/
f
A u A. f /k.n/ u ž for all n ½ N .u; ž/:
.n/
.n/
Since limm!1 A. f /m u D A f u for all n 2 N and u 2 D.A/, we may choose a natural
number M.u; ž; n/ ½ N .u; ž/ such that
.n/
.n/
. f /l
f
. f .n/ /m . f .n/ /m u
A
u
C
A
u
A
u
A
2ž for all m; l ½ M.u; ž; n/:
Enlarging k.n/ such that k.n/ ½ M.u; ž; n/, if necessary, we find
.n/ f
A u A f u
A f u
A. f
.n/ /
k.n/
.n/
u C A. f /k.n/ u
A. f
.n/ /
m
.n/
u C A. f /m u
.n/ A f u
3ž
if n ½ N .u; ž/ and k.n/; m ½ M.u; ž; n/, and the claim follows as n ! 1.
[28]
Subordination in the sense of Bochner and a functional calculus
395
Acknowledgement
Financial support by DFG post-doctoral fellowship Schi 419/1–1 is gratefully
acknowledged. Part of this work was done when visiting the University of Warwick,
Coventry, U.K., and the Université d’Evry–Val d’Essonne, Evry, France. I would
like to thank Professors K. D. Elworthy, Coventry, and F. Hirsch, Evry, for their kind
invitations. I am deeply indebted to Professor R. deLaubenfels, Ohio, for his helpful
comments and remarks.
Note added in proof
All results of Section 2 and those of Section 4 that do not rely upon the log-convexity
of the cone C BF —it is used only in Corollary 4.4—hold also for BF and not only
for the subclass CRBF . In order to see this, we remark that for any g 2 BF with
g.x/ D a C bx C .0;1/ .1 e sx / ¼.ds/ we have
a
g.x/
D CbC
x
x
Z
.0;1/
1
e
x
sx
a
¼.ds/ D C b C
x
Z
e
.0;1/
rx
¼.[r; 1// dr 2 C M ;
and thus, see [3, Exercise 9.10], g Ž f = f 2 C M for all f 2 BF . We use now the
Yosida approximation y.nI x/ D nx=.n C x/, n 2 N, which is itself in BF —see
Remark 4.2(d) for details—and find
lim y.nI f .x// D f .x/
n!1
and
y.nI f .x//
Db
n .x/ 2 C M
f .x/
with positive sub-probability measures n . By Lévy’s continuity theorem, n ! Ž0
(the argument of Lemma 2.4, 2.5 applies), hence the conditions of Remark 2.11 are
met (with obvious changes in the notation, for example, f n of Section 2 has to be
redefined as f n :D y.nI f .Ð// etc.).
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