© 2018. Shawgy Hussein, Simon Joseph, Ahmed Sufyan, Murtada Amin, Ranya Tahir & Hala Taha. This is a research/review paper, distributed under the terms of the Creative Commons AttributionNoncommercial 3.0 Unported License http://creativecommons.org/licenses/bync/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the srcinal work is properly cited.
Global Journal of Science Frontier Research:
F
Mathematics and Decision Sciences
Volume 18 Issue 5 Version 1.0 Year 2018 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN:
22494626
& Print ISSN:
09755896
Functional Calculus for the Series of Semigroup Generators Via Transference
By Shawgy Hussein, Simon Joseph, Ahmed Sufyan, Murtada Amin, Ranya Tahir & Hala Taha
Upper Nile University
Keywords: functional calculus, transference, operator semigroup, fourier multiplier,  boundedness. GJSFRF Classification: FOR Code: MSC 2010: 47A60
FunctionalCalculusfortheSeriesofSemigroupGeneratorsViaTransference
St rictly as per the compliance and regulations of:
Abstract
We apply an established transference principle in order to obtain the boundedness of certain functional calculifor sequence of semigroup generators. In particular, it is proved that by Markus Haase and Jan Rozendaal [20] if – bethe sequence generates semigroups on a Hilbert space, then for each the sequence of operators hasbounded calculus for the closed ideal of bounded holomorphic functions on (sufficiently large) right half –plane thatsatisfies the sequence of . The bounded of this calculusgrows at mostlogarithmically as . As a consequence decay at . Then we show that each sequence of semigroupgenerator has a socalled (strong) mbounded calculus for all a
nd that this property characterize thesequence of semigroup generators. Similar results
are obtained if the underlying Banach space is a UMD space. Uponrestriction to socalled semigroups, the Hilbertspaceresults actually hold in general Banach spaces.
)=
m
Functional Calculus for the Series of Semigroup Generators Via Transference
bstract
We apply an established transference principle in order to obtain the boundedness of certain functional calculi for sequence of semigroup generators. In particular, it is proved that by Markus Haase and Jan Rozendaal [20] if – be the sequence generates semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on sufficiently large) right half –plane that
satisfies the sequence of . The bounded of this calculus grows at most logarithmically as As a consequence decay at . Then we show that each sequence of semigroup generator has a socalled strong) mbounded calculus for all , nd that this property characterize the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to socalled emigroups the Hilbert space results actually hold in general Banach spaces.
)=
m
K e y wo rds:f u nct io n a l c a lculus, transference, operator semigroup, fourier multiplier, boundedness.
I.
I
ntroduction
A functional calculus for a (possibly unbounded) sequence of operators on a Banach space X is a methodof associating a closed sequence of operators to each taken from a set of functions in such a way that formulae valid for the functions turn into valid formulae for the operators upon replacing the independent variables by . A common way to establish such a calculus is to start with an algebra
(
)
=
(
)
ofgood functions where definitions of as bounded sequence of operators are more or less straightforward, and then extend this primaryor elementary calculusby means of multiplicative regularization(see [7,Chapter 1] and [3] ). It is then natural to ask which of the so constructed closed sequence of operators are actually
bounded, a questionparticularly relevant in applications, e.g., to evolution equations,
see for instance [1,11].
(
)
(
)
The latter question links functional calculus theory to the theory of vectorvalued singular integrals, best seen in the theory of sectorial (or striptype) operators with a bounded 
calculus, see for instance [13]. It appears there that in order to obtain
nontrivial results the underlying Banach space must allow for singular integrals to
N
otes
Shawgy Hussein
,Simon Joseph
,Ahmed Sufyan
, Murtada Amin
,Ranya Tahir
¥
&Hala Taha
§
Author
: Sudan University of Science and Technology, College of Science, Department of Mathematics, Sudan. email: shawgy2020@gmail.com Author
:Upper Nile University, Faculty of Education, Department of Mathematics, South Sudan. email: s.j.u.khafiir@gmail.com Author
:Ministry of Education, Department ofMathematics , Sultanate Oman.email: shibphdoman@gmail.com Author
:Ministry ofEducation and Higher Education, Department ofMathematics, Qatar.email: mlgandi2@gmail.com Author
¥
: Jazan University, Faculty of Science, Department of Mathematics, Kingdom of Saudi Arabia.email: rayya2004@hotmail.com Author
§
:Princess Nourah bint Abdulrahman University, College of Science, Department of Mathematics, Kingdom of Saudi Arabia.email:hala_taha2011@hotmail.com
1
G l o b a l J o u r n a l o f S c i e n c e F r o n t i e r R e s e a r c h V o l u m e X V I I I I s s u e e r s i o n I V V
Y e a r
2 0 1 8
37
( F )
©2018 Global Journals
[10]
—
show that the presence ofa group of operators does warrant the boundedness of certain
calculi. In [8] the underlying structure of these results was brought to light,
namely a transference principle, a factorization of the sequence of operators in terms of vectorvalued Fourier multiplier operators. Finally, in [9] it was shown that semigroups also allow for such transference principles.


(
)
(a) For
< 0 and
(
) one has
(
)
(1+
)
(H ) with
Markus Haase and Jan Rozendaal [20] developed this approach further. They apply the general form of the transference principle for semigroups given in [9] in order to obtain bounded functional calculi for the sequence of generators of semigroups. These results, in particular Theorems 3.3, 3.7, and 4.3,are proved for general Banach spaces. However, they make use of (subalgebras of ) the analytic ; X) Fourier
multiplier algebra (see (2.1) below for a definition), and hence are useful only if the
underlying Banach space has a geometry that allows for nontrivial Fourier multiplier operators. In case X = H is a Hilbert space one o
btains particularly nice results, which we want to summarize here (see [20]).

(
Let
−
be the sequence of generators of bounded
–
semigroups on a Hilbert space H with M := . Then the following assertions hold.
∑
(
)
∑‖
‖
where c(1+
) = O(
 
) as
0 , and c(1+
) = O(1) as
→ ∞.(b) For
< 0 <
and
with Re
< 0 there is
1such that
∑
∑‖
‖
for all
(
) . In particular, dom(
)
⊆
dom(
(
)).(c)
has strongmbounded
calculus of type 0 for each m
.(See Corollary 3.10 for (a) and (b) and Corollary 4.4 for (c).)
When X is a UMD space one can derive similar results , we extend the Hilbert space results to general Banach spaces by replacing the assumption of boundedness of
the semigroup by its

boundedness, a concept strongly put forward by Kalton and Weis
[12]. Inparticular, Theorem 1.1 holds true for bounded semigroups on arbitrary Banach spaces with M being the  bound of the semigroups .
We stress the fact that in contrast to [7], where sectorial operators and, accordingly, functional calculi on sectors, were considered, we deals with general sequence of semigroup generators and with functional calculi on halfplanes . The abstract theory of (holomorphic) functional calculi on halfplanes can be found in [3 corollary 6.5and 7.1]
Banach space is a Hilbert space, it turns out that simple resolvent estimates are notenough for the boundedness of an calculus [7, Section 9].
However, some of the central positive results in that theory
—
McIntosh
’
s theorem [15], the Boyadzhiev
–
deLaubenfels theorem [4] and the Hieber
–
Pr
ü
ss theorem converge, i.e., be a UMD space (or better ,a Hilbert space). Furthermore, even if the
Functional Calculus for the Series of Semigroup Generators Via Transference
1
G l o b a l J o u r n a l o f S c i e n c e F r o n t i e r R e s e a r c h V o l u m e X V I I I I s s u e e r s i o n I V V
Y e a r
2 0 1 8
38
( F )
©2018 Global Journals
N
otes
In [9] and its sequal paper [17] the functional calculus for a semigroup generator is constructed in a rather unconventional way using ideas from systems theory. However, a closer inspection reveals that transference (i.e., the factorization over a Fourier multiplier) is present there as well, hidden in the very construction of the functional calculus.
N
otes
a)
Notation and terminology
We write for the natural numbers and for the nonnegative reals.The letters X and Y are used to denote Banach spaces over the complex number field. The space of bounded linear operators on X is denoted by
For a closed sequence of operators
on X
their domains are denoted by dom
( )and their ranges by ran( ). The spectrums of are
( )and the resolvent sets For all the operatorsis the resolvents of at .
:= {1, 2, . . .}
+
:= [0, ∞)
(X)
ρ(
) :=
\σ(
)
ρ(
)
,
) := (
−
)
−1
(X)
For
,
(
; X) is the Bochner space of equivalence classes of Xvalued (1+
–Lebesgue integrable functions on
.The Hölder
conjugate of (1+
) is
(
)
. The norm
is usually denoted by
‖‖
on
For and , we let
By M
( )(resp.
, we denote the
space of complexvalued Borel measures on with the total variation norm, and we write for the distributions on of the form for some Then
is a Banach algebra under convolution with the series of
norms
(
):=
+
)) M(
(resp.
+
)
(
+
)
+
(ds) =
(ds)
M(
+
).
(
+
).
∑
∑
For
(
+
) , we let supp(
)
be the topological support of
, functions
such that
(
+
) are
usually identiﬁed with its associated measure
s
(
+
) given by
(ds)=
(s) ds
. Functions and measures deﬁned on
+
are identiﬁed with their extensions to
by setting them equal to zero outside
+
. For an open subset Ω ≠
∅
of
we , let
(Ω) be the space of bounded holomorphic functions on Ω, unital Banach algebra with respect to theseries ofnorms
∑
∑
∑

We shall mainly consider the case where Ω is equal to a right halfplanes
{

}
for some
(we write
+
for
).For convenience we abbreviate the coordinate functions
simply by the letters
. Under this convention,
=
(
) for functions
deﬁned on some domain Ω
⊆
.
The starting point of the present work was the article [19] by Hans Zwart, in
particular[19,Theorem 2.5.2]. There is shown that one has an estimate (1.1) with
as
. (The case
) in (1.2) is an immediate consequence , however , that case is essentiallytrivial )
Functional Calculus for the Series of Semigroup Generators Via Transference
1
G l o b a l J o u r n a l o f S c i e n c e F r o n t i e r R e s e a r c h V o l u m e X V I I I I s s u e e r s i o n I V V
Y e a r
2 0 1 8
39
( F )
©2018 Global Journals
For
and
(
+
) , we let
̂
(
) ∩ C(
̅
),
̂(
)
∫
be the Laplace–Stieltjes transforms of
.
II.
F
ourier
M
ultipliers And
F
unctional
C
alculus
We briefly discuss some of the concepts thatwill be used in what follows (see [20]).
a)
Fourier multipliers
We shall need results from Fourier analysis as collected in [7.Appendix E]. Fix a Banach space X and let and . Then m is a bounded Fourier multiplier if thereexists such that
(
;
(X))
(
; X)
1
(
)
(
)
(
)
for each Xvalued Schwartz functions . In this case the mappings, extends uniquely to bounded sequence of operators on if and on if
=
∞
. We let be the norms of the operators , and let (X) be the unital Banach algebra of all bounded Fourier multipliers, endowed with the norm For and we let
(
; X)
< ∞
(
;X)
‖‖
(
; X)
‖‖
.
,


be the analytic
(
; X)Fourier multiplier algebrason
, endowed theseries ofnorms
Here
(
+
)
(
) denotes the trace of the holomorphic functions
on the boundary∂
=
+ i
. By classical Hardy space theories,
(
)
́
(́
)
exists for almost all s
, with
∑
=
∑
.
Remark 2.1
(Important ). To simplify notation we sometimes omit the reference to the Banach space X and write instead of whenever it is convenient .
(
)
(
) ,
∫
The Fourier transform of an Xvalued tempered distribution Φ on
is denoted by
Φ.For instance, if
M(
) then
(
) aregiven by
Functional Calculus for the Series of Semigroup Generators Via Transference
1
G l o b a l J o u r n a l o f S c i e n c e F r o n t i e r R e s e a r c h V o l u m e X V I I I I s s u e e r s i o n I V V
Y e a r
2 0 1 8
40
( F )
©2018 Global Journals
N
otes