Legal forms

Functional Calculus for the Series of Semigroup Generators Via Transference

Description
Functional Calculus for the Series of Semigroup Generators Via Transference
Categories
Published
of 29
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  © 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 Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/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: 2249-4626  & Print ISSN: 0975-5896   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. GJSFR-F 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 so-called (strong) m-bounded 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 so-called 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 so-called strong) m-bounded 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 so-called 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 vector-valued singular integrals, best seen in the theory of sectorial (or strip-type) 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. e-mail: shawgy2020@gmail.com  Author     :Upper Nile University, Faculty of Education, Department of Mathematics, South Sudan. e-mail: s.j.u.khafiir@gmail.com Author    :Ministry of Education, Department ofMathematics , Sultanate Oman.e-mail: shibphdoman@gmail.com Author    :Ministry ofEducation and Higher Education, Department ofMathematics, Qatar.e-mail: mlgandi2@gmail.com Author   ¥  : Jazan University, Faculty of Science, Department of Mathematics, Kingdom of Saudi Arabia.e-mail: rayya2004@hotmail.com Author   § :Princess Nourah bint Abdulrahman University, College of Science, Department of Mathematics, Kingdom of Saudi Arabia.e-mail: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 vector-valued 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 strongm-bounded   -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 half-planes . The abstract theory of (holomorphic) functional calculi on half-planes 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 X-valued (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 complex-valued 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 half-planes    {  |    } 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 X-valued 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 X-valued 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
Search
Similar documents
View more...
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks