Algebraic Analysis
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For all R comma R to2000 the powerMathematics of prime Subject in Classification : 1 - 0 , 1 A 55 , 1 A 60 , 0 A 20 . nnoindenthline factKey words that and the phrases differential: algebraic analysis operator , right invertible $ D = operatorn f r a , c operationalf d gf d calculus t g .$ The is right invertible in several function spaces . 2000 Mathematicspaper is Subject in final formClassification and no version : 1 hyphen of it will 0 commabe published 1 A 55 elsewhere comma . 1 A 60 comma 0 A 20 period n hspaceKey words∗fn f and i l l phrasesg Foundations : algebraic of analysis Algebraic comma rightAnalysis invertible are operator the following comma operational : Let calculus $ L period ( X ) $ [47] beThe the paper set is of in finalall form linear and no version of it will be published elsewhere period open square bracket 47 closing square bracket nnoindent operators with domains and ranges in a linear space $ X ( $ in general , without any topology ) nnoindent over a field $ F $ of scalars with characteristic zero and let $ L f 0 g ( X ) = nf A n in L ( X ) :$dom$A =$ nnoindent $ X ng . $ n h f i l l Let $R ( X ) $ be the set of all right invertible operators in $ L ( X ) . $ n h f i l l Let $ D n in R ( X ) . $ n h f i l l Let nnoindent $ R f D g nsubset L f 0 g ( X ) $ be the set of all right inverses for $D ,$ i .e $. DR = I ($ identityoperator)if nnoindent $ R n in R f D g ( $ i . e . n h f i l l the Leibniz − Newton formula holds $ : n f r a c f d gf d t g n int ^f t g f a g f ( s ) ds = f ( t )$ forallfunctions $ f $ nnoindent from the space under consideration ) . n h f i l l Moreover , dom $D = RX noplus $ ker $ D . $ n h f i l l Forall $R , R^f nprime g n in $ n begin f a l i g n ∗g n r u l e f3emgf0.4 pt g nendf a l i g n ∗g n centerline f2000 Mathematics Subject Classification : 1 − 0 ,1A55 ,1A60 ,0A20 . g nnoindent Key words and phrases : algebraic analysis , right invertible operator , operational calculus . The paper is in final form and no version of it will be published elsewhere . n [ [ 47 ] n ] 48 .. D period PRZEWORSKA hyphen ROLEWICZ nnoindentR sub D comma48 nquad x in XD comma . PRZEWORSKA Rx minus R− toROLEWICZ the power of prime x in ker D comma i period e period the difference of two primitives of x is a constant period Let nnoindentF sub D = open$ R bracef D Fg in L, sub x 0 openn in parenthesisX , X closing Rx parenthesis− R ^f : n Fprime to theg powerx of 2n in = F$ semicolon ker $ FX D =, kernel $ i D .