U.P.B. Sci. Bull., Series A, Vol. 74, Iss. 3, 2012 ISSN SCALAR OPERATORS. Mariana ZAMFIR 1, Ioan BACALU 2

Size: px
Start display at page:

Download "U.P.B. Sci. Bull., Series A, Vol. 74, Iss. 3, 2012 ISSN SCALAR OPERATORS. Mariana ZAMFIR 1, Ioan BACALU 2"

Transcription

1 U.P.B. ci. Bull., eries A, Vol. 74, Iss. 3, 212 IN A CALAR OPERATOR Mariana ZAMFIR 1, Ioan BACALU 2 În această lucrare studiem o clasă nouă de operatori numiţi -scalari. Aceştia apar în mod natural, ca o generalizare a operatorilor A -scalari introduşi în [4] şi se definesc cu ajutorul homomorfismelor spectrale (funcţii - spectrale), care, la rândul lor, sunt generalizări ale homomorfismelor spectrale (funcţii A -spectrale) din [4]. Pe parcursul lucrării sunt prezentate unele proprietăţi ale acestor operatori, dintre care, cea mai semnificativă este aceea cum că sunt -decompozabili, în sensul din [1]. This paper is devoted to the study of a new class of operators, called - scalar operators, naturally appearing as a generalization of the A -scalar operators [4]. This study uses the concept of -spectral homomorphism which is also the generalization of the spectral homomorphism ( A -spectral function) studied in [4]. Furthermore, we prove some properties concerning the -scalar operators; their main quality is that of being decomposable [1]. Keywords: scalar ( A -scalar); spectral ( A -spectral); A -scalar operator; spectral function; restriction and quotient of an operator Introduction Let X be a Banach space, let B ( X ) be the algebra of all linear bounded operators on X and let C be the complex plane. If T B ( X) and Y X is a (closed) subspace invariant to T, let us denote by T Y the restriction of T to Y, respectively by T the operator induced by T in the quotient space X = X / Y. In what follows, by subspace of X we understand a closed linear manifold of X. Recall that Y is a spectral maximal space of T if it is an invariant subspace to T such that for any other subspace Z X, also invariant to T, the inclusion 1 Assistant Prof., Departament of Mathematics and Computer cience, Technical University of Civil Engineering of Bucharest, Romania, zamfirvmariana@yahoo.com 2 Associated Prof., Faculty of Applied ciences, University POLITEHNICA of Bucharest, Romania

2 9 Mariana Zamfir, Ioan Bacalu σ ( T Z) σ( T Y) implies Z Y i n is i ([6]). A family of open sets G { G } = 1 n said to be an covering of the closed set σ if G Gi σ and i = 1 Gi = ( i = 1,2,..., n) (where is also closed) ([11]). T X σ T compact) The operator B ( ) is decomposable (where ( ) if for any finite open covering G { G i} n i = 1 of σ ( T ), there is a system n Y { Yi} i = 1 of spectral maximal spaces of T such that σ ( T Y) G, n σ ( T Yi) Gi ( i = 1,2,..., n) and X = Y + Yi ([1]). If =, then T is i= 1 decomposable ([6]). An open set Ω is said to be a set of analytic uniqueness for T B ( X) if for any open set ω Ω and any analytic function f : ω X satisfying the equation ( I T) f ( ), it follows that f ( ) in ω ([1]). For T B ( X ) there is a unique maximal open set Ω T of analytic uniqueness (2.1., [1]). We denote by T = Ω T = \ ΩT the analytic spectral residuum of T. For x X, a point is in δ T ( x ) if in a neighborhood V of there is at least an analytic X -valued function f x (called T -associated to x ) such that ( μi T) fx ( μ) x, for all μ V. We shall put γt ( x) = δt ( x), ρt ( x) = δt ( x) Ω T, σt ( x) = ρt ( x) = γt ( x) T and XT ( F) = { x X; σ T ( x) F}, where T F ([1], [11]). An operator T B ( X) is said to have the single-valued extension property if for any analytic function f : ω X (where ω open), with ( I T) f ( ), it results that ( ) extension property if and only if T = ; then we have σt( x) γt( x) is in ρt ( x) = δt ( x) an unique analytic function ( ) any x X ([1]). We recall that if T B ( X), T, T F and XT ( ) closed, for F closed, then XT ( ) σ ( T X ( F) ) F ([1], Propositions 2.4. and 3.4.). T f ([5]). T has the single-valued = and there x, T associated to x, for F is F is a spectral maximal space of T and

3 scalar operators 91 We remind that a set A is of dimension (totally disconnected) if any subset of it is both open and closed in the relative topology of A, or, equivalently, if any connected component of A is reduced to a single point ([9]). 2. Preliminaries Definition 2.1. Let Ω be a set of the complex plane C and let Ω be a compact subset. An algebra A of C -valued functions defined on Ω is called n -normal if for any finite open -covering G { G i} i = 1 of Ω, there are the functions f, fi A ( 1 i n) such that: 1) f ( Ω) [,1], fi ( Ω) [,1] ( 1 i n) ; 2) supp( f ) G, supp( fi) Gi ( 1 i n) ; n 3) f + fi = 1 on Ω, i= 1 f A is defined as: ( ) { ( ) } where the support of supp f = μ Ω; f μ. Definition 2.2. An algebra A of C -valued functions defined on Ω is called -admissible if: 1),1 A (where,1 denote the functions f ( ), f ( ) 1); 2) A is -normal; 3) for any f A and any ξ supp( f ), the function f ξ : Ω C belongs to A. ( ) f,for Ω\ f ξ ( ) = ξ, for Ω { ξ} { ξ} A -scalar if there are : f on, Definition 2.3. An operator T B ( X) is said to be A and an algebraic homomorphism U A B ( X) an -admissible algebra such that U 1 = I and U T respectively is the identical function ( ) f = (where 1 is the function ( ) 1 on ). The application U is called -spectral homomorphism ( -spectral function or calculus) for T. A -functional

4 92 Mariana Zamfir, Ioan Bacalu If =, then we put A = A and we obtain an A -spectral function and an A -scalar operator ([4]). Definition 2.4. A subspace Y of X is said to be invariant with respect to an -spectral function U : B ( X) if U f Y Y, for any f A. Definition 2.5. The support of an -spectral function U is denoted by supp( U ) and it is defined as the smallest closed subset of Ω such that U f = for any f A with supp( f ) supp( U ) =. We recall several important proprieties of an A -spectral function U (see [4]), because we want to obtain similar properties for an -spectral function: 1) supp( U) = σ ( ), where is the identical function f ( ) U 2) U has the single-valued extension property; 3) σu ( U f x) supp( f ), for any f A and x X ; 4) σ U ( x) supp( f ) = U f ( x) = ; 5) x XU ( F) U f ( x) =, for any f A with property: supp( f ) F =, F Ω closed; 6) U is decomposable. 3. -spectral functions and -scalar operators ; Theorem 3.1. Let T ( X) B be an -scalar operator and let U be an -spectral function for T. Then we have the relations: supp U σ T σ T supp U. ( ) ( ) and ( ) ( ) Proof. Let us consider f A such that supp( f ) ( ( T) ) ξ supp( f ) and is the identical function f ( ), then we have ( ξi T) U f = ( ξi U ) U f = U( ξ ) f = U f whence U f ξ ξ ξ =R ( ξ, T) U ξ f, for ξ ρ( T) supp( f ) The function F : C B ( X ). σ =. If

5 scalar operators 93 ( ξ, T) U f, for ξ ρ( T), for ξ supp( f ) R F ( ξ ) = U fξ is entire and lim F ( ξ ) =, ξ therefore F. It follows that U f ξ = on supp( f ) and U f =, accordingly supp( U) σ ( T). Let now ξ supp( U), let let W be an open neighborhood of supp( U) algebra A being -normal, there is a function f μ W and f ( μ ) = for whence μ V ξ V ξ be an open neighborhood of ξ and such that Vξ W =. The. Therefore ( f ) ( supp( U) ) supp 1 =, U1 f =, hence U f = I. A with ( ) 1 f μ = for It follows that U f ( ξi T) = U f ( ξi U ) ξ ξ = = ( ξ I U ) U f == U ( ) f U f I ξ ξ = = ξ therefore we finally have ξ σ( U ) = σ( T) and hence σ ( T) supp( U). Lemma 3.1. If ( I U ) x =, with x and f A with f ( ) = = c, for G Ω, where G is an open neighborhood of, then U f x = cx. Proof. From the equality U x = x, with x, it results that is the eigenvalue of U corresponding to the eigenvector x, hence σ p ( U ) σ( U ) supp( U ) Ω whence G Ω (where σ p ( U ) is the point spectrum of U, i.e. the set of all eigevalues of U ). If we denote g = f c, then we can write U f x cx Ugx U x U g I U x and consequently U f x = cx. ( ) g ( )

6 94 Mariana Zamfir, Ioan Bacalu A -spectral function for T B ( X) Theorem 3.2. If U is an, then T. Moreover, if dim( ) 1, we have T = (i.e. T has the single-valued extension property). Proof. Let f : Gf X ( G f C open, Gf = ) be an analytic function such that ( ξi T) f ( ξ). Let us suppose that there is ξ G Gf ( G is a connected component of G f ) with f ( ξ ). Then f ( ξ ), for ξ D G, where D is a disk with center in ξ. If D = D( ξ, r) and D = D ( ξ, r ), < r < r, the algebra A being -normal, it results that there is a function g A such that 1, for ξ Ω D g ( ξ ) =., for ξ Ω\ ( Ω D ) According to Lemma 3.1, we have f ( ξ),forξ G D Ug f ( ξ ) =., for ξ G \ D By analytic extension, it results that Ug f ( ξ ) =, for ξ G, hence ( ξ) ( ξ) f = Ug f =, for ξ G and thus we have obtained a contradiction. Consequently, f ( ξ ) = on f Proposition 3.1. If U is an ( U f x) supp( f ) G, therefore ΩT γ T, for any f A and x X., i.e. T. A -spectral function for T B ( X), then Moreover, if supp( f ) σ T U f x supp f. Proof. For any ξ supp( f ), we have f ξ A and the X -valued, then ( ) ( ) function ξ U f x is analytic. Consequently, ξ ( ξ ) ( ξ ) I T U f x = I U U f x = U f x, therefore ξ δt ( U f x), hence T ( U f x) supp( f ) Moreover, for ξ γ. f A with supp( f ) ξ, it follows that ( U x) = ( U x) ( U x) supp( f ) σ γ γ T f T T f T f.

7 scalar operators 95 B be an -scalar operator having an -spectral function U and let Y be a spectral maximal space of T such that Y = XT ( F), for F C closed, F. Then T Y is an -scalar operator. Proof. A spectral maximal space Y of T is also an ultrainvariant subspace to T, therefore Y is invariant to U f, for any f A. It is easy to prove that U Y : B ( Y) is an -spectral function for T Y B ( Y), hence T Y is -scalar. Theorem 3.3. Let T B ( X) be an -scalar operator and let U be an -spectral function for T. Then T is an -decomposable operator. Poof. According to some results studied in [1] and [2], it is enough to show that T is ( 1, )-decomposable, i.e. for any open ( 1, ) -covering { G, G 1 } of the complex plane C, there is a system { Y, Y 1 } of invariant subspaces to T such that σ ( T Y) G, σ ( T Y1) G1 and X = Y + Y1. ( 1, ) -covering { G, G 1 } of the complex plane C is also an -covering of Ω, thus there are the functions f, f1 A such that f ( ), f1( ) 1, supp( f ) G, supp( f1) G1, f + f1 = 1 on Ω. For every x X we have the relation not x = Ux 1 = U f x+ U f x= y 1 + y1. Let F C be closed such that F = or F and E ( F) = { ker U f ; f, supp( f ) F = } = = x X; f, supp( f ) F = U f x =. Proposition 3.2. Let T ( X) { } Obviously, E ( F ) are closed subspaces of X invariant to T. Let us show that σ ( T E ( F) ) F. For ξ \ F f A such that f = 1 on F neighborhood V of ξ (when F, there is a function Ω and f = on V Ω, for an open suitable = and ξ, we must take V ). Therefore supp( 1 f ) F Ω=, U1 f x ( I U f ) x consequently U f x = x or U f E ( F) = I. But ( ξ ) fξ ( ) = f ( ) ( Ω ), hence = =, for any x ( F ) E ;

8 96 Mariana Zamfir, Ioan Bacalu U ( ξ ) f = U f = ( ξ I T ) U ξ f. ξ In the last equality, if we consider the restriction to E ( F ), it results that ρ( T E ( F) ) and thus σ ( T E ( F) ) F. Then, for any x X, it follows ξ that E ( supp( )) E ( ) f ( supp( )) ( 1) 1 ( supp( )) ( supp( )) y = U f x f G y1 = U x E f1 E G and X = E f + E f1 and on account of the above results we deduce that T is -decomposable. Remark 3.1. If T B ( X) is an -scalar operator and U is an - spectral function for T, then T is -decomposable, hence σ ( T) and on supp U σ T. account of Theorem 3.1, it results that ( ) ( ) Example 3.1. Let T B ( X) be an A -scalar operator and let U : A B ( X) be its A -spectral function; let also Y be a closed invariant subspace to T, which is not invariant with respect to U (i.e. there is a function f A such that Y is not invariant to U f ). Then the quotient operator T induced by T in the quotient space X = X / Y is an A -scalar operator, where = σ ( T Y) and A is the subalgebra of A composed by all functions f A which have one of the following properties: supp f = ; (1) ( ) (2) supp( f ) and U f Y Y. If the functions f A satisfy the condition (1), then for y Y, we have the relation supp( f ) σ T ( y) =. Accordingly U f y = and U f makes sense, where U f is the operator induced by U f B ( X) in the quotient space X = X / Y. If the functions f A satisfy the condition (2), then the functions g = 1 f verify the relation supp( g) =. Therefore for y Y we have Ug y =, i.e. U f y y =. ince for ξ supp ( f ) σ( T Y), Y is invariant to the resolvent

9 scalar operators 97 function R(, T ) ξ of T, it results that f ξ A and hence the operators U f ξ make sense also in this case. It is easy to verify that the application U : ( X ) f A, is an -spectral function for T B ( X ) ( ) = f, U f U -scalar. U f, A B defined by, therefore T is Example 3.2. Let T B ( X ) be an A -scalar operator, let U : A B ( X) be its A -spectral function and let also Y be a closed invariant subspace to T, which is not invariant to U. Then the restriction T Y is an -scalar operator, where = σ ( T ) and A is an -admissible subalgebra of A composed by all functions f which have one of the following properties: supp f = ; (1) ( ) (2) supp( f ). The functions f A with the properties supp( f ) not belong to the algebra A. and supp( f ), do It can be easily to verify that the restriction U : B ( Y) defined by U ( f ) = U f A, is an A -spectral function for T Y B ( Y), A f, therefore T Y is -scalar. Remark 3.2. If T B ( X ) is an A -scalar operator, U : A B ( X) is an A -spectral function for T and Y is a closed subspace invariant to both T and U, then the restriction T Y and the quotient T induced by T in the quotient space X = X / Y are A -scalar operators. If σ ( T) is totally disconnected (i.e. dim = ), then the topology τ of induces a topology τ of countable metric space on. It is shown that τ has a basis composed by spectral sets for T. According to the Lindelöf theorem, we obtain for τ a countable basis ( δ n ) of spectral sets for T (a set δ is n δ σ T σ T ) and said to be a spectral set for T if ( ) is both open and closed in ( ) we define the projector E ( δ ) by the relation

10 98 Mariana Zamfir, Ioan Bacalu 1 E( δ ) = R(, T) d 2π i C for any δ in the field Σ of all spectral sets, where C is a Jordan curve δ σ T and does not contain inside another (admissible contour) containing ( ) points of σ ( T ). Example 3.3. If an operator T B ( X ) has the spectrum σ ( T) with dim 1 and δ = σ( T) \, with dimδ = for T, then, according to Example 1.2, Chapter 3, [4], the operator T E( ) A -scalar, with σ ( T E( δ) X) = δ and the operator ( ) equal to. It is easy to verify that T is -scalar. = δ,, and δ being spectral sets δ X is T E X has the spectrum R E F E R E N C E [1]. I. Bacalu, ome properties of decomposable operators, Rev. Roum. Math. Pures et Appl., 21, 1976, pp [2]. I. Bacalu, Descompuneri spectrale reziduale (Residually spectral decompositions), t. Cerc. Mat. I (198), II (198), III (1981). [3]. I. Bacalu, pectral Decompositions, Ed. Politehnica Press, Bucharest, 28. [4]. I. Colojoară and C. Foiaş, Theory of generalized spectral operators, Gordon Breach, cience Publ., New York-London-Paris, [5]. N. Dunford and J.T. chwartz, Linear Operators, Interscience Publishers, New-York, Part I (1958), Part II (1963), Part III (1971). [6]. C. Foiaş, pectral maximal spaces and decomposable operators in Banach paces, Archiv der Math., 14, 1963, pp [7]. K.B. Laursen and M.M. Neumann, An Introduction to Local pectral Theory, London Math. oc. Monographs New eries, Oxford Univ. Press., New-York, 2. [8]. B. Nagy, Residually spectral operators, Acta Math. Acad. ci. Hungar, 35, 198, pp [9]. M. Nicolescu, Funcţii reale şi elemente de analiză funcţională (Real functions and functional analysis theory), Ed. Did. şi Ped., Bucureşti, [1]. F.H. Vasilescu, Residually decomposable operators in Banach spaces, Tôhoku Mat. Jurn., 21, 1969, pp [11]. F.H. Vasilescu, Analytic Functional Calculus and pectral Decompositions, D. Reidel Publishing Company, Dordrecht; Editura Academiei, Bucharest, 1982.

Sung-Wook Park*, Hyuk Han**, and Se Won Park***

Sung-Wook Park*, Hyuk Han**, and Se Won Park*** JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS Sung-Wook Park*, Hyuk Han**,

More information

THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS THOMAS L. MILLER AND VLADIMIR MÜLLER Abstract. Let T be a bounded operator on a complex Banach space X. If V is an open subset of the complex plane,

More information

Spectrally Bounded Operators on Simple C*-Algebras, II

Spectrally Bounded Operators on Simple C*-Algebras, II Irish Math. Soc. Bulletin 54 (2004), 33 40 33 Spectrally Bounded Operators on Simple C*-Algebras, II MARTIN MATHIEU Dedicated to Professor Gerd Wittstock on the Occasion of his Retirement. Abstract. A

More information

LINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO

LINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (757 763) 757 LINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO Hassane Benbouziane Mustapha Ech-Chérif Elkettani Ahmedou Mohamed

More information

Local Spectral Theory for Operators R and S Satisfying RSR = R 2

Local Spectral Theory for Operators R and S Satisfying RSR = R 2 E extracta mathematicae Vol. 31, Núm. 1, 37 46 (2016) Local Spectral Theory for Operators R and S Satisfying RSR = R 2 Pietro Aiena, Manuel González Dipartimento di Metodi e Modelli Matematici, Facoltà

More information

z -FILTERS AND RELATED IDEALS IN C(X) Communicated by B. Davvaz

z -FILTERS AND RELATED IDEALS IN C(X) Communicated by B. Davvaz Algebraic Structures and Their Applications Vol. 2 No. 2 ( 2015 ), pp 57-66. z -FILTERS AND RELATED IDEALS IN C(X) R. MOHAMADIAN Communicated by B. Davvaz Abstract. In this article we introduce the concept

More information

arxiv: v1 [math.sp] 29 Jan 2018

arxiv: v1 [math.sp] 29 Jan 2018 Essential Descent Spectrum Equality Abdelaziz Tajmouati 1, Hamid Boua 2 arxiv:1801.09764v1 [math.sp] 29 Jan 2018 Sidi Mohamed Ben Abdellah University Faculty of Sciences Dhar El Mahraz Laboratory of Mathematical

More information

ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction

ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction J. Korean Math. Soc. 41 (2004), No. 4, pp. 617 627 ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION In Ho Jeon and B. P. Duggal Abstract. Let A denote the class of bounded linear Hilbert space operators with

More information

On the simplest expression of the perturbed Moore Penrose metric generalized inverse

On the simplest expression of the perturbed Moore Penrose metric generalized inverse Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated

More information

SVEP and local spectral radius formula for unbounded operators

SVEP and local spectral radius formula for unbounded operators Filomat 28:2 (2014), 263 273 DOI 10.2298/FIL1402263A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat SVEP and local spectral radius

More information

arxiv: v1 [math.rt] 14 Nov 2007

arxiv: v1 [math.rt] 14 Nov 2007 arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof

More information

Almost Invariant Half-Spaces of Operators on Banach Spaces

Almost Invariant Half-Spaces of Operators on Banach Spaces Integr. equ. oper. theory Online First c 2009 Birkhäuser Verlag Basel/Switzerland DOI 10.1007/s00020-009-1708-8 Integral Equations and Operator Theory Almost Invariant Half-Spaces of Operators on Banach

More information

LOCAL SPECTRA: CONTINUITY AND RESOLVENT EQUATION

LOCAL SPECTRA: CONTINUITY AND RESOLVENT EQUATION LOCAL SPECTRA: CONTINUITY AND RESOLVENT EQUATION S. V. Djordjevic and José A. Canavati Comunicación Técnica No I-02-07/23-04-2002 (MB/CIMAT) Local spectra: continuity and resolvent equation S.V. Djordjević

More information

Totally supra b continuous and slightly supra b continuous functions

Totally supra b continuous and slightly supra b continuous functions Stud. Univ. Babeş-Bolyai Math. 57(2012), No. 1, 135 144 Totally supra b continuous and slightly supra b continuous functions Jamal M. Mustafa Abstract. In this paper, totally supra b-continuity and slightly

More information

DOMINANT TAYLOR SPECTRUM AND INVARIANT SUBSPACES

DOMINANT TAYLOR SPECTRUM AND INVARIANT SUBSPACES J. OPERATOR THEORY 61:1(2009), 101 111 Copyright by THETA, 2009 DOMINANT TAYLOR SPECTRUM AND INVARIANT SUBSPACES C. AMBROZIE and V. MÜLLER Communicated by Nikolai K. Nikolski ABSTRACT. Let T = (T 1,...,

More information

1994, Phd. in mathematics, Faculty of mathematics and informatics, University of Bucharest.

1994, Phd. in mathematics, Faculty of mathematics and informatics, University of Bucharest. DUMITRU POPA Date of birth: 21. 07. 1956, com. Cosimbesti, jud. Ialomita. Studies: 1976-1981, Faculty of mathematics and informatics, University of Bucharest. 1994, Phd. in mathematics, Faculty of mathematics

More information

ON µ-compact SETS IN µ-spaces

ON µ-compact SETS IN µ-spaces Questions and Answers in General Topology 31 (2013), pp. 49 57 ON µ-compact SETS IN µ-spaces MOHAMMAD S. SARSAK (Communicated by Yasunao Hattori) Abstract. The primary purpose of this paper is to introduce

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

Fréchet algebras of finite type

Fréchet algebras of finite type Fréchet algebras of finite type MK Kopp Abstract The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional.

More information

MATRIX LIE GROUPS AND LIE GROUPS

MATRIX LIE GROUPS AND LIE GROUPS MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either

More information

Spectral theory for compact operators on Banach spaces

Spectral theory for compact operators on Banach spaces 68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy

More information

SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR

SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR E. A. ALEKHNO (Belarus, Minsk) Abstract. Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ ew (T ) of the

More information

ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović

ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS Srdjan Petrović Abstract. In this paper we show that every power bounded operator weighted shift with commuting normal weights is similar to a contraction.

More information

SOME ABSOLUTELY CONTINUOUS REPRESENTATIONS OF FUNCTION ALGEBRAS

SOME ABSOLUTELY CONTINUOUS REPRESENTATIONS OF FUNCTION ALGEBRAS Surveys in Mathematics and its Applications ISSN 1842-6298 Volume 1 (2006), 51 60 SOME ABSOLUTELY CONTINUOUS REPRESENTATIONS OF FUNCTION ALGEBRAS Adina Juratoni Abstract. In this paper we study some absolutely

More information

s P = f(ξ n )(x i x i 1 ). i=1

s P = f(ξ n )(x i x i 1 ). i=1 Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological

More information

Upper triangular forms for some classes of infinite dimensional operators

Upper triangular forms for some classes of infinite dimensional operators Upper triangular forms for some classes of infinite dimensional operators Ken Dykema, 1 Fedor Sukochev, 2 Dmitriy Zanin 2 1 Department of Mathematics Texas A&M University College Station, TX, USA. 2 School

More information

Compact operators on Banach spaces

Compact operators on Banach spaces Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact

More information

JOINT TOPOLOGICAL ZERO DIVISORS FOR A REAL BANACH ALGEBRA H. S. Mehta 1, R.D.Mehta 2, A. N. Roghelia 3

JOINT TOPOLOGICAL ZERO DIVISORS FOR A REAL BANACH ALGEBRA H. S. Mehta 1, R.D.Mehta 2, A. N. Roghelia 3 Mathematics Today Vol.30 June & December 2014; Published in June 2015) 54-58 ISSN 0976-3228 JOINT TOPOLOGICAL ZERO DIVISORS FOR A REAL BANACH ALGEBRA H. S. Mehta 1, R.D.Mehta 2, A. N. Roghelia 3 1,2 Department

More information

Lemma 1.3. The element [X, X] is nonzero.

Lemma 1.3. The element [X, X] is nonzero. Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

More information

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem 56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi

More information

arxiv: v1 [math.fa] 14 Jul 2018

arxiv: v1 [math.fa] 14 Jul 2018 NOTES ON (THE BIRMAK KREĬN VISHIK THEORY ON) SELFADJOINT EXTENSIONS OF SEMIBOUNDED SYMMETRIC OPERATORS arxiv:1807.05363v1 [math.fa] 14 Jul 018 TIBERIU CONSTANTINESCU AND AURELIAN GHEONDEA Abstract. We

More information

Generalized metric properties of spheres and renorming of Banach spaces

Generalized metric properties of spheres and renorming of Banach spaces arxiv:1605.08175v2 [math.fa] 5 Nov 2018 Generalized metric properties of spheres and renorming of Banach spaces 1 Introduction S. Ferrari, J. Orihuela, M. Raja November 6, 2018 Throughout this paper X

More information

Some icosahedral patterns and their isometries

Some icosahedral patterns and their isometries Some icosahedral patterns and their isometries Nicolae Cotfas 1 Faculty of Physics, University of Bucharest, PO Box 76-54, Postal Office 76, Bucharest, Romania Radu Slobodeanu 2 Faculty of Physics, University

More information

On α-embedded subsets of products

On α-embedded subsets of products European Journal of Mathematics 2015) 1:160 169 DOI 10.1007/s40879-014-0018-0 RESEARCH ARTICLE On α-embedded subsets of products Olena Karlova Volodymyr Mykhaylyuk Received: 22 May 2014 / Accepted: 19

More information

WEYL S THEOREM FOR PAIRS OF COMMUTING HYPONORMAL OPERATORS

WEYL S THEOREM FOR PAIRS OF COMMUTING HYPONORMAL OPERATORS WEYL S THEOREM FOR PAIRS OF COMMUTING HYPONORMAL OPERATORS SAMEER CHAVAN AND RAÚL CURTO Abstract. Let T be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property dim

More information

Bulletin of the Iranian Mathematical Society

Bulletin of the Iranian Mathematical Society ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Special Issue of the Bulletin of the Iranian Mathematical Society in Honor of Professor Heydar Radjavi s 80th Birthday Vol 41 (2015), No 7, pp 155 173 Title:

More information

AN EXTENSION OF THE NOTION OF ZERO-EPI MAPS TO THE CONTEXT OF TOPOLOGICAL SPACES

AN EXTENSION OF THE NOTION OF ZERO-EPI MAPS TO THE CONTEXT OF TOPOLOGICAL SPACES AN EXTENSION OF THE NOTION OF ZERO-EPI MAPS TO THE CONTEXT OF TOPOLOGICAL SPACES MASSIMO FURI AND ALFONSO VIGNOLI Abstract. We introduce the class of hyper-solvable equations whose concept may be regarded

More information

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5 MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have

More information

FEEDBACK DIFFERENTIAL SYSTEMS: APPROXIMATE AND LIMITING TRAJECTORIES

FEEDBACK DIFFERENTIAL SYSTEMS: APPROXIMATE AND LIMITING TRAJECTORIES STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLIX, Number 3, September 2004 FEEDBACK DIFFERENTIAL SYSTEMS: APPROXIMATE AND LIMITING TRAJECTORIES Abstract. A feedback differential system is defined as

More information

Characterization of half-radial matrices

Characterization of half-radial matrices Characterization of half-radial matrices Iveta Hnětynková, Petr Tichý Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8, Czech Republic Abstract Numerical radius r(a) is the

More information

Gaussian automorphisms whose ergodic self-joinings are Gaussian

Gaussian automorphisms whose ergodic self-joinings are Gaussian F U N D A M E N T A MATHEMATICAE 164 (2000) Gaussian automorphisms whose ergodic self-joinings are Gaussian by M. L e m a ńc z y k (Toruń), F. P a r r e a u (Paris) and J.-P. T h o u v e n o t (Paris)

More information

Remarks on the Spectrum of Bounded and Normal Operators on Hilbert Spaces

Remarks on the Spectrum of Bounded and Normal Operators on Hilbert Spaces An. Şt. Univ. Ovidius Constanţa Vol. 16(2), 2008, 7 14 Remarks on the Spectrum of Bounded and Normal Operators on Hilbert Spaces M. AKKOUCHI Abstract Let H be a complex Hilbert space H. Let T be a bounded

More information

ON OPERATORS WHICH ARE POWER SIMILAR TO HYPONORMAL OPERATORS

ON OPERATORS WHICH ARE POWER SIMILAR TO HYPONORMAL OPERATORS Jung, S., Ko, E. and Lee, M. Osaka J. Math. 52 (2015), 833 847 ON OPERATORS WHICH ARE POWER SIMILAR TO HYPONORMAL OPERATORS SUNGEUN JUNG, EUNGIL KO and MEE-JUNG LEE (Received April 1, 2014) Abstract In

More information

H-convex Riemannian submanifolds

H-convex Riemannian submanifolds H-convex Riemannian submanifolds Constantin Udrişte and Teodor Oprea Abstract. Having in mind the well known model of Euclidean convex hypersurfaces [4], [5] and the ideas in [1], many authors defined

More information

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify

More information

A generic property of families of Lagrangian systems

A generic property of families of Lagrangian systems Annals of Mathematics, 167 (2008), 1099 1108 A generic property of families of Lagrangian systems By Patrick Bernard and Gonzalo Contreras * Abstract We prove that a generic Lagrangian has finitely many

More information

A note on the σ-algebra of cylinder sets and all that

A note on the σ-algebra of cylinder sets and all that A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In

More information

Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach

Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach Alberto Bressan Department of Mathematics, Penn State University University Park, Pa 1682, USA e-mail: bressan@mathpsuedu

More information

Moore-Penrose Inverse and Operator Inequalities

Moore-Penrose Inverse and Operator Inequalities E extracta mathematicae Vol. 30, Núm. 1, 9 39 (015) Moore-Penrose Inverse and Operator Inequalities Ameur eddik Department of Mathematics, Faculty of cience, Hadj Lakhdar University, Batna, Algeria seddikameur@hotmail.com

More information

On Dense Embeddings of Discrete Groups into Locally Compact Groups

On Dense Embeddings of Discrete Groups into Locally Compact Groups QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 4, 31 37 (2003) ARTICLE NO. 50 On Dense Embeddings of Discrete Groups into Locally Compact Groups Maxim S. Boyko Institute for Low Temperature Physics and Engineering,

More information

Assignment #10 Morgan Schreffler 1 of 7

Assignment #10 Morgan Schreffler 1 of 7 Assignment #10 Morgan Schreffler 1 of 7 Lee, Chapter 4 Exercise 10 Let S be the square I I with the order topology generated by the dictionary topology. (a) Show that S has the least uppper bound property.

More information

Peak Point Theorems for Uniform Algebras on Smooth Manifolds

Peak Point Theorems for Uniform Algebras on Smooth Manifolds Peak Point Theorems for Uniform Algebras on Smooth Manifolds John T. Anderson and Alexander J. Izzo Abstract: It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if

More information

Stone-Čech compactification of Tychonoff spaces

Stone-Čech compactification of Tychonoff spaces The Stone-Čech compactification of Tychonoff spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto June 27, 2014 1 Completely regular spaces and Tychonoff spaces A topological

More information

Operators shrinking the Arveson spectrum

Operators shrinking the Arveson spectrum Operators shrinking the Arveson spectrum A. R. Villena joint work with J. Alaminos and J. Extremera Departamento de Análisis Matemático Universidad de Granada BANACH ALGEBRAS 2009 STEFAN BANACH INTERNATIONAL

More information

Topological groups with dense compactly generated subgroups

Topological groups with dense compactly generated subgroups Applied General Topology c Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 85 89 Topological groups with dense compactly generated subgroups Hiroshi Fujita and Dmitri Shakhmatov Abstract.

More information

Isomorphism for transitive groupoid C -algebras

Isomorphism for transitive groupoid C -algebras Isomorphism for transitive groupoid C -algebras Mădălina Buneci University Constantin Brâncuşi of Târgu Jiu Abstract We shall prove that the C -algebra of the locally compact second countable transitive

More information

MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

More information

1 Invariant subspaces

1 Invariant subspaces MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another

More information

The projectivity of C -algebras and the topology of their spectra

The projectivity of C -algebras and the topology of their spectra The projectivity of C -algebras and the topology of their spectra Zinaida Lykova Newcastle University, UK Waterloo 2011 Typeset by FoilTEX 1 The Lifting Problem Let A be a Banach algebra and let A-mod

More information

Orthogonal Pure States in Operator Theory

Orthogonal Pure States in Operator Theory Orthogonal Pure States in Operator Theory arxiv:math/0211202v2 [math.oa] 5 Jun 2003 Jan Hamhalter Abstract: We summarize and deepen existing results on systems of orthogonal pure states in the context

More information

arxiv: v1 [math.fa] 5 Nov 2012

arxiv: v1 [math.fa] 5 Nov 2012 ONCE MORE ON POSITIVE COMMUTATORS ROMAN DRNOVŠEK arxiv:1211.0812v1 [math.fa] 5 Nov 2012 Abstract. Let A and B be bounded operators on a Banach lattice E such that the commutator C = AB BA and the product

More information

THE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS. Carlos Biasi Carlos Gutierrez Edivaldo L. dos Santos. 1. Introduction

THE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS. Carlos Biasi Carlos Gutierrez Edivaldo L. dos Santos. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 177 185 THE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS Carlos Biasi Carlos Gutierrez Edivaldo L.

More information

On the Local Spectral Properties of Weighted Shift Operators

On the Local Spectral Properties of Weighted Shift Operators Syracuse University SURFACE Mathematics Faculty Scholarship Mathematics 3-5-2003 On the Local Spectral Properties of Weighted Shift Operators Abdellatif Bourhim Abdus Salam International Centre for Theoretical

More information

VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES Bull. Austral. Math. Soc. 78 (2008), 487 495 doi:10.1017/s0004972708000877 VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES CAROLYN E. MCPHAIL and SIDNEY A. MORRIS (Received 3 March 2008) Abstract

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

SOME APPLICATIONS OF THE HILBERT TRANSFORM

SOME APPLICATIONS OF THE HILBERT TRANSFORM U.P.B. Sci. Bull. Series A, Vol. 71, Iss. 3, 2009 ISSN 1223-7027 SOME APPLICATIONS OF THE HILBERT TRANSFORM Ştefania Constantinescu 1 În acest articol, se dau unele proprietăţi ale transformării Hilbert,

More information

SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS. 1. Preliminaries

SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS. 1. Preliminaries Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 10.2478/v10127-011-0026-0 Tatra Mt. Math. Publ. 49 (2011), 71 79 SUMS AND PRODUCTS OF EXTRA STRONG ŚWIA TKOWSKI FUNCTIONS Paulina Szczuka ABSTRACT. In this paper we present

More information

BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction

BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of

More information

here, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional

here, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional 15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:

More information

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms (February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops

More information

COUNTABLY HYPERCYCLIC OPERATORS

COUNTABLY HYPERCYCLIC OPERATORS COUNTABLY HYPERCYCLIC OPERATORS NATHAN S. FELDMAN Abstract. Motivated by Herrero s conjecture on finitely hypercyclic operators, we define countably hypercyclic operators and establish a Countably Hypercyclic

More information

On a theorem of Ziv Ran

On a theorem of Ziv Ran INSTITUTUL DE MATEMATICA SIMION STOILOW AL ACADEMIEI ROMANE PREPRINT SERIES OF THE INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY ISSN 0250 3638 On a theorem of Ziv Ran by Cristian Anghel and Nicolae

More information

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 18. Category O for rational Cherednik algebra We begin to study the representation theory of Rational Chrednik algebras. Perhaps, the class of representations

More information

ON THE STABILITY OF THE QUASI-LINEAR IMPLICIT EQUATIONS IN HILBERT SPACES

ON THE STABILITY OF THE QUASI-LINEAR IMPLICIT EQUATIONS IN HILBERT SPACES Khayyam J. Math. 5 (219), no. 1, 15-112 DOI: 1.2234/kjm.219.81222 ON THE STABILITY OF THE QUASI-LINEAR IMPLICIT EQUATIONS IN HILBERT SPACES MEHDI BENABDALLAH 1 AND MOHAMED HARIRI 2 Communicated by J. Brzdęk

More information

Mañé s Conjecture from the control viewpoint

Mañé s Conjecture from the control viewpoint Mañé s Conjecture from the control viewpoint Université de Nice - Sophia Antipolis Setting Let M be a smooth compact manifold of dimension n 2 be fixed. Let H : T M R be a Hamiltonian of class C k, with

More information

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying

More information

Comparison between some Kurzweil-Henstock type integrals for Riesz-space-valued functions

Comparison between some Kurzweil-Henstock type integrals for Riesz-space-valued functions Comparison between some Kurzweil-Henstock type integrals for Riesz-space-valued functions A. Boccuto V. A. Skvortsov ABSTRACT. Some kinds of integral, like variational, Kurzweil-Henstock and (SL)-integral

More information

Convergence results for solutions of a first-order differential equation

Convergence results for solutions of a first-order differential equation vailable online at www.tjnsa.com J. Nonlinear ci. ppl. 6 (213), 18 28 Research rticle Convergence results for solutions of a first-order differential equation Liviu C. Florescu Faculty of Mathematics,

More information

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f)) 1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

More information

Generalized directional derivatives for locally Lipschitz functions which satisfy Leibniz rule

Generalized directional derivatives for locally Lipschitz functions which satisfy Leibniz rule Control and Cybernetics vol. 36 (2007) No. 4 Generalized directional derivatives for locally Lipschitz functions which satisfy Leibniz rule by J. Grzybowski 1, D. Pallaschke 2 and R. Urbański 1 1 Faculty

More information

2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2

2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2 1 A Good Spectral Theorem c1996, Paul Garrett, garrett@math.umn.edu version February 12, 1996 1 Measurable Hilbert bundles Measurable Banach bundles Direct integrals of Hilbert spaces Trivializing Hilbert

More information

Operator Theory and Related Topics, Université de Lille 1, 2010 An application of generalized spectral operators

Operator Theory and Related Topics, Université de Lille 1, 2010 An application of generalized spectral operators Operator Theory and Related Topics, Université de Lille 1, 2010 An application of generalized spectral operators Definition (Dunford) spectral operator : U B(X) (X = complex Banach space) spectral measure

More information

BOUNDARY VALUE PROBLEMS IN KREĬN SPACES. Branko Ćurgus Western Washington University, USA

BOUNDARY VALUE PROBLEMS IN KREĬN SPACES. Branko Ćurgus Western Washington University, USA GLASNIK MATEMATIČKI Vol. 35(55(2000, 45 58 BOUNDARY VALUE PROBLEMS IN KREĬN SPACES Branko Ćurgus Western Washington University, USA Dedicated to the memory of Branko Najman. Abstract. Three abstract boundary

More information

NORMAL FAMILIES OF HOLOMORPHIC FUNCTIONS ON INFINITE DIMENSIONAL SPACES

NORMAL FAMILIES OF HOLOMORPHIC FUNCTIONS ON INFINITE DIMENSIONAL SPACES PORTUGALIAE MATHEMATICA Vol. 63 Fasc.3 2006 Nova Série NORMAL FAMILIES OF HOLOMORPHIC FUNCTIONS ON INFINITE DIMENSIONAL SPACES Paula Takatsuka * Abstract: The purpose of the present work is to extend some

More information

March 25, 2010 CHAPTER 2: LIMITS AND CONTINUITY OF FUNCTIONS IN EUCLIDEAN SPACE

March 25, 2010 CHAPTER 2: LIMITS AND CONTINUITY OF FUNCTIONS IN EUCLIDEAN SPACE March 25, 2010 CHAPTER 2: LIMIT AND CONTINUITY OF FUNCTION IN EUCLIDEAN PACE 1. calar product in R n Definition 1.1. Given x = (x 1,..., x n ), y = (y 1,..., y n ) R n,we define their scalar product as

More information

SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT

SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NIT-Karnataka,

More information

Topologies, ring norms and algebra norms on some algebras of continuous functions.

Topologies, ring norms and algebra norms on some algebras of continuous functions. Topologies, ring norms and algebra norms on some algebras of continuous functions. Javier Gómez-Pérez Javier Gómez-Pérez, Departamento de Matemáticas, Universidad de León, 24071 León, Spain. Corresponding

More information

are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication 7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

More information

On hyperconnected topological spaces

On hyperconnected topological spaces An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) Tomul LXII, 2016, f. 2, vol. 1 On hyperconnected topological spaces Vinod Kumar Devender Kumar Kamboj Received: 4.X.2012 / Accepted: 12.XI.2012 Abstract It

More information

A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM

A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 1, March 23 A MAXIMUM PRINCIPLE FOR A MULTIOBJECTIVE OPTIMAL CONTROL PROBLEM Rezumat. Un principiu de maxim pentru o problemă vectorială de

More information

On polynomially -paranormal operators

On polynomially -paranormal operators Functional Analysis, Approximation and Computation 5:2 (2013), 11 16 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac On polynomially

More information

arxiv: v1 [math.fa] 23 Dec 2015

arxiv: v1 [math.fa] 23 Dec 2015 On the sum of a narrow and a compact operators arxiv:151.07838v1 [math.fa] 3 Dec 015 Abstract Volodymyr Mykhaylyuk Department of Applied Mathematics Chernivtsi National University str. Kotsyubyns koho,

More information

Hilbert space methods for quantum mechanics. S. Richard

Hilbert space methods for quantum mechanics. S. Richard Hilbert space methods for quantum mechanics S. Richard Spring Semester 2016 2 Contents 1 Hilbert space and bounded linear operators 5 1.1 Hilbert space................................ 5 1.2 Vector-valued

More information

arxiv: v1 [math.sp] 29 Jan 2018

arxiv: v1 [math.sp] 29 Jan 2018 Problem of descent spectrum equality Abdelaziz Tajmouati 1 and Hamid Boua 2 arxiv:1801.09752v1 [math.sp] 29 Jan 2018 Sidi Mohamed Ben Abdellah University Faculty of Sciences Dhar El Mahraz Laboratory of

More information

Spectral theory for linear operators on L 1 or C(K) spaces

Spectral theory for linear operators on L 1 or C(K) spaces Spectral theory for linear operators on L 1 or C(K) spaces Ian Doust, Florence Lancien, and Gilles Lancien Abstract It is known that on a Hilbert space, the sum of a real scalar-type operator and a commuting

More information

On the Maximal Ideals in L 1 -Algebra of the Heisenberg Group

On the Maximal Ideals in L 1 -Algebra of the Heisenberg Group International Journal of Algebra, Vol. 3, 2009, no. 9, 443-448 On the Maximal Ideals in L 1 -Algebra of the Heisenberg Group Kahar El-Hussein Department of Mathematics, Al-Jouf University Faculty of Science,

More information

THE JORDAN-BROUWER SEPARATION THEOREM

THE JORDAN-BROUWER SEPARATION THEOREM THE JORDAN-BROUWER SEPARATION THEOREM WOLFGANG SCHMALTZ Abstract. The Classical Jordan Curve Theorem says that every simple closed curve in R 2 divides the plane into two pieces, an inside and an outside

More information

Decompositions in Hilbert spaces

Decompositions in Hilbert spaces LECTURE 6 Decompositions in Hilbert spaces In this lecture we discuss three fundamental decompositions for linear contractions on Hilbert spaces: the von eumann decomposition, the rational spectrum decomposition

More information

SOLVABILITY OF NONLINEAR EQUATIONS

SOLVABILITY OF NONLINEAR EQUATIONS SOLVABILITY OF NONLINEAR EQUATIONS PETRONELA CATANĂ Using some numerical characteristics for nonlinear operators F acting between two Banach spaces X and Y, we discuss the solvability of a linear ecuation

More information

The Cayley-Hamilton Theorem and the Jordan Decomposition

The Cayley-Hamilton Theorem and the Jordan Decomposition LECTURE 19 The Cayley-Hamilton Theorem and the Jordan Decomposition Let me begin by summarizing the main results of the last lecture Suppose T is a endomorphism of a vector space V Then T has a minimal

More information