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