aa r X i v : . [ m a t h . F A ] M a y TENSOR PRODUCTS AND THESEMI-BROWDER JOINT SPECTRA
ENRICO BOASSO
Abstract.
Given two complex Banach spaces X and X , a tensor productof X and X , X ˜ ⊗ X , in the sense of J. Eschmeier ([5]), and two finite tuplesof commuting operators, S = ( S , . . . , S n ) and T = ( T , . . . , T m ), defined on X and X respectively, we consider the ( n + m )-tuple of operators defined on X ˜ ⊗ X , ( S ⊗ I, I ⊗ T ) = ( S ⊗ I, . . . , S n ⊗ I, I ⊗ T , . . . , I ⊗ T m ), and we givea description of the semi-Browder joint spectra introduced by V. Kordula, V.M¨uller and V. Rakoˇ c evi´ c in [7] and of the split semi-Browder joint spectra (seesection 3), of the ( n + m )-tuple ( S ⊗ I, I ⊗ T ), in terms of the correspondingjoint spectra of S and T . This result is in some sense a generalization of aformula obtained for other various Browder spectra in Hilbert spaces and fortensor products of operators and for tuples of the form ( S ⊗ I, I ⊗ T ). Inaddition, we also describe all the mentioned joint spectra for a tuple of leftand right multiplications defined on an operator ideal between Banach spacesin the sense of [5]. Introduction
Given a complex Banach space X , V. Kordula, V. M¨uller and V. Rakoˇ c evi´ c ex-tended in [7] the notion of upper and lower semi-Browder spectrum of an operatorto n -tuples of commuting operators, and they proved the main spectral proper-ties for this joint spectra, i.e., the compactness, nonemptiness, the projectionproperty and the spectral mapping property.On the other hand, there are other many joint Browder spectra, for example,we may consider the one introduced by R. E. Curto and A. T. Dash in [2], σ b , andthe joint Browder spectra defined by A. T. Dash in [3], σ b , σ b and σ Tb . By theobservation which follows Definition 4 in [3] and the Example in [7], we have thatthe Browder spectra of V. Kordula, V. M¨uller and V. Rakoˇ c evi´ c , σ B + and σ B − ,differ, in general, from the other mentioned joint Browder spectra. However, ifwe consider two complex Hilbert spaces H and H , and S and T two operatorsdefined on H and H respectively, by [2] and [3, Theorem 7] we have that thejoint Browder spectra σ b , σ b , σ b and σ Tb of the tuple of operators ( S ⊗ I, I ⊗ T )defined on H ⊗ H , coincide with the set σ b ( S ) × σ ( T ) ∪ σ ( S ) × σ b ( T ) , where σ and σ b denote, respectively, the usual and the Browder spectrum of anoperator.Moreover, if S = ( S , . . . , S n ), respectively T = ( T , . . . , T m ), is an n -tuple,respectively an m -tuple, of commuting operators defined on the Hilbert space H , respectively H , R. E. Curto and A. T. Dash computed in [2] the Browder spectum of the ( n + m )-tuple ( S ⊗ I, I ⊗ T ) = ( S ⊗ I, . . . , S n ⊗ I, I ⊗ T , . . . , I ⊗ T m ),and they obtained the formula σ b ( S ⊗ I, I ⊗ T ) = σ b ( S ) × σ T ( T ) ∪ σ T ( T ) × σ b ( T ) , where σ T denotes the Taylor joint spectrum (see [9]).In this article we give in some sense a generalization of the above formulasfor commutative tuples of Banach spaces operators and for the semi-Browderjoint spectra. Indeed, we consider two complex Banach spaces, X and X , atensor product between X and X in the sense of J. Eschmeier ([5]) X ˜ ⊗ X , S and T , two commuting tuples of Banach space operators defined on X and X respectively, and we describe the semi-Browder joint spectra introduced in[7], σ B + and σ B − , and the split semi-Browder joint spectra sp B + and sp B − (seesection 3) of the tuple ( S ⊗ I, I ⊗ T ), in terms of the corresponding semi-Browderjoint spectra and of the defect and the approximate point spectra of S and T .The results that we have obtained extend in same way the above formulas, seesection 5. Furthermore, since for our objective we need to know the Fredholmjoint spectra of J.J. Buoni, R. Harte and T. Wickstead of ( S ⊗ I, I ⊗ T ) ([1]) andits split versions ([4]) we also descrive in section 4 these joint spectra.In addition, by similar arguments we describe in section 6 all the mentionedjoint spectra for a tuple of left and right multiplications defined on an operatorideal between Banach spaces in the sense of [5].However, in order to give our descriptions, we need to introduce the split semi-Browder joint spectra of a tuple of commuting Banach space operators, and toprove their main spectral properties (see section 3).The article is organized as follows. In section 2 we recall several definitionsand results which we need for our work. In section 3 we introduce the splitsemi-Browder joint spectra and prove their main spectral properties. In section4 we compute the semi-Fredholm joint spectra of ( S ⊗ I, I ⊗ T ). In section 5we compute the semi-Browder joint spectra of ( S ⊗ I, I ⊗ T ), and in section 6,the semi-Fredholm and the semi-Browder joint spectra of a tuple of left and rightmultiplications defined on an operator ideal between Banach spaces in the senseof [5]. 2. Preliminaries
Let us begin our work by recalling the definitions of the lower semi-Fredholm andof the lower semi-Browder joint spectra of a finite tuple of operators denfined ona complex Banach space, for a complete exposition see [1] and [7].Let T = ( T , . . . , T n ) be an n -tuple of commuting operators defined on a Banachspace X , and for k ∈ N define M k ( T ) = R ( T k ) + . . . + R ( T kn ). Clearly X ⊇ M ( T ) ⊇ M ( T ) ⊇ . . . ⊇ M k ( T ) ⊇ . . . Let us set R ∞ ( T ) = ∩ ∞ k =1 M k ( T ). We nowmay recall the definition of the lower semi-Browder joint spectrum (see [7]).We say that T = ( T , . . . , T n ) is lower semi-Browder if codim R ∞ ( T ) < ∞ .The set of all lower semi-Browder n -tuples is denoted by B ( n ) − ( X ), and the lowersemi-Browder spectrum is the set σ B − ( T ) = { λ ∈ C n : T − λ / ∈ B ( n ) − ( X ) } , EMI-BROWDER JOINT SPECTRA 3 where λ = ( λ , . . . , λ n ) and T − λ = ( T − λ I, . . . , T n − λ n I ).As usual (see [1]), we say that T = ( T , . . . , T n ) is lower semi-Fredholm, i.e., T ∈ Φ ( n ) − ( X ), ifcodim M ( T ) = codim ( R ( T ) + . . . + R ( T n )) < ∞ , equivalently, if the operator ˆ T : X n → X defined by ˆ T ( x , . . . , x n ) = T ( x )+ . . . + T n ( x n ) is lower semi-Fredholm, i.e., R ( ˆ T ) is closed and has finite codimension.The lower semi-Fredholm spectrum is the set σ Φ − ( T ) = { λ ∈ C n : T − λ / ∈ Φ ( n ) − ( X ) } . An easy calculation shows that σ Φ − ( T ) ⊆ σ B − ( T ) ⊆ σ δ ( T ) , where σ δ ( T ) is the defect spectrum of T , i.e., σ δ ( T ) = { λ ∈ C n : codim M ( T − λ ) = 0 } . Moreover, it is easy to see that the lower semi-Browder spectrum may bedecomposed as the disjoint union of two sets, σ B − ( T ) = σ Φ − ( T ) ∪ A ( T ) , where A ( T ) = { λ ∈ C n : ∀ k ∈ N , ≤ codim M k ( T − λ ) < ∞ , codim M k ( T − λ ) −−−→ k →∞ ∞} . Now, we recall the definition of the upper semi-Fredholm and the upper semi-Browder joint spectra, as above, for a complete exposition see [1] and [7].If T is an n -tuple of commuting operators defined on a Banach space X , then T is said upper semi-Fredholm, i.e., T ∈ Φ ( n )+ ( X ), if the map ˜ T : X → X n definedby ˜ T ( x ) = ( T ( x ) , . . . , T n ( x )) is upper semi-Fredholm, equivalently, if ˜ T has finitedimensional null space and closed range. Moreover, T is said upper semi-Browder, i.e., T ∈ B ( n )+ ( X ), if T ∈ Φ ( n )+ ( X ) and dim N ∞ ( T ) < ∞ , where N ∞ ( T ) = ∪ k ∈ N [ N ( T k ) ∩ . . . ∩ N ( T kn )] . As above, the upper semi-Fredholm spectrum is the set σ Φ + ( T ) = { λ ∈ C n : T − λ / ∈ Φ ( n )+ ( X ) } , and the upper semi-Browder spectrum is the set σ B + ( T ) = { λ ∈ C n : T − λ / ∈ B ( n )+ ( X ) } . In addition, it is easy to see that σ Φ + ( T ) ⊆ σ B + ( T ) ⊆ σ π ( T ) , where σ π ( T ) denotes the approximate point spectrum of T , σ π ( T ) = { λ ∈ C n : N ( ˜ T − λ ) = 0 or R ( ˜ T − λ ) is not closed } . Moreover, it is easy to see that the upper semi-Browder spectrum may bedecomposed as the disjoint union of two sets, σ B + ( T ) = σ Φ + ( T ) ∪ D ( T ) , ENRICO BOASSO where D ( T ) = { λ ∈ C n : ∀ k ∈ N , ≤ dim N k ( ˜ T − λ ) < ∞ , R ( ˜ T − λ ) isclosed, and dim N k ( ˜ T − λ ) −−−→ k →∞ ∞} , where N k ( ˜ T − λ ) = N ( ˜( T − λ ) k ) and( T − λ ) k = (( T − λ ) k , . . . , ( T n − λ n ) k ).Let us recall that the semi-Fredholm and the semi-Browder joint spectra arecompact nonempty subsets of C n , which also satisfy the projection property andthe analytic spectral mapping theorem for tuples of holomorphic functions definedon a neighborhood of the Taylor joint spectrum [9] (see [4] and [7]).On the other hand, in order to prove our main results, we have to recall theaxiomatic tensor product between Banach spaces introduced by J. Eschmeier in[5]. This notion will be central in this work. For a complete exposition see [5].We proceed as follows.A pair < X, ˜ X > of Banach spaces will be called a dual pairing, if( A ) ˜ X = X ′ or ( B ) X = ˜ X ′ . In both cases, the canonical bilinear mapping is denoted by X × ˜ X → C , ( x, u ) → < x, u > . If < X, ˜ X > is a dual pairing, we consider the subalgebra L ( X ) of L( X )consisting of all operators T ∈ L( X ) for which there is an operator T ′ ∈ L( ˜ X )with < T x, u > = < x, T ′ u >, for all x ∈ X and u ∈ ˜ X . It is clear that if the dual pairing is < X, X ′ > , then L ( X ) = L( X ), and that if the dual pairing is < X ′ , X > , then L ( X ) = { T ∗ : T ∈ L( ˜ X ) } . In particular, each operator of the form f y,v : X → X, x → < x, v > y, is contained in L ( X ), where y ∈ X and v ∈ ˜ X .We now recall the definition of the tensor product given by J. Eschmeier in [5].Given two dual pairings < X, ˜ X > and < Y, ˜ Y > , a tensor product of theBanach spaces X and Y relative to the dual pairings < X, ˜ X > and < Y, ˜ Y > , isa Banach space Z together with continuous bilinear mappings X × Y → Z, ( x, y ) → x ⊗ y ; L ( X ) × L ( Y ) → L( Z ) , ( T, S ) → T ⊗ S, which satisfy the following conditions,(T1) k x ⊗ y k = k x kk y k ,(T2) T ⊗ S ( x ⊗ y ) = ( T x ) ⊗ ( Sy ),(T3) ( T ⊗ S ) ◦ ( T ⊗ S ) = ( T T ) ⊗ ( S S ) , I ⊗ I = I ,(T4) Im( f x,u ⊗ I ) ⊆ { x ⊗ y : y ∈ Y } , Im( f y,v ⊗ I ) ⊆ { x ⊗ y : x ∈ X } .In this work, as in [5], instead of Z we shall often write X ˜ ⊗ Y . In addition, asin [5], we shall have two applications of this definition of tensor product. Firstof all, the completion X ˜ ⊗ α Y of the algebraic tensor product of Banach spaces X and Y with respect to a quasi-uniform crossnorm α (see [6]) and an operatorideal between Banach spaces (see [5] and section 6). EMI-BROWDER JOINT SPECTRA 5
In section 4 and 5, given two complex Banach spaces X and X , and two tuplesof Banach spaces operators, S and T , defined on X and X respectively, weshall describe the semi-Fredholm and the semi-Browder joint spectra of the tuple( S ⊗ I, I ⊗ T ), whose operators, S i ⊗ I and I ⊗ T j , i = 1 , . . . , n and j = 1 , . . . , m ,are defined on X ˜ ⊗ X , a tensor product of X and X relative to < X , X ′ > and < X , X ′ > . However, in the following section, we first introduce the splitsemi-Browder joint spectra, which will be necessary for our description.3. The split semi-Browder joint spectra
In this section we introduce the upper and lower split semi-Browder joint spectra.We also prove their main spectral properties.Let us consider, as in section 2, a complex Banach space X and T = ( T , . . . , T n )a commuting tuple of operators defined on X . We say that T is lower split semi-Browder if R ∞ ( T ) has finite codimension and N ( ˆ T ) has a direct complementin X n , where ˆ T : X n → X is the map considered in section 2. We denote by SB ( n ) − ( X ) the set of all lower split semi-Browder n -tuples, and the lower splitsemi-Browder spectrum is the set sp B − ( T ) = { λ ∈ C n : T − λ / ∈ S B ( n ) − ( X ) } . It is clear that sp B − ( T ) = σ B − ( T ) ∪ C − ( T ) , where C − ( T ) = { λ ∈ C n : N ( ˆ T − λ ) has not a direct complement in X n } . Inparticular, sp B − ( T ) is a nonempty set.On the other hand, if we consider the split defect spectrum and the essentialsplit defect spectrum of T introduced in [4], sp δ ( T ) and sp δe ( T ) respectively, setsthat by [4, Theorem 2.7] may be presented as sp δ ( T ) = σ δ ( T ) ∪ C − ( T ) , sp δe ( T ) = σ Φ − ( T ) ∪ C − ( T ) , then we have that sp δe ( T ) ⊆ sp B − ( T ) ⊆ sp δ ( T ) . In addition, if we consider the set ˜ A ( T ) = { λ ∈ C n : λ / ∈ sp δe ( T ) , ∀ k ∈ N , ≤ codim M k ( T − λ ) < ∞ , codim M k ( T − λ ) −−−→ k →∞ ∞} , then it is clear that˜ A ( T ) ⊆ A ( T ) ⊆ σ B − ( T ) ⊆ sp B − ( T ) . In particular sp δe ( T ) ∪ ˜ A ( T ) ⊆ sp B − ( T ) . On the other hand, let us consider λ ∈ sp B − ( T ), and let us decompose thelower split semi-Browder spectrum of T as sp B − ( T ) = σ B − ( T ) ∪ C − ( T ) = σ Φ − ( T ) ∪ A − ( T ) ∪ C − ( T ) . Now, if λ ∈ σ Φ − ( T ) ∪ C − ( T ), then λ ∈ sp δe ( T ). Moreover, if λ ∈ A ( T ) \ ( σ Φ − ( T ) ∪C − ( T )), then λ ∈ A ( T ) \ sp δe ( T ) = ˜ A ( T ). Thus, we have that sp B − ( T ) = sp δe ( T ) ∪ ˜ A ( T ) . We now introduce the upper split semi-Browder spectrum.
ENRICO BOASSO If X and T = ( T , . . . , T n ) are as above, then we say that T is upper split semi-Browder if it is upper semi-Browder and R ( ˜ T ) has a direct complement in X n ,where ˜ T : X → X n is the map considered in section 2. We denote by SB ( n )+ ( X )the set of all upper split semi-Browder n -tuples, and the upper split semi-Browderspectrum is the set sp B + ( T ) = { λ ∈ C n : T − λ / ∈ S B ( n )+ ( X ) } . It is clear that sp B + ( T ) = σ B + ( T ) ∪ C + ( T ) , where C + ( T ) = { λ ∈ C n : R ( ˜ T − λ ) has not a direct complement in X n } . In par-ticular, sp B + ( T ) is a nonempty set.On the other hand, if we consider the split approximate point spectrum andthe essential split approximate point spectrum of T (see [4]), sp π ( T ) and sp πe ( T )respectively, i.e., the sets sp π ( T ) = σ π ( T ) ∪ C + ( T ) , sp πe ( T ) = σ Φ + ( T ) ∪ C + ( T ) , then we have that sp πe ( T ) ⊆ sp B + ( T ) ⊆ sp π ( T ) . In addition, if we consider the set ˜ D ( T ) = { λ ∈ C n : λ / ∈ sp πe ( T ) , ∀ k ∈ N , ≤ dim N k ˜( T − λ ) < ∞ , dim N k ˜( T − λ ) −−−→ k →∞ ∞} , then it is clear that˜ D ( T ) ⊆ D ( T ) ⊆ σ B + ( T ) ⊆ sp B + ( T ) . In particular sp πe ( T ) ∪ ˜ D ( T ) ⊆ sp B + ( T ) . On the other hand, let us consider λ ∈ sp B + ( T ), and let us decompose theupper split semi-Browder spectrum of T as sp B + ( T ) = σ B + ( T ) ∪ C + ( T ) = σ Φ + ( T ) ∪ D + ( T ) ∪ C + ( T ) . Now, if λ ∈ σ Φ + ( T ) ∪ C + ( T ), then λ ∈ sp πe ( T ). Moreover, if λ ∈ D ( T ) \ ( σ Φ + ( T ) ∪C + ( T )), then λ ∈ D ( T ) \ sp πe ( T ) = ˜ D ( T ). Thus, we have that sp B + ( T ) = sp πe ( T ) ∪ ˜ D ( T ) . We now see that the sets that we have introduced satisfy the main spectralproperties.
Proposition 3.1.
Let X be a complex Banach space and T = ( T , . . . , T n ) acommuting tuple of bounded linear operators defined on X . Then the sets sp B − ( T ) and sp B + ( T ) are compact subsets of C n .Proof. Since sp B − ( T ) = sp δe ( T ) ∪ ˜ A ( T ) ⊆ sp δe ( T ) ∪ σ B − ( T ), we have that sp B − ( T )is a bounded subset of C n .On the other hand, let us consider a sequence ( λ n ) n ∈ N ⊆ sp B − ( T ), and λ ∈ C n such that λ n −−−→ n →∞ λ . If there exists a subsequence ( λ n k ) k ∈ N ⊆ sp δe ( T ), then λ ∈ sp δe ( T ) ⊆ sp B − ( T ). Thus, we may suppose that there is n ∈ N such that forall n ∈ N , n ≥ n , λ n ∈ ˜ A ( T ). Moreover, we may also suppose that λ / ∈ sp δe ( T ). EMI-BROWDER JOINT SPECTRA 7
In particular, there is an open neighborhood of λ , U , such that U ∩ sp δe ( T ) = ∅ ,and there is n ∈ N such that λ n ∈ U , for all n ≥ n .However, since for all n ≥ n , λ n ∈ ˜ A ( T ) ⊆ A ( T ) ⊆ σ B − ( T ), then λ ∈ σ B − ( T ).But λ / ∈ σ Φ − ( T ), for σ Φ − ( T ) ⊆ sp δe ( T ). Then, λ ∈ A ( T ) \ sp δe ( T ) = ˜ A ( T ) ⊆ sp B − ( T ).By means of a similar argument, it is possible to see that sp B + ( T ) is a compactsubset of C n . (cid:3) Proposition 3.2.
Let X be a complex Banach space and T = ( T , . . . , T n , T n +1 ) a commuting tuple of bounded linear operators defined on X . If π : C n +1 → C n denotes the projection onto the first n -coordinate, then we have that (i) π ( sp B − ( T , . . . , T n , T n +1 )) = sp B − ( T , . . . , T n ) , (ii) π ( sp B + ( T , . . . , T n , T n +1 )) = sp B + ( T , . . . , T n ) . Proof.
By [7, Corollary 7] we now that π ( σ B − ( T , . . . , T n , T n +1 )) = σ B − ( T , . . . , T n ) ⊆ sp B − ( T , . . . , T n ). Moreover, since C − ( T , . . . , T n , T n +1 ) ⊆ sp δe ( T , . . . , T n , T n +1 ),by [4, Corollary 2.6] we have that π ( C − ( T , . . . , T n , T n +1 )) ⊆ π ( sp δe ( T , . . . , T n , T n +1 )) = sp δe ( T , . . . , T n ) ⊆ sp B − ( T , . . . , T n ) . Thus, we have that π ( sp B − ( T , . . . , T n , T n +1 )) ⊆ sp B − ( T , . . . , T n ) . On the other hand, by [7, Corollary 7] we also have that σ B − ( T , . . . , T n ) = π ( σ B − ( T , . . . , T n , T n +1 )) ⊆ π ( sp B − ( T , . . . , T n , T n +1 )) . Furthermore, since C − ( T , . . . , T n ) ⊆ sp δe ( T , . . . , T n ), by [4, Corollary 2.6] wealso have that C − ( T , . . . , T n ) ⊆ sp δe ( T , . . . , T n ) = π ( sp δe ( T , . . . , T n , T n +1 )) ⊆ π ( sp B − ( T , . . . , T n , T n +1 )). Thus, sp B − ( T , . . . , T n ) ⊆ π ( sp B − ( T , . . . , T n , T n +1 )) , i.e., we have proved the first statement of the proposition.By means of a similar argument it is possible to see the second statement. (cid:3) In the following proposition we shall see that the split semi-Browder jointspectra satisfy the analytic spectral mapping theorem.
Proposition 3.3.
Let X be a complex Banach space and T = ( T , . . . , T n ) a com-muting tuple of bounded linear operators defined on X . Then, if f ∈ O ( sp ( T )) m ,we have that (i) f ( sp B − ( T , . . . , T n )) = sp B − ( f ( T , . . . , T n )) , (ii) f ( sp B + ( T , . . . , T n )) = sp B + ( f ( T , . . . , T n )) , where sp ( T ) denotes the split spectrum of T .Proof. By [4, Corollary 2.6], the split sectrum of T , sp ( T ), satisfies the analyticspectral mapping theorem, i.e., there is an algebra morphismΦ : O ( sp ( T )) → L ( X ) , f → f ( T ) , ENRICO BOASSO such that 1( T ) = I , z i ( T ) = T i , 1 ≤ i ≤ n , where z i denotes the projection of C n onto the i -th coordinate, and such that the equality sp ( f ( T )) = f ( sp ( T )), holdsfor all f ∈ O ( sp ( T )) m .Now, as in [4], let us consider the algebra A = Φ( O ( sp ( T ))) ⊆ L ( X ) . Then, we have that the split spectrum is a spectral system on A , in the sense of[4, section 1].In order to show this claim, since the split spectrum is a compact set whichalso satisfies the projection property ([4, Corollary 2.6]), we have only to see thatif a = ( a , . . . , a n ) is a tuple of commuting operators such that a i ∈ A , then sp ( a ) ⊆ σ Ajoint ( a ) (the usual joint spectrum of a with respect to the algebra A , see[4, section 1]).In fact, if λ = ( λ , . . . , λ n ) ∈ sp ( a ) \ σ Ajoint ( a ) , then there are B , . . . , B n ∈ A such that P ni =1 B i ( a i − λ i I ) = I , where I denotes the identity map of X . Inparticular, n X i =1 L B i ( L a i − λ i I L ( X ) ) = I L ( X ) . Then, λ / ∈ σ ( L a ), the Taylor joint spectrum of the tuple of left multiplication, L a = ( L a , . . . , L a n ), defined on L ( X ). However, by [4, Corollary 2.5], λ / ∈ sp ( a ),which is impossible by our assumption.Now, since sp B − ( T , . . . , T n ) and sp B + ( T , . . . , T n ) are contained in sp ( T ), byPropositions 3.1 and 3.2, sp B − ( T , . . . , T n ) and sp B + ( T , . . . , T n ) are spectral sys-tems on A contained in sp ( T ). Then, by [4, Theorem 1.2] and [4, Corollary 1.3],since the split spectrum is a spectral system on A which satisfy the analyticspectral mapping theorem, sp B − ( T , . . . , T n ) and sp B + ( T , . . . , T n ) also satisfy theanalytic spectral mapping theorem defined on O ( sp ( T )). (cid:3) In the following section we give a description of the semi-Fredholm joint spectraof the system ( S ⊗ I, I ⊗ T ), which will be a central step for one of the maintheorems of the present article.4. The semi-Fredholm joint spectra
In this section we consider two complex Banach spaces X and X , two tuplesof bounded linear operators defined on X and X , S = ( S , . . . , S n ) and T =( T , . . . , T n ) respectively, and we describe the semi-Fredholm joint spectra of the( n + m )-tuple ( S ⊗ I, I ⊗ T ) defined on X ˜ ⊗ X , a tensor product between X and X relative to < X , X ′ > and < X , X ′ > , where ( S ⊗ I, I ⊗ T ) = ( S ⊗ I, . . . , S n ⊗ I, I ⊗ T , . . . , I ⊗ T m ).We recall that if K , K and K are the Koszul complexes associated to thetuples S , T and ( S ⊗ I, I ⊗ T ) respectively (see [9]), i.e., K = ( X ⊗ ∧ C n , d ), K = ( X ⊗ ∧ C m , d ) and K = ( X ˜ ⊗ X ⊗ ∧ C n + m , d ), then, by [5, section 3] wehave that K is isomorphic to the total complex of the double complex obtainedfrom the tensor product of the complexes K and K ; we denote this total complexby K ˜ ⊗ K . Moreover, if we consider the differential spaces associated to K , K , EMI-BROWDER JOINT SPECTRA 9 K , and K ˜ ⊗ K , which we denote, by K , K , K , and K ˜ ⊗K respectively, thenwe have that K ∼ = K ˜ ⊗K , and if the boundary of these differential spaces are, ∂ , ∂ , ∂ , and ∂ respectively, then we have that ∂ = ∂ ⊗ I + η ⊗ ∂ , where η isthe map, η : K → K , η | X ⊗ ∧ m C = ( − m I (for a complete exposition see [5,section 3]).In the following proposition we describe the defect, the approximate pointspectrum, and the split version of these spectra for the tuple ( S ⊗ I, I ⊗ T ). Thisresult is necessary for our description of the semi-Fredholm joint spectra. Proposition 4.1.
Let X and X be two complex Banach spaces, and X ˜ ⊗ X a tensor product of X and X relative to < X , X ′ > and < X , X ′ > . Letus consider two tuples of commuting operators defined on X and X , S and T respectively. Then, for the tuple ( S ⊗ I, I ⊗ T ) , defined on X ˜ ⊗ X , we have that (i) σ δ ( S ) × σ δ ( T ) ⊆ σ δ ( S ⊗ I, I ⊗ T ) ⊆ sp δ ( S ⊗ I, I ⊗ T ) ⊆ sp δ ( S ) × sp δ ( T ) , (ii) σ π ( S ) × σ π ( T ) ⊆ σ π ( S ⊗ I, I ⊗ T ) ⊆ sp π ( S ⊗ I, I ⊗ T ) ⊆ sp π ( S ) × sp π ( T ) .In addition, if X and X are Hilbert spaces, the above inclusions are equalities.Proof. Let us consider λ ∈ C n , µ ∈ C m and the Koszul complexes associated to S − λ , T − µ and ( S ⊗ I, I ⊗ T ) − ( λ, µ ) = (( S − λ ) ⊗ I, I ⊗ ( T − µ )), which wedenote by K , K and K . By the previous observation we have that K ∼ = K ˜ ⊗ K .Moreover, if we consider the differential spaces associated to these complexes, K K , and K , then we have that K ∼ = K ˜ ⊗K .Now, we may apply [5, Theorem 2.2] to the differential spaces K , K , and K ˜ ⊗K . However, by the definition of the map ϕ in [5, Theorem 2.2], the gradingof the differential spaces K , K , and K ˜ ⊗K , and of the isomorphism K ∼ = K ˜ ⊗K , we have the left hand side inclusion of the first statement.The middle inclusion is clear.Let us now suppose that ( λ, µ ) / ∈ sp δ ( S ) × sp δ ( T ). Then, either λ / ∈ sp δ ( S )or µ / ∈ sp δ ( T ). We shall see that if λ / ∈ sp δ ( S ), then ( λ, µ ) / ∈ sp δ ( S ⊗ I, I ⊗ T ).By means of a similar argumet it is possible to see that if µ / ∈ sp δ ( T ) then( λ, µ ) / ∈ sp δ ( S ⊗ I, I ⊗ T ).Now, if λ / ∈ sp δ ( S ), there is a bounded linear operator h : X → X ⊗ ∧ n C suchthat d ◦ h = I, where d : X ⊗ ∧ C n → X is the chain map of the Koszul complex K at level p = 1.Let us consider the map H : X ˜ ⊗ X → X ⊗ ∧ C n ˜ ⊗ X , H = h ⊗ I. Then, by the properties of the tensor product introduced in [5], H is a well definedmap which satisfies d ◦ H = d ◦ h ⊗ I = I ⊗ I = I, where d is the chain map of the complex K ˜ ⊗ K at level p = 1. Since K ∼ = K ˜ ⊗ K , we have that ( λ, µ ) / ∈ sp δ ( S ⊗ I, I ⊗ T ).The second statement may be proved by means of a similar argument. (cid:3) In the following proposition we state our description of the semi-Fredholm jointspectra.
Proposition 4.2.
Let X and X be two complex Banach spaces, and X ˜ ⊗ X a tensor product of X and X relative to < X , X ′ > and < X , X ′ > . Letus consider two tuples of commuting operators defined on X and X , S and T respectively. Then, for the tuple ( S ⊗ I, I ⊗ T ) , defined on X ˜ ⊗ X , we have that (i) σ Φ − ( S ) × σ δ ( T ) ∪ σ δ ( S ) × σ Φ − ( T ) ⊆ σ Φ − ( S ⊗ I, I ⊗ T ) ⊆ sp δe ( S ⊗ I, I ⊗ T ) ⊆ sp δe ( S ) × sp δ ( T ) ∪ sp δ ( S ) × sp δe ( T ) , (ii) σ Φ + ( S ) × σ π ( T ) ∪ σ π ( S ) × σ Φ + ( T ) ⊆ σ Φ + ( S ⊗ I, I ⊗ T ) ⊆ sp πe ( S ⊗ I, I ⊗ T ) ⊆ sp πe ( S ) × sp π ( T ) ∪ sp π ( S ) × sp πe ( T ) . In addition, if X and X are Hilbert spaces, the above inclusions are equalities.Proof. First of all, let us observe that we use the same notations of Proposition4.1.With regard to the first statement, in order to prove the left hand side inclu-sion, it is enough to adapt for this case the argument that we have developed inProposition 4.1 for the corresponding inclusion.The middle inclusion is clear.Let us denote by E the set E = sp δe ( S ) × sp δ ( T ) ∪ sp δ ( S ) × sp δe ( T ), andlet us consider ( λ, µ ) ∈ sp δe ( S ⊗ I, I ⊗ T ) \ E . Then, by Proposition 4.1, since( λ, µ ) ∈ sp δ ( S ⊗ I, I ⊗ T ) ⊆ sp δ ( S ) × sp δ ( T ), we have that λ ∈ sp δ ( S ) \ sp δe ( S ) and µ ∈ sp δ ( T ) \ sp δe ( T ). In particular, there are two linear bounded maps h : X → X ⊗ ∧ C n , g : X → X ⊗ ∧ C m , and two compact operators k : X → X and k : X → X such that d ◦ h = I − k , d ◦ g = I − k , where d is the boundary map of the complex K at level p = 1. Moreover, by anargument similar to [4, Theorem 2.7] or [5, Proposition 2.1], the maps k i , i = 1,2, may be chosen as finite rank projectors.In addition, by the properties of the tensor product introduced in [5], we mayconsider the well defined map H : X ˜ ⊗ X → ( K ˜ ⊗ K ) , H = ( h ⊗ I, I ⊗ g ) . Now, an easy calculation shows that d ◦ H = I − k ⊗ k , where d denotes thechain map of the complex K ˜ ⊗ K at level p = 1. However, it is not difficult tosee, using in particular [5, Lemma 1.1], that k ⊗ k is a finite rank projector whoserange coincide with R ( k ) ⊗ R ( k ). In particular, k ⊗ k is a compact operator.Thus, since K ∼ = K ˜ ⊗ K , ( λ, µ ) / ∈ sp δe ( S ⊗ I, I ⊗ T ), which is impossible by ourassumptions.By means of a similar argument it is possible to see the second statement. (cid:3) EMI-BROWDER JOINT SPECTRA 11 The semi-Browder joint spectra
In this section we give our description of the semi-Browder joint spectra of thetuple ( S ⊗ I, I ⊗ T ). The following theorem is one of the main results of thepresent article. Theorem 5.1.
Let X and X be two complex Banach spaces, and X ˜ ⊗ X atensor product of X and X relative to < X , X ′ > and < X , X ′ > . Letus consider two tuples of commuting operators defined on X and X , S and T respectively. Then, for the tuple ( S ⊗ I, I ⊗ T ) , defined on X ˜ ⊗ X , we have that (i) σ B − ( S ) × σ δ ( T ) ∪ σ δ ( S ) × σ B − ( T ) ⊆ σ B − ( S ⊗ I, I ⊗ T ) ⊆ sp B − ( S ⊗ I, I ⊗ T ) ⊆ sp B − ( S ) × sp δ ( T ) ∪ sp δ ( S ) × sp B − ( T ) , (ii) σ B + ( S ) × σ π ( T ) ∪ σ π ( S ) × σ B + ( T ) ⊆ σ B + ( S ⊗ I, I ⊗ T ) ⊆ sp B + ( S ⊗ I, I ⊗ T ) ⊆ sp B + ( S ) × sp π ( T ) ∪ sp π ( S ) × sp B + ( T ) , In addition, if X and X are Hilbert spaces, the above inclusions are equalities.Proof. First of all, as in Proposition 4.2, we use the notations of Proposition 4.1.Let us consider ( λ, µ ) ∈ σ B − ( S ) × σ δ ( T ). If λ ∈ σ Φ − ( S ), then, by Proposition4.2, ( λ, µ ) ∈ σ Φ − ( S ) × σ δ ( T ) ⊆ σ Φ − ( S ⊗ I, I ⊗ T ) ⊆ σ B − ( S ⊗ I, I ⊗ T ).Now, if λ ∈ A ( S ), since µ ∈ σ δ ( T ), by the definition of the map ϕ in [5,Theorem 2.2], the grading of the complex K , K , and K ˜ ⊗ K , and by theisomorphism K ∼ = K ˜ ⊗ K , we have that dim H ( K ) = dim H ( K ˜ ⊗ K ) ≥ dim H ( K ) × dim H ( K ) ≥
1. In particular, ( λ, µ ) ∈ σ δ ( S ⊗ I, I ⊗ T ).Moreover, if dim H ( K ) = ∞ , then ( λ, µ ) ∈ σ Φ − ( S ⊗ I, I ⊗ T ) ⊆ σ B − ( S ⊗ I, I ⊗ T ).On the other hand, if we suppose that ( λ, µ ) / ∈ σ Φ − ( S ⊗ I, I ⊗ T ). Then,we consider the tuples of operators ( S − λ ) l = (( S − λ ) l , . . . , ( S n − λ n ) l ) and( T − µ ) l = (( T − µ ) l , . . . , ( T m − µ m ) l ), and we denote by K l and K l the Koszulcomplexes associated to the tuples ( S − λ ) l and ( T − µ ) l , respectively. Moreover,if we denote by K l the Koszul complex associated to the tuple (( S − λ ) l ⊗ I, I ⊗ ( T − µ ) l ), as above, K l is isomorphic to the total complex of the double complexof the tensor product of K l and K l , i.e., K l ∼ = K l ˜ ⊗ K l .In addition, as we have seen for the complexes K , K , K , and K ˜ ⊗ K , we havethat dim H ( K l ) = dim H ( K l ˜ ⊗ K l ) ≥ dim H ( K l ) × dim H ( K l ). Now, since µ ∈ σ δ ( T ), by the analytic spectral mapping theorem for the defect spectrum (see[4, Corollary 2.1], we have that dim H ( K l ) = 0. In addition, since dim H ( K l ) =codim M l ( S − λ ), and since λ ∈ A ( S ), then dim H ( K l ) −−−→ l →∞ ∞ . However,dim H ( K l ) = codim M l (( S − λ ) ⊗ I, I ⊗ ( T − µ )). In particular, ( λ, µ ) ∈ A ( S ⊗ I, I ⊗ T ) ⊆ σ B − ( S ⊗ I, I ⊗ T ).By means of a similar argument it is possible to see that σ δ ( S ) × σ B − ( T ) ⊆ σ B − ( S ⊗ I, I ⊗ T ). The middle inclusion is clear.In order to see the right hand inclusion, let us consider ( λ, µ ) ∈ sp B − ( S ⊗ I, I ⊗ T ). If ( λ, µ ) ∈ sp δe ( S ⊗ I, I ⊗ T ), then by Proposition 4.2, ( λ, µ ) ∈ sp δe ( S ) × sp δ ( T ) ∪ sp δ ( S ) × sp δe ( T ) ⊆ sp B − ( S ) × sp δ ( T ) ∪ sp δ ( S ) × sp B − ( T ).On the other hand, if ( λ, µ ) ∈ ˜ A ( S ⊗ I, I ⊗ T ), since by Proposition 4.1 sp B − ( S ⊗ I, I ⊗ T ) ⊆ sp δ ( S ⊗ I, I ⊗ T ) ⊆ sp δ ( S ) × sp δ ( T ), if ( λ, µ ) / ∈ ( sp B − ( S ) × sp δ ( T ) ∪ sp δ ( S ) × sp B − ( T )), then λ / ∈ sp B − ( S ) and µ / ∈ sp B − ( T ). In particular, λ / ∈ sp δe ( S )and µ / ∈ sp δe ( T ), and there is l ∈ N such that for all r ≥ l , dim H ( K r ) =dim H ( K l ) and dim H ( K r ) = dim H ( K l ).In addition, by the analytic spectral mapping theorem of the essential splitdefect spectrum ([4, Corollary 2.6]), the complex K r and K r are Fredholm splitfor all r ∈ N at level p = 0. In particular, for all r ∈ N there are bounded linearmaps h r : X → X ⊗ ∧ C n and g r : X → X ⊗ ∧ C m , and finite rank projectors(see Proposition 4.2), k r : X → X and k r : X → X , such that d r ◦ h r = I − k r , d r ◦ g r = I − k r , where d r and d r are the chain maps of the complex K r and K r at level p = 1,respectively.Moreover, since the complexes K r and K r are Fredholm split at level p = 0, by[4, Theorem 2.7] the complexes K r and K r are Fredholm at level p = 0 and N ( d r )and N ( d r ) have direct complements in X ⊗ ∧ C n and X ⊗ ∧ C m respectively.Now, by an argument similar to [4, Theorem 2.7] or [5, Proposition 2.1], wehave that the maps h r , g r k r and k r may be chosen in the following way. If N r and N r are finite dimensional subspaces of X and X respectively, such that R ( d r ) ⊕ N r = X and R ( d r ) ⊕ N r = X and L r and L r are closed linear subspacesof X ⊗ ∧ C n and X ⊗ ∧ C m respectively, such that N ( d r ) ⊕ L r = X ⊗ ∧ C n ,and N ( d r ) ⊕ L r = X ⊗ ∧ C m , then, k r , respectively k r , may be chosen asthe projector onto N r , respectively N r , whose null space coincide with R ( d r ),respectively R ( d r ), and the map h r , respectively g r , may be chosen such that h r ◦ d r = I | L r , respectively g r ◦ d r = I | L r , h r | N r = 0, respectively g r | N r = 0. In particular, R ( k r ) ∼ = H ( K r ) and R ( k r ) ∼ = H ( K r ).Now, as in Proposition 4.2, for all r ∈ N we have a well defined map H r : X ˜ ⊗ X → ( K r ˜ ⊗ K r ) such that d r ◦ H r = I − k r ⊗ k r , where d r is the boundary map of the complex K r ˜ ⊗ K r at level p = 1.Then, since for all r ∈ N , R ( k r ⊗ k r ) = R ( k r ) ⊗ R ( k r ) (see Proposition 4.2),for all r ≥ l we have thatdim H ( K r ) = dim H ( K r ˜ ⊗ K r ) ≤ dim R ( k r ⊗ k r )= dim R ( k r ) × dim R ( k r ) = dim H ( K r ) × dim H ( K r )= dim H ( K l ) × dim H ( K l ) , which is impossible for ( λ, µ ) ∈ ˜ A ( S ⊗ I, I ⊗ T ) and dim H ( K r ) = codim M r (( S − λ ) ⊗ I, I ⊗ ( T − µ )). EMI-BROWDER JOINT SPECTRA 13
By means of a similar argument it is possible to see the second statement. (cid:3) Operator ideals between Banach spaces
In this section we extend our descriptions of the semi-Fredholm joint spectra andthe semi-Browder joint spectra for tuples of left and right multiplications definedon an operator ideal between Banach spaces in the sense of [5]. We first recallthe definition of such an ideal and then we introduce the tuples with which weshall work. For a complete exposition see [5].An operator ideal J between Banach spaces X and X will be a linear subspace of L( X , X ), equiped with a space norm α such that(i) x ⊗ x ′ ∈ J and α ( x ⊗ x ′ ) = k x kk x k ,(ii) SAT ∈ J and α ( SAT ) ≤k S k α ( A ) k T k ,where x ∈ X , x ′ ∈ X ′ , A ∈ J , S ∈ L( X ), T ∈ L( X ), and x ⊗ x ′ is theusual rank one operator X → X , x → < x , x ′ > x .Examples of this kind of ideals are given in [5, section 1].Let us recall that such operator ideal J is naturally a tensor product relativeto < X , X ′ > and < X ′ , X > , with the bilinear mappings X × X ′ → J, ( x , x ′ ) → x ⊗ x ′ , L ( X ) × L ( X ′ ) → L( J ) , ( S, T ′ ) → S ⊗ T ′ , where S ⊗ T ′ ( A ) = SAT .On the other hand, if X is a Banach space and U ∈ L( X ), we denote by L U and R U the operators of left and right multiplication in L( X ), respectively, i.e.,if V ∈ L( X ), then L U ( V ) = U V and R U ( V ) = V U .Now, if S = ( S , . . . , S n ) and T = ( T , . . . , T m ) are tuples of commuting oper-ators defined on X and X respectively, if J is seen as a tensor product of X and X relative to < X , X ′ > and < X ′ , X > , then the tuple of left and rightmultilications ( L S , R T ) defined on L ( J ), ( L S , R T ) = ( L S , . . . , L S n , R T , . . . , R T m ),may be identified with the ( n + m )-tuple ( S ⊗ I, I ⊗ T ′ ) defined on X ˜ ⊗ X ′ , where T ′ = ( T ′ , . . . , T m ′ ) and for all i = 1 , . . . , m , T i ′ is the adjoint map associated to T i (see [5, Theorem 3.1]).In addition, if λ ∈ C n and µ ∈ C m , and if we denote by K and K ′ the Koszulcomplexes associated to S and λ and T ′ and µ respectively, then the total complexof the double complex obtained from the tensor product of K and K ′ , K ˜ ⊗ K ′ is isomorphic to ˜ K , the Koszul complex associated to ( S ⊗ I, I ⊗ T ′ ) and ( λ, µ )on X ˜ ⊗ X , which is naturally isomorphic to the Koszul complex of ( L S , R T ) and( λ, µ ) on L( J ), see [5, section 3].In order to state our description of the semi-Fredholm and the semi-Browderjoint spectra of the tuple ( L S , R T ), as we have done in section 4, we first describethe defect and the approximate point spectra of the mentioned tuple. Proposition 6.1.
Let X and X be two complex Banach spaces, and J andoperator ideal between X and X in the sense of [5]. Let us consider two tuplesof commuting operators defined on X and X , S and T respectively. Then, if ( L S , R T ) is the tuple of left and right multiplications defined on L ( J ) , we havethat (i) σ δ ( S ) × σ π ( T ) ⊆ σ δ ( L S , R T ) ⊆ sp δ ( L S , R T ) ⊆ sp δ ( S ) × sp π ( T ) , (ii) σ π ( S ) × σ δ ( T ) ⊆ σ π ( L S , R T ) ⊆ sp π ( L S , R T ) ⊆ sp π ( S ) × sp δ ( T ) . In addition, if X and X are Hilbert spaces, the above inclusions are equalities.Proof. As we have said, J may be seen as the tensor product of X and X ′ , X ˜ ⊗ X ′ , relative to < X , X ′ > and < X , X ′ > , and ( L S , R T ) may be identifiedwith the tuple ( S ⊗ I, I ⊗ T ′ ). Moreover, if K and K ′ denote the differential spaceassociated to K and K ′ respectively, then ˜ K , the differentiable space associatedto ˜ K , is isomorphic to K ˜ ⊗K ′ (see [5, section 3]).In addition, since for all i = 1 , . . . , n S i ∈ L( X ) and for all j = 1 , . . . , m T j ∈ L( X ), the differential spaces K and K ′ satisfy the conditions of [5, Theorem2.2], and by means of an argument similar to the one of Proposition 4.1 we havethat σ δ ( S ) × σ δ ( T ′ ) ⊆ σ δ ( S ⊗ I, I ⊗ T ′ ) = σ δ ( L S , R T ) . However, by [8, Theorem 2.0], σ π ( T ) = σ δ ( T ′ ). Thus, we have proved the lefthand side inclusion of the first statement.The middle inclusion is clear.In order to see the right hand inclusion, let us first observe that if µ / ∈ sp π ( T ),then µ / ∈ sp δ ( T ′ ).In fact, if K is split at level p = m , then by [8, Lemma 2.2] K ′ is split at level p = 0.Now, by the isomorphism of [8, Lemma 2.2], if we think the homotopy operatorwhich gives the splitting for the complex K ′ at level p = 0 as a matrix, theneach component of the matrix is an adjoint operator. In particular, by meansof the properties of the tensor product of [5], it is possible to adapt the proofof the corresponding inclusion of Proposition 4.1 in order to see that if ( λ, µ ) / ∈ sp δ ( S ) × sp π ( T ), then ( λ, µ ) / ∈ sp δ ( S ⊗ I, I ⊗ T ′ ) = sp δ ( L S , R T ).The second statement may be proved by means of a similar argument. (cid:3) In the following proposition we give our description of the semi-Fredholm jointspectra of the tuple ( L S , R T ). Proposition 6.2.
Let X and X be two complex Banach spaces, and J andoperator ideal between X and X in the sense of [5]. Let us consider two tuplesof commuting operators defined on X and X , S and T respectively. Then, if ( L S , R T ) is the tuple of left and right multiplications defined on L ( J ) , we havethat (i) σ Φ − ( S ) × σ π ( T ) ∪ σ δ ( S ) × σ Φ + ( T ) ⊆ σ Φ − ( L S , R T ) ⊆ sp δe ( L S , R T ) ⊆ sp δe ( S ) × sp π ( T ) ∪ sp δ ( S ) × sp πe ( T ) , (ii) σ Φ + ( S ) × σ δ ( T ) ∪ σ π ( S ) × σ Φ − ( T ) ⊆ σ Φ + ( L S , R T ) ⊆ sp πe ( L S , R T ) ⊆ sp πe ( S ) × sp δ ( T ) ∪ sp π ( S ) × sp δe ( T ) . EMI-BROWDER JOINT SPECTRA 15
In addition, if X and X are Hilbert spaces, the above inclusions are equalities.Proof. By means of an argument similar to the one of Proposition 4.2, adaptedas we have done in Poposition 6.1, it is possible to see that σ Φ − ( S ) × σ δ ( T ′ ) ∪ σ δ ( S ) × σ Φ − ( T ′ ) ⊆ σ Φ − ( S ⊗ I, I ⊗ T ′ ) = σ Φ − ( L S , R T ) . However, by [8, Theorem 2.0] σ δ ( T ′ ) = σ π ( T ), and by elementary propertiesof the adjoint of an operator it is easy to see that σ Φ + ( T ) ⊆ σ Φ − ( T ′ ). Thus, wehave seen the left hand side inclusion of the first statement.The middle inclusion is clear.Let us consider ( λ, µ ) ∈ sp δe ( L S , R T ) \ ( sp δe ( S ) × sp π ( T ) ∪ sp δ ( S ) × sp πe ( T )).By Proposition 6.1 we have that λ ∈ sp δ ( S ) \ sp δe ( S ) and µ ∈ sp π ( T ) \ sp πe ( T ).However, by [8, Lemma 2.2] and elementary properties of the adjoint of an oper-ator we have that µ ∈ sp δ ( T ′ ) \ sp δe ( T ′ ). Then, as in Proposition 4.2, there aretwo linear bounded maps h : X → X ⊗ ∧ C n , g ′ : X ′ → X ′ ⊗ ∧ C m , and twofinite rank projectors k : X → X and k ′ : X → X such that d ◦ h = I − k , d ′ ◦ g ′ = I − k ′ , where d ′ is the boundary map of the complex K ′ at level p = 1.Now, by the isomorphism of [8, Lemma 2.2], if we think the map g ′ as a matrix,then each component of the matrix is an adjoint operator. Then, by the propertiesof the tensor product introduced in [5], we may consider the well defined map H : X ˜ ⊗ X ′ → ( K ˜ ⊗ K ′ ) , H = ( h ⊗ I, I ⊗ g ′ ) . Now, by an argument similar to the one of Proposition 4.2, it is easy to seethat ( λ, µ ) / ∈ sp δe ( S ⊗ I, I ⊗ T ′ ) = sp δe ( L S , R T ), which is impossible by ourassumptions.By means of a similar argument it is possible to see the second statement. (cid:3) We now give our description of the semi-Browder joint spectra of the tuple ofleft and right multiplications ( L S , R T ) defined on L( J ). Theorem 6.3.
Let X and X be two complex Banach spaces, and J and op-erator ideal between X and X in the sense of [5]. Let us consider two tuplesof commuting operators defined on X and X , S and T respectively. Then, if ( L S , R T ) is the tuple of left and right multiplications defined on L ( J ) , we havethat (i) σ B − ( S ) × σ π ( T ) ∪ σ δ ( S ) × σ B + ( T ) ⊆ σ B − ( L S , R T ) ⊆ sp B − ( L S , R T ) ⊆ sp B − ( S ) × sp π ( T ) ∪ sp δ ( S ) × sp B + ( T ) , (ii) σ B + ( S ) × σ δ ( T ) ∪ σ π ( S ) × σ B − ( T ) ⊆ σ B + ( L S , R T ) ⊆ sp B + ( L S , R T ) ⊆ sp B + ( S ) × sp δ ( T ) ∪ sp π ( S ) × sp B − ( T ) , In addition, if X and X are Hilbert spaces, the above inclusions are equalities. Proof.
In order to see the first statement, let us observe that if K r is the Koszulcomplex associated to the tuple ( S − λ ) r = (( S − λ ) r , . . . , ( S n − λ n ) r ), and if K ′ r is the Koszul complex associated to the tuple ( T ′ − µ ) r = (( T ′ − µ ) r , . . . , ( T m − µ m ) r ), then ˜ K r , the Koszul complex associated to the tuple (( S − λ ) r ⊗ T, I ⊗ ( T ′ − µ ) r ), is isomorphic to the total complex obtained from the double complexof the tensor product of K r and K ′ r , i.e., ˜ K r ∼ = K r ˜ ⊗ K ′ r (see [5, section 3]).Now, we may adapt the proof of the left hand inclusion of Theorem 5.1, aswe have done in Proposition 6.1, using in particular Proposition 6.2 instead ofProposition 4.2, in order to see that σ B − ( S ) × σ δ ( T ′ ) ⊆ σ B − ( S ⊗ I, I ⊗ T ′ ) = σ B − ( L S , R T ). However, by [8, Theorem 2.0], σ π ( T ) = σ δ ( T ′ ). Thus, σ B − ( S ) × σ π ( T ) ⊆ σ B − ( L S , R T )A similar argument, using in particular that σ B − ( T ′ ) = σ B + ( T ) (see [7, Theorem11]), gives us that σ δ ( S ) × σ B + ( T ) ⊆ σ B − ( L S , R T ).The middle inclusion is clear.In order to see the right hand inclusion, it is possible to adapt the proof of thecorresponding part of Theorem 5.1.Indeed, if we use Proposition 6.2 instead of Proposition 4.2, we have that sp δe ( L S , R T ) ⊆ sp B − ( S ) × sp π ( T ) ∪ sp δ ( S ) × sp B + ( T ). On the other hand, if wesuppose that ( λ, µ ) ∈ ˜ A ( L S , R T ) \ ( sp B − ( S ) × sp π ( T ) ∪ sp δ ( S ) × sp B + ( T )), then itis possible to adapt the argument of Theorem 5.1 in order to get a contradiction.However, in order to adapt this part of the proof, we have to observe the followingfacts.First, by [8, Lemma 2.2], if µ / ∈ sp πe ( T ), then µ / ∈ sp δe ( T ′ ). Moreover, ifthere exists l ∈ N such that for all r ≥ l dim H m ( K r ) = dim H m ( K l ), thenby [7, Theorem 11] it is easy to see that dim H ( K ′ r ) = dim H ( K ′ l ), for all r ≥ l . In addition, if µ / ∈ sp δe ( T ′ ), by the analytic spectral mapping theorem forthe essential split defect spectrum, the complex K ′ r are Fredholm split for all r ∈ N , i.e., there are operators g r ′ : X ′ → X ′ ⊗ ∧ C m and finite rank projectors k r ′ : X ′ → X ′ such that d r ′ ◦ g r ′ = I − k r ′ , where d r ′ denotes the chain mapof the complex K ′ r at level p = 1. Furthermore, by [8, Lemma 2.2], if for r ∈ N we think the map g r ′ as matrix, then each component of the matrix is an adjointoperator, and by elementary properties of the adjoint of an operator, the maps g r ′ and k r ′ may be chosen with the same properties of the maps g r and k r ofTheorem 5.1. With all this observations it is possible to conclude the proof ofthe right hand side inclusion of the first statement.The second statement may be proved by means of a similar argument. (cid:3) References
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EMI-BROWDER JOINT SPECTRA 17
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