Spectrum of Weighted Composition Operators Part VII Essential spectra of weighted composition operators on C(K). The case of non-invertible homeomorphisms
aa r X i v : . [ m a t h . F A ] F e b SPECTRUM OF WEIGHTED COMPOSITIONOPERATORSPART VIIESSENTIAL SPECTRA OF WEIGHTEDCOMPOSITION OPERATORS ON C ( K ) . THE CASE OFNON-INVERTIBLE HOMEOMORPHISMS. ARKADY KITOVER AND MEHMET ORHON
Abstract.
We provide a complete description of the spectrumand the essential spectra of weighted composition operators T = wT ϕ on C ( K ) in the case when the map ϕ is a non-invertiblehomeomorphism of K into itself. Introduction
Let K be a compact Hausdorff space and C ( K ) be the space of allcomplex-valued continuous functions on K . A weighted compositionoperator T on C ( K ) is an operator of the form( T f )( k ) = w ( k ) f ( ϕ ( k ) , k ∈ K, f ∈ C ( K ) , where ϕ is a continuous map of K into itself and w ∈ C ( K ).The spectrum of arbitrary weighted composition operators on C ( K )was investigated by the first named author in [3, Theorems 3.10, 3.12,and 3.23]. On the other hand, the full description of essential spectra(in particular Fredholm and semi-Fredholm spectra) of such operatorsis, as far as we are informed, still not known. In a special case, whenthe map ϕ is a homeomorphism of K onto itself, such a descriptionwas obtained in [4, Theorems 2.7 and 2.11]. In this paper we providea description of the spectrum (Theorem 3.1) and the essential spectra(Theorems 5.1, 5.2, and 5.6) of a weighted composition operator T = wT ϕ in the case when ϕ is a non-surjective homeomorphism of K into itself. Date : Thursday 11 th February, 2021.2010
Mathematics Subject Classification.
Primary 47B33; Secondary 47B48, 46B60.
Key words and phrases.
Spectrum, Fredholm spectrum, essential spectra. Preliminaries
In the sequel we use the following standard notations. N is the semigroup of all natural numbers. Z is the ring of all integers. R is the field of all real numbers. C is the field of all complex numbers. T is the unit circle. We use the same notation for the unit circleconsidered as a subset of the complex plane and as the group of allcomplex numbers of modulus 1. U is the open unit disc. D is the closed unit disc.All the linear spaces are considered over the field C of complex num-bers.The algebra of all bounded linear operators on a Banach space X isdenoted by L ( X ).Let E be a set and ϕ : E → E be a map of E into itself. Then ϕ n , n ∈ N , is the n th iteration of ϕ , ϕ ( e ) = e, e ∈ E ,If F ⊆ E then ϕ ( − n ) ( F ) means the full n th preimage of F , i.e. ϕ ( − n ) ( F ) = { e ∈ E : ϕ n ( e ) ∈ F } .If the map ϕ is imjective then ϕ − n , n ∈ N , is the n th iteration of theinverse map ϕ − . In this case we will write ϕ − n ( F ) instead of ϕ ( − n ) ( F ).Let w be a complex-valued function on E . Then w = 1 and w n = w ( w ◦ ϕ ) . . . ( w ◦ ϕ n − ), n ∈ N .Recall that an operator T ∈ L ( X ) is called semi-Fredholm if its range R ( T ) is closed in X and either dim ker T < ∞ or codim R ( T ) < ∞ .The index of a semi-Fredholm operator T is defined asind T = dim ker T - codim R ( T ).The subset of L ( X ) consisting of all semi-Fredholm operators is de-noted by Φ.Φ + = { T ∈ Φ : null ( T ) = dim ker T < ∞} is the set of all uppersemi-Fredholm operators in L ( X ).Φ − = { T ∈ Φ : def ( T ) = codim R ( T ) < ∞} is the set of all lowersemi-Fredholm operators in L ( X ). F = Φ + ∩ Φ − is the set of all Fredholm operators in L ( X ). W = { T ∈ F : ind T = 0 } is the set of all Weyl operators in L ( X ).Let T be a bounded linear operator on a Banach space X . As usual,we denote the spectrum of T by σ ( T ) and its spectral radius by ρ ( T ).We will consider the following subsets of σ ( T ). σ p ( T ) = { λ ∈ C : ∃ x ∈ X \ { } , T x = λx } .σ a.p. ( T ) = { λ ∈ C : ∃ x n ∈ X, k x n k = 1 , T x n − λx n → } . pectrum of weighted composition operators. VII 3 σ r ( T ) = σ ( T ) \ σ a.p. ( T ) == { λ ∈ σ ( T ) : the operator λI − T has the left inverse } . Remark 2.1.
It is clear that σ a.p. ( T ) is the union of the point spectrum σ p ( T ) and the approximate point spectrum σ a ( T ) of T , while σ r ( T ) isthe residual spectrum of T . We have to notice that the definition ofthe residual spectrum varies in the literature. Remark 2.2.
If needed to avoid an ambiguity, we will use notations σ ( T, X ), ρ ( T, X ), et cetera.Following [2] we consider the following essential spectra of T . σ ( T ) = { λ ∈ C : λI − T Φ } is the semi-Fredholm spectrum of T . σ ( T ) = { λ ∈ C : λI − T Φ + } is the upper semi-Fredholm spectrumof T . σ ( T ′ ) = { λ ∈ C : λI − T Φ − } is the lower semi-Fredholm spectrumof T . σ ( T ) = { λ ∈ C : λI − T
6∈ F } is the Fredholm spectrum of T . σ ( T ) = { λ ∈ C : λI − T
6∈ W} is the Weyl spectrum of T . σ ( T ) = σ ( T ) \ { ζ ∈ C : there is a component C of the set C \ σ ( T )such that ζ ∈ C and the intersection of C with the resolvent set of T is not empty } is the Browder spectrum of T .The Browder spectrum was introduced in [1] as follows: λ ∈ σ ( T ) \ σ ( T ) if and only if λ is a pole of the resolvent R ( λ, T ). It is not difficultto see ( [2, p. 40]) that the definition of σ ( T ) cited above is equivalentto the original definition of Browder.It is well known (see e.g. [2]) that the sets σ i ( T ) , i ∈ [1 , . . . ,
5] arenonempty closed subsets of σ ( T ) and that σ i ( T ) ⊆ σ j ( T ) , ≤ i < j ≤ , where all the inclusions can be proper. Nevertheless all the spectralradii ρ i ( T ) , i = 1 , ..., ρ e ( T ), (see [2,Theorem I.4.10]) which is called the essential spectral radius of T . Itis also known (see [2]) that the spectra σ i ( T ) , i = 1 , . . . , σ ( T ) in general is not.It is immediate to see that σ ( T ) = σ ( T ) ∩ σ ( T ′ ) and that σ ( T ) = σ ( T ) ∪ σ ( T ′ ).Let us recall that a sequence x n of elements of a Banach space X iscalled singular if it does not contain any norm convergent subsequence.We will use the following well known characterization of σ ( T ) (seee.g. [2]). The following statements are equivalent(a) λ ∈ σ ( T ).(b) There is a singular sequence x n such that k x n k = 1 and λx n − T x n → Arkady Kitover and Mehmet Orhon The spectrum of T = wT ϕ Let K be a compact Hausdorff space, ϕ be a homeomorphism of K into itself, and w ∈ C ( K ). We consider the weighted compositionoperator T = wT ϕ on C ( K ) defined as( T f )( k ) = w ( k ) f ( ϕ ( k )) , f ∈ C ( K ) , k ∈ K. (1)By the reasons outlined in the introduction we will always assumethat ϕ ( K ) $ K. (2)We have to introduce some additional notations. L = ∞ \ n =0 ϕ n ( K ) , M = K \ Int K L, N = L \ Int K L. (3)Obviously, ϕ is a homeomorphism of L and N onto themselves and (1)defines the action of T on the spaces C ( L ), C ( M ), and C ( N ). Theorem 3.1.
Let K be a compact Hausdorff space, ϕ be a homeo-morphism of K into itself, and w ∈ C ( K ) . Let T be the operator on C ( K ) defined by (1). Assume (2) and notations in (3). Then(I) σ ( T, C ( M )) is either the disk ρ ( T, C ( M )) U or the singleton { } .(II) σ ( T ) = σ ( T, C ( M )) ∪ σ ( T, C ( L )) .Proof. (I) follows from (2) and Theorems 3.10 and 3.12 in [3].The proof of (II) will be divided into several steps.Step 1. We will prove the inclusion σ ( T, C ( M )) j σ ( T ). Assume tothe contrary that there is a λ ∈ C , λ ∈ σ ( T, C ( M )) \ σ ( T ). Because0 ∈ σ ( T ), we can assume without loss of generality that λ = 1. Then( I − T ) C ( K ) = C ( K ) and because ϕ ( Int K L ) = Int K L = ϕ − ( Int K L )we also have ( I − T ) C ( M ) = C ( M ). Because 1 ∈ σ ( T, C ( M )) thereis an f ∈ C ( M ) such that f = 0 and T f = f . Then it follows fromLemma 3.6 in [3] that there is a point k ∈ M such that | w n ( k ) | ≥ , | w n ( ϕ − n ) | ≤ , n ∈ N . (4)The point k is either not ϕ -periodic or, in virtue of (2), a limit point ofthe set of all non ϕ -periodic points in K . It follows from the proof ofTheorem 3.7 in [3] that T ⊂ σ ( T ), in contradiction with our assump-tion.Step 2. On this step we prove the inclusion σ ( T, C ( L )) j σ ( T ). Let λ ∈ σ ( T, C ( L ) \ σ ( T ). We can assume that λ = 1, and like on theprevious step ( I − T ) C ( K ) = C ( K ) implies that ( I − T ) C ( L ) = C ( L ).Therefore there is an f ∈ C ( L ), f = 0, such that T f = f . Considertwo possibilities. pectrum of weighted composition operators. VII 5 (a) f L \ Int K L . Let k ∈ L \ Int K L be such that | f ( k ) | =max L \ Int K L | f | . Then like on step 1 we see that T ⊆ σ ( T ).(b) f ≡ L \ Int K L . We will define the function ˜ f ∈ C ( K ) as˜ f ( k ) = ( f ( k ) if k ∈ L k ∈ K \ L .
Then T ˜ f = ˜ f , and 1 ∈ σ ( T ) contrary to our assumption.Combining steps 1 and 2 we see that σ ( T, C ( M )) ∪ σ ( T, C ( L )) j σ ( T ).Step 3. We prove the inclusion σ ( T ) j σ ( T, C ( M )) ∪ σ ( T, C ( L )). Let λ ∈ σ ( T ). If λ = 0 then λ ∈ D and therefore without loss of generalitywe can assume that λ = 1.Consider first the case when 1 ∈ σ ap ( T ). Then there is a sequence f n ∈ C ( K ), k f n k = 1 and f n → n →∞
0. But then clearly either k f n k C ( L ) k f n k C ( M )
0, and therefore1 ∈ σ ap ( T, C ( L ) ∪ σ ap ( T, C ( M )) ⊆ D ∪ σ ( T, C ( L )) . If on the other hand 1 ∈ σ r ( T, C ( K )) then there is a regular nonzeroBorel measure µ on K , µ ∈ C ( K ) ′ , such that T ′ µ = µ . It is easy to seethat supp ( µ ) ⊆ L whence 1 ∈ σ ( T, C ( L )). (cid:3) Some axillary results
To obtain a description of the essential spectra of T we will need aseries of lemmas. In the statements of all of the lemmas we will assume,without mentioning it explicitly, that T is an operator on C ( K ) definedby (1), that ϕ is a homeomorphism of K into itself, and that (2) holds.We will also assume notations from (3). Lemma 4.1.
Assume that T is invertible on C ( L ) and that < | λ | < /ρ ( T − , C ( L )) . Then ( λI − T ) C ( K ) = C ( K ) .Proof. It is enough to prove that the operator λI − T ′ is bounded frombelow, where T ′ is the Banach dual of T . Assume to the contrary thatthere is a sequence µ n ∈ ( C ( K )) ′ such that k µ n k = 1 and T ′ µ n − λµ n →
0. Because the operator T ′ preserves disjointness (see e.g. [4,Lemma 5.13]) we have | T ′ || µ n | − | λ || µ n | →
0. Let µ ∈ C ( K ) ′ be a limitpoint of the set {| µ n |} in the weak ⋆ topology. Then µ is a probabilitymeasure on K . Because the operator | T ′ | = | T | ′ is weak ⋆ continuous wehave | T ′ | µ = | λ | µ . But then supp ( µ ) j L whence | λ | ∈ σ ( | T | , C ( L )).The last statement involves a contradiction because the operator | T | isinvertible on C ( L ) and ρ ( | T | − , C ( L )) = ρ ( T − , C ( L )). (cid:3) Arkady Kitover and Mehmet Orhon
Lemma 4.2. (1) Let λ ∈ σ ap ( T, C ( N )) . Then λ T ⊆ σ ( T ) .(2) Let λ ∈ σ ap ( T ′ , C ′ ( N )) . Then λ T ⊆ σ ( T ′ ) .Proof. We divide the proof into four steps.(I) Let λ = 0 ∈ σ ap ( T, C ( N )). Then the weight w takes value 0 on N . It follows from the definition of N that there are pairwise distinctpoints k n ∈ K such that | w ( k n ) | ≤ /n . Let u n be the characteristicfunction of the singleton { k n } . Then u n ∈ C ′′ ( K ), k u n k = 1, thesequence u n is singular, and T ′′ u n →
0. Thus 0 ∈ σ ( T ′′ ) = σ ( T ).(II) Let 0 ∈ σ ap ( T ′ , C ′ ( N )). Because T ′ = ( T ϕ ) ′ w ′ and ( T ϕ ) ′ is anisometry of C ′ ( N ) the weight w takes value 0 on N . Let k n be as inpart (I) of the proof and δ n be the Dirac measure corresponding to thepoint k n . Then the sequence δ n is singular and T ′ δ n → λ ∈ σ ap ( T, C ( N )) and λ = 0. Without loss of generalitywe can assume that | λ | = 1. Recall that the restriction of ϕ on N is ahomeomorphism of N onto itself. Therefore by [3, Lemma 3.6] there isa point k ∈ N such that | w n ( k ) | ≥ | w n ( ϕ − n ( k )) | ≤ n ∈ N . Letus fix an m ∈ N . From the definition of the set N follows that there isa net { k α } of points in K \ L convergent to ϕ − m ( k ). From this trivialobservation and from the fact that K \ L does not contain ϕ -periodicpoints easily follows the existence of points k n ∈ K \ L, n ∈ N with theproperties.(a) The points ϕ i ( k n ) , − n − ≤ i ≤ n + 1 are pairwise distinct.(b) The sets A n = { ϕ i ( k n ) , − n − ≤ i ≤ n + 1 } are pairwise disjoint.(c) For any n ∈ N the following inequalities hold | w i ( k n ) | ≥ / | w i ( ϕ − i ( k n )) | ≤ . (5)Let u n be the characteristic function of the singleton { ϕ n ( k n ) } . Then u n ∈ C ′′ ( K ). Let us fix α ∈ C such that | α | = 1. Consider F n ∈ C ′′ ( K ), F n = n X i =0 (cid:0) − √ n (cid:1) | i − n | α − i ( T ′′ ) i u n . (6)It follows from (5) and 6 by the means of a simple estimate (see also [3,Proof of Theorem 3.7]) that k T ′′ F n − αF n k = o ( k F n k ) , n → ∞ . (7)Condition (b) guarantees that the sequence F n is singular and there-fore (7) implies that α ∈ σ ( T ′′ ) = σ ( T ) . (IV) Let λ ∈ σ ap ( T ′ , C ′ ( N )). It was proved in [3] that there is apoint k ∈ N such that | w n ( k ) | ≤ | w n ( ϕ − n ( k )) | ≥
1. Then we pectrum of weighted composition operators. VII 7 can find points k n ∈ K \ L satisfying conditions (a) and (b) above andalso the following condition | w i ( k n ) | ≤ | w i ( ϕ − i ( k n )) | ≥ / , n ∈ N . (8)Let ν n be the Dirac measure δ ϕ − n ( k n ) , α ∈ T , and µ n = n X i =0 (cid:0) − √ n (cid:1) | i − n | α − i ( T ′ ) i ν n , (9)It follows from (8) and (9 that k T ′ µ n − αµ n k = o ( k µ n k ) , n → ∞ (10)Condition (b) guarantees that the sequence µ n is singular, and there-fore (10) implies that α ∈ σ ( T ′ ). (cid:3) Lemma 4.3. σ ( T, C ( L )) j σ ( T ) and σ ( T ′ , C ′ ( L )) j σ ( T ′ ) .Proof. Let λ ∈ σ ( T, C ( L )). Then there is a singular sequence f n ∈ C ( L ) such that k f n k = 1 and T f n − λf n →
0. We have to consider twopossibilities.(1) k f n k C ( N )
0. Then λ ∈ σ ( T ) by Lemma 4.2 (1).(2) k f n k C ( N ) →
0. Then we can find g n ∈ C ( L ) such that f n − g n → g n ≡ N . Clearly, the sequence g n is singular in C ( L ). Wedefine the function h n ∈ C ( K ) as follows h n ( k ) = ( g n ( k ) if k ∈ L k ∈ K \ L .
The sequence h n is singular in C ( K ) and T h n − λh n →
0. Therefore λ ∈ σ ( T ).The second inclusion is trivial. (cid:3) Lemma 4.4.
Let | λ | > ρ ( T, C ( N )) and λ σ ( T, C ( L )) . Then λ σ ( T ) .Proof. Assume to the contrary that there is a singular sequence f n ∈ C ( K ) such that k f n k = 1 and T f n − λf n →
0. Because | λ | >ρ ( T, C ( N )) and ρ ( T, C ( M )) = ρ ( T, C ( N )) ( see e.g. [3, Theorem 3.23]),we have k f n k C ( M ) →
0. Therefore, if g n is the restriction of f n on L then the sequence g n is singular in C ( L ), k g n k →
1, and
T g n − λg n → λ ∈ σ ( T, C ( L )), a contradiction. (cid:3) Lemma 4.5.
Let T be invertible on C ( N ) and | λ | < /ρ ( T − , C ( N )) .Assume also that λ σ ( T, C ( L )) . Then the following statements areequivalent.(1) λ ∈ σ ( T ) .(2) card ( K \ ϕ ( K )) = ∞ . Arkady Kitover and Mehmet Orhon
Proof.
By Theorem 3.1 we have λ ∈ σ ( T ) and by Lemma 4.1 ( λI − T ) C ( K ) = C ( K ). Therefore λ σ ( T ) if and only if dim ker ( λI − T ) = dim ker ( λI − T ′′ ) < ∞ .Assume that card ( K \ ϕ ( K ) < ∞ . This condition combined with λ σ ( T, C ( N )) provides that dim ker (( λI − T ) , C ( M )) < ∞ . Combiningit with the condition λ σ ( T, C ( L )) we see that dim ker ( λI − T ) < ∞ .Thus, (1) ⇒ (2).Assume next that card ( K \ ϕ ( K )) = ∞ . Then clearly dim ker ( λI − T ) = ∞ and therefore λ ∈ σ ( T ). (cid:3) Lemma 4.6.
The set σ ( T, C ( M )) is rotation invariant and for a λ ∈ σ ( T, C ( M )) , λ = 0 , the following conditions are equivalent.(1) λ T ∩ σ ( T, C ( M )) = ∅ .(2) M is the union of two clopen (in M ) subsets M and M such that(a) M = ∅ ,(b) ϕ ( M i ) j M i , i = 1 , ,(c) If M = ∅ then ρ ( T, C ( M )) < | λ | ,(d) T is invertible on C ( N ) and | λ | < /ρ ( T − , C ( N )) where N = ∞ T n =0 ϕ n ( M ) ,(e) card ( M \ ϕ ( M )) < ∞ .Proof. The implication (2) ⇒ (1) follows from Lemmas 4.4 and 4.5.To prove that (1) ⇒ (2) notice that if λ ∈ σ ( T, C ( M )) \ σ ( T, C ( M ))then by Lemma 4.2 we have λ T ∩ σ ap ( T, C ( N )) = ∅ . We have toconsider two possibilities.(I) λ T ∩ σ ( T, C ( N )) = ∅ . Then (see [3]) N is the union of two clopen(in N ) subsets N and N (one of them might be empty), such that ϕ ( N i ) = N i , i = 1 , ,ρ ( T, C ( N )) < | λ | ,T is invertible on C ( N ) and | λ | < /ρ ( T − , C ( N )) . It follows from the definition of N that M is the union of two clopen(in M ) subsets M and M such that N i = ∞ T n =0 ϕ n ( M i ) , i = 1 ,
2. Itremains to apply Lemmas 4.4 and 4.5.(II) λ T ⊂ σ r ( T, C ( N )). Then (see [3, Theorem 3.29]) N is the unionof three pairwise disjoint nonempty subsets N , N , and O such that( α ) N i , i = 1 , N ,( β ) ϕ ( N i ) = N i , i=1,2,( γ ) ρ ( T, C ( N )) < | λ | ,( δ ) The operator T is invertible on C ( N ) and | λ | < /ρ ( T, C ( N )), pectrum of weighted composition operators. VII 9 ( ε ) If V and V are open neighborhoods in N of N and N , respectively,then there is an n ∈ N , such that for any m ≥ n we have ϕ m ( N \ V ) ⊆ V .We need to consider two subcases.( IIa ) For any open (in M ) neighborhood V of N there is an infinitesubset E of M \ ϕ ( M ) such that ∀ k ∈ E ∃ n = n ( k ) ∈ N such that ϕ n ( k ) ∈ V. It follows from ( δ ) that there are a positive number ε and open (in M )neighborhoods V n , n ∈ N of N such that | w n ( t ) | ≥ ( | λ | + ε ) n , t ∈ V n . (11)By our assumption there are pairwise distinct points k n , n ∈ N andpositive integers m n such that k n ∈ M \ ϕ ( M ) and u n = ϕ m n ( k n ) ∈ V n .We define f n ∈ C ′′ ( M ) as follows. f n ( u n ) = 1 ,f n ( ϕ − l ( u n )) = w l ( ϕ − l ( u n ) λ l , l = 1 , . . . , m n ,f n ( ϕ l ( u n )) = λ l w l ( u n ) , l = 1 , . . . , n,f n ( k ) = 0 otherwise . It follows from the definition of f n and (11) that k f n k ≥ T ′′ f n − λf n →
0. Because the sequence f n is singular we get λ ∈ σ ( T ′′ , C ′′ ( M )) = σ ( T, C ( M )), a contradiction.( IIb ) There is an open (in M ) neighborhood V of N such that the set F = { k ∈ M \ ϕ ( M ) : ∃ n ∈ N such that ϕ n ( k ) ∈ V } is at most finite. It follows from the definition of N that F cannot beempty. Clearly F consists of points isolated in M . We will bring theassumption that F is finite to a contradiction. It is not difficult to seefrom ( ε ) that there is a k ∈ F such that the intersection of cl { ϕ n ( k ) : n ∈ N } with each of the sets N , N , and O is not empty. Thereforewe can assume without loss of generality that M = cl { ϕ n ( k ) : n ∈ N } .Let W be an open neighborhood of N in M such that clW ∩ N = ∅ .It follows from ( ε ) that there is an m ∈ N such that ϕ m ( W ) ⊆ W .Considering, if necessary, the operator T m instead of T we can assumethat m = 1. There is a p ∈ N such that ϕ p ( k ) ∈ W . Then ϕ n ( k ) ∈ W for any n ≥ p , a contradiction. (cid:3) Lemma 4.7. σ ( T ′ , C ′ ( M )) ∪ { } = σ ( T ′ , C ′ ( N )) T ∪ { } . Proof.
The inclusion σ ( T ′ , C ′ ( N )) T ∪ { } ⊆ σ ( T ′ , C ′ ( M )) ∪ { } fol-lows from Lemma 4.2.To prove the converse inclusion consider λ ∈ σ ap ( T ′ , C ′ ( N )) \ { } .The proof of Lemma 4.1 shows that | λ | ∈ σ ap ( | T ′ | , C ′ ( N )). But then(see [4]) λ ∈ σ ap ( T ′ , C ′ ( N )) T . (cid:3) Description of essential spectra of T = wT ϕ Finally we can provide a complete description of essential spectraof weighted composition operators on C ( K ) induced by non-surjectivehomeomorphisms. The statements of Theorems 5.1 and 5.2 below fol-low from the previous lemmas. Theorem 5.1.
Let K be a compact Hausdorff space, ϕ be a homeomor-phism of K into itself, and w ∈ C ( K ) . Let T be the operator on C ( K ) defined by (1). Assume (2) and notations in (3). Let λ ∈ σ ( T ) \ { } .The operator λI − T is upper semi-Fredholm if and only if the followingconditions are satisfied(a) The operator λI − T is upper semi-Fredholm on C ( L ) .(b) The set M is the union of two ϕ -invariant disjoint closed subsets M and M such that(c) if M = ∅ then ρ ( T, C ( M )) < | λ | ,(d) if M = ∅ then T is invertible on C ( N ) , where N = ∞ T n =0 ϕ n ( M ) , | λ | < /ρ ( T − , C ( N )) , and the set M \ ϕ ( M ) is finite.Moreover, dim ker ( λI − T ) = dim ker ( λI − T, C ( L )) + card ( M \ ϕ ( M )) . Theorem 5.2.
Let K be a compact Hausdorff space, ϕ be a homeomor-phism of K into itself, and w ∈ C ( K ) . Let T be the operator on C ( K ) defined by (1). Assume (2) and notations in (3). Let λ ∈ σ ( T ) \ { } .The operator λI − T is lower semi-Fredholm if and only if the followingconditions are satisfied(a) The operator λI − T is lower semi-Fredholm on C ( L ) .(b) λ T ⊆ σ r ( T ′ , C ′ ( N )) .Moreover, def ( λI − T ) = def ( λI − T ) , C ( L )) . Corollary 5.3.
Assume conditions of Theorem 5.1. Let λ ∈ σ ( T ) \{ } .The operator λI − T is Fredholm if and only if it is Fredholm on C ( L ) and conditions (b) - (d) from the statement of Theorem 5.1 are satisfied.Moreover ind ( λI − T ) = ind ( λI − T, C ( L ) + card ( M \ ϕ ( M ) . In particular, if λ σ ( T, C ( L )). pectrum of weighted composition operators. VII 11 Corollary 5.4.
Assume conditions of Theorem 5.1. Assume addition-ally that the set of all ϕ -periodic points is of first category in K . Thenthe spectrum σ ( T ) and the essential spectra σ i ( T ) , i = 1 , · · · , are ro-tation invariant. Corollary 5.5.
Assume conditions of Theorem 3.1.(1) If the set of all isolated ϕ -periodic points is empty, then σ ( T ) = σ ( T ) .(2) If K has no isolated points (in particular, if Int K L = ∅ ), then σ ( T ) = σ ( T ) Proof.
The proof follows from Theorems 3.1 and 5.1, and from [4, The-orems 2.7 and 2.11]. (cid:3)
To finish our description of essential spectra of T it remains to lookat the case λ = 0; Theorem 5.6.
Assume conditions of Theorem 3.1. Then (1)
The operator T is upper semi-Fredholm if and only if the followingtwo conditions are satisfied(a) The set Z ( w ) = { k ∈ K : w ( k ) = 0 } is either empty or all of itspoints are isolated in K ,(b) the set K \ ϕ ( K ) is finite.Moreover, dim ker T = card (( K \ ϕ ( K )) ∪ Z ( w )) . (2) The operator T is lower semi-Fredholm if and only if the set Z ( w ) = { k ∈ K : w ( k ) = 0 } is either empty or all of its points are isolated in K .Moreover, def T = card Z ( w ) . (3) The operator T is Fredholm if and only if it is semi-Fredholm and dim ker T < ∞ . (4) The operator T is Fredholm and ind T = 0 if and only if T isFredholm and w ≡ on K \ ϕ ( K ) . (5) 0 ∈ σ ( T ) .Proof. (1) Assume that T is semi-Fredholm and that dim ker T < ∞ .Then the same is true for T ′′ . If k ∈ K \ ϕ ( K ) then T ′′ χ k = 0 where χ k ∈ C ( K ) ′′ is the characteristic function of the singleton { k } . Therefore card ( K \ ϕ ( K ) < ∞ .Similarly, if k ∈ ϕ ( K ) and w ( k ) = 0 then T ′′ χ ϕ ( k ) = 0 whence Z ( w )is finite or empty. Assume now that w ( k ) = 0 but k is not isolatedin K . Then there is a sequence of pairwise distinct points k n ∈ K such that | w ( k n ) | ≤ /n . The sequence χ ϕ ( k n ) is singular in C ( K ) ′′ and T ′′ χ ϕ ( k n ) → ∈ σ ( T ). Conversely, assume conditions ( a ) and ( b ). Assume also, contrary tothe statement of the theorem that there is a singular sequence f n ∈ C ( K ) such that k f n k = 1 and T f n →
0. It is immediate to see that f n → E = ϕ ( K ) \ ϕ ( Z ( w )). Because the set K \ E is finite the sequence f n contains a convergent subsequence and thuscannot be singular.Finally, if T f = 0 then supp f ⊆ K \ ϕ ( K )) ∪ Z ( w ) whence dim ker T = card (( K \ ϕ ( K )) ∪ Z ( w )).(2) Assume that T is semi-Fredholm and that def T < ∞ . If k ∈ Z ( w )then T ′ δ k = 0 whence Z ( w ) is either finite or empty.Conversely, if card Z ( w ) < ∞ , k µ n k = 1, and T ′ µ n → T ′ preserves disjointness) | T ′ || µ n | = | T ′ µ n | →
0. Let ν n and ν n be the restrictions of the measure | µ n | on Z ( w ) and K \ Z ( w ),respectively. Because Z ( w ) is finite there is a positive constant c suchthat | w | > c on K \ Z ( w ). Therefore ν n → µ n . Therefore, 0 σ ( T ′ ).It is immediate to see that if T ′ µ = 0 then supp µ ⊆ Z ( w ) whence def T = card Z ( w ).(3) and (4) follow immediately from (1) and (2).(5) If σ ( T, C ( M )) is a disk of positive radius then it follows directlyfrom the definition of σ ( T ) that 0 ∈ σ ( T ). On the other hand, if ρ ( T, C ( M )) = 0 then there is a point k ∈ N such that w ( k ) = 0.Because k is not an isolated point of K we see that 0 ∈ σ ( T ) ∩ σ ( T ′ ) = σ ( T ) ⊆ σ ( T ). (cid:3) Example 5.7.
Let
T f ( x ) = f ( x/ , f ∈ C [0 , , x ∈ [0 , σ ( T ) = σ ( T ) = σ ( T ) = σ ( T ) = σ ( T ) = D .(2) σ ( T ′ ) = σ ( T ) = T . References [1] Browder F.E., On the spectral theory of elliptic differential operators I, Math.Ann. 142 , 22-130 (1961)[2] Edmunds D.E. and Evans W.D., Spectral Theory and Differential Operators,Clarendon Press, Oxford (1987)[3] Kitover A.K., Spectrum of weighted composiiton operators: part 1. Spectrumof weighted composition operators on C ( K ) and uniform algebras, Positivity,15, 639 - 659 (2011)[4] Kitover A.K., Spectrum of Weighted Composition Operators Part III: EssentialSpectra of Some Disjointness Preserving Operators on Banach Lattices, In:Ordered Structures and Applications: Positivity VII, Trends in Mathematics,Springer International Publishing, 233–261 (2016) pectrum of weighted composition operators. VII 13 Community College of Philadelphia, 1700 Spring Garden St., Philadel-phia, PA, USA
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