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Bounded Linear) Operator on a Complex Banach Space X, and F[T ]X−→Is Defined by Means of a Local Functional Calculus

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Bounded Linear) Operator on a Complex Banach Space X, and F[T ]X−→Is Defined by Means of a Local Functional Calculus PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 2, February 1997, Pages 417–425 S 0002-9939(97)03477-1 STABILITY OF THE LOCAL SPECTRUM TERESA BERMUDEZ,´ MANUEL GONZALEZ,´ AND ANTONIO MARTINON´ (Communicated by Palle E. T. Jorgensen) Abstract. We give some conditions implying the equality of local spectra σ(x, T )=σ(f[T]x, T ), where T : X X is a (bounded linear) operator on a complex Banach space X, and f[T ]x−→is defined by means of a local functional calculus. Moreover, we give conditions implying the stability of the local spectrum for the holomorphic and the meromorphic functional calculi. 1. Introduction Let X be a complex Banach space and let T be a (bounded linear) operator defined on X. For every x X, the operator T has a local spectrum σ(x, T )which is a useful tool in the study∈ of the structure of the spectrum and the invariant subspaces of T. The problem we address is the detection of vectors y which have the same local spectrum as a fixed vector x, namely (1) σ(x, T )=σ(y,T). This problem has deserved the attention of several authors. In [2], Erdelyi and Lange prove that if T is an operator satisfying the Single Valued Extension Property (hereafter referred to as SVEP) and xT is the local resolvent function of T in x, then (2) σ(xT (λ)b,T)=σ(x, T ) for all λ C σ(x, T ). Moreover, if A is an operator which commutes with an operator T∈ satisfying\ the SVEP,b then (3) σ(Ax, T ) σ(x, T ), ⊂ for all x X. In particular, if A has an inverse, then the expression (3) turns into ∈ an equality. It also follows, from the results derived by Bartle [1], that given λ C ∈ and n N,wehave ∈ (4) σ((λ T )nx, T ) σ(x, T ) σ((λ T )nx, T ) λ . − ⊂ ⊂ − ∪{ } Received by the editors May 18, 1995. 1991 Mathematics Subject Classification. Primary 47A11, 47A60. Key words and phrases. Local spectrum, holomorphic functional calculus, meromorphic func- tional calculus. Supported in part by DGICYT Grant PB 91-0307 (Spain). c 1997 American Mathematical Society 417 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 418 TERESA BERMUDEZ,´ MANUEL GONZALEZ,´ AND ANTONIO MARTINON´ Hence if λ/σ(x, T ), then ∈ σ(x, T )=σ((λ T )nx, T ). − Finally McGuire [7] shows that if T is an operator in a complex separable Hilbert space H with an empty point spectrum, and f is an analytic function on an open set (f) containing σ(x, T ), not identically zero on any component of (f), then 4 4 σ(f[T ]x, T )=σ(x, T ), where f[T ]x is defined by using the “Cauchy formula” with the local resolvent of T in x (see below). In this paper we give conditions implying the equality σ(x, T )=σ(Ax, T ) for certain operators A obtained from T using the meromorphic functional calculus or the local functional calculus. Our results include those of [1], [2] and [7]. 2. Preliminaries Let X be a complex Banach space. We denote by L(X) the class of all (bounded linear) operators on X, and by C(X) the class of all closed operators T with domain D(T )andrange R(T )inX. Given T C(X), we have that λ belongs to ρ(T), the resolvent set of T, if there ∈ 1 1 exists (λ T )− L(X) such that R((λ T )− )=D(T)andforeveryx Xwe have − ∈ − ∈ 1 (λ T )(λ T )− x = x. − − We denote by σ(T ):=C ρ(T)thespectrum set of T. Note that the set ρ(T )is \ 1 open and the resolvent function λ ( λ T ) − is analytic in ρ(T ). Likewise, for every x X the local−→ spectral− theory is defined as follows. We say that λ ρ(x, T ), the local∈ resolvent set of T in x, if there exists an analytic function w : U ∈ X defined on a neighborhood U of λ, which satisfies the equation −→ (µ T )w(µ)=x, − for every µ U. Wedenotebyσ(x, T ):=C ρ(x, T )thelocal spectrum set of T in x.Since∈ wis not necessarily unique, a property\ is introduced to avoid this problem. A closed linear operator T : D(T ) X X satisfies the SVEP if for every ⊂ −→ analytic function h : (h) X definedonanopenset (h) C, the condition (λ T )h(λ) 0 implies4 h −→0. If T satisfies the SVEP, then4 for⊂ every x X there − ≡ ≡ ∈ exists a unique maximal analytic function xT : ρ(x, T ) X such that −→ (λI T )xT (λ)=x, − b for every λ ρ(x, T ). The function xT is called the local resolvent function of T at x. See [2], [3]∈ and [6] for further details.b In the following proposition we recallb from [2] some basic properties for operators satisfying the SVEP. Proposition 1. Let T L(X ) satisfy the SVEP and let x X. Then the following assertions hold: ∈ ∈ (i) If λ ρ(x, T ),thenσ(xT(λ),T)=σ(x, T ). ∈ b License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STABILITY OF THE LOCAL SPECTRUM 419 (ii) If S L(X) commutes with T and y = Sx,thenSxT(λ)=yT(λ)for λ ρ(x, T∈).Inparticular,σ(Sx,T) σ(x, T ). ∈ ⊂ For T L(X ), the holomorphic functional calculus is definedb as followsb [9]. Let f be an analytic∈ function defined on an open set (f) containing σ(T ). The operator f(T ) L(X) is defined by the “Cauchy formula”4 ∈ 1 f(T ):= f(λ)R(λ, T )dλ, 2πi ZΓ where Γ is the boundary of a Cauchy domain D such that σ(T ) D (f). This definition may be extended to meromorphic functions.⊂ Let f⊂4be a mero- morphic function in an open set (f) containing σ(T ), such that the poles of f are 4 not in the point spectrum σp(T ), and let α1,...,αk be the poles of f in σ(T ), with multiplicity n1,...,nk, respectively. We consider the polynomial p given by k ni p(λ)= (αi λ) . − i=1 Y Note that g(λ):=f(λ)p(λ) is an analytic function. In [4], Gindler defines a mero- morphic functional calculus by 1 f T := g(T )p(T )− . { } In this way he obtains an operator f T C(X).Clearly, the meromorphic calculus is an extension of the holomorphic calculus.{ }∈ 3. The local functional calculus Let f be an analytic function defined on an open set (f). For H a Hilbert space and T L(H) an operator with empty point spectrum,4 McGuire [7] introduces a local functional∈ calculus in which he defines f[T ]x,forx Hwith σ(x, T ) (f), by ∈ ⊂4 1 (5) f [T ]x = f(λ)x (λ)dλ, 2πi T ZΓ where Γ is the boundary of a Cauchy domain Db such that σ(x, T ) D (f). Using this idea, for any T L(X) satisfying the SVEP we define⊂ an operator⊂4 ∈ f[T ]:D(f[T]) X X ⊂ −→ with domain D(f[T ]) := x X : σ(x, T ) (f) { ∈ ⊂4 } and f[T ]x given by (5) for x D(f[T ]). ∈ Proposition 2. Let T L(X ) satisfy the SVEP and let f be an analytic function in (f). Then D(f[T ])∈is a linear subspace of X and f[T ] is a linear operator. 4 Proof. It follows from σ(x + y,T) σ(x, T ) σ(y,T) [2, Proposition 1.5] and the definition. ⊂ ∪ Next, we prove some results concerning the local functional calculus. Proposition 3. Let T L(X ) satisfy the SVEP and let f be an analytic function in (f). Then the following∈ assertions hold: 4 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 420 TERESA BERMUDEZ,´ MANUEL GONZALEZ,´ AND ANTONIO MARTINON´ (i) If S L(X) commutes with T ,thenScommutes with f[T ]; i.e., SD(f[T]) D(f[∈T ]) and Sf[T]x = f[T]Sx for all x D(f[T ]). ⊂ ∈ (ii) If x D(f[T ]) and y := f[T ]x,thenf[T]xT =yT in ρ(x, T ), hence σ(f[T ]x, T ) σ(∈x, T ). ⊂ Proof. (i) Let x D(f[T ]). By Proposition 1b we haveb σ(Sx,T) σ(x, T ), and so ∈ ⊂ Sx D(f[T]). Moreover, SxT is a restriction of the local resolvent of Sx. Then we have∈ b 1 Sf[T]x = S f(λ)x (λ)dλ 2πi T ZΓ 1 = f(λ)Sx b(λ)dλ 2πi T ZΓ = f[T ]Sx. b (ii) We prove that f[T ]xT is analytic at every point λ ρ(x, T ). By Proposition ∈ 1 and using the expression for the local resolvent of xT (λ) in [2, Proposition 1.5] we have b 1 xT (µ) xT (λ)b f [T ]xT (λ)= f(µ) − dµ 2πi λ µ ZΓ − 1 f(µ)x b(µ) b 1 f(µ)x (λ) b = T dµ T dµ. 2πi λ µ − 2πi λ µ ZΓ − ZΓ − b b For every x∗ X ∗ , the first integral satisfies ∈ 1 f(µ)x (µ) 1 f(µ) x T dµ = x (x (µ))dµ. ∗ 2πi λ µ 2πi λ µ ∗ T ZΓ − ZΓ − b So it is analytic by [8, Theorem 10.7].
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