© Hindawi Publishing Corp. ON THE LARGEST ANALYTIC SET FOR CYCLIC OPERATORS

We describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space ℋ; some related consequences are discussed. Furthermore, we show that two densely similar cyclic Banach-space operators possessing Bishop's property (β) have equal approximate point spectra.


Introduction.
In the present paper, all Banach spaces are complex.Let ᐄ be a Banach space and let ᏸ(ᐄ) denote the algebra of all linear bounded operators on ᐄ.For an operator T ∈ ᏸ(ᐄ), let T * , σ (T ), ρ(T ) := C\σ (T ), σ p (T ), σ ap (T ), Γ (T ), ker T , and ranT denote the adjoint operator acting on the dual space ᐄ * , the spectrum, the resolvent set, the point spectrum, the approximate point spectrum, the compression spectrum, the kernel, and the range, respectively, of T .For an operator T ∈ ᏸ(ᐄ), let (T ) denote the open set of complex numbers λ ∈ C for which there exists a nonzero analytic function φ : ᐂ → ᐄ on some open disc ᐂ centered at λ such that (T − µ)φ(µ) = 0 ∀µ ∈ ᐂ. (1.1) The operator T is said to have the single-valued extension property if (T ) is empty.Equivalently if, for every open subset U of C, the only analytic solution φ : U → ᐄ of the equation (T − λ)φ(λ) = 0 (λ ∈ U) is the identically zero function φ ≡ 0 on U. Recall that the operator T is called cyclic with cyclic vector x ∈ ᐄ if the finite linear combinations of the vectors x, T x, T 2 x,... are dense in ᐄ.For a subset F of C, let F := {z : z ∈ F } denote the conjugate set of F .Let T be a cyclic linear bounded operator on a Hilbert space Ᏼ with cyclic vector x.A point λ ∈ C is said to be a bounded point evaluation for T if there is a constant M > 0 such that p(λ) ≤ M p(T )x (1.2) for every polynomial p.The set of all bounded point evaluations for T will be denoted by B(T ).Note that it follows from Riesz Representation theorem that λ ∈ B(T ) if and only if there exists a unique vector k(λ) ∈ Ᏼ such that p(λ) = p(T )x, k(λ) (1.3) for every polynomial p.An open subset O of C is said to be an analytic set for T if it is contained in B(T ) and if for every y ∈ Ᏼ, the complex-valued function y defined on B(T ) by y(λ) = y,k(λ) is analytic on O.The largest analytic set for T will be denoted by B a (T ) and every point of it will be called an analytic bounded point evaluation for T .This paper has been divided into three sections.In Section 2, we give a complete description of the largest analytic set for cyclic Hilbert-space operators and explain more about bounded point evaluations from the point of view of local spectral theory.In Section 3, we prove that (T * ) = σ (T )\σ ap (T ) for every cyclic Banach-space operator possessing Bishop's property (β).Therefore, we use this result and show that densely similar cyclic Banach-space operators possessing Bishop's property (β) have the same approximate point spectra; this result generalizes [12,Theorem 4].
We will need to introduce some notions from the local spectral theory.Suppose that ᐄ is a Banach space.Let T ∈ ᏸ(ᐄ); the local resolvent set ρ T (x) of T at a point x ∈ ᐄ is the union of all open subsets U ⊂ C for which there is an analytic ᐄ-valued function φ on U such that (T − λ)φ(λ) = x for every λ ∈ U .The complement in C of ρ T (x) is called the local spectrum of T at x and will be denoted by σ T (x); it is a closed subset contained in σ (T ).It is well known that T has the single-valued extension property if and only if zero is the only element x of ᐄ for which σ T (x) = ∅.For a closed subset F of C, let ᐄ T (F ) := {x ∈ ᐄ : σ T (x) ⊂ F } be the corresponding analytic spectral subspace; it is a T -hyperinvariant subspace, generally nonclosed in ᐄ.The operator T is said to satisfy Dunford's condition (C) if for every closed subset F of C, the linear subspace ᐄ T (F ) is closed.For every open subset U of C, we let ᏻ(U, ᐄ) denote the space of analytic ᐄ-valued functions defined on U .It is a Fréchet space when endowed with the topology of uniform convergence on compact subsets of U .Recall also that the operator T is said to possess Bishop's property (β) provided that for every open subset U of C, the mapping T U : ᏻ(U, ᐄ) → ᏻ(U, ᐄ), given by (T U f )(λ) = (T − λ)f (λ) for every f ∈ ᏻ(U, ᐄ) and for λ ∈ U, is injective and has a closed range.It is known that the Bishop's property (β) implies the Dunford's condition (C) and it turns out that the single-valued extension property follows from the Dunford's condition (C).Stampfli [14] and Radjabalipour [10] have shown that hyponormal operators satisfy Dunford's condition (C), and Putinar [9] has shown that hyponormal operators, M-hyponormal operators, and more generally subscalar operators have Bishop's property (β).For thorough presentations of the local spectral theory, we refer to [5,7].

Analytic bounded point evaluations for cyclic operators.
Throughout this section, let Ᏼ be a Hilbert space and T ∈ ᏸ(Ᏼ) be a cyclic operator with cyclic vector x ∈ Ᏼ.For λ ∈ B(T ), let k(λ) denote the vector of Ᏼ given by (1.3) for every polynomial p.
Proof.It is clear that (iv) and (v) are equivalent since for every λ ∈ B(T ), we have On the other hand, the implications (ii)⇒(iii) and (iii)⇒(iv) are trivial.So, it suffices to establish the implications (i)⇒(ii) and (iv)⇒(i).
Assume that O is an analytic set for T .Let By uniform boundedness principle, sup Hence, there is a constant M > 0 such that and the implication (i)⇒(ii) is proved.Now, suppose that O ⊂ B(T ) and the function λ k(λ) is bounded on compact subsets of O. Let y ∈ Ᏼ, then there is a sequence of polynomials (p n ) n such that lim n→+∞ p n (T )x − y = 0.It follows from the Cauchy-Schwartz inequality that for every compact subset K of O, we have (2.8) Hence, the function y is a uniform limit on O of a sequence of polynomials.By Montel's theorem, y is an analytic function on O. Therefore, O is an analytic set for T ; so, the implication (iv)⇒(i) holds.
The following result gives a complete description of B a (T ); it is a simple but useful result from which one can derive many known results as immediate consequences.
Proof.First of all, note that if (T −λ) * u = 0 for some u ∈ Ᏼ, then for every polynomial p, we have p(T )x, u = p(λ) x, u . (2.9) Let λ ∈ (T * ); there is a nonzero analytic Ᏼ-valued function φ : ᐂ → Ᏼ on some open disc ᐂ centered at λ such that (2.10) Using the fact that a nonzero analytic Ᏼ-valued function has isolated zeros, one can assume that the function φ has no zeros in ᐂ.Hence, ᐂ ⊂ σ p (T * ) = B(T ); and therefore, it follows from (2.9) that x, φ(µ) for every µ ∈ ᐂ. (2.12) We will show that the function φ is analytic on O. Indeed, for every y ∈ Ᏼ and for every λ 0 ∈ O, we have lim (2.13) Hence, for every y ∈ Ᏼ, the function λ φ(λ), y is analytic on O; therefore, the function φ is analytic on O. On the other hand, the function φ is without zeros on O and satisfies the following equation: ( This gives O = B a (T ) ⊂ (T * ), and the proof is completed.
Corollary 2.3.The following identities hold: (2.15) Proof.Since for every λ ∈ B a (T ), λ is a simple eigenvalue for T * with corresponding eigenvector k(λ), the proof follows by combining Theorem 2. In 1982, Raphael showed that quasisimilar cyclic subnormal operators have the same analytic bounded point evaluations (see [12]).In 1994, Williams proved that general quasisimilar cyclic Hilbert-space operators have the same analytic bounded point evaluations (see [15,Theorem 1.5]).In view of Theorem 2.2, one can see immediately that general densely similar cyclic Hilbert-space operators have the same analytic bounded point evaluations.
To end this section, we will be mainly concerned with two interesting open problems related to the bounded point evaluations for cyclic hyponormal operators.Recall that an operator R ∈ ᏸ(Ᏼ) is said to be subnormal if it has a normal extension.The operator R is said to be hyponormal if R * y ≤ Ry for every y ∈ Ᏼ.Note that every subnormal operator is hyponormal with converse false (see [6]).Recall also that the operator R is said to be pure if {0} is the only reducing subspace M such that R |M is normal.
Combining Theorem 2.2 and [6, Theorem VIII.4.3], we see that the cyclic operator T is normal if and only if T is a subnormal operator and T * has the single-valued extension property.So, one may ask if this result remains valid for noncyclic subnormal cases.Unfortunately, this result is no longer valid; an example of a nonnormal, decomposable, subnormal operator is constructed by Radjabalipour (see [11]).However, we do not know if a similar result remains valid for the case of cyclic hyponormal operators; this suggests the following question.
Question 2.5.Suppose that T is a cyclic hyponormal operator and T * has the single-valued extension property.Is T a normal operator?
The next problem is of some interest in view of the fact that if it has a positive answer, then one can deduce immediately that every hyponormal operator has a proper closed invariant subspace.Question 2.6.Suppose that T is a pure cyclic hyponormal operator.Do we have that B a (T ) ≠ ∅?

Densely similarity and approximate point spectra for cyclic operators possessing Bishop's property (β).
We first need to give some notations and definitions.Let ᐄ be a Banach space; recall that an operator T ∈ ᏸ(ᐄ) is said to be semi-Fredholm if ran T is closed and dim(ker T ) < +∞ or codim(ran T ) < +∞.Moreover, if ran T is closed and both dim(ker T ) and codim(ran T ) are finite, then the operator T is said to be Fredholm.If T is semi-Fredholm, then the index of T is defined by ind(T ) := dim(ker T ) − codim(ran T ).For an operator T ∈ ᏸ(ᐄ), define σ lr e (T ) := {λ ∈ C : T − λ is not semi-Fredholm}, ρ e (T ) := {λ ∈ C : T − λ is Fredholm}.These are called the Wolf spectrum and Fredholm domain, respectively, of T .
Let T ∈ ᏸ(Ᏼ) be a cyclic operator on a Hilbert space Ᏼ.It is shown in [3] that if T possesses Bishop's property (β), then B a (T ) = Γ (T )\σ ap (T ) if and only if B a (T ) ∩ σ p (T ) = ∅ and was derived from this result that if T is hyponormal, M-hyponormal, or p-hyponormal operator, then B a (T ) = Γ (T )\σ ap (T ) (see also [2]).However, using generalized spectral theory, it is proved in [8] that B a (T )\σ lr e (T ) = Γ (T )\σ g (T ), where σ g (T ) denotes the generalized spectrum of T .Therefore, the localized version of Bishop's property (β) allowed to show that B a (T )\σ β (T ) = Γ (T )\σ ap (T ), (3.1)where σ β (T ) is the set of points λ ∈ C on which T fails to have Bishop's property (β).As a consequence, it is obtained that if T possesses Bishop's property (β), then B a (T ) = Γ (T )\σ ap (T ).In view of Theorem 2.2, one may ask whether a similar description of (T * ) can be obtained if T is a cyclic Banachspace operator possessing Bishop's property (β).In fact, we will prove that (T * ) = σ (T )\σ ap (T ) for every cyclic Banach-space operator T possessing Bishop's property (β).The idea behind a part of our proof comes from the proof of [2, Theorem In order to prove the reverse inclusion of (3.3), it suffices to show that T − λ is injective and has a closed range for every λ ∈ (T * ).First, we prove that for every λ ∈ (T * ), we have codim(ran(T − λ)) = 1; in particular, ran(T − λ) is closed (see [7,Lemma 3.1.2]).Indeed, let λ 0 ∈ (T * ); there is an analytic function without zeros, Λ : ᐂ → ᐄ * , on some open disc ᐂ centered at λ 0 such that Set Φ(λ) = Λ(λ)/ x, Λ(λ) , λ ∈ ᐂ, where x ∈ ᐄ is a cyclic vector for T , and the symbol •, • designs the duality map between ᐄ and ᐄ * .Note that for every polynomial p, we have Let y ∈ ᐄ; there is a sequence of polynomials (p n ) n≥0 such that (p n (T )x) n≥0 converges to y in ᐄ.Define analytic ᐄ-valued functions on ᐂ by For every compact subset K of ᐂ, we have In particular, we have This shows that codim(ran(T − λ 0 )) = 1.
Next, suppose for the sake of contradiction that there is By [1, Corollary 2.7], we deduce that ind(T − λ 0 ) = 0; and so, T − λ 0 is a Fredholm operator for which dim(ker is an open set and the index is a constant function on the components of ρ e (T ), there is δ > 0 such that As codim(ran(T − λ)) = 1, for every λ ∈ (T * ), we have dim ker(T − λ) = 1 for every λ ∈ B λ 0 ,δ ; (3.11) in particular, B(λ 0 ,δ) ⊂ σ p (T ).We have a contradiction to [1, Theorem 2.6] since T has the single-valued extension property.Thus, T −λ 0 is injective, and the proof is completed.
Note that (3.3) holds for arbitrary Banach-space operator T not necessarily cyclic.The following example shows that this inclusion may not be reversed in general even if the operator T possesses Bishop's property (β).
We conclude this paper by mentioning that one can show with no extra effort that Theorem 3.1 and Proposition 3.3 remain valid for rationally cyclic Banach-space operators.

Call for Papers
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2 and [ 1 ,Remark 2 . 4 .
Theorem 1.9].In view of Theorem 2.2, the following are immediate consequences:(i) B a (T ) is independent of the choice of cyclic vector for T (see[15,  Proposition 1.4]); (ii) B a (T ) = ∅ if and only if T * has the single-valued extension property.In particular, if T is a cyclic normal operator, then B a (T ) = ∅.Suppose that ᐄ and ᐅ are Banach spaces.Recall that the two operators R ∈ ᏸ(ᐄ) and S ∈ ᏸ(ᐅ) are said to be densely similar (quasisimilar ) if there exist two bounded linear transformations X : ᐄ → ᐅ and Y : ᐅ → ᐄ having dense range (having dense range and injectives) such that XR = SX, RY = Y S.(2.16)