����������� ������ 54 (2002), 57–70

UDK 517.984.3

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ASCENT, DESCENT, QUASI-NILPOTENT PART AND ANALYTIC CORE OF OPERATORS

Abstract. This paper concerns a localized version of the single valued extension property of a bounded operator T ∈ L(X), where X is a Banach space, at a point λ_{0} ∈ ℂ. We shall relate this property to the ascent and the descent of λ_{0}I − T, as well as to some spectral subspaces as the quasi-nilpotent part and the analytic core of λ_{0}I − T. We shall also describe all these notions in the setting of an abstract shift condition, and in particular for weighted right shift operators on ℓ^{p}(ℕ), where 1 ≤ p < ∞ .

1. The single-valued extension property

One of basic properties in local spectral theory is the so-called single valued extension property for bounded operators on Banach spaces. This property is enjoyed by several classes of operators as the decomposable operators, as well as other classes of operators; we refer to the excellent monograph by Laursen and Neumann [16] for a modern treatment of the theory of decomposable operators.

In this paper we shall consider the following local version of this property, introduced by Finch [13] and studied later by several authors [18, 19, 26, 1, 2, 3, 5, 6].

Definition 1.1. Let X be a complex Banach space and T ∈ L(X). The operator T is said to have the single valued extension property at λ_{0} ∈ ℂ (abbreviated SVEP at λ_{0}), if for every open disc 𝔻_{λ_{0}} centered at λ_{0}, the only analytic function f: 𝔻_{λ_{0}} → X which satisfies the equation

(λI − T)f(λ) = 0  (1)

is the function f ≡ 0.

AMS Subject Classification: 47 A 10, 47 A 11, 47 A 53, 47 A 55

Keywords and phrases: Single valued extension property, quasi-nilpotent part and analytic core, property (Q), weighted right shift operators.

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

The research was supported by the International Cooperation Project between the University of Palermo (Italy) and Conicit-Venezuela.

An operator T ∈ L(X) is said to have the SVEP if T has the SVEP at every point λ ∈ ℂ.

The SVEP of T ∈ L(X) may be also defined as follows: Let U be an open subset of ℂ and let H(U, X) denote the space of X-valued functions on U equipped with the topology of uniform convergence on compact subsets of U. Then H(U, X) is a Fréchet space and every T ∈ L(X) induces a continuous mapping T_{U} on H(U, X), defined by

T_{U}(f)(λ) := (λI − T)f(λ) for all f ∈ H(U, X).  (2)

The operator T ∈ L(X) has the SVEP precisely when T_{U} is injective.

The SVEP may be characterized by means of some typical tools originating from the local spectral theory. Recall that, for a bounded operator T ∈ L(X), the local resolvent set of T at the point x ∈ X, is defined as the union of all open subsets U of ℂ such that there exists an analytic function f: U → X which satisfies

(λI − T)f(λ) = x for all λ ∈ U .  (3)

The local spectrum σ_{T} (x) of T at x is the set defined by σ_{T} (x) := ℂ \ ρ_{T} (x) and obviously σ_{T} (x) ⊆ σ(T), where σ(T) denotes the spectrum of T.

Clearly, any analytic function which verifies (3) on this union is a local extension of the analytic function R(λ, T)x := (λI − T)^{−1}x defined on the resolvent set ρ(T) of T. Generally, the analytic solutions of (3) are not uniquely determined. It is clear from the definition that, if T has the SVEP at λ_{0}, then the analytic solution of (3) is uniquely determined in an open disc centered at λ_{0}.

For every subset F of ℂ, let us denote by X_{T} (F) the analytic spectral subspace of T associated with Ω:

X_{T} (F) := { x ∈ X : σ_{T} (x) ⊆ F }.

For an arbitrary operator T ∈ L(X) and a closed subset F of ℂ, the glocal spectral subspace X_{T} (F) is defined as the set of all x ∈ X for which there exists an analytic function f: ℂ\F → X which satisfies the identity (λI−T)f(λ) = x for all λ ∈ ℂ\F. Note that T has SVEP if and only if X_{T} (F) = X_{T} (F) for all closed sets F ⊆ ℂ, see Proposition 3.3.2 of [16].

The SVEP, as well as the SVEP at a point λ_{0} ∈ ℂ, may be characterized in a very simple way:

Theorem 1.2. Let T ∈ L(X), X a Banach space. Then

(i) T has the SVEP at λ_{0} if and only if ker (λ_{0}I − T) ∩ X_{T} (∅) = {0} [1, Theorem 1.9];

(ii) T has the SVEP if and only if X_{T} (∅) = {0}, and this is the case if and only if X_{T} (∅) is closed; see [16, Proposition 1.2.16].

The basic role of SVEP arises in local spectral theory, since every decomposable operator enjoys this property. Recall that a bounded operator T ∈ L(X) is said to have the Bishop’s property (β) if for every open set U the operator T_{U} defined in (2) is injective and has closed range, while T ∈ L(X) is said to have the decomposition property (δ) if X = X_{T} (\overline{U}) + X_{T} (\overline{V}) for every open cover \{U, V \} of ℂ. The decomposability of T ∈ L(X) may be defined in several ways, for instance as the union of the property (β) and the property (δ), see [16, Theorem 2.5.19]. Note that the property (β) implies that T has SVEP, while the property (δ) implies SVEP for T^{⋆}, see [16, Theorem 2.5.19]. The class of decomposable operators contains, for instance, all normal operators on Hilbert spaces, all spectral operators, all operators with a non-analytic functional calculus and any operator with a totally disconnected spectrum, [16]. Examples of non-decomposable operators which have the SVEP may be found among the class of all multipliers of a commutative semi-prime Banach algebra, [16, Proposition 4.2.1]

We shall now introduce two important subspaces in local spectral theory and in Fredholm theory:

Definition 1.3. Let X be a Banach space and T ∈ L(X). The analytic core of T is the set K(T) of all x ∈ X such that there exists a sequence (u_{n}) ⊂ X and δ > 0 for which:

(a) x = u_{0}, and T u_{n+1} = u_{n} for every n ∈ ℕ.

(b) ∥u_{n}∥ ≤ δ^{n}∥x∥ for every n ∈ ℕ.

It easily follows, from the definition, that K(T) is a linear subspace of X and that T (K(T)) = K(T). In general, K(T) is not closed and K(T) ⊆ T^{∞}(X), where T^{∞}(X) := ⋂_{n=1}^{∞} T^{n}(X) is the hyperrange of T. Furthermore, if T is quasi-nilpotent then K(T) = {0}, see [18].

Definition 1.4. Let T ∈ L(X), X a Banach space. The quasi-nilpotent part of T is the set

H_{0}(T) := { x ∈ X : lim_{n→∞} ∥T^{n}x∥^{1/n} = 0 }.

Also H_{0}(T) is a linear subspace of X, generally not closed. Furthermore, N^{∞}(T) ⊆ H_{0}(T), where N^{∞}(T) := ⋂_{n=1}^{∞} ker T^{n} is the hyperkernel of T, and T is quasi-nilpotent if and only if H_{0}(T) = X, [27, Theorem 1.5].

The systematic investigation of the spaces K(T) and H_{0}(T) was initiated by Mbekhta [18], after an earlier work of Vrbová [27]. In particular, these authors established the following local spectral characterizations of K(T) and H_{0}(T).

Theorem 1.5. For a bounded operator T ∈ L(X), X a Banach space we have (i) K(λ_{0}I − T) = X_{T} (ℂ \ {λ_{0}}). (ii) H_{0}(λ_{0}I − T) = X_{T} ({λ_{0}}), so, if T has SVEP, H_{0}(λ_{0}I − T) = X_{T} ({λ_{0}}).

Note that, for every λ_{0} ∈ ℂ, the following inclusions hold:

X_{T} (∅) ⊆ X_{T} (ℂ \ {λ_{0}}) = K(λ_{0}I − T) ⊆ (λ_{0}I − T)^{∞}(X)  (4)

and

ker (λ_{0}I − T) ⊆ N^{∞}(λ_{0}I − T) ⊆ H_{0}(λ_{0}I − T) ⊆ X_{T} ({λ_{0}}).  (5)

Two important notions in Fredholm theory are those of the ascent and the descent of an operator. The ascent of an operator T is the smallest non-negative integer p := p(T) such that ker T^{p} = ker T^{p+1}. If such integer does not exist we put p(T) := ∞. The descent of an operator T is the smallest non-negative integer q := q(T) such that T^{q}(X) = T^{q+1}, and if such integer does not exist we put q(T) := ∞. It is well-known that, if p(T) and q(T) are both finite, then p(T) = q(T), [14, Proposition 38.3]. Furthermore, if λ_{0} belongs to the spectrum σ(T), then 0 < p(λ_{0}I − T) = q(λ_{0}I − T) < ∞ if and only if λ_{0} is a pole of the resolvent R(λ, T) := (λI − T)^{−1}, [14, Proposition 50.23]. Obviously, in this case λ_{0} is an isolated point of σ(T).

Recall that T ∈ L(X), X is said to be semi-Fredholm if T (X) is closed and at least one of the two defects α(T) := dim ker T or β(T) := codim T (X) is finite.

Definition 1.6. An operator T ∈ L(X), X a Banach space, is said to be semi-regular if T (X) is closed and ker T ⊆ T^{∞}(X).

An operator T ∈ L(X) is said to admit a generalized Kato decomposition, abbreviated GKD, if there exists a pair of T -invariant closed subspaces (M, N) such that X = M ⊕ N, the restriction T |M is semi-regular and T |N is quasinilpotent.

An important case is obtained if we assume in the definition above that T |N is nilpotent. In this case T is said to be of Kato type, [17]. If N is finite-dimensional then T is said to be essentially semi-regular, see Rakočević [24] or Müller [22]. Obviously, any semi-regular operator is of Kato type. Note that if T is of Kato type then T^{∞}(X) = K(T) and K(T) is closed, see [2, Theorem 2.3 and Theorem 2.4]. An important class of operators of Kato type is given by the class of all semi-Fredholm operators, see West [29]. Furthermore, taking M = {0} and N = X, we see that every quasi-nilpotent operator is of Kato type.

In the sequel by M^{⊥} we shall denote the annihilator of the subset M ⊆ X; and by ^{⊥}N the pre-annihilator of the subset N ⊆ X^{⋆}.

Theorem 1.7. For a bounded operator T ∈ L(X), where X is a Banach space, the following implications hold:

(i) H_{0}(λ_{0}I − T) closed ⇒ H_{0}(λ_{0}I − T) ∩ K(λ_{0}I − T) closed ⇒ H_{0}(λ_{0}I − T) ∩ K(λ_{0}I − T) = {0} ⇒ T has SVEP at λ_{0}.

(ii) X = H(λ_{0}I − T) + K(λ_{0}I − T) ⇒ T^{⋆} has SVEP at λ_{0}.

Moreover, if λ_{0}I − T is of Kato type, then all these implications are equivalences.

Proof. Without loss of generality, we may consider λ_{0} = 0.

(i) Assume that H_{0}(T) is closed and let T̃ denote the restriction of T to the Banach space H_{0}(T). Obviously, H_{0}(T) = H_{0}(T̃), so that T̃ is quasi-nilpotent and hence K(T̃) = {0}. It is easy to see that H_{0}(T) ∩ K(T) = K(T̃). This shows the first implication. The second implication of (i) is an immediate consequence of Theorem 1.5. Indeed, we have

ker (λ_{0}I − T) ∩ X_{T} (∅) ⊆ H_{0}(λ_{0}I − T) ∩ K(λ_{0}I − T),

so, if the last intersection is {0}, then T has the SVEP at λ_{0}, by Theorem 1.2.

(ii) From [17, Proposition 1.8] we know that H_{0}(T) ⊆^{⊥} K(T^{⋆}) and hence K(T^{⋆}) ⊆ H_{0}(T)^{⊥}. We also have H_{0}(T^{⋆}) ⊆ K(T)^{⊥}. Indeed, let φ ∈ H_{0}(T^{⋆}) and consider an arbitrary element x ∈ K(T). According to the definition of K(T), there is a sequence (u_{n}) ⊂ X, and a δ > 0, such that u_{0} = x, T u_{n+1} = u_{n} and ∥u_{n}∥ ≤ δ^{n}∥x∥ for every n ∈ ℕ. Clearly, T^{n}u_{n} = x for every n ∈ ℕ. Consequently,

|φ(x)| = |φ(T^{n}u_{n})| = |(T^{⋆n}φ)(u_{n})| ≤ ∥u_{n}∥∥T^{⋆n}φ∥ ≤ δ^{n}∥T^{⋆n}φ∥,

and hence |φ(x)|^{\frac{1}{n}} ≤ δ∥T^{⋆n}φ∥^{\frac{1}{n}} for every n ∈ ℕ. The last term converges to 0 as n → ∞, since φ ∈ H_{0}(T^{⋆}), and from this it follows that φ(x) = 0, i.e. φ ∈ K(T)^{⊥}. Finally, if X = H_{0}(T) + K(T) then {0} = H_{0}(T)^{⊥} ∩ K(T)^{⊥} ⊇ H_{0}(T^{⋆}) ∩ K(T^{⋆}), thus, by part (i), T^{⋆} has the SVEP at 0.

For the last assertion see Theorem 2.6 of [3].

An example, given in [5], of a bilateral right shift T defined in the Hilbert space L^{2}(ω), where ω := (ω_{n})_{n∈ℤ} is a suitable weight sequence, shows that the SVEP at a point λ_{0} does not, in general, implies that H_{0}(λ_{0}I − T) ∩ K(λ_{0}I − T) = {0}.

Also the finiteness of the ascent and the descent has important consequences on the SVEP. In fact we have the following implications.

Theorem 1.8. For a bounded operator T on a Banach space X the following implications hold:

(i) p(λ_{o}I −T) < ∞ ⇒ N^{∞}(λ_{o}I −T)∩(λ_{o}I −T)^{∞}(X) = {0} ⇒ T has SVEP at λ_{o}. (ii) q(λ_{o}I −T) < ∞ ⇒ X = N^{∞}(λ_{o}I −T)+(λ_{o}I −T)^{∞}(X) ⇒ T^{⋆} has SVEP at λ_{o}.

Moreover, if λ_{0}I − T is of Kato type, then all these implications are equivalences.

Proof. (i) There is no loss of generality in assuming λ_{o} = 0.

Let p := p(T) < ∞. Then N^{∞}(T) = ker T^{p} and hence, by [14, Proposition 38.1], N^{∞}(T) ∩ T^{p}(X) = {0}. From T^{∞}(X) ⊆ T^{p}(X) we obtain that N^{∞}(T) ∩ T^{∞}(X) = {0}. The second implication is a consequence of Theorem 1.2, since, from the inclusions (4), we obtain that

ker T ∩ X_{T} (∅) ⊆ N^{∞}(T) ∩ T^{∞}(X) = {0}.

(ii) Also here we may assume that λ_{o} = 0. Let q := q(T) < ∞. Then T^{∞}(X) = T^{q}(X) and X = T^{n}(X) + ker T^{q} for every n ∈ ℕ, by [14, Proposition 38.2]. From this it easily follows that X = N^{∞}(T)+T^{∞}(X), so the first implication of (ii) is proved.

In order to show the second implication of (ii), we first note that, if X = N^{∞}(T) + T^{∞}(X), then N^{∞}(T)^{⊥} ∩ T^{∞}(X)^{⊥} = {0}. Now, let us consider an element x^{⋆} ∈ ker T^{⋆} ∩ X_{T^{⋆}}(∅). Clearly,

x^{⋆} ∈ ker T^{⋆} ⊆ ker (T^{⋆})^{n} = ker (T^{n})^{⋆} = \overline{T^{n}(X)}^{⊥} ⊆ T^{n}(X)^{⊥},

for every n ∈ ℕ and therefore x^{⋆} ∈ T^{∞}(X)^{⊥}. On the other hand, from σ_{T^{⋆}}(x^{⋆}) = ∅ we obtain, by Theorem 1.5, that

x^{⋆} ∈ K(T^{⋆}) ⊆ (T^{⋆})^{n}(X^{⋆}) = (T^{n})^{⋆}(X^{⋆}) ⊆ (ker T^{n})^{⊥}

for every n ∈ ℕ. From this it follows that x^{⋆} ∈ N^{∞}(T)^{⊥} and therefore x^{⋆} ∈ N^{∞}(T)^{⊥} ∩ T^{∞}(X)^{⊥}, which implies that x^{⋆} = 0. Again, from Theorem 1.2 we conclude that T^{⋆} has the SVEP at 0.

For a proof of the last assertion see Corollary 2.7 of [2].

Hence each one of the two conditions p(λ_{0}I − T) < ∞ or H_{0}(λ_{0}I − T) closed implies the SVEP at λ_{0}. The next two examples show that in general these two conditions are independent.

Example 1.9. Let T : ℓ^{2}(ℕ) → ℓ^{2}(ℕ) be defined by

T x := (\frac{x_{2}}{2} , ⋯, \frac{x_{n}}{n} , ⋯), where x = (x_{1}, ⋯, x_{n}, ⋯).

It is easily seen that ∥T^{k}∥ = \frac{1}{(k+1)!}, from which it follows that T is quasi-nilpotent and therefore H_{0}(T) = ℓ^{2}(ℕ). Obviously, p(T) = ∞.

Example1.10. In [3] it has been given an example of a direct sum of unilateral weighted shifts for which p(T) = 0 and H_{0}(T) is not closed. The following simpler example is taken from [10]. Let 1 ≤ p < ∞ be given and denote by ω = (ω_{n})_{n∈ℕ} a bounded sequence of positive real numbers. Let us consider the corresponding weighted unilateral right shift T on ℓ^{p}(ℕ), defined by

T x := ∑_{n=1}^{∞} ω_{n}x_{n}e_{n+1} for all x = (x_{n})_{n∈ℕ} ∈ ℓ^{p}(ℕ),

where (e_{n}) stands for the canonical basis of ℓ^{p}(ℕ). This operator has SVEP, since T has no eigenvalues and hence p(λI − T) = 0 for all λ ∈ ℂ.

A routine calculation shows that the norm of T^{n} is given by

∥T^{n}∥ = sup_{k∈ℕ}(ω_{k}… ω_{k+n−1}) for all n ∈ ℕ.

Suppose now that (ω_{n})_{n∈ℕ} is defined by

ω_{n} := \{ \begin{matrix}0 & if n is a square of an integer \\ 1 & otherwise \end{matrix}

It is easily seen that ∥T^{n}∥ = 1 for all n ∈ ℕ, so that T is not quasi-nilpotent. This excludes that H_{0}(T) is closed, see next Theorem 2.7. It is easy to see that K(T) = {0}. Therefore, this example shows that the implication H_{0}(T) closed ⇒ K(T) ∩ H_{0}(T) = {0}, noted in Theorem 1.7, cannot in general be reversed.

From Theorem 1.8 it follows that the finite ascent property for an operator T ∈ L(X) defined as:

p(λI − T) < ∞ for every λ ∈ ℂ

implies that T has SVEP. There are many examples of operators for which the condition p(λI − T) < ∞ holds for every λ ∈ ℂ. For instance, every multiplier of a semi-prime Banach algebra verifies this property, see [16, p. 406], and in particular every convolution operator on a group algebra L^{1}(G), where G is a locally compact Abelian group. Other examples of operators of this type are the generalized scalar operators, see [28], as well as several other classes of operators studied in [15]. As noted by Barnes [10], a class of operators which have this ascent property is given by the class P (X) of all bounded operators on a Banach space X which satisfy a polynomial growth condition, where T ∈ L(X) is said to satisfy a polynomial growth condition, if there exists a K > 0, a δ > 0 for which

∥exp(iλT)∥ ≤ K(1 + |λ|^{δ}) for all λ ∈ ℝ.

The finite descent property for an operator T ∈ L(X) is defined as:

q(λI − T) < ∞ for every λ ∈ ℂ.

This property is obviously satisfied by every operator for which every spectral point is a pole of the resolvent.

Theorem 1.11. Let T ∈ L(X), where X is a Banach space. Then T has the finite descent property precisely when σ(T) is finite set of poles of the resolvent R(λ, T).

Proof. Clearly, if σ(T) is finite set of poles of R(λ, T) then q(λI − T) < ∞ for every λ ∈ ℂ. Conversely, suppose that q(λI − T) < ∞ for all λ ∈ ℂ. Then q(λI − T) < ∞ for all λ ∈ ∂σ(T), ∂σ(T) the boundary of σ(T). Since T ha SVEP at every λ ∈ ∂σ(T) then the condition q(λI − T) < ∞ entails that every λ ∈ ∂σ(T) is a pole of R(λ, T), see Corollary 1 of [25]. Clearly, this implies that σ(T) = ∂σ(T), so that the spectrum σ(T) is a finite set of poles of R(λ, T).

Therefore, the finite descent property implies that both T and T^{∗} have SVEP (actually we have more, T is decomposable since it has finite spectrum). It should be noted that the proof of Theorem 1.11 shows that the finite descent property is equivalent to the apparently weaker condition q(λI − T) < ∞ for all λ ∈ ∂σ(T).

Theorem 1.7 suggests in a very natural way the following concept, introduced in [3]:

Definition 1.12. A bounded operator T ∈ L(X), X a Banach space, is said to have property (Q) if H_{0}(λI − T) is closed for every λ ∈ ℂ.

Clearly, every quasi-nilpotent operator has property (Q), since H_{0}(λI − T) = {0} for all λ ≠ 0 and H_{0}(T) = X. More generally,

σ(T) finite ⇒ T has property (Q).  (6)

Indeed, if λ ∈ σ(T) is isolated then H_{0}(λI − T) coincides with the range of the spectral projection associated with the singleton set {λ}, see [14, Proposition 49.1]. In particular, the implication

T has finite descent property ⇒ T has property (Q).  (7)

holds. Since every multiplier of a semi-simple Banach algebra has property (Q), see Theorem 1.8 of [3], we see that any multiplier with a non-finite spectrum provides an example of operator which has property (Q), but not satisfies the finite descent property.

Recall that a bounded operator T ∈ L(X), X a Banach space, is said to have Dunford’s property (C), shortly property (C), if the analytic subspace X_{T} (Ω) is closed for every closed subset Ω ⊆ ℂ. It should be noted that property (β) implies property (C), see [16, Proposition 1.2.19] and it turns out, by part (ii) of Theorem 1.2, that property (C) implies that T has SVEP.

An obvious consequence of part (ii) of Theorem 1.5 is that if T has property (C) then H_{0}(λI − T) = X_{T} ({λ}) is closed for every λ ∈ ℂ, so that the following implications hold:

T has property (C) ⇒ T has property (Q) ⇒ T has SVEP.  (8)

Note that neither of the implications (8) may be reversed in general. A first counter-example, of an operator which has SVEP but not property (Q) is given by the operator T defined in Example 1.10, see also next Theorem 2.7. An example of an operator which shows that the first implication is not reversed in general, may be found among the convolution operators T_{μ} of group algebras L^{1}(G), since these operators have property (Q), see [3], while may have not property (C), see Theorem 4.11.8 and Theorem 4.1.12 of [16].

2. An abstract shift condition

In this section we shall consider operators T ∈ L(X) on a Banach space X for which T^{∞}(X) = {0}. This condition may be viewed, in a certain sense, as an abstract shift condition, since it is satisfied by every weighted right shift T on ℓ^{p}(ℕ). Clearly, the condition T^{∞}(X) = {0} entails that T is non-surjective and hence 0 ∈ σ(T). Moreover, this condition also implies that K(T) = {0}, since K(T) is a subset of T^{∞}(X), but the quasi-nilpotent Volterra operator V on the Banach space X := C[0, 1], defined by

(V f)(t) := ∫_{0}^{t} f(s)ds for all f ∈ C[0, 1] and t ∈ [0, 1],

shows that, in general, the converse is not true. Indeed, V is quasi-nilpotent and hence K(V) = {0}, while

V^{∞}(X) = {f ∈ C^{∞}[0, 1] : f^{(n)}(0) = 0 n ∈ ℤ_{+}},

thus V^{∞}(X) is not closed and, consequently, strictly larger than K(V) = {0}.

It is easily seen that the condition T^{∞}(X) = {0} has some other important consequences, for instance:

T^{∞}(X) = {0} ⇒ p(λI − T) = 0 for all λ ≠ 0.

and

q(λI − T) = ∞ for all λ ∈ σ(T) \ {0}.  (9)

Indeed, ker (λI − T) = {0} for all 0 ≠ λ ∈ ℂ, since ker (λI − T) ⊆ T^{∞}(X) for all λ ≠ 0. This implies that q(λI − T) = ∞ for all λ ∈ σ(T) \ {0} otherwise, if were q(λI − T) < ∞, then q(λI − T) = p(λI − T) = 0 and hence λ ∉ σ(T), which is impossible.

For an operator T ∈ L(X), let

k(T) := inf{∥T x∥ : x ∈ X and ∥x∥ = 1}

be the lower bound of T and define

i(T) := lim k(T^{n})^{1/n} = sup_{n∈ℕ} k(T^{n})^{1/n}.

Clearly, if r(T) is the spectral radius of T ∈ L(X) then i(T) ≤ r(T). In the sequel by D(0, i(T)) we shall denote the closed disc centered at 0 and radius i(T).

Theorem 2.1. If T^{∞}(X) = {0} then T has SVEP. Moreover, the following statements hold:

(i) σ_{T} (x) is connected and

D(0, i(T)) ⊆ σ_{T} (x) for all 0 ≠ x ∈ X.  (10)

(ii) H_{0}(λI − T) = {0} for all λ ≠ {0}. Consequently, T has property (Q) if and only if H_{0}(T) is closed.

(iii) If i(T) > 0, then T has property (Q). (iv) If i(T) = r(T), then T has property (C).

Proof. The SVEP may be proved in several ways, for instance from Theorem 1.2, since ker (λI − T) ∩ K(λI − T) = {0} for every λ ∈ ℂ. Moreover, the local spectrum σ_{T} (x) is connected, by Theorem 1 of [26]. The proof of the inclusion (10) is proved in [16, Theorem 1.6.3]. To show the statement (ii) observe first that the SVEP for T implies, by part (ii) of Theorem 1.5, that

H_{0}(λI − T) = {x ∈ X : σ_{T} (x) ⊆ {λ}} for all λ ∈ ℂ.

Now, let λ ≠ 0 and suppose that there is 0 ≠ x ∈ H_{0}(λI − T). Since T has SVEP, by part (ii) of Theorem 1.2, we obtain that σ_{T} (x) ≠ ∅, so that σ_{T} (x) = {λ}, which is impossible since 0 ∈ σ_{T} (x), by part (i). Therefore H_{0}(λI − T) = {0} for all λ ≠ 0.

To prove (iii) it suffices to prove, by part (ii), that H_{0}(T) = {0}. Since,

H_{0}(T) = {x ∈ X : σ_{T} (x) ⊆ {0}},

from the inclusion (10) we infer that the condition i(T) > 0 entails that each x ≠ 0 cannot belong to H_{0}(T). Therefore, H_{0}(T) = {0}.

The assertion (iv) has been proved in Proposition 1.6.5 of [16]. We give the simple proof for sake of completeness. Suppose now that i(T) = r(T). Then D(0, r(T)) is contained in σ_{T} (x) for all non-zero x ∈ X, and hence

σ_{T} (x) = D(0, r(T)) = σ(T) for all non-zero x ∈ X.

This implies that X_{T} (Ω) = X for every closed set Ω which contains D(0, r(T)), while X_{T} (Ω) = {0} otherwise.

Let σ_{w}(T) denote the Weyl spectrum, i.e. the complement of the set of all λ ∈ ℂ for which λI − T is a Fredholm operator with index ind T := α(T) − β(T) = 0. The Browder spectrum σ_{b}(T) is defined as the complement of all λ ∈ ℂ for which λI − T ∈ Φ(X) and p(λI − T) = q(λI − T) < ∞. Note that σ_{w}(T) ⊆ σ_{b}(T) and this inclusion is in general proper.

Theorem 2.2. Let T ∈ L(X), X an infinite-dimensional Banach space, and suppose that T^{∞}(X) = {0}. Then σ(T) is connected and

σ(T) = σ_{w}(T) = σ_{b}(T).  (11)

Proof. By Proposition 1.3.2 of [16] we have, since T has SVEP,

σ(T) = σ_{su}(T) = ⋃_{x∈X} σ_{T} (x),

where σ_{su}(T) denotes the surjectivity spectrum of T. Since the local spectra σ_{T} (x) are connected then σ(T) is connected. The equality (11) has been established in [26]. We give here a simpler proof.

By Corollary 2.8 of [7], we have σ_{w}(T) = σ_{b}(T), since T has SVEP. We show that σ_{b}(T) = σ(T). The inclusion σ_{b}(T) ⊆ σ(T) holds for all T ∈ L(X), so it remains to establish that σ(T) ⊆ σ_{b}(T). Observe that, if the spectral point λ ∈ ℂ is not isolated in σ(T), then λ ∈ σ_{b}(T).

Suppose first that T is quasi-nilpotent. Then σ_{b}(T) = σ(T) = {0}, since σ_{b}(T) is non-empty whenever X is infinite-dimensional. Suppose that T is not quasinilpotent and let 0 ≠ λ ∈ σ(T). Since σ(T) is connected and 0 ∈ σ(T), then neither 0 or λ are isolated points in σ(T). Hence σ(T) ⊆ σ_{b}(T).

Let ρ_{kt}(T) denote the Kato type resolvent of T, defined as

ρ_{kt}(T) := {λ ∈ ℂ : λI − T is of Kato type}.

The set ρ_{kt}(T) is an open subset of ℂ, see [4], so it may be decomposed in maximal connected open components.

The following two results generalize Theorem 2 of Schmoeger [26].

Theorem 2.3. Suppose that T ∈ L(X), where X is a Banach space, is of Kato type. Let Ω denote the connected component of ρ_{kt}(T) which contains 0. Then T^{∞}(X) = {0} if and only if p := p(T) < ∞ and

Ω ⊆ σ_{T} (x) for all x ∉ ker T^{p}.  (12)

Furthermore, if T^{∞}(X) = {0}, then T has property (Q).

Proof. Suppose that T^{∞}(X) = {0}. Since T has SVEP then p(T) < ∞, by Theorem 1.8, and H_{0}(T) = ker T^{p}, by Corollary 2.7 of [3]. The SVEP of T also ensures, by Theorem 1.7, that H_{0}(λI −T) is closed for all λ ∈ Ω. Since the mapping

λ ∈ Ω ↦ \overline{H_{0}(λI − T)} + K(λI − T) = H_{0}(λI − T) + K(λI − T)

is constant on Ω, see [20] or [8], we then have

ker T^{p} = H_{0}(T) + K(T) = H_{0}(λI − T) + K(λI − T)

for all λ ∈ Ω. Now, by part (iii) of Theorem 2.1, H_{0}(λI − T) = {0} for all λ ≠ 0 and hence

ker T^{p} = K(λI − T) = {x ∈ X : λ ∉ σ_{T} (x)},

for all λ ∈ Ω \ {0}. Thus, for x ∉ ker T^{p} we obtain Ω \ {0} ⊆ σ_{T} (x) and, by part (ii) of Theorem 2.1, 0 ∈ σ_{T} (x) for all x ≠ 0. This implies that Ω ⊆ σ_{T} (x) for all x ∉ ker T^{p}.

Conversely, assume that p = p(T) < ∞ and Ω ⊆ σ_{T} (x) for all x ∉ ker T^{p}. The condition p := p(T) < ∞ entails that T has SVEP at 0 and, by Theorem 1.8, N^{∞}(T) ∩ T^{∞}(X) = {0}. Assume that there exists 0 ≠ x ∈ T^{∞}(X). Then x ∉ N^{∞}(T) = ker T^{p}, thus 0 ∈ σ_{T} (x). On the other hand, since T^{∞}(X) = K(T), see Theorem 2.4 of [2], then x ∈ K(T), so that 0 ∉ σ_{T} (x), by Theorem 1.5; a contradiction. Therefore, T^{∞}(X) = {0}.

To show the last assertion, observe that H_{0}(λI − T) = {0} for all λ ≠ 0, while the SVEP, by Theorem 1.7, implies that also H_{0}(T) is closed.

It should be noted that the previous result extends, in a sense, Theorem 2.1. In fact, if σ_{ap}(T) denotes the approximate point spectrum of T, then

σ_{ap}(T) ⊆ {λ ∈ ℂ : i(T) ≤ |λ| ≤ r(T)},

see Proposition 1.6.2 of [16]. Therefore the condition i(T) > 0 entails that T is bounded below and hence is of Kato type.

Theorem 2.4. Suppose that T ∈ L(X) is of Kato type and T^{∞}(X) = {0}. If T is not quasi-nilpotent then T^{∗} does not have the SVEP. Moreover, T is not decomposable.

Proof. We know that q(λ_{0}I − T) = ∞ for all 0 ≠ λ ∈ σ(T). Indeed, if for some λ_{0} ∈ σ(T) \ {0} we have q(λ_{0}I − T) < ∞. Let Ω be the component of σ_{kt}(T) containing 0. Since σ(T) is connected and T is not quasi-nilpotent, then Ω ∩ (σ(T) \ {0}) ≠ ∅. Let μ ∈ Ω ∩ (σ(T) \ {0}). Since μI − T is of Kato type, the condition q(μI − T) = ∞ entails that T^{∗} does not have the SVEP at μ, see Theorem 2.9 of [3].

The last assertion is clear, since the decomposability of T implies that T^{⋆} is decomposable and hence has SVEP, by Theorem 2.5.19 of [16].

The previous results apply to isometries, since i(T) = r(T) = 1 for every isometry T ∈ L(X). Note that for every isometry σ_{ap}(T) is contained in the unit circle, so that λI − T is bounded below and hence upper semi-Fredholm for every |λ| < 1, see Proposition 1.6.2 of [16]. Hence every isometry T has property (C), by Theorem 2.1 (actually we have much more, T has property (β), by Proposition 1.6.7 of [16]). In the case that an isometry T is non-invertible, for instance in the case that T^{∞}(X) = {0}, the spectrum is the entire closed unit disc, while σ_{ap}(T) is the unit circle. Furthermore, by Corollary 2.9 of [7], we have σ_{ap}(T) = σ_{kt}(T).

An isometry T ∈ L(X) for which the condition T^{∞}(X) = {0} is satisfied is said to be a semi-shift. Proposition 1.6.8 of [16] shows that T is a semi-shift if and only if T has fat local spectra, i.e. the equality σ_{T} (x) = σ(T) holds for every x ≠ 0, see also [23]. Examples of semi-shifts are the unilateral right shift operators of arbitrary multiplicity on ℓ^{p}(ℕ), as well as every right translation operator on L^{p}([0, ∞)), see Section 1.6 of [16]. From Theorem 2.4 it follows that every semi-shift operator is not decomposable.

The following result has been established in [21]. We give an alternative proof.

Theorem 2.5. If K(T) = {0}, the following assertions are equivalent:

(i) T is decomposable;  (ii) T has property (δ);
(iii) T is quasi-nilpotent;  (iv) 0 is an isolated point of σ(T);

(v) q(λI − T) < ∞ for all λ ≠ 0.

Proof. Clearly (i) ⇒ (ii). To establish the implication (ii) ⇒ (iii), we show first that the surjectivity spectrum σ_{su}(T) is {0}. Suppose that λ ≠ 0 and choose ε > 0 such the closed disc D(λ, ε) does not contains 0. Let U := 𝔻(0, ε/2) and V := ℂ \ D(0, ε), where 𝔻(0, ε/2) is the open disc centered at 0 and radius ε/2 . Clearly, {U, V } is an open cover of ℂ, so, taking into account that T has SVEP, the property (δ) implies that X = X_{T} (D(0, ε)) + X_{T} (\overline{ℂ \ D(0, ε)}). From the inclusion \overline{ℂ \ D(0, ε)}) ⊆ ℂ \ \{0\} we infer

X_{T} (\overline{ℂ \ D(0, ε)}) ⊆ X_{T} (ℂ \ \{0\}) = K(T) = \{0\},

so that X = X_{T} (D(0, ε)). On the other hand, by Proposition 1.2.16 of [16], we know that

(μI − T)(X_{T} (D(0, ε)) = X_{T} (D(0, ε)) for all |μ| > ε,

thus λ ∉ σ_{su}(T). Hence, σ_{su}(T) = {0}. On the other hand the point spectrum σ_{p}(T) is contained in {0}, so that σ(T) = σ_{p}(T) ∪ σ_{su}(T) = {0}.

Clearly, (iii) ⇒ (i), so the statements (i), (ii) and (iii) are equivalent. Obviously, (iii) ⇒ (iv). We prove that (iv) ⇒ (iii). Suppose 0 isolated in σ(T). Then K(T) = ker P and H_{0}(T) = P (X), where P is the spectral projection associated with the spectral set {0}. From K(T) = {0} we deduce that H_{0}(T) = X, so T is quasi-nilpotent. It is evident that (iii) ⇒ (v). To show the opposite implication, assume that T is not quasi-nilpotent. Then q(λI − T) = ∞ for all 0 ≠ λ ∈ σ(T), by (9), so the proof is complete.

It is easy to see that, if T^{∞}(X) = {0}, then

T has the finite descent condition ⇔ T is nilpotent.

In fact, if q := q(T) < ∞, then T^{∞}(X) = T^{p}(X) = {0}, while the converse is obvious.

Example 2.6. In [16, p.89] an example is given of a right shift T on ℓ^{p}(ℕ) such that T has property (C), but not property (β). Therefore, T also provides an example of a shift which has property (Q), but not quasi-nilpotent.

However, the following result, established in [10], shows that the property (Q) for a right shift T is equivalent to be T quasi-nilpotent in some special case.

Theorem 2.7. Suppose that infinitely many weights ω_{n} are zero. Then for the corresponding right shift T on ℓ^{p}(ℕ), 1 ≤ p < ∞, the following statements are equivalent:

(i) T is quasi-nilpotent;  (ii) T is decomposable;
(iii) T has property (δ);  (iv) T has property (β);

(v) T has property (C);  (vi) T has property (Q);

(vii) H_{0}(T) is closed.

Proof. The equivalences (i) ⇔ (ii) ⇔ (iii) have been proved in Theorem 2.5. The implications (ii) ⇒ (iv) ⇒ (v) ⇒ (vi) are satisfied by every bounded operator.

(vi) ⇒ (i) Suppose that H_{0}(T) is closed. Since T e_{n} = ω_{n}e_{n+1} for all n ∈ ℕ, if ω_{n} = 0 then e_{n} ∈ ker T ⊆ H_{0}(T). Suppose that ω_{n} ≠ 0 and let k be the smallest integer such that ω_{n+k} = 0 . It is easy to check that

T^{k+2}e_{n} = ω_{n}ω_{n+1} ⋯ ω_{n+k} e_{n+k+1} = 0,

so e_{n} ∈ ker T^{k+2} ⊆ H_{0}(T). This shows that H_{0}(T) = ℓ^{p}(ℕ) and hence T is quasinilpotent. Therefore (vi) ⇔ (i). Since the equivalence (vi) ⇔ (vii) has been proved in part (ii) of Theorem 2.1, the proof is complete.

We conclude this paper by mentioning a characterization of property (Q), established by Bourhim [10, Proposition 4.5], in the case that the weighted right shift T is injective. It is easily seen that, if (ω_{n})_{n∈ℕ} is the weight sequence, then T is injective if and only if none of the weights ω_{n} is zero; so that Theorem 2.7 does not apply to this case.

Theorem 2.8. Suppose that the right shift operator T on ℓ^{p}(ℕ), for some 1 ≤ p < ∞, is injective. Then the following statements are equivalent:

(i) T has property (Q);

(ii) Either T is quasi-nilpotent or d(T) := lim_{n→∞} sup(ω_{1} ⋯ ω_{n})^{1/n} > 0.

Observe that the quantity i(T) for a right shift operator T with weight (ω_{n})_{n∈ℕ} may be computed as

i(T) = lim_{n→∞} inf_{k∈ℕ}(ω_{k} ⋯ ω_{k+n−1})^{1/n}

and i(T) ≤ d(T), so, to prove the preceding theorem, we cannot apply the result from Theorem 2.1.

Acknowldgement. The first author thanks the organizers of the Symposium V. Rakočević and D. -Dor - devic for this fruitful session.

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(received 29.10.2002)

P. Aiena, Dipartimento di Matematica ed Applicazioni, Facoltà di Ingegneria, Università di Palermo, Viale delle Scienze, I-90128 Palermo (Italy) E-mail: paiena@mbox.unipa.it M. T. Biondi, Departamento de Matemáticas, Facultad de Ciencias, Universidad UCLA de Barquisimeto (Venezuela), E-mail: mtbiondi@hotmail.com