Spectrum of Class wF ( p , r , q ) Operators

This paper discusses some spectral properties of class wF(p,r,q) operators for p > 0, r > 0, p + r ≤ 1, and q ≥ 1. It is shown that if T is a class wF(p,r,q) operator, then the Riesz idempotent Eλ of T with respect to each nonzero isolated point spectrum λ is selfadjoint and Eλ = ker(T − λ)= ker(T − λ)∗ . Afterwards, we prove that every class wF(p,r,q) operator has SVEP and property (β), and Weyl’s theorem holds for f (T) when f ∈H(σ(T)).


Introduction
A capital letter (such as T) means a bounded linear operator on a complex Hilbert space Ᏼ. For p > 0, an operator T is said to be p-hyponormal if (T * T) p ≥ (TT * ) p , where T * is the adjoint operator of T. An invertible operator T is said to be log-hyponormal if log(T * T) ≥ log(TT * ). If p = 1, T is called hyponormal, and if p = 1/2 T is called semihyponormal. Log-hyponormality is sometimes regarded as 0-hyponormal since (X p − 1)/ p → logX as p → 0 for X > 0. See Martin and Putinar [1] and Xia [2] for basic properties of hyponormal and semihyponormal operators. Log-hyponormal operators were introduced by Tanahashi [3], Aluthge and Wang [4], and Fujii et al. [5] independently. Aluthge [6] introduced phyponormal operators.
In Section 2, we prove that Riesz idempotent E λ of T with respect to each nonzero isolated point spectrum λ is selfadjoint and E λ Ᏼ = ker(T − λ) = ker(T − λ) * . In Section 3, we will show that each class wF(p,r, q) operator has SVEP (single-valued extension property) and Bishop's property (β). In Section 4, we show that Weyl's theorem holds for class wF(p,r, q).
If λ ∈ σ iso (T), the Riesz idempotent E λ of T with respect λ is defined by where Ᏸ is an open disk which is far from the rest of σ(T) and ∂Ᏸ means its boundary. Stampfli [24] showed that if T is hyponormal, then E λ is selfadjoint and E λ Ᏼ = ker(T − λ) = ker(T − λ) * . The recent developments of this result are shown in [16,17,20,22], and so on. In this section, it is shown that when λ = 0, this result holds for class wF(p,r, q) with p + r ≤ 1 and q ≥ 1. It is always assumed that λ ∈ σ iso (T) when the idempotent E λ is considered.
Reference [21] gave an example that the operator T is w-hyponormal, E 0 is not selfadjoint, and kerT = ker T * .
An operator T is said to be isoloid if σ iso (T) ⊆ σ p (T), is said to be reguloid if (T − λ)Ᏼ, is closed for each λ ∈ σ iso (T).
Theorem 2.2. If T belongs to class wF(p,r, q) with p + r ≤ 1, then T is isoloid and reguloid.
To give proofs, we prepare the following results. Theorem 2.3 (see [14]). Let λ = 0, and let {x n } be a sequence of vectors. Then the following assertions are equivalent. ( Theorem 2.4 (see [8]). If T is a class wF(p,r, q) operator for p + r ≤ 1 and q ≥ 1, then the following assertions hold.
Proof. We only need to show that σ a (T(t)) = τ t (σ a (T)) by homotopy property of the spectrum [2, page 19].
J. Yuan and Z. Gao 5 Proof. Let x ∈ E λ , by the formula above we have Hence |T| p x ∈ E λ (p,r)Ᏼ.
Proof. Since T(p,r) is α 0 -hyponormal by Theorem 2.5, we only need to prove that E 0 Ᏼ ⊆ E 0 (p,r)Ᏼ for E 0 Ᏼ ⊇ E 0 (p,r)Ᏼ holds by Lemma 2.6 and Theorem 2.10. We may also assume that p + r = 1 by Lemma 2.6.
It follows from Lemma 2.11 that (2.11), and E 0 (p,r)Ᏼ is reduced by |T| p .
Proof of Theorem 2.1. We only need to prove (1) for (2) holds by Lemma 2.12.

Proof of Theorem 2.2.
We only need to prove that T is reguloid for T being isoloid follows by Theorem 2.1 easily.

SVEP and Bishop's property (β)
where H(G) means the space of all analytic functions on G.
When T have SVEP at each λ ∈ Ꮿ, say that T has SVEP. This is a good property for operators. If T has SVEP, then for each λ ∈ Ꮿ, λ − T is invertible if and only if it is surjective (cf. [29,18]). When T has Bishop's property (β) at each λ ∈ Ꮿ, simply say that T has property (β).
This is a generalization of SVEP and it is introduced by Bishop [30] in order to develop a general spectral theory for operators on Banach space.

, then T has SVEP if and only if T(p,r) has SVEP, T has property (β) if and only if T(p,r) has property (β). In particular, every class wF(p,r, q) operator T with p + r ≤ 1 has SVEP and property (β).
This result is a generalization of [18]. Lemma 3.4

and the relations between T and its transformation T(p,r) are important:
T(p,r)|T| p = |T| p U|T| r |T| p = |T| p T, U|T| r T(p,r) = U|T| r |T| p U|T| r = TU|T| r . (3.1) Lemma 3.4 (see [18]). Let G be open subset of complex plane Ꮿ and let f n ∈ H(G) be functions such that μ f n (μ) → 0 uniformly on every compact subset of G, then f n (μ) → 0 uniformly on every compact subset of G.
Proof of Theorem 3.3. We only prove that T has property (β) if and only if T(p,r) has property (β) because the assertion that T has SVEP if and only if T(p,r) has SVEP can be proved similarly. Suppose that T(p,r) has property (β). Let G be an open neighborhood of λ and let f n ∈ H(G) be functions such that (μ − T) f n (μ) → 0 uniformly on every compact subset of G. By (3.1), (T(p,r) − μ)|T| p f n (μ) = |T| p (T − μ) f n (μ) → 0 uniformly on every compact subset of G. Hence T f n (μ) = U|T| r |T| p f n (μ) → 0 uniformly on every compact subset of G for T(p,r) has property (β), so that μ f n (μ) → 0 uniformly on every compact subset of G, and T having property (β) follows by Lemma 3.4.
Suppose that T has property (β). Let G be an open neighborhood of λ and let f n ∈ H(G) be functions such that (μ − T(p,r)) f n (μ) → 0 uniformly on every compact subset of G. By (3.1), (μ − T)(U|T| r f n (μ)) = U|T| r (μ − T(p,r)) f n (μ) → 0 uniformly on every compact subset of G. Hence T(p,r) f n (μ) → 0 uniformly on every compact subset of G for T has property (β) so that μ f n (μ) → 0 uniformly on every compact subset of G, and T(p,r) having property (β) follows by Lemma 3.4.

Weyl spectrum
For a Fredholm operator T, indT means its (Fredholm) index. A Fredholm operator T is said to be Weyl if indT = 0.
Let σ e (T), σ w (T), and π 00 (T) mean the essential spectrum, Weyl spectrum, and the set of all isolated eigenvalues of finite multiplicity of an operator T, respectively (cf. [28,17]).
According to Coburn [31], we say that Weyl's theorem holds for an operator T if σ(T) − σ w (T) = π 00 (T). Very recently, the theorem was shown to hold for several classes of operators including w-hyponormal operators and paranormal operators (cf. [17,32,20]).
In this section, we will prove that Weyl's theorem and Weyl spectrum mapping theorem hold for class wF(p,r, q) operator T with p + r ≤ 1. We also assume that p + r = 1 because of the inclusion relations among class wF(p,r, q) [9]. (1) Weyl's theorem holds for T. (

3) Weyl's theorem holds for f (T) when f ∈ H(σ(T)).
This is a generalization of the related assertions of [17].
Theorem 4.2. Let T belong to class wF(p,r, q) with p + r = 1, then the following assertions hold.
Theorem 4.2(1) is a generalization of [26] and (2) is a generalization of [24]. To give proofs, the following results are needful.
Theorem 4.3 [9]. Let p > 0, r > 0, and q ≥ 1, s ≥ p, t ≥ r. If T is a class wF(p,r, q) operator and T(s,t) is normal, then T is normal.
If T belongs to class wF(p,r, q) with p + r = 1 and is Fredholm, then indT ≤ 0.
This result can be regarded as a good complement of Theorem 2.1.