International Journal of Applied Mathematics

International Journal of Applied Mathematics ————————————————————– Volume 24 No. 3 2011, 349-360

ALGEBRAICALLY ABSOLUTE-(p, r)-PARANORMAL OPERATORS D. Senthilkumar1 , P. Maheswari Naik2 § 1,2 Post

Graduate and Research Department of Mathematics Government Arts College (Autonomous) Coimbatore, 641 018, Tamil Nadu, INDIA 1 e-mail: [email protected] 2 e-mail: [email protected]

Abstract: An operator T ∈ B(H) is said to be absolute-(p, r)-paranormal operator if |T |p |T ∗ |r xr x ≥ |T ∗ |r xp+r for all x ∈ H and for positive real number p > 0 and r > 0, where T =U |T | is the polar decomposition of T . An operator T ∈ absolute-(p, r)-paranormal operator is said to be algebraically absolute-(p, r)-paranormal operator if there exists a non constant complex polynomial p such that p(T ) ∈ absolute-(p, r)-paranormal operator. In this paper, it is proved that for operators T ∈ algebraically absolute-(p, r)-paranormal operator, both T and T ∗ satisfy Weyl’s theorem, furthermore, if also either ind(T − μ) ≥ 0 or ind(T − μ) ≤ 0 for all complex μ for which T −μ is Fredholm, then f (T ) and f (T ∗ ) satisfy Weyl’s theorem for every function f which is analytic on an open neighbourhood U of σ(T ) and which is non constant on each of the connected components of U. Assuming that the operator T ∈ algebraically absolute-(p, r)-paranormal operator has the single valued extension property, it is shown that T ∗ satisfies a-Weyl’s theorem. AMS Subject Classification: 47A11, 47A10, 47A13 Key Words: absolute-(p, r)-paranormal operator, Weyl’s theorem, single valued extension property, isoloid 1. Introduction and Preliminaries Let H be an infinite dimensional complex Hilbert space and B(H) denote the Received:

January 27, 2011

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algebra of all bounded linear operators acting on H. Every operator√T can be decomposed into T =U |T | with a partial isometry U , where |T |= T ∗ T . In this paper, T =U |T | denotes the polar decomposition satisfying the kernel condition N (U )=N (|T |). An operator T is said to be hyponormal if T ∗ T ≥ T T ∗ . p-hyponormal operators and log-hyponormal operators are defined as an extension of hyponormal operators. An operator T is said to be p -hyponormal operator for p > 0 if (T ∗ T )p ≥ (T T ∗ )p . An operator T is said to be log-hyponormal operator if T is invertible and logT ∗ T ≥ logT T ∗ . An operator T is said to be paranormal operator if T 2 x ≥ T x2 for every unit vector x. Paranormal operators have been studied by many authors [4], [7] and [13]. In [4], Ando showed that every log-hyponormal operator is paranormal. An operator T is said to be class A if |T 2 | ≥ |T |2 , where |T | = 1 (T ∗ T ) 2 . Furuta, Ito and Yamazaki [10] introduced class A(k) and absolute-kparanormal operators for k > 0 as generalizations of class A and paranormal operators, respectively. An operator T belongs to class A(k) if 1 (T ∗ |T |2k T ) k+1 ≥ |T |2 and T is said to be absolute-k-paranormal operator if |T |k T x ≥ T xk+1 for every unit vector x. On other hand Fujii, Izumino and Nakamoto [8] introduced p-paranormal operators for p > 0 as another generalization of paranormal operators. An operator T is said to be p-paranormal operator if |T |p U |T |p x ≥ |T |p x2 for every unit vector x, where the polar decomposition of T is T =U |T |. Fujii, Jung, S. H. Lee, M. Y. Lee and Nakamoto [9] introduced class A(p, r) as a further generalization of class A(k). An operator T ∈ A(p, r) r for p > 0 and r > 0 if (|T ∗ |r |T |2p |T ∗ |r ) p+r ≥ |T ∗ |2r and class AI(p, r) is class of all invertible operators which belong to class A(p, r). Yamazaki and Yanagida [24] introduced absolute-(p, r)-paranormal operator. It is a further generalization of the classes of both absolute-k-paranormal operators and p-paranormal operators as a parallel concept of class A(p, r). An operator T is said to be absolute-(p, r)-paranormal operator if |T |p |T ∗ |r xr ≥ |T ∗ |r xp+r for every unit vector x or equivalently |T |p |T ∗ |r xr x ≥ |T ∗ |r xp+r for all x ∈ H and for positive real numbers p > 0 and r > 0. It is also proved that T =U |T | is absolute-(p, r)-paranormal operator for p > 0 and r > 0 if and only if r|T |r U ∗ |T |2p U |T |r − (p + r)λp|T |2r + pλp+r I ≥ 0 for all real λ. An operator T is said to be normaloid if T = r(T ). An operator T ∈ absolute-(p, r)-paranormal operator is said to be algebraically absolute-(p, r)-paranormal operator if there exists a non constant complex polynomial p such that p(T ) ∈ absolute-(p, r)-paranormal operator. In this paper, it is proved that for operators T ∈ algebraically absolute-(p, r)-

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paranormal operator, both T and T ∗ satisfy Weyl’s theorem, furthermore, if also either ind(T −μ) ≥ 0 or ind(T −μ) ≤ 0 for all complex μ for which T −μ is Fredholm, then f (T ) and f (T ∗ ) satisfy Weyl’s theorem for every function f which is analytic on an open neighbourhood U of σ(T ) and which is non constant on each of the connected components of U. Assuming that the operator T ∈ algebraically absolute-(p, r)-paranormal operator has the single valued extension property, it is shown that T ∗ satisfies a-Weyl’s theorem. If T ∈ B(H) , we write N (T ) and R(T ) for null space and range of T , respectively. Let α(T ) = dimN (T ) = dim (T −1 (0)), β(T ) = dimN (T ∗ ) = dim (H/T (H)), σ(T ) denote the spectrum and σa (T ) denote the approximate point spectrum. An operator T ∈ B(H) is called Fredholm if it has closed range, finite dimensional null space and its range has finite co-dimension. The index of a Fredholm operator is given by i(T ) = α(T ) − β(T ). The ascent of T , asc(T ), is the least non-negative integer n such that T −n (0) = T −(n+1) (0) and the descent of T , dsc(T ), is the least non-negative integer n such that T n (H) = T (n+1) (H). We say that T is of finite ascent (resp., finite descent) if asc(T − λI) < ∞ (resp., dsc(T − λI) < ∞) for all complex numbers λ. T is called Weyl if it is Fredholm of index zero and Browder if it is Fredholm of finite ascent and descent. Let C denote the set of complex numbers. The Weyl spectrum σw (T ) and the Browder spectrum σb (T ) of T are the sets σw (T ) = {λ ∈ C : T −λ is not Weyl} and σb (T ) = {λ ∈ C : T −λ is not Browder}. Let π0 (T ) denote the set of Riesz points of T (i.e., the set of λ ∈ C such that T −λ is Fredholm of finite ascent and descent [5]) and let π00 (T ) denote the set of eigen values of T of finite geometric multiplicity. The operator T ∈ B(H) is said to satisfy Browder’s theorem if σ(T )\σw (T ) = π0 (T ) and T is said to satisfy Weyl’s theorem if σ(T )\σw (T ) = π00 (T ). In [11], Weyl’s theorem for T implies Browder’s theorem for T , and Browder’s theorem for T is equivalent to Browder’s theorem for T ∗ is proved. The essential spectrum σe (T ) of T ∈ B(H) is the set σe (T ) = {λ ∈ C : T − λ is not Fredholm}. Let accσ(T ) denote the set of all accumulation points of σ(T ), then σe (T ) ⊆ σw (T ) ⊆ σb (T ) ⊆ σe (T ) ∪ accσ(T ). Let πa0 (T ) be the set of λ ∈ C such that λ is an isolated point of σa (T ) and 0 < α(T − λ) < ∞, where σa (T ) denotes the approximate point spectrum of the operator T . Then π0 (T ) ⊆ π00 (T ) ⊆ πa0 (T ). We say that a-Weyl’s theorem holds for T if σaw (T )=σa (T )\πa0 (T ) ) denotes the essential approximate point spectrum of T (i.e., where σaw (T σaw (T ) = {σa (T + K) : K ∈ K(H)} with K(H) denoting the ideal of compact operators on H). Let Φ+ (H) = {T ∈ B(H) : α(T ) < ∞ and T (H)

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is closed} and Φ− (H) = {T ∈ B(H) : β(T ) < ∞} denote the semigroup of upper semi Fredholm and lower semi Fredholm operators in B(H) and let Φ− + (H) = {T ∈ Φ+ (H) : ind(T ) ≤ 0}. Then σaw (T ) is the complement in C of all those λ for which (T − λ) ∈ Φ− + (H) [19]. The concept of a-Weyl’s theorem was introduced by Rakocvic [21]. The concept states that a-Weyl’s theorem for T ⇒ Weyl’s theorem for T , but the converse is generally false. Let σab (T ) denote the Browder essential approximate point spectrum of T , σab (T ) = {σa (T + K) : T K = K T and K ∈ K(H)} = {λ ∈ C : T − λ ∈ / Φ− + (H) or asc(T − λ) = ∞}, then σaw (T ) ⊆ σab (T ). We say that T satisfies a-Browder’s theorem if σab (T ) = σaw (T ), see [19]. An operator T ∈ B(H) has the single valued extension property at λ0 ∈ C, if for every open disc Dλ0 centered at λ0 the only analytic function f : Dλ0 → H which satisfies (T − λ)f (λ)=0 for all λ ∈ Dλ0 is the function f ≡ 0. Trivially, every operator T has SVEP at points of the resolvent ρ(T ) = C/σ(T ); also T has SVEP at λ ∈ is σ(T ). We say that T has SVEP if it has SVEP at every λ ∈ C. The Quasinilpotent part H0 (T − λ) and the analytic core K(T − λ) of (T − λ) are defined by 1 H0 (T − λ) = {x ∈ H : limn→∞ (T − λ)n x n = 0} and K(T − λ) = {x ∈ H: there exists a sequence {xn } ⊂ H and δ > 0 for which x = x0 , (T − λ)(xn+1 ) = xn and xn ≤ δn x for all n =1, 2, 3, ...}.

We note that H0 (T − λ) and K(T − λ) are non-closed hyperinvariant subspaces of (T −λ) such that (T −λ)−q (0) ⊆ H0 (T −λ) for all q = 0, 1, 2, 3, ... and (T − λ)K(T − λ) = K(T − λ), see [17]. The operator T ∈B(H) is said to be semi-regular if T (H) is closed and T −1 (0) ⊂ T ∞ (H) = n∈N T n (H); T admits a generalized kato decomposition, GKD for short, if there exists a pair of T - invariant closed subspaces (M, N ) such that H = M ⊕ N , the restriction T |M is quasinilpotent and T |N is semi-regular. An operator T ∈ H has a GKD at every λ ∈ isoσ(T ), namely H = H0 (T − λ) ⊕ K(T − λ). We say that T is of kato type at a point λ if (T − λ)|M is nilpotent in the GKD for (T − λ). If T − λ is kato type, then K(T − λ) = (T − λ)∞ (H), see [1]. Fredholm (also semi-Fredholm) operators are kato type, see [14].

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The operators T ∈ B(H) satisfying property H(q), (i.e., H(q) = H0 (T − λ) = (T − λ)−q (0)) for some integer q ≥ 1, are Kato type at isolated points of σ(T ) [3] (but not every Kato type operator T satisfies property H(q)). It is easily seen that if T satisfies property H(q), then T has finite ascent (and hence also SVEP).

2. Main Results In this paper, we prove that if an operator T ∈ algebraically absolute(p, r)-paranormal operator, then f (T ) satisfies Weyl’s theorem for every f ∈ H(σ(T )), where H(σ(T )) denotes the set of function f : U → C which are analytic on an open neighbourhood U of σ(T ) and which is non constant on each of the connected components of U. Lemma 2.1. Let T be invertible absolute-(p, r)-paranormal operator, λ ∈ C and assume that σ(T ) = {λ}, then T = λI. Proof. Case (i): λ = 0. Since T is absolute-(p, r)-paranormal operator, T is normaloid [24, Theorem 8]. Therefore T = 0. Case (ii): λ = 0. Since T is invertible and T is absolute-(p, r)-paranormal operator, we have T is normaloid by [24, Theorem 8]. But T −1 is absolute(r, p)-paranormal operator by [24, Proposition 4]. Therefore T −1 is also normaloid by [24, Theorem 8]. But σ(T −1 )={ λ1 } then T T −1 =|λ|| λ1 |=1. Then by [18], T is convexoid. So w(T )={λ}. Therefore T =λ. Lemma 2.2. Let T be invertible algebraically absolute-(p, r)-paranormal operator and assume that σ(T ) = {λ}, then T − λ is nilpotent. Proof. If σ(T ) = {λ}, then σ(p(T )) = {p(λ)} and p(T ) = p(λ)I (By Lemma 2.1). Letting 0 = p(T ) − p(λ)I = c(T − λ)m ni=1 (T − λi ) for some scalars c, λi (1 ≤ i ≤ n) and integer m ≥ 1, it is seen that T − λi is invertible for all 1 ≤ i ≤ n. Hence (T − λ)m = 0. Definition 2.3. An operator T is said to be isoloid if the isolated points of the spectrum of T are eigen values of T .

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Lemma 2.4. Let T be invertible algebraically absolute-(p, r)-paranormal operator, then T is isoloid. Proof. Let λ ∈ isoσ(T ) then H = H0 (T − λ) ⊕ K(T − λ) / σ(T2 ). and T = T1 ⊕ T2 = T |H0 (T −λ) ⊕ T |K(T −λ) , where σ(T1 ) = {λ} and λ ∈ Since T1 ∈ algebraically absolute-(p, r)-paranormal operator, it follows from Lemma 2.2, that there exists an integer m ≥ 1, such that (T1 − λ)m = 0. But then H0 (T − λ) = (T1 − λ)−m (0) = (T − λ)−m (0) and (T − λ)m (H) = 0 ⊕ (T − λ)m K(T − λ) = K(T − λ) ⇒ H = (T − λ)−m (0) ⊕ (T − λ)m (H) ⇒ λ is a pole of the resolvent of T Hence λ is an eigen value of T . Lemma 2.5. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator, then π00 (T ) ⊆ σ(T )\σw (T ) and π00 (p(T )) ⊆ σ(p(T ))\σw (p(T )). Proof. If λ ∈ π00 (T ), then λ ∈ isoσ(T ) and 0 < α(T − λ) < ∞. The hypothesis λ ∈ isoσ(T ) implies that asc(T − λ) = dsc(T − λ) < ∞. Hence α(T − λ) = β(T − λ) [12, Proposition 38.6], which implies that T − λ ∈ Φ(H) and ind(T − λ) = 0. Hence λ ∈ σ(T )\σw (T ). Similarly we can prove the other inclusion. Lemma 2.6. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator, then p(T ) has SVEP at all points λ ∈ σ(p(T ))\σw (p(T )). Proof. Let λ ∈ σ(p(T ))\σw (p(T )). Then p(T )−λ is a Fredholm operator of index 0 (which implies that T − λ is Kato type [2, Remark 2.2 (iv)]. Next we prove that the point spectrum of p(T ) does not cluster at λ, then this implies that p(T ) has SVEP at λ [2, Theorem 2.6]. Let N (S) and γ(S) denote the null space and the minimal modulus function of an operator S ∈ B(H), and let d(x, N (S)) = inf y∈N (S) x − y denote the distance of x ∈ H from N (S). Then from [6, pg. 93], we know that a subspace M of H is said to be orthogonal to a subspace N of H, if m ≤ m + n for all m ∈ M and n ∈ N . Assume to the contrary that the point spectrum of p(T ) clusters at λ. Then there exists a sequence {λn } of non zero eigenvalues of p(T ) converging to λ. Then from [23, Lemma 2.6], we have that eigen spaces corresponding to distinct non zero eigen values of an absolute-(p, r)-paranormal opera-

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tor are orthogonal, furthermore, it follows from [23, Lemma 2.6], that if one of the eigen values is 0 and the other eigen value is α = 0, then the eigen spaces corresponding to the eigen value α is orthogonal to the eigen space corresponding to the eigen value 0. Choose a non zero eigen value λn from the sequence converging to λ. Then d(xn , N (p(T ) − λ)) ≥ 1 for all xn ∈ N (p(T ) − λn ) such that xn = 1. We have, δ(λn , λ) = sup{d(xn , N (p(T ) − λ)) : xn ∈ N (p(T ) − λn ), xn = 1} ≥ 1 for all n, which implies that . |λn − λ|/δ(λn , λ) → 0 as n → ∞ But then, . γ(p(T ) − λ) = |λn − λ|/δ(λn , λ) → 0 as n → ∞ Since (p(T ) − λ)(x) is closed, this is a contradiction [12, Proposition 36.1]. Consequently, points λ ∈ σ(p(T ))\σw (p(T )) are isolated in the point spectrum of p(T ). Theorem 2.7. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator, then p(T ) satisfies Weyl’s theorem. Proof. It is sufficient if we prove σ(p(T ))\σw (p(T )) ⊆ π00 (p(T )). The other inclusion follows from Lemma 2.5. If λ ∈ σ(p(T ))\σw (p(T )), then p(T ) − λ ∈ Φ(H), ind(p(T ) − λ) = 0 and p(T ) has SVEP at λ (Lemma 2.5). Hence λ ∈ π00 (p(T )) ([2, Corollary 2.10]). We know from [15, Theorem 3.3.9], that for an operator T ∈ B(H), SVEP for T implies SVEP for f (T ) for every f which is analytic on an open neighbourhood of σ(T ); conversely, if f ∈ H(σ(T )), then SVEP for f (T ) implies SVEP for T . Theorem 2.8. (see [2]) For every T ∈ B(H) and f ∈ H(σ(T )), f (T ) has SVEP at a λ ∈ c if and only if T has SVEP at every μ ∈ σ(T ) such that f (μ) = λ. Lemma 2.9. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator, then T has SVEP at every λ ∈ σ(T )\σw (T ). Proof. Since π00 (T ) ⊆ σ(T )\σw (T ) (By Lemma 2.5), we have σw (T ) ⊆ σ(T )\π00 (T ). Also, since operators T ∈ invertible algebraically absolute(p, r)-paranormal are isoloid, p(σ(T )\π00 (T )) = σ(p(T ))\π00 (p(T )), see [16].

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Since p(T ) satisfies Weyl’s theorem (By Theorem 2.7), p(σw (T )) ⊆ p(σ(T )\π00 (T )) = σ(p(T ))\π00 (p(T )) = σw (p(T )) ⊆ p(σw (T )), which implies that p(σw (T )) = σw (p(T )). Choose a μ ∈ σ(T ) such that / p(σw (T )) = σw (p(T )), which μ ∈ / σw (T ), and set p(μ) = λ. Then λ ∈ implies that p(T ) has SVEP at λ (By Lemma 2.6). Consequently, T has SVEP at μ (By Theorem 2.8). Theorem 2.10. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator, then both T and T ∗ satisfies Weyl’s theorem. Proof. We know that π00 (T ) ⊆ σ(T )\σw (T ). For reverse inclusion, choose a μ ∈ σ(T )\σw (T ). Then T − μ ∈ Φ(H) and ind(T − μ) = 0. P at μ ( By Lemma 2.9), μ ∈ isoσ(T ) ([1, Corollary 2.10]). Hence μ ∈ π00 (T ), and T satisfies Weyl’s theorem. Since T satisfies Weyl’s theorem, both T and T ∗ satisfy Browder’s theorem. In particular, σ(T ∗ )\σw (T ∗ ) = π0 (T ∗ ) ⊆ π00 (T ∗ ). Let λ ∈ π00 (T ∗ ); then λ ∈ isoσ(T ), and there exists an integer m ≥ 1 such that H = (T − λ)−m (0) ⊕ (T − λ)m (H) ( By Lemma 2.4) ⇒ H∗ =(T ∗ − λI ∗ )m (H∗ ) ⊕ (T ∗ − λI ∗ )−m (0). ∗ Hence asc(T − λI ∗ ) = dsc(T ∗ − λI ∗ ) < ∞, which, since 0 < α(T ∗ − λI ∗ ) < ∞, implies that ind(T ∗ − λI ∗ ) =0 ([12, Proposition 38.6]). Thus λ ∈ σ(T ∗ )\σw (T ∗ ), and T ∗ satisfies Weyl’s theorem. Theorem 2.11. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator such that either ind(T − μ) ≥ 0 or ind(T − μ) ≤ 0 for all μ for which T − μ ∈ Φ(H), f (T ) and f (T ∗ ) satisfy Weyl’s theorem for every f ∈ H(σ(T )). Proof. From the previous theorem, we have that T satisfies Weyl’s theorem. Then using isoloid property of T (i.e., Lemma 2.6), it follows that f (σw (T )) = f (σ(T )\π00 (T )) = σ(f (T ))\π00 (f (T )). From [11], we know that f (σw (T )) ⊆ σw (f (T )) for every T ∈ B(H) and f ∈ H(σ(T )). Thus to prove that f (T ) satisfies Weyl’s theorem it will suffice to prove the reverse inclusion. Let λ ∈ / σ w (f (T )), and let f (T ) − λI = cg(T ) ni=1 (T − μi ) since f (T )−λI ∈ for some scalars c, μi (1 ≤ i ≤ n) and invertible g(T ). Then, Φ(H) and ind(f (T ) − λI) =0, each T − μi ∈ Φ(H) and ni=1 ind(T − μi )

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=0. Thus, since either ind(T − μ) ≥ 0 or ind(T − μ) ≤ 0 for all μ for which T − μ ∈ Φ(H), T − μi ∈ Φ(H) and ind(T − μi ) = 0 for all 1 ≤ i ≤ n. Hence / f (σw (T )). Now we show μi ∈ σ(T )\σw (T ) for all 1 ≤ i ≤ n, and λ = f (μi ) ∈ that f (T ∗ ) satisfies Weyl’s theorem. Since f (T ) satisfies Weyl’s theorem, it implies f (T ∗ ) satisfies Browder’s theorem, σ(f (T ∗ ))\σw (f (T ∗ )) = π0 (f (T ∗ )) ⊆ π00 (f (T ∗ )) Let λ ∈ π00 (f (T ∗ )). Then, since σ(f (T ∗ )) = σ(f (T )∗ ) = σ(f (T )), λ ∈ isoσ(f (T )). Hence each μi , 1 ≤ i ≤ n, in the representation f (T ) − λI = cg(T ) ni=1 (T − μi ) is isolated in σ(T ), and therefore an eigenvalue of T (by Lemma 2.4). Consequently, λ ∈ π00 (f (T )). Since f (T ) satisfies Weyl’s theorem, λ is a pole of the resolvent of f (T ) ⇒ λ is a pole of the resolvent of f (T )∗ . Thus π0 (f (T ∗ )) = π00 (f (T ∗ )). Theorem 2.12. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator such that T (resp., T ∗ ) has SVEP, then T ∗ (resp., T ) satisfies a-Weyl’s theorem. Proof. We already know that T ∗ satisfies Weyl’s theorem. If T has SVEP, then σ(T ∗ ) = σa (T ∗ ) and σa (T ∗ )\σw (T ∗ ) = πa0 (T ∗ ). But we know that σea (T ∗ ) = σe (T ∗ ) ⊆ σw (T ∗ ). To prove the reverse inclusion, take a λ ∈ σea (T ∗ ). Then T ∗ −λI ∗ ∈ Φ+ (H∗ ) and ind(T ∗ −λI ∗ ) ≤ 0, which implies that T −λ ∈ Φ− (H) and ind(T −λ) ≥ 0. Combining T −λ ∈ Φ− (H) with the fact that T has SVEP, it follows from [1, Theorem 2.6] that asc(T − λ) < ∞. But then ind(T − λ) ≤ 0. Hence ind(T − λ) = ind(T ∗ − λI ∗ ) = 0, and λ∈ / σw (T ∗ ). Now we observe that if T ∗ has SVEP, then σ(T ) = σa (T ) [15, pg. 35] and σw (T ) = σea (T ). Since T satisfies Weyl’s theorem, π00 (T ) = σ(T )\σw (T ) = σa (T )\σea (T ) = πa0 (T ). Corollary 2.13. If T ∈ invertible algebraically absolute-(p, r)-paranormal operator has SVEP, then f (T ∗ ) satisfies a-Weyl’s theorem for every f ∈ H(σ(T )). Proof. Since T has SVEP implies f (T ) has SVEP, and since f (T ∗ ) satisfies Weyl’s theorem (by Theorem 2.11), the proof follows from the previous theorem. Theorem 2.14. If either T ∈ invertible algebraically absolute-(p, r)paranormal operator is such that ind(T − λ) ≥ 0 or T ∈ invertible absolute(p, r)-paranormal operator is such that ind(T − λ) ≥ 0 for all λ for which

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T −λ ∈ Φ+ (H), then f (T ) satisfies a-Browder’s theorem for all f ∈ H(σ(T )). Proof. The hypothesis ind(T −λ) ≤ 0 or ind(T −λ) ≥ 0 for all λ such that T − λ ∈ B(H) ∩ Φ+ (H) implies that σea (f (T )) = f (σea (T )) [22, Theorem 2]. Since f (σea (T )) ⊆ f (σab (T )) = σab (f (T )) for every T ∈ B(H) [20, Theorem 3.4], to prove the Theorem it will suffice to prove that σab (f (T )) ⊆ / σea (f (T )). Then f (T ) − λ ∈ Φ− σea (f (T )). Suppose that λ ∈ + (H). Letting f (T ) − λI = cg(T ) ni=1 (T − μi ) for some scalars for some scalars c, μi (1 ≤ i ≤ n) and invertible g(T ), it then follows that T −μi ∈ Φ+ (H) for all 1 ≤ i ≤ n and ni=1 ind(T −μi ) ≤ 0. But then either p(T ) ∈ absolute-(p, r)-paranormal operator, ind(T − μi ) = 0 and T − μi ∈ Φ(H) or T ∈ absolute-(p, r)-paranormal operator, ind(T − μi ) ≤ 0 and T ∈ Φ+ (H) for all 1 ≤ i ≤ n. Thus T has SVEP at μi for all 1 ≤ i ≤ n. Hence f (T ) has SVEP at λ. Since f (T ) ∈ Φ+ (H) , asc(f (T ) − λ) < ∞ [12, Theorem 2.6], and λ ∈ σa (T )\σab (T ) [20, Theorem 2.1]. References [1] P. Aiena, O. Monsalve, The single valued extension property and the generalized kato decomposition property, Acta Sci. Math. (Szeged), 67 (2001), 461-477. [2] P. Aiena, T.L. Miller, M.M. Neumann, On a localised single valued extension property, Math. Proc. Royal Irish Acad., 104A, No. 1 (2004), 17-34. [3] P. Aiena, F. Villafane, Weyl’s theorem of some classes of operators, Integr. Equation Op. Th., 53, No. 4 (2005), 453-466. [4] T. Ando, Operators with a norm condition, Acta Sci. Math. (Szeged), 33 (1972), 169-178. [5] S.R. Caradus, W.E. Pfaffenberger, Y. Bertram, Calkin Algebras and Algebras of Operators on Banach Spaces, Marcel Dekker, New York (1974). [6] N.J. Dunford, T. Schwartz, Linear Operators, Part I, Interscience, New York (1964). [7] T. Furuta, On the class of paranormal operators, Proc. Japan Acad., 43 (1967), 594-598.

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