- Email: [email protected]
Riesz idempotent of (n, k)-quasi-*-paranormal operators
Acta Mathematica Scientia 2016,36B(5):1487–1491 http://actams.wipm.ac.cn
RIESZ IDEMPOTENT OF (n, k)-QUASI-∗-PARANORMAL OPERATORS∗
Q²)
Qingping ZENG (
College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China E-mail : [email protected]
¨~#)
Huaijie ZHONG (
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China E-mail : [email protected] Abstract A bounded linear operator T on a complex Hilbert space H is called (n, k)-quasi∗-paranormal if kT 1+n (T k x)k1/(1+n) kT k xkn/(1+n) ≥ kT ∗ (T k x)k
for all x ∈ H,
where n, k are nonnegative integers. This class of operators has many interesting properties and contains the classes of n-∗-paranormal operators and quasi-∗-paranormal operators. The aim of this note is to show that every Riesz idempotent Eλ with respect to a non-zero isolated spectral point λ of an (n, k)-quasi-∗-paranormal operator T is self-adjoint and satisfies ranEλ = ker(T − λ) = ker(T − λ)∗ . Key words
∗-class A operator; ∗-paranormal operator; Riesz idempotent
2010 MR Subject Classification
1
47A10; 47B20
Introduction
There is a growing interest concerning nonhyponormal operators. Let L(H) be the C ∗ algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H. Below we list some of these nonhyponormal operators. Recall that an operator T ∈ L(H) is said to be • ∗-class A if |T 2 | ≥ |T ∗ |2 (see [5]); • quasi-∗-class A if T ∗ |T 2 |T ≥ T ∗ |T ∗ |2 T (see [11]); • k-quasi-∗-class A if T ∗k |T 2 |T k ≥ T ∗k |T ∗ |2 T k (see [9]); • ∗-paranormal if kT 2 xk1/2 kxk1/2 ≥ kT ∗ xk for all x ∈ H (see [2]); • quasi-∗-paranormal if kT 2 (T x)k1/2 kT xk1/2 ≥ kT ∗ (T x)k for all x ∈ H (see [10]); • n-∗-paranormal if kT 1+n xk1/(1+n) kxkn/(1+n) ≥ kT ∗xk for all x ∈ H (see [8]), ∗ Received
October 16, 2014; revised March 12, 2016. This work has been supported by National Natural Science Foundation of China (11301077, 11301078, 11401097, 11501108) and Natural Science Foundation of Fujian Province (2015J01579, 2016J05001).
1488
ACTA MATHEMATICA SCIENTIA
Vol.36 Ser.B
here and henceforth, n, k denote nonnegative integers. As an extension of the above operator classes, we introduced and studied in [21] the following definition. Definition 1.1 An operator T ∈ L(H) is said to be (n, k)-quasi-∗-paranormal if kT 1+n (T k x)k1/(1+n) kT k xkn/(1+n) ≥ kT ∗(T k x)k for all x ∈ H. The class of (n, k)-quasi-∗-paranormal operators has many interesting properties (see [21]), such as inclusion relations, SVEP (single valued extension property), matrix representation, joint point spectrum, and so on. In the present note, we continue to investigate the properties of (n, k)-quasi-∗-paranormal operators. We show that every Riesz idempotent Eλ with respect to a non-zero isolated spectral point λ of an (n, k)-quasi-∗-paranormal operator T is self-adjoint and satisfies ranEλ = ker(T − λ) = ker(T − λ)∗ .
2
Riesz Idempotent of (n, k)-Quasi-∗-Paranormal Operators
The self-adjointness of Riesz idempotent with respect to the isolated spectral point of an operator was investigated by a number of mathematicians around the world. For an isolated spectral point λ ∈ isoσ(T ), the Riesz idempotent Eλ with respect to λ is defined by Z 1 Eλ := (z − T )−1 dz, 2πi ∂D where D is a closed disk with center λ and its radius is small enough such that D ∩ σ(T ) = {λ}. In general, the Riesz idempotent Eλ is not orthogonal and Eλ is orthogonal if and only if Eλ is self-adjoint. Stampfli [12] showed that the Riesz idempotent Eλ for an isolated spectral point λ of a hyponormal operator T is self-adjoint. Stampfli’s result was extended to p-hyponormal operators and log-hyponormal operators by Ch¯ o and Tanahashi [4], to M -hyponormal operators by Ch¯o and Han [3], to ∗-paranormal operators by Tanahashi and Uchiyama [14]. In the case λ 6= 0, Stampfli’s result was extended to p-quasihyponormal operators by Tanahashi and Uchiyama [15], to (p, k)-quasihyponormal operators by Tanahashi, Uchiyama and Ch¯o [16], to w-hyponormal operators by Han, Lee and Wang [6], to class A operators by Uchiyama and Tanahashi [18], to quasi-class A operators by Jeon and Kim [7], to quasi-class (A, k) operators by Tanahashi, Jeon, Kim and Uchiyama [13], to paranormal operators by Uchiyama [17], to k-quasi-∗-class A operators by Mecheri [9]. In this section, we will extend Stampfli’s result to n-∗-paranormal operators and, to (n, k)quasi-∗-paranormal operators in the case λ 6= 0. Theorem 2.1 (1) Let T be (n, k)-quasi-∗-paranormal and 0 6= λ ∈ isoσ(T ). Then the Riesz idempotent Eλ is self-adjoint and ranEλ = ker(T − λ) = ker(T − λ)∗ . (2) Let T be n-∗-paranormal and λ ∈ isoσ(T ). Then the Riesz idempotent Eλ is self-adjoint and ranEλ = ker(T − λ) = ker(T − λ)∗ .
No.5
Q.P. Zeng & H.J. Zhong: RIESZ IDEMPOTENT OF (n, k)-QUASI-∗-PARANORMAL
1489
It is worth to note that Theorem 2.1 (1) does not hold even for (0, 1)-quasi-∗-paranormal (that is, quasi-hyponormal) operators when λ = 0 (see [15, Example 6]). To give the proof of Theorem 2.1, we prepare the following lemmas. Recall that an operator T ∈ L(H) is said to be n-paranormal if kT 1+nxk1/(1+n) kxkn/(1+n) ≥ kT xk for all x ∈ H (see [19, 20]). Lemma 2.2 Let T be n-∗-paranormal and λ ∈ isoσ(T ). Then λ is a pole of order one of the resolvent of T . Proof
Let λ ∈ isoσ(T ). Then H = ranEλ ⊕ kerEλ .
By [1, Theorem 3.74], it follows that ranEλ = H0 (T − λ) and kerEλ = K(T − λ), where H0 (T − λ) := {x ∈ H : lim k(T − λ)n xk1/n = 0} n→∞
and K(T − λ) :={x ∈ H : there exist a sequence {xn }n≥1 in H and a constant δ > 0 such that (T − λ)x1 = x, (T − λ)xn+1 = xn and kxn k ≤ δ n kxk for all n ≥ 1}. Since T is n-∗-paranormal, it follows from [21] that T is (n + 1)-paranormal. Hence by [14, Theorem 2] we have that ranEλ = ker(T − λ). Since H = ranEλ ⊕ kerEλ , ran(T − λ) = (T − λ)kerEλ = (T − λ)K(T − λ) = K(T − λ) = kerEλ . Consequently, H = ker(T − λ) ⊕ ran(T − λ). That is, λ is a pole of order one.
Suppose that ranT k is not dense. Let T1 T2 on H = ranT k ⊕ ker(T ∗k ), T = 0 T3
Lemma 2.3 (see [21])
where ranT k is the closure of ranT k . If T is (n, k)-quasi-∗-paranormal, then T1 is n-∗-paranormal, T3k = 0 and σ(T ) = σ(T1 ) ∪ {0}. Lemma 2.4 If T is (n, k)-quasi-∗-paranormal and quasi-nilpotent, then T k+1 = 0. Proof Assume that ranT k is dense, then T is n-∗-paranormal, hence by [21], T is (n + 1)paranormal. Thus by [14, Corollary 1], T = 0. So we may suppose that ranT k is not dense. Hence by Lemma 2.3, we can write T1 T2 on H = ranT k ⊕ ker(T ∗k ), T = 0 T3
1490
ACTA MATHEMATICA SCIENTIA
Vol.36 Ser.B
where T1 is n-∗-paranormal, T3k = 0 and σ(T ) = σ(T1 ) ∪ {0}. Since σ(T1 ) is not empty and 0 T2 . An easy σ(T ) = {0}, we see that σ(T1 ) = {0}. Hence T1 = 0, and so T = 0 T3 k 0 T T 2 3 = 0. computation yields that T k+1 = 0 T3k+1
Lemma 2.5 Let T be (n, k)-quasi-∗-paranormal and λ ∈ isoσ(T ). Then λ is a pole of the resolvent of T , and the order of λ is no more than k + 1 when λ = 0, and 1 when λ 6= 0. Proof
Let λ ∈ isoσ(T ). Then H = ranEλ ⊕ kerEλ .
Write T = T1 ⊕ T2 with respect to this direct decomposition. Since σ(T1 ) = {λ} and σ(T2 ) = σ(T )\{λ}, T1 − λ is quasi-nilpotent and T2 − λ is invertible. Case I If ranT1k is dense, then T1 is n-∗-paranormal, hence by Lemma 2.2, λ is a pole of order one of the resolvent of T1 . Consequently, λ is a pole of order one of the resolvent of T . Case II If ranT1k is not dense, then by Lemma 2.3, we can write T11 T12 on H = ranT1k ⊕ ker(T1∗k ), T1 = 0 T13 k where T11 is n-∗-paranormal, T13 = 0 and σ(T1 ) = σ(T11 ) ∪ {0}. Then σ(T1 ) = {λ} implies that λ = 0. It follows from Lemma 2.4 that T1k+1 = 0, and hence λ = 0 is a pole of order no more than k + 1.
Lemma 2.6 (see [21])
Let T be (n, k)-quasi-∗-paranormal and 0 6= λ ∈ C. Then ker(T − λ) ⊆ ker(T ∗ − λ).
Proof of Theorem 2.1 (1) By Lemma 2.5, λ is an eigenvalue of T , ranEλ = ker(T − λ) and kerEλ = ran(T − λ). Since ker(T − λ) ⊆ ker(T ∗ − λ) by Lemma 2.6, ranEλ = ker(T − λ) is a reducing subspace of T . Write T = λ ⊕ T1 with respect to the orthogonal decomposition H = ker(T − λ) ⊕ ker(T − λ)⊥ , where T1 is (n, k)-quasi-∗-paranormal with ker(T1 − λ) = {0}. Since λ ∈ isoσ(T ) and σ(T ) = {λ}∪σ(T1 ), it then follows from Lemma 2.5 that T1 −λ is invertible. Consequently, ker(T −λ) = ker(T ∗ − λ). Since ker(T − λ) ⊆ ker(T ∗ − λ), we have ker(T − λ)⊥ran(T − λ), and hence ranEλ ⊥kerEλ . That is, Eλ is self-adjoint. (2) It needs only to consider the case when λ = 0. Applying the above argument and using Lemma 2.2 and the fact that ker(T ) ⊆ ker(T ∗ ), the assertion follows. References [1] Aiena P. Fredholm and Local Spectral Theory, with Application to Multipliers. Dordrecht: Kluwer Academic Publishers, 2004 [2] Arora S C, Thukral J K. On a class of operators. Kyngpook Math J, 1986, 21(41): 381–386 [3] Ch¯ o M, Han Y M. Riesz idempotent and algebraically M -hyponormal operators. Integral Equations Operator Theory, 2005, 53(3): 311–320
No.5
Q.P. Zeng & H.J. Zhong: RIESZ IDEMPOTENT OF (n, k)-QUASI-∗-PARANORMAL
1491
[4] Ch¯ o M, Tanahashi K. Isolated point of spectrum of p-hyponormal, log-hyponormal operator. Integral Equations Operator Theory, 2002, 43(4): 379–384 [5] Duggal B P, Jeon I H, Kim I H. On ∗-paranormal contractions and properties for ∗-class A operators. Linear Algebra Appl, 2012, 436(5): 954–962 [6] Han Y M, Lee J I, Wang D. Riesz idempotent and Weyl’s theorem for w-hyponormal operators. Integral Equations Operator Theory, 2005, 53(1): 51–60 [7] Jeon I H, Kim I H. On operators satisfying T ∗ |T 2 |T ≥ T ∗ |T |2 T . Linear Algebra Appl, 2006, 418(2/3): 854–862 [8] Lee M Y, Lee S H, Rhoo C S. Some remark on the structure of k∗-paranormal operators. Kyngpook Math J, 1995, 35(2): 205–211 [9] Mecheri S. Isolated points of spectrum of k-quasi-∗-class A operators. Studia Math, 2012, 208(1): 87–96 [10] Mecheri S. On quasi-∗-paranormal operators. Ann Funct Anal, 2012, 3(1): 86–91 [11] Shen J L, Zuo F, Yang C S. On operators satisfying T ∗ |T 2 |T ≥ T ∗ |T ∗ |2 T . Acta Math Sinica, Eng Ser, 2010, 26(11): 2109–2116 [12] Stampfli J G. Hyponormal operators and spectral density. Trans Amer Math Soc, 1965, 117: 469–476 [13] Tanahashi K, Jeon I H, Kim I H, Uchiyama A. Quasinilpotent part of class A or (p, k)-quasihyponormal operators//Oper Theory Adv Appl. Birkh¨ auser: Basel, 2009, 187: 199–210 [14] Tanahashi K, Uchiyama A. A note on ∗-paranormal operators and related classes of operators. Bull Korean Math Soc, 2014, 51(2): 357–371 [15] Tanahashi K, Uchiyama A. Isolated point of spectrum of p-quasihyponormal operators. Linear Algebra Appl, 2002, 341(1/3): 345–350 [16] Tanahashi K, Uchiyama A, Ch¯ o M. Isolated points of spectrum of (p, k)-quasihyponormal operators. Linear Algebra Appl, 2004, 382: 221–229 [17] Uchiyama A. On the isolated points of the spectrum of paranormal operators. Integral Equations Operator Theory, 2006, 55(1): 145–151 [18] Uchiyama A, Tanahashi K. On the Riesz idempotent of class A operators. Math Inequal Appl, 2002, 5(2): 291–298 [19] Yuan J T, Gao Z S. Weyl spectrum of class A(n) and n-paranormal operators. Integral Equations Operator Theory, 2008, 60(2): 289–298 [20] Yuan J T, Ji G X. On (n, k)-quasiparanormal operators. Studia Math, 2012, 209(3): 289–301 [21] Zeng Q P, Zhong H J. On (n, k)-quasi-∗-paranormal operators. Bull Malays Math Sci Soc, 2016, DOI: 10.1007/s40840-015-0119-z
RIESZ IDEMPOTENT OF (n, k)-QUASI-∗-PARANORMAL OPERATORS∗
Q²)
Qingping ZENG (
College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China E-mail : [email protected]
¨~#)
Huaijie ZHONG (
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China E-mail : [email protected] Abstract A bounded linear operator T on a complex Hilbert space H is called (n, k)-quasi∗-paranormal if kT 1+n (T k x)k1/(1+n) kT k xkn/(1+n) ≥ kT ∗ (T k x)k
for all x ∈ H,
where n, k are nonnegative integers. This class of operators has many interesting properties and contains the classes of n-∗-paranormal operators and quasi-∗-paranormal operators. The aim of this note is to show that every Riesz idempotent Eλ with respect to a non-zero isolated spectral point λ of an (n, k)-quasi-∗-paranormal operator T is self-adjoint and satisfies ranEλ = ker(T − λ) = ker(T − λ)∗ . Key words
∗-class A operator; ∗-paranormal operator; Riesz idempotent
2010 MR Subject Classification
1
47A10; 47B20
Introduction
There is a growing interest concerning nonhyponormal operators. Let L(H) be the C ∗ algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H. Below we list some of these nonhyponormal operators. Recall that an operator T ∈ L(H) is said to be • ∗-class A if |T 2 | ≥ |T ∗ |2 (see [5]); • quasi-∗-class A if T ∗ |T 2 |T ≥ T ∗ |T ∗ |2 T (see [11]); • k-quasi-∗-class A if T ∗k |T 2 |T k ≥ T ∗k |T ∗ |2 T k (see [9]); • ∗-paranormal if kT 2 xk1/2 kxk1/2 ≥ kT ∗ xk for all x ∈ H (see [2]); • quasi-∗-paranormal if kT 2 (T x)k1/2 kT xk1/2 ≥ kT ∗ (T x)k for all x ∈ H (see [10]); • n-∗-paranormal if kT 1+n xk1/(1+n) kxkn/(1+n) ≥ kT ∗xk for all x ∈ H (see [8]), ∗ Received
October 16, 2014; revised March 12, 2016. This work has been supported by National Natural Science Foundation of China (11301077, 11301078, 11401097, 11501108) and Natural Science Foundation of Fujian Province (2015J01579, 2016J05001).
1488
ACTA MATHEMATICA SCIENTIA
Vol.36 Ser.B
here and henceforth, n, k denote nonnegative integers. As an extension of the above operator classes, we introduced and studied in [21] the following definition. Definition 1.1 An operator T ∈ L(H) is said to be (n, k)-quasi-∗-paranormal if kT 1+n (T k x)k1/(1+n) kT k xkn/(1+n) ≥ kT ∗(T k x)k for all x ∈ H. The class of (n, k)-quasi-∗-paranormal operators has many interesting properties (see [21]), such as inclusion relations, SVEP (single valued extension property), matrix representation, joint point spectrum, and so on. In the present note, we continue to investigate the properties of (n, k)-quasi-∗-paranormal operators. We show that every Riesz idempotent Eλ with respect to a non-zero isolated spectral point λ of an (n, k)-quasi-∗-paranormal operator T is self-adjoint and satisfies ranEλ = ker(T − λ) = ker(T − λ)∗ .
2
Riesz Idempotent of (n, k)-Quasi-∗-Paranormal Operators
The self-adjointness of Riesz idempotent with respect to the isolated spectral point of an operator was investigated by a number of mathematicians around the world. For an isolated spectral point λ ∈ isoσ(T ), the Riesz idempotent Eλ with respect to λ is defined by Z 1 Eλ := (z − T )−1 dz, 2πi ∂D where D is a closed disk with center λ and its radius is small enough such that D ∩ σ(T ) = {λ}. In general, the Riesz idempotent Eλ is not orthogonal and Eλ is orthogonal if and only if Eλ is self-adjoint. Stampfli [12] showed that the Riesz idempotent Eλ for an isolated spectral point λ of a hyponormal operator T is self-adjoint. Stampfli’s result was extended to p-hyponormal operators and log-hyponormal operators by Ch¯ o and Tanahashi [4], to M -hyponormal operators by Ch¯o and Han [3], to ∗-paranormal operators by Tanahashi and Uchiyama [14]. In the case λ 6= 0, Stampfli’s result was extended to p-quasihyponormal operators by Tanahashi and Uchiyama [15], to (p, k)-quasihyponormal operators by Tanahashi, Uchiyama and Ch¯o [16], to w-hyponormal operators by Han, Lee and Wang [6], to class A operators by Uchiyama and Tanahashi [18], to quasi-class A operators by Jeon and Kim [7], to quasi-class (A, k) operators by Tanahashi, Jeon, Kim and Uchiyama [13], to paranormal operators by Uchiyama [17], to k-quasi-∗-class A operators by Mecheri [9]. In this section, we will extend Stampfli’s result to n-∗-paranormal operators and, to (n, k)quasi-∗-paranormal operators in the case λ 6= 0. Theorem 2.1 (1) Let T be (n, k)-quasi-∗-paranormal and 0 6= λ ∈ isoσ(T ). Then the Riesz idempotent Eλ is self-adjoint and ranEλ = ker(T − λ) = ker(T − λ)∗ . (2) Let T be n-∗-paranormal and λ ∈ isoσ(T ). Then the Riesz idempotent Eλ is self-adjoint and ranEλ = ker(T − λ) = ker(T − λ)∗ .
No.5
Q.P. Zeng & H.J. Zhong: RIESZ IDEMPOTENT OF (n, k)-QUASI-∗-PARANORMAL
1489
It is worth to note that Theorem 2.1 (1) does not hold even for (0, 1)-quasi-∗-paranormal (that is, quasi-hyponormal) operators when λ = 0 (see [15, Example 6]). To give the proof of Theorem 2.1, we prepare the following lemmas. Recall that an operator T ∈ L(H) is said to be n-paranormal if kT 1+nxk1/(1+n) kxkn/(1+n) ≥ kT xk for all x ∈ H (see [19, 20]). Lemma 2.2 Let T be n-∗-paranormal and λ ∈ isoσ(T ). Then λ is a pole of order one of the resolvent of T . Proof
Let λ ∈ isoσ(T ). Then H = ranEλ ⊕ kerEλ .
By [1, Theorem 3.74], it follows that ranEλ = H0 (T − λ) and kerEλ = K(T − λ), where H0 (T − λ) := {x ∈ H : lim k(T − λ)n xk1/n = 0} n→∞
and K(T − λ) :={x ∈ H : there exist a sequence {xn }n≥1 in H and a constant δ > 0 such that (T − λ)x1 = x, (T − λ)xn+1 = xn and kxn k ≤ δ n kxk for all n ≥ 1}. Since T is n-∗-paranormal, it follows from [21] that T is (n + 1)-paranormal. Hence by [14, Theorem 2] we have that ranEλ = ker(T − λ). Since H = ranEλ ⊕ kerEλ , ran(T − λ) = (T − λ)kerEλ = (T − λ)K(T − λ) = K(T − λ) = kerEλ . Consequently, H = ker(T − λ) ⊕ ran(T − λ). That is, λ is a pole of order one.
Suppose that ranT k is not dense. Let T1 T2 on H = ranT k ⊕ ker(T ∗k ), T = 0 T3
Lemma 2.3 (see [21])
where ranT k is the closure of ranT k . If T is (n, k)-quasi-∗-paranormal, then T1 is n-∗-paranormal, T3k = 0 and σ(T ) = σ(T1 ) ∪ {0}. Lemma 2.4 If T is (n, k)-quasi-∗-paranormal and quasi-nilpotent, then T k+1 = 0. Proof Assume that ranT k is dense, then T is n-∗-paranormal, hence by [21], T is (n + 1)paranormal. Thus by [14, Corollary 1], T = 0. So we may suppose that ranT k is not dense. Hence by Lemma 2.3, we can write T1 T2 on H = ranT k ⊕ ker(T ∗k ), T = 0 T3
1490
ACTA MATHEMATICA SCIENTIA
Vol.36 Ser.B
where T1 is n-∗-paranormal, T3k = 0 and σ(T ) = σ(T1 ) ∪ {0}. Since σ(T1 ) is not empty and 0 T2 . An easy σ(T ) = {0}, we see that σ(T1 ) = {0}. Hence T1 = 0, and so T = 0 T3 k 0 T T 2 3 = 0. computation yields that T k+1 = 0 T3k+1
Lemma 2.5 Let T be (n, k)-quasi-∗-paranormal and λ ∈ isoσ(T ). Then λ is a pole of the resolvent of T , and the order of λ is no more than k + 1 when λ = 0, and 1 when λ 6= 0. Proof
Let λ ∈ isoσ(T ). Then H = ranEλ ⊕ kerEλ .
Write T = T1 ⊕ T2 with respect to this direct decomposition. Since σ(T1 ) = {λ} and σ(T2 ) = σ(T )\{λ}, T1 − λ is quasi-nilpotent and T2 − λ is invertible. Case I If ranT1k is dense, then T1 is n-∗-paranormal, hence by Lemma 2.2, λ is a pole of order one of the resolvent of T1 . Consequently, λ is a pole of order one of the resolvent of T . Case II If ranT1k is not dense, then by Lemma 2.3, we can write T11 T12 on H = ranT1k ⊕ ker(T1∗k ), T1 = 0 T13 k where T11 is n-∗-paranormal, T13 = 0 and σ(T1 ) = σ(T11 ) ∪ {0}. Then σ(T1 ) = {λ} implies that λ = 0. It follows from Lemma 2.4 that T1k+1 = 0, and hence λ = 0 is a pole of order no more than k + 1.
Lemma 2.6 (see [21])
Let T be (n, k)-quasi-∗-paranormal and 0 6= λ ∈ C. Then ker(T − λ) ⊆ ker(T ∗ − λ).
Proof of Theorem 2.1 (1) By Lemma 2.5, λ is an eigenvalue of T , ranEλ = ker(T − λ) and kerEλ = ran(T − λ). Since ker(T − λ) ⊆ ker(T ∗ − λ) by Lemma 2.6, ranEλ = ker(T − λ) is a reducing subspace of T . Write T = λ ⊕ T1 with respect to the orthogonal decomposition H = ker(T − λ) ⊕ ker(T − λ)⊥ , where T1 is (n, k)-quasi-∗-paranormal with ker(T1 − λ) = {0}. Since λ ∈ isoσ(T ) and σ(T ) = {λ}∪σ(T1 ), it then follows from Lemma 2.5 that T1 −λ is invertible. Consequently, ker(T −λ) = ker(T ∗ − λ). Since ker(T − λ) ⊆ ker(T ∗ − λ), we have ker(T − λ)⊥ran(T − λ), and hence ranEλ ⊥kerEλ . That is, Eλ is self-adjoint. (2) It needs only to consider the case when λ = 0. Applying the above argument and using Lemma 2.2 and the fact that ker(T ) ⊆ ker(T ∗ ), the assertion follows. References [1] Aiena P. Fredholm and Local Spectral Theory, with Application to Multipliers. Dordrecht: Kluwer Academic Publishers, 2004 [2] Arora S C, Thukral J K. On a class of operators. Kyngpook Math J, 1986, 21(41): 381–386 [3] Ch¯ o M, Han Y M. Riesz idempotent and algebraically M -hyponormal operators. Integral Equations Operator Theory, 2005, 53(3): 311–320
No.5
Q.P. Zeng & H.J. Zhong: RIESZ IDEMPOTENT OF (n, k)-QUASI-∗-PARANORMAL
1491
[4] Ch¯ o M, Tanahashi K. Isolated point of spectrum of p-hyponormal, log-hyponormal operator. Integral Equations Operator Theory, 2002, 43(4): 379–384 [5] Duggal B P, Jeon I H, Kim I H. On ∗-paranormal contractions and properties for ∗-class A operators. Linear Algebra Appl, 2012, 436(5): 954–962 [6] Han Y M, Lee J I, Wang D. Riesz idempotent and Weyl’s theorem for w-hyponormal operators. Integral Equations Operator Theory, 2005, 53(1): 51–60 [7] Jeon I H, Kim I H. On operators satisfying T ∗ |T 2 |T ≥ T ∗ |T |2 T . Linear Algebra Appl, 2006, 418(2/3): 854–862 [8] Lee M Y, Lee S H, Rhoo C S. Some remark on the structure of k∗-paranormal operators. Kyngpook Math J, 1995, 35(2): 205–211 [9] Mecheri S. Isolated points of spectrum of k-quasi-∗-class A operators. Studia Math, 2012, 208(1): 87–96 [10] Mecheri S. On quasi-∗-paranormal operators. Ann Funct Anal, 2012, 3(1): 86–91 [11] Shen J L, Zuo F, Yang C S. On operators satisfying T ∗ |T 2 |T ≥ T ∗ |T ∗ |2 T . Acta Math Sinica, Eng Ser, 2010, 26(11): 2109–2116 [12] Stampfli J G. Hyponormal operators and spectral density. Trans Amer Math Soc, 1965, 117: 469–476 [13] Tanahashi K, Jeon I H, Kim I H, Uchiyama A. Quasinilpotent part of class A or (p, k)-quasihyponormal operators//Oper Theory Adv Appl. Birkh¨ auser: Basel, 2009, 187: 199–210 [14] Tanahashi K, Uchiyama A. A note on ∗-paranormal operators and related classes of operators. Bull Korean Math Soc, 2014, 51(2): 357–371 [15] Tanahashi K, Uchiyama A. Isolated point of spectrum of p-quasihyponormal operators. Linear Algebra Appl, 2002, 341(1/3): 345–350 [16] Tanahashi K, Uchiyama A, Ch¯ o M. Isolated points of spectrum of (p, k)-quasihyponormal operators. Linear Algebra Appl, 2004, 382: 221–229 [17] Uchiyama A. On the isolated points of the spectrum of paranormal operators. Integral Equations Operator Theory, 2006, 55(1): 145–151 [18] Uchiyama A, Tanahashi K. On the Riesz idempotent of class A operators. Math Inequal Appl, 2002, 5(2): 291–298 [19] Yuan J T, Gao Z S. Weyl spectrum of class A(n) and n-paranormal operators. Integral Equations Operator Theory, 2008, 60(2): 289–298 [20] Yuan J T, Ji G X. On (n, k)-quasiparanormal operators. Studia Math, 2012, 209(3): 289–301 [21] Zeng Q P, Zhong H J. On (n, k)-quasi-∗-paranormal operators. Bull Malays Math Sci Soc, 2016, DOI: 10.1007/s40840-015-0119-z