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Research | Open | Published:

Weyl-type theorems and k-quasi-M-hyponormal operators

Abstract

In this paper, we show that if E is the Riesz idempotent for a non-zeroisolated point λ of the spectrum of ak-quasi-M-hyponormal operator T, then E isself-adjoint, and R(E)=N(Tλ)=N ( T λ ) . Also, we obtain that Weyl-type theorems hold foralgebraically k-quasi-M-hyponormal operators.

MSC: 47B20, 47A10.

1 Introduction

Let T be a bounded linear operator on a complex Hilbert space H,write it for TB(H), take a complex number λ in , and,henceforth, shorten TλI to Tλ. One of recent trends in operator theory is studying naturalextensions of normal operators. We introduce some of these operators as follows.

T is said to be a hyponormal operator if T TT T ;

T is M-hyponormal [1] if there exists a real positive number M such that

M 2 ( T λ ) (Tλ)(Tλ) ( T λ ) for all λC;

T is quasi-M-hyponormal [2] if there exits a real positive number M such that

T ( M 2 ( T λ ) ( T λ ) ) T T (Tλ) ( T λ ) Tfor all λC;

T is k-quasi-M-hyponormal [3] if there exists a real positive number M such that

T k ( M 2 ( T λ ) ( T λ ) ) T k T k (Tλ) ( T λ ) T k for all λC,

where k is a natural number.

It is clear that hyponormalM-hyponormalk-quasi-M-hyponormal.

We give the following example to indicate that there exists an M-hyponormaloperator, which is not hyponormal.

Example 1.1 Consider the unilateral weighted shift operator as aninfinite-dimensional Hilbert space operator. Recall that given a bounded sequence ofpositive numbers α: α 1 , α 2 , α 3 , (called weights), the unilateral weighted shift W α associated with α is the operator on H= l 2 defined by W α e n := α n e n + 1 for all n1, where { e n } n = 1 is the canonical orthogonal basis for l 2 . It is well known that W α is hyponormal if and only if α is monotonicallyincreasing. Also, W α is M-hyponormal if and only if α iseventually increasing. Hence, if we take the weights α such that α 1 =2, α 2 =1, α 3 =2, α 3 = α 4 = , then W α is an M-hyponormal operator, but it is nothyponormal.

Next, we give a 2-quasi-M-hyponormal operator, which is notM-hyponormal.

Example 1.2 Let T= ( 0 1 0 0 ) defined on C 2 . Then by simple calculations, we see that T is a2-quasi-M-hyponormal operator, but is not M-hyponormal.

If TB(H), we shall write N(T) and R(T) for the null space and the range space of T. Also, let α(T):=dimN(T), β(T):=dimN( T ), σ(T) and isoσ(T) for the spectrum and the isolated points of the spectrum ofT, respectively.

Let λisoσ(T). The Riesz idempotent E of T with respect toλ is defined by E= 1 2 π i D ( μ T ) 1 dμ, where D is a closed disk, centered at λ,which contains no other points of σ(T). It is well known that the Riesz idempotent satisfies E 2 =E, ET=TE, σ(T | R ( E ) )={λ}, and N(Tλ)R(E). Stampfli [4] showed that if T satisfies the growth condition G 1 , then E is self-adjoint and R(E)=N(Tλ). Recently, Chō and Tanahashi [5] obtained an improvement of Stampfli’s result top-hyponormal operators or log-hyponormal operators. Furthermore, Chōand Han extended it to M-hyponormal operators as follows.

Proposition 1.3 [[6], Theorem 4]

Let T be an M-hyponormal operator, and let λ be an isolated point ofσ(T). If E is the Riesz idempotent for λ, then E is self-adjoint, andR(E)=N(Tλ)=N ( T λ ) .

2 Isolated point of spectrum of k-quasi-M-hyponormaloperators

Lemma 2.1 Let T be a k-quasi-M-hyponormal operator. If0λC, and assume thatσ(T)={λ}, thenT=λI.

Proof If λ0 and σ(T)={λ}, then T is invertible, so T is anM-hyponormal operator, and hence, T=λI by [6]. □

Lemma 2.2 Let T be a k-quasi-M-hyponormal operator and0λC. ThenTx=λximplies that T x= λ ¯ x.

Proof Suppose that Tx=λx. Since T is a k-quasi-M-hyponormaloperator, M(Tα) T k y ( T α ) T k y for all vectors yH and αC. In particular, M(Tλ) T k x ( T λ ) T k x. Since Tx=λx, 0=M | λ | k (Tλ)x=M(Tλ) T k x ( T λ ) T k x= | λ | k ( T λ ) x. |λ|0, therefore ( T λ ) x=0. □

Theorem 2.3 Let T be a k-quasi-M-hyponormal operator, and let λ be a non-zero isolated point ofσ(T). Then the Riesz idempotent E for λ is self-adjoint, and

R(E)=N(Tλ)=N ( T λ ) .

Proof We can derive the result from Lemma 2.2, [[3], Theorem 2.5] and [[7], Lemma 5.2]. □

3 Weyl-type theorems of algebraically k-quasi-M-hyponormaloperators

We say that T is an algebraically k-quasi-M-hyponormaloperator if there exists a nonconstant complex polynomial p such that p(T) is a k-quasi-M-hyponormal operator. From thedefinition above, T is an algebraicallyk-quasi-M-hyponormal operator, then so is Tλ for each λC.

An operator T is called Fredholm if R(T) is closed, and both N(T) and N( T ) are finite-dimensional. The index of a Fredholm operatorT is given by i(T)=α(T)β(T). An operator T is called Weyl if it is Fredholm of indexzero. The Weyl spectrum of T[8] is defined by w(T):={λC:Tλ is not Weyl}. Following [9], we say that Weyl’s theorem holds for T if σ(T)w(T)= π 00 (T), where π 00 (T):={λisoσ(T):0<α(Tλ)<}.

More generally, Berkani investigated the B-Fredholm theory (see [1012]). We define TSB F + (H) if there exists a positive integer n such that R( T n ) is closed, T [ n ] :R( T n )xTxR( T n ) is upper semi-Fredholm (i.e., R( T [ n ] )=R( T n + 1 ) is closed, dimN( T [ n ] )=dimN(T)R( T n )<) and i( T [ n ] )0[12]. We define σ S B F + (T)={λC:TλSB F + (H)}. Let E a (T) denote the set of all isolated points λ of σ a (T) with 0<α(Tλ). We say that generalized a-Weyl’s theorem holds forT if σ a (T) σ S B F + (T)= E a (T).

We know that Weyl’s theorem holds for hermitian operators [13], which have been extended to hyponormal operators [14], algebraically hyponormal operators by [15], algebraically M-hyponormal operators [6] and algebraically quasi-M-hyponormal operators [2], respectively. In this section, we obtain that generalizeda-Weyl’s theorems hold for algebraicallyk-quasi-M-hyponormal operators.

Lemma 3.1[3]

LetTB(H)be a k-quasi-M-hyponormal operator, let the rangeof T k be not dense and

T=( T 1 T 2 0 T 3 )onH= R ( T k ) ¯ N ( T k ) .

Then T 1 is M-hyponormal, T 3 k =0andσ(T)=σ( T 1 ){0}.

Theorem 3.2 Let T be a quasinilpotent algebraically k-quasi-M-hyponormal operator. Then T is nilpotent.

Proof We first assume that T is ak-quasi-M-hyponormal operator. Consider two cases, Case I: If therange of T k has dense range, then it is an M-hyponormal operator.Hence, by [[6], Lemma 8], T is nilpotent. Case II: If T does nothave dense range, then by Lemma 3.1, we can represent T as the uppertriangular matrix

T=( T 1 T 2 0 T 3 )onH= R ( T k ) ¯ N ( T k ) ,

where T 1 :=T| R ( T k ) ¯ is an M-hyponormal operator. Since T isquasinilpotent, σ(T)={0}. But σ(T)=σ( T 1 ){0}, hence, σ( T 1 )={0}. Since T 1 is an M-hyponormal operator, T 1 =0. Since T 3 k =0, simple computation shows that

T k + 1 =( 0 T 2 T 3 k 0 T 3 k + 1 )=0.

Now, suppose that T is an algebraicallyk-quasi-M-hyponormal operator. Then there exists a nonconstantpolynomial p such that p(T) is a k-quasi-M-hyponormal operator. If p(T) has dense range, then p(T) is an M-hyponormal operator. Thus T is analgebraically M-hyponormal operator. It follows from [[6], Lemma 8] that it is nilpotent. If ( p ( T ) ) k does not have a dense range, then by Lemma 3.1, we canrepresent p(T) as the upper triangular matrix

p(T)=( A B 0 C )onH= R ( ( p ( T ) ) k ) ¯ N ( ( p ( T ) ) k ) ,

where A:=p(T)| R ( ( p ( T ) ) k ) ¯ is an M-hyponormal operator. Since σ(T)={0} and σ(p(T))=p(σ(T))={p(0)}, the operator p(T)p(0) is quasinilpotent. But σ(p(T))=σ(A){0}, thus σ(A){0}={p(0)}. So p(0)=0, and hence, p(T) is quasinilpotent. Since p(T) is a k-quasi-M-hyponormal operator, by theprevious argument p(T) is nilpotent. On the other hand, since p(0)=0, p(z)=c z m (z λ 1 )(z λ 2 )(z λ n ) for some natural number m. p(T)=c T m (T λ 1 )(T λ 2 )(T λ n ). p(T) is nilpotent, therefore, T isnilpotent. □

Recall that an operator T is said to be isoloid if every isolated point of σ(T) is an eigenvalue of T and polaroid if every isolatedpoint of σ(T) is a pole of the resolvent of T. In general, ifT is polaroid, then it is isoloid. However, the converse is not true.In [6], it is showed that every algebraically M-hyponormal operator isisoloid, we can prove more.

Theorem 3.3 Let T be an algebraically k-quasi-M-hyponormal operator. Then T is polaroid.

Proof Suppose that T is an algebraicallyk-quasi-M-hyponormal operator. Then p(T) is a k-quasi-M-hyponormal operator for somenonconstant polynomial p. Let λisoσ(T) and E λ be the Riesz idempotent associated to λ defined by E λ := 1 2 π i D ( μ T ) 1 dμ, where D is a closed disk of center λ,which contains no other point of σ(T). We can represent T as the direct sum in the followingform:

T=( T 1 0 0 T 2 ),

where σ( T 1 )={λ} and σ( T 2 )=σ(T){λ}. Since T 1 is an algebraically k-quasi-M-hyponormaloperator, so is T 1 λ. But σ( T 1 λ)={0}, it follows from Theorem 3.2 that T 1 λ is nilpotent, thus T 1 λ has finite ascent and descent. On the other hand, since T 2 λ is invertible, clearly, it has finite ascent and descent. Tλ has finite ascent and descent, and hence, λ is apole of the resolvent of T, therefore, T ispolaroid. □

Corollary 3.4 Let T be an algebraically k-quasi-M-hyponormal operator. Then T is isoloid.

We say that T has the single valued extension property (abbreviated SVEP)if, for every open set U of , the only analytic solution f: UH of the equation

(Tλ)f(λ)=0for all λU

is a zero function on U.

Theorem 3.5 Let T be an algebraically k-quasi-M-hyponormal operator. Then T has SVEP.

Proof Suppose that T is an algebraicallyk-quasi-M-hyponormal operator. Then p(T) is a k-quasi-M-hyponormal operator for somenonconstant complex polynomial p, and hence, p(T) has SVEP by [[3], Theorem 2.1]. Therefore, T has SVEP by [[16], Theorem 3.3.9]. □

In the following theorem, H(σ(T)) denotes the space of functions analytic in an open neighborhood of σ(T).

Theorem 3.6 Let T or T be an algebraically k-quasi-M-hyponormal operator. ThenWeyl’s theorem holds forf(T)for everyfH(σ(T)).

Proof Firstly, suppose that T is an algebraicallyk-quasi-M-hyponormal operator. We first show that Weyl’stheorem holds for T. Using the fact [[17], Theorem 2.2] that if T is polaroid, then Weyl’stheorem holds for T if and only if T has SVEP at points of λσ(T)w(T). We have that T is polaroid by Theorem 3.3, andT has SVEP by Theorem 3.5. Hence, T satisfiesWeyl’s theorem.

Next, suppose that T is an algebraically k-quasi-M-hyponormaloperator. Now we show that Weyl’s theorem holds for T. We use thefact [[18], Theorem 3.1] that if T or T has SVEP, then Weyl’s theorem holds for T if andonly if π 00 (T)= p 00 (T). Since T has SVEP, it is sufficient to show that π 00 (T)= p 00 (T). p 00 (T) π 00 (T) is clear, so we only need to prove π 00 (T) p 00 (T). Let λ π 00 (T). Then λ is an isolated point of σ(T). Hence, λ is a pole of the resolvent of T,since T is polaroid by Theorem 3.3, that is, p(λT)=q(λT)<. By assumption, we have α(λT)<, so β(λT)<. Hence, we conclude that λ p 00 (T). Therefore, Weyl’s theorem holds for T.

Finally, we can derive the result by Theorem 3.5 and [[17], Theorem 2.4]. □

Following [[19], Theorem 3.12], we obtain the following result.

Theorem 3.7 Let f be an analytic function onσ(T), and f is not constant on each connected component of the open set U containingσ(T).

  1. (i)

    If T is an algebraically k-quasi-M-hyponormal operator, then f(T) satisfies a generalized a-Weyl’s theorem.

  2. (ii)

    If T is an algebraically k-quasi-M-hyponormal operator, then f( T ) satisfies a generalized a-Weyl’s theorem.

References

  1. 1.

    Wadhwa BL: M -hyponormal operators. Duke Math. J. 1974, 41: 655–660. 10.1215/S0012-7094-74-04168-4

  2. 2.

    Han YM, Son JH: On quasi- M -hyponormal operators. Filomat 2011, 25: 37–52.

  3. 3.

    Mecheri S: On k -quasi- M -hyponormal operators. Math. Inequal. Appl. 2013, 16: 895–902.

  4. 4.

    Stampfli J: Hyponormal operators and spectral density. Trans. Am. Math. Soc. 1965, 117: 469–476.

  5. 5.

    Chō M, Tanahashi K: Isolated point spectrum of p -hyponormal, log-hyponormaloperators. Integral Equ. Oper. Theory 2002, 43: 379–384. 10.1007/BF01212700

  6. 6.

    Chō M, Han YM: Riesz idempotent and algebraically M -hyponormal operators. Integral Equ. Oper. Theory 2005, 53: 311–320. 10.1007/s00020-004-1314-8

  7. 7.

    Yuan JT, Ji GX:On (n,k)-quasiparanormal operators. Stud. Math. 2012, 209: 289–301. 10.4064/sm209-3-6

  8. 8.

    Harte RE: Invertibility and Singularity for Bounded Linear Operators. Dekker, New York; 1988.

  9. 9.

    Harte RE, Lee WY: Another note on Weyl’s theorem. Trans. Am. Math. Soc. 1997, 349: 2115–2124. 10.1090/S0002-9947-97-01881-3

  10. 10.

    Berkani M, Arroud A: Generalized Weyl’s theorem and hyponormal operators. J. Aust. Math. Soc. 2004, 76: 291–302. 10.1017/S144678870000896X

  11. 11.

    Berkani M, Koliha JJ: Weyl type theorems for bounded linear operators. Acta Sci. Math. 2003, 69: 359–376.

  12. 12.

    Berkani M, Sarih M: On semi B -Fredholm operators. Glasg. Math. J. 2001, 43: 457–465.

  13. 13.

    Weyl H: Über beschränkte quadratische Formen, deren Dierenz vollsteigist. Rend. Circ. Mat. Palermo 1909, 27: 373–392. 10.1007/BF03019655

  14. 14.

    Coburn LA: Weyl’s theorem for nonnormal operators. Mich. Math. J. 1966, 13: 285–288.

  15. 15.

    Han YM, Lee WY: Weyl’s theorem holds for algebraically hyponormal operators. Proc. Am. Math. Soc. 2000, 128: 2291–2296. 10.1090/S0002-9939-00-05741-5

  16. 16.

    Laursen KB, Neumann MM: Introduction to Local Spectral Theory. Clarendon, Oxford; 2000.

  17. 17.

    Duggal BP: Polaroid operators satisfying Weyl’s theorem. Linear Algebra Appl. 2006, 414: 271–277. 10.1016/j.laa.2005.10.002

  18. 18.

    Aiena P: Classes of operators satisfying a-Weyl’s theorem. Stud. Math. 2005, 169: 105–122. 10.4064/sm169-2-1

  19. 19.

    Aiena P, Aponte E, Balzan E: Weyl type theorems for left and right polaroid operators. Integral Equ. Oper. Theory 2010, 66: 1–20. 10.1007/s00020-009-1738-2

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Acknowledgements

We wish to thank the referee for careful reading and valuable comments for theorigin draft. This work is supported by the Basic Science and TechnologicalFrontier Project of Henan Province (No. 132300410261). This work is partiallysupported by the National Natural Science Foundation of China (No. 11271112,11201126).

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Correspondence to Hongliang Zuo.

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The authors declare that they have no competing interests.

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Both authors have given equal contributions in this paper.

Fei Zuo and Hongliang Zuo contributed equally to this work.

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Keywords

  • k-quasi-M-hyponormal operators
  • Riesz idempotent operators
  • Weyl-type theorems