Weyl and Browder theorems for operators with or without SVEP at zero

The study of operators having some special spectral properties like Weyl's theorem, Browder's theorem and the SVEP has been of important interest for some time now. The SVEP is very useful in the study of the local spectral theory. In this paper, we explore the single-valued extension property (SVEP) for some operators on Hilbert spaces. We characterize operators with or without SVEP at zero and those where Weyl's and Browder's theorems hold. It is shown that if a Fredholm operator has no SVEP at zero, then zero is an accumulation point of the spectrum of the operator. It is also shown that quasi similar Fredholm operators have equal Weyl spectrum.

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Introduction

In this paper H will denote a complex separable Hilbert space and () BH will denote the Banach algebra of bounded linear operators. We denote by () GH the subalgebra of ()

where iso K is the set of isolated points of K and acc K is the set of accumulation points of . K By definition, an accumulation point of a set K is a limit point of .

K It can lie in the set K or can be a boundary point of . K Clearly every isolated point of a set K is in a boundary point of .

K

We denote by iso () T  the isolated point spectrum of T (i.e. the set of isolated eigenvalues of T of finite multiplicity). Clearly iso ( ) ( ). TT  and the point 0  is an isolated point of ( ).

 

 

TT

TT

 

From the definitions, we have that a necessary and sufficient condition for 00 ()

It is well known (see [16], Theorem 3) that if T From the definition, it is clear that () b T  is the largest part of () T  that remains unchanged under compact perturbations in the commutant of . T Clearly, every invertible operator as well as well as every Browder operator is a Weyl operator. In fact

and is non-empty if the Hilbert space H is infinite-dimensional (see [12]

 

This concept was introduced in 1997 by Harte and Lee [1]. It is known that for any ()

The investigation of operators satisfying Weyl's theorem was initiated by H. Weyl [19] in 1909 (see also [18]) who studied the spectra of all compact perturbations TK  of a Hermitian operator T on a Hilbert space H and showed that   belongs to () TK   for every compact operator K precisely when  is not an isolated point of finite multiplicity in ( ). T  Weyl [19] proved that for every Hermitian operator T on a complex Hilbert space ,

result describing the largest subset of the spectrum remaining invariant under arbitrary compact perturbation was extended by [18] to several classes of operators including p  hyponormal and M  hyponormal operators.

 

This is also equivalent to saying that every isolated point of ()

Clearly every polaroid operator is isoloid, but the converse is not true in general.

An operator T is said to be decomposable if for any open covering 12 { , } UU of the complex plane , there are two closed

An operator T is said to have the SVEP if it has the SVEP at every .   We denote by ( , ) O U H the Frechet space of H  valued analytic functions on the open subset U of , equipped with the topology of uniform convergence on compact subsets of .

() T B H 

has the SVEP at every point of the resolvent. Moreover from the identity theorem for analytic functions it is easily seen that T has the SVEP at every point of the boundary () T   of the spectrum. In particular, T has the SVEP at every isolated point of the spectrum that is iso

T has Bishop's property (  ) then T has the SVEP.

The SVEP was introduced by Dunford (see [6], [7]) and it plays an important role in local spectral theory and Fredholm theory, especially Weyl's theorem of operators and its generalizations. Note every operator has SVEP. Finch [8] has proved that if an operator is surjective but not injective, then it does not have SVEP. It has been shown in [10]

Using the SVEP of Theorem 2.2 says that the SVEP is stable under quasisimilarity. Since unitary equivalence and similarity imply quasisimilarity, we conclude also that the SVEP is invariant under unitary equivalence and similarity of operators. We need the following basic results.

Lemma 2.3:

 is a countable set and has no accumulation point except possibly 0. In this case, every non-zero number in () T

 is an eigenvalue of T and also a pole of the resolvent of . T

 

On the other hand, if  is an eigenvalue of infinite multiplicity, then ( ) . IT     By definition this means that ( ). e T

 

It is well known that quasisimilar operators need not have the same spectrum. The following result shows that if at least one of these operators has an isolated spectrum and has the SVEP, then they have the same spectrum.

  This proves that T has the SVEP at 0  . Proposition 2.6 asserts that any operator T has the SVEP on its resolvent set ( )

Remark:

The converse of Theorem 2.21 also holds for injective operators. If T has the SVEP at 0 then 0 either belongs to ( ), T iso

Ascent, Descent, Index, Isolated points of the Spectrum, Poles of the Resolvent and the SVEP It is also known (see [1] , Theorem 3.1) that if both ascent and descent of T are finite, then they are equal.

Proposition 3.1: Let ( ). T B H 

Then ( ) 0 asc T  if and only if T is injective and ( ) 0 dsc T  if and only if T is surjective.

This equivalent to saying that T is injective.

On the other hand, ( ) 0 dsc T  is equivalent to saying that 0 ( ) ( ) .

Ran T Ran T H 

This equivalent to saying that T is surjective.

Theorem 3.2: ( [1] , Theorem 3.8) for ()

The following results show the connection between isolated points and the SVEP.

T B H 

Then the following assertions are equivalent.

(i). 0 is a pole of the resolvent of T .

T has the SVEP at 0 and ( ) . dsc T  Proof: (i)  (ii): Suppose 0 is a pole of the resolvent of . T Then 0 is an isolated point in ( ), T  and so T has the SVEP at 0.

Using the definition of a pole of the resolvent, we conclude that ( ) . dsc T 

. (ii)  (i): Suppose that T has the SVEP at 0 and ( ) . (i). Weyl's theorem holds for . T (ii). T satisfies Browder's theorem and 0 00 ( ) ( ). TT   Proof: (i)  (ii): Weyl's theorem implies Browder's theorem was proved in Theorem 2.11. (ii)  (i): Suppose Browder's theorem and 0 00 ( ) ( ).

TT  

TT  

 

we have that (i)  (ii)  (iii).

Since int( ( )) , T     we have that (iii)  (iv). To prove the implication (iv)  (iii)  (ii), we use the fact that since by the Punctured neighbourhood theorem we have ()

we conclude that (ii)  (iii).

A necessary and sufficient condition for T to satisfy Browder's theorem is that T has SVEP at ( ). w T ( ) .

 

e T   Thus by Proposition 3.9 we conclude that in finite dimensional spaces, 0 00 ( ) ( ).

TT  

However, this condition is also enjoyed by some operators acting on infinite dimensional Hilbert spaces. ( , , , ) ( , , , ). 23

Clearly T is compact, normal and hence satisfies Weyl's theorem and hence Browder's theorem. For this operator, int( ( ))

p T   and so T has the SVEP. Here, () T  is at most countable.

Recall that a subset M of X is dense if .

MX 

A subset M of X is nowhere dense if int( ) . M  A set M is said to be somewhere dense if it is not nowhere dense. We note that in Example 1, the spectrum of T is discrete, and hence nowhere dense in . Remark: We note that Theorem 3.11 gives a necessary but not sufficient condition when an operator T has SVEP. There exist operator such that T such that () T  is "somewhere" dense and have the SVEP. The unilateral shift operator on

T

T



From this characterization, we see that every Browder has the SVEP at 0.

Spectral Relations and the SVEP

We now give some relationship between some special of ( ). T (see [1]).

Acknowledgement

AcknowledgementsThe author wishes to pay tribute to the Eastern Africa University Mathematics Programme (EAUMP) for funding this research.

1. Aiena P. Fredholm and local spectral theory with applications to multipliers, Kluwer Acad. Publishers, 2004.
2. Aiena P. Browder's theorems and the spectral mapping theorem, Divulgaciones Mathematics. 2007; 15(2):207226.
3. Aiena P, Rosas E. Single-valued extension property at the points of the approximate point spectrum, J Math. Anal. 2003; 279:180-188.
4. Duggal B.P., Kim H. Generalized Browder, Weyl spectra and Polaroid property under compact perturbation, J Korean Math. Soc. 2017; 54(1):281-302.
5. Duggal BP, Kubrusly CS. Quasisimilar k-paranormal operators, Operators and Matrices. 2011; 5:417-423.
6. N. Dunford, Spectral theory I, Resolution of the Identity, Pacific J. Math. 1952; 2:559-614.
7. Dunford Spectral operators, Pacific J Math. 1954; 4:321354.
8. Finch JK. The single valued extension property on a Banach space, Pacific J Math. 1975; 58:61-69.
9. Halmos P.R. A Hilbert space problem book, SpringerVerlag, New York, 1982.
10. Han YM, Son JH. On quasi-m-hyponormal operators, Filomat. 2011; 25(1):37-52.
11. Harte R. Woo Young Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 1997; 349:2115-2124.
12. Kubrusly C.S. Fredholm theory in Hilbert space-A concise introductory exposition, Bulletin of the Belgian Mathematical Society-Simon Stevin. 2008; 15:153-177.
13. Kubrusly C.S. Spectral theory of operators on Hilbert spaces, Birkhauser, New York, 2010.
14. Laursen K.B, Neumann MM. An introduction to local spectral theory, Clarendon Press, Oxford, 2000.
15. Nzimbi B.M, Pokhariyal G.P., Moindi S. K., A note on metric equivalence of some operators, Far East J of Math. Sci.(FJMS). 2013; 75(2):301-318.
16. Schmoeger C. On isolated points of the spectrum of a bounded linear operator, Proc. Amer. Math. Soc. 1993; 117:715-719.
17. Tajmouati A, Karmouni M, Abkari M. Pseudo semi BFredholm and generalized Drazin invertible operators through localized SVEP, Italian Journal of Pure and Applied Mathematics. 2017; 37:301-314.
18. Warner K. K. A note on a theorem of Weyl, Proc. Amer. Math. Soc. 1969; 23:469-471.
19. Weyl H. Uber beschrankte quadratische Formen, deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo. 1909; 27:373-392.
20. Jong-Kwang Yoo, Equality of spectra of quasisimilar operators with Property (C), Far East J Math Sci (FJMS). 2017; 101(4):773-783.
21. Jong-Kwang Yoo, Hyuk Han. Spectra of asymptotically quasisimilar subdecomposable operators, J of the Chungcheong Math. Soc. 2009; 22(2):271{279.