Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular ad Browder s theorem hold for T, we show that the spectrum σ is cotiuous at T if ad oly if the essetial spectrum σ e is cotiuous at T. 1. Itroductio Let L (H ) deote the algebra of bouded liear operators o a separable ifiite dimesioal complex Hilbert space H. Let K deote the set of all compact subsets of the complex plae C. Equippig K with the Hausdorff metric, oe may cosider the spectrum σ as a fuctio σ : L (H ) K mappig operators T L (H ) ito their spectrum σ(t ). Newburgh [12] may be the first to have systematically ivestigated the cotiuity of the spectrum. He showed that the spectrum of a elemet of a Baach algebra is upper semicotiuous ad that the spectrum is cotiuous at ay elemet with totally discoected spectrum. I additio, he showed that the spectrum is cotiuous o a abelia Baach algebra ad o the class of operators satisfyig G 1 -coditio. Studies idetifyig sets C of operators for which σ becomes cotiuous whe restricted to C has bee carried out by a umber authors (see, for example, [5, 7, 8, 10]). O the other had, Coway ad Morrel [2, 3] Received October 15, 2015. Revised September 16, 2015. Accepted Sptember 18, 2015. 2010 Mathematics Subject Classificatio: 47A10, 47A53. Key words ad phrases: spectral cotiuity, biquasitriagular. This work was supported by the Icheo Natioal Uiversity Research Grat i 2013. c The Kagwo-Kyugki Mathematical Society, 2015. This is a Ope Access article distributed uder the terms of the Creative commos Attributio No-Commercial Licese (http://creativecommos.org/liceses/by -c/3.0/) which permits urestricted o-commercial use, distributio ad reproductio i ay medium, provided the origial work is properly cited.
602 I. H. Jeo ad I. H. Kim have udertake a detailed study o the cotiuity of various spectra i L (H ). Give a operator T L (H ), let α(t ) = dim ker(t ) ad β(t ) = dim ker(t ). T is upper semi-fredholm if T H is closed ad α(t ) < ad T is lower semi-fredholm if T H is closed ad β(t ) <. If T is semi-fredholm, the the idex of T, id(t ), is defied by id(t ) = α(t ) β(t ). T is said to be Fredholm if T H is closed ad the deficiecy idices α(t ) ad β(t ) are (both) fiite. Let F deote the set of Fredholm operators ad S F the set of semi-fredholm operators. Let P (T ) = {λ σ(t ) : T λ S F ad id(t λ) = } for Z {± } ad let P ± (T ) = {P (T ) : 0} ad P (T ) = {P (T ) : 1}. Recall that if C is the ideal of compact operators o H ad (1.1) π : L (H ) L (H )/C is the caoical map, the the essetial spectrum of T is defied by σ e (T ) = σ(π(t )). I the followig, we shall deote the set of accumulatio poits (resp. isolated poits) of σ(t ) by accσ(t )(resp. isoσ(t )) ad write σ 0 p(t ) for the isolated eigevalues of fiite multiplicity. Recall that the Weyl spectrum σ w (T ) ad the Browder spectrum σ b (T ) of T are the sets σ w (T ) = {λ C : T λ S F ad σ b (T ) = σ e (T ) accσ(t), respectively. Let π 0 (T ) := σ(t ) \ σ b (T ) deote the set of Riesz poits of T. We say that Browder s theorem holds for T if σ(t ) \ σ w (T ) = π 0 (T ). It is well kow [9, Theorem 2; 9] that each of the followig coditios is equivalet to Browder s theorem for T : (1.2) σ(t ) = σ w (T ) σ 0 p(t ); (1.3) σ w (T ) = σ b (T ); (1.4) accσ(t ) σ w (T ). I [6] Djordjevic ad Ha proved that the followig;
A ote o spectral cotiuity 603 Propositio 1.1. If Browder s theorem holds for T L (H ), the the followigs are equivalet: (1) σ is cotiuous at T ; (2) σ w is cotiuous at T ; (3) σ b is cotiuous at T. I the preset ote, provided T L (H ) is biquasitriagular ad Browder s theorem hold for T, we show that σ is cotiuous at T if ad oly if σ e is cotiuous at T. 2. Mai results A operator T L (H ) is called quasitriagular if there exists a sequece {F } of projectios of fiite rak that F 1 weakly ad F T F T F 0. Also, a operator T L (H ) is called biquasitriagular if T ad T are quasitriagular. It is well kow [1] that a operator is quasitriagular if ad oly if for each λ C such that T λ semi-fredholm, id(t λ) 0. It follows that a operator T is biquasitriagular if ad oly if for each λ C such that T λ semi-fredholm, id(t λ) = 0. Cosequetly, (2.1) T is biquasitriagular if ad oly if P ± (T ) =. We begi with the result of Coway ad Morrel; Lemma 2.1. [3, Corollary 4.3] If T L (H ) is biquasitriagular, the σ e is cotiuous at T if ad oly if for each λ σ e (T ) ad ɛ > 0, the ɛ-eighborhood of λ cotais a compoet of σ e. Theorem 2.2. Let T L (H ) be biquasitriagular. If σ is cotiuous at T, the σ e is cotiuous at T. Proof. Assume that the spectrum σ is cotiuous at T. Sice T is biquasitriagular, [11, Theorem 3] imply that σ(t ) = σ e (T ) σ 0 p ad σ(t ) is the closure of its isolated poits. Also, σ e (T ) is the closure of its trivial compoets. Thus Lemma 2.1 implies that σ e is cotiuous at T. Now we cosider the iverse of Theorem 2.2.
604 I. H. Jeo ad I. H. Kim Theorem 2.3. If Browder s theorem holds for T L (H ) ad σ e is cotiuous at T, the σ is cotiuous at T. Proof. To prove the theorem it would suffice to prove that if {T } L (H ) is a sequece of operators such that lim T T = 0 for some operator T L (H ), the (2.2) accσ(t ) lim if σ(t ). because it is well kow [12] that the fuctio σ is upper semi-cotiuous ad (2.3) isoσ(t ) lim if σ(t ). Asuume that λ accσ(t ). First, let λ accσ(t ) σ e (T ). Sice the fuctio σ e is cotiuous at T, the λ lim if σ e(t ) lim if σ(t ). Secod, let λ accσ(t ) \ σ e (T ). The T λ is Fredholm. Assume to the cotrary that λ / lim if σ(t ). The there exists a δ > 0, a eighbourhood N δ (λ) of λ ad a subsequece {T k } of {T } such that σ(t k ) N δ (λ) = for every k 1. Evidetly, T k λ is Fredholm, with id(t k λ) = 0, ad lim (T k λ) (T λ) = 0. It follows from the cotiuity of the idex that id(t λ) = 0, ad so T λ is Weyl. This is a cotradictio to (1.4) because Browder s theorem holds for T. Corollary 2.4. Let T L (H ) be biquasitriagular ad Browder s theorem holds for T. The the essetial spectrum σ e is cotiuous at T if ad oly if the spectrum σ is cotiuous at T. Proof. It is immediately follows from combiig Theorem 2.2 ad Theorem 2.3.
A ote o spectral cotiuity 605 Refereces [1] C. Apostol, C. Foias ad D. Voiculescu, Some results o o-quasitriagular operators. IV, Rev. Roum. Math. Pures Appl. 18 (1973), 487 514. [2] J. B. Coway ad B. B. Morrel, Operators that are poits of spectral cotiuity, Itegral equatios ad operator theory, 2 (1979), 174-198. [3] J. B. Coway ad B. B. Morrel, Operators that are poits of spectral cotiuity II, Itegral equatios ad operator theory 4 (1981), 459-503. [4] N. Duford, Spectral operators, Pacific J. Math. 4 (1954), 321 354. [5] S. V. Djordjević ad B. P. Duggal, Weyl s theorem ad cotiuity of spectra i the class of p-hypoormal operators, Studia Math. 143 (2000), 23 32. [6] S. V.Djordjević ad Y. M. Ha, Browder s theorem ad spectral cotiuity, Glasgow Math. J. 42 (2000), 479 486. [7] S. V.Djordjević, Cotiuity of the essetial spectrum i the class of quasihypoormal operators, Vesik Math. 50 (1998) 71 74. [8] B. P. Duggal, I. H. Jeo, ad I. H. Kim, Cotiuity of the spectrum o a class of upper triagular operator matrices, Jour. Math. Aal. Appl. 370 (2010), 584 587. [9] R. Harte ad W. Y. Lee, Aother ote o Weyl s theorem, Tras. Amer. Math. Soc. 349 (1997), 2115 2124. [10] I. S. Hwag ad W. Y. Lee, The spectrum is cotiuous o the set of p- hypoormal operators, Math. Z. 235 (2000), 151 157. [11] R. Lage, Biquasitriagularity ad spectral cotiuity, Glasgow Math. J. 26 (1985), 177 180. [12] J. D. Newburgh, The variatio of spectra, Duke Math. J. 18 (1951), 165 176. I Ho Jeo Departmet of Mathematics Educatio Seoul Natioal Uiversity of Educatio Seoul 137-742, Korea E-mail: jihmath@sue.ac.kr I Hyou Kim Departmet of Mathematics Icheo Natioal Uiversity Icheo 406-772, Korea E-mail: ihkim@iu.ac.kr