Weyl s Theorem and Property (Saw)

Transcription

1 International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, HIKARI Ltd, Weyl s Theorem and Property (Saw) N. Jayanthi Government Arts College (Autonomous) Coimbatore, Tamilnadu, India Copyright c 2018 N. Jayanthi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In [8], Sanabria et al. have shown that Weyl s theorem holds for operators satisfying property(saw). In this paper, it is shown that its converse is not true with examples and the required condition is derived. Mathematics Subject Classification: 47A10, 47A53 Keywords: Weyl s theorem and property(saw) 1 Introduction and Preliminaries Let B(H) be the Banach algebra of all bounded linear operators on a non-zero complex Hilbert space H. By an operator T, we mean an element in B(H). If T lies in B(H), then T denotes the adjoint of T in B(H). The ascent of T denoted by p(t ), is the least non-negative integer n such that ker T n = ker T n1. The descent of T denoted by q(t ), is the least non-negative integer n such that ran(t n ) = ran(t n1 ). T is said to be of finite ascent if p(t λ) <, for all λ C. If p(t) and q(t) are both finite, then p(t)=q(t)([5], Proposition 38.3). Moreover, 0 < p(λi T ) = q(λi T ) < precisely when λ is a pole of the resolvent of T. An operator T is called a Fredholm operator if the range of T denoted by ran(t ) is closed and both ker T and ker T are finite dimensional and is denoted by T Φ(H). An operator T is called upper semi-fredholm operator, T Φ (H), if ran(t ) is closed and ker T is finite dimensional. An operator T is called lower semi-fredholm operator, T Φ (H), if ker T is finite

2 434 N. Jayanthi dimensional. The index of a semi-fredholm operator is an integer defined as ind(t ) = dim ker T dim ker T. An upper semi-fredholm operator, with index less than or equal to 0 is called upper semi-weyl operator and is denoted by T Φ (H). A lower semi-fredholm operator with index greater than or equal to 0 is called lower semi-weyl operator and is denoted by T Φ (H). A Fredholm operator of index 0 is called Weyl operator. The spectrum of T is denoted by σ(t ), where σ(t ) = {λ C : T λi is not invertible}. The set of all isolated eigenvalues of finite multiplicity of T is denoted by π 00 (T ). The set of all isolated eigenvalues of finite multiplicity of T in σ a (T ) is denoted by π a 00(T ) The Weyl spectrum of T is defined as σ w (T ) = {λ C : T λi is not Weyl}. The upper semi-weyl spectrum of T is defined as σ SF (T ) = {λ C : T λi is not upper semi-weyl}. For an operator T and a non-negative integer n, define T [n] to be the restriction of T to R(T n ) viewed as a map from R(T n ) into R(T n ). In particular, T [0] = T. If for some integer n, R(T n ) is closed and T [n] is an upper(resp. a lower) semi-fredholm operator, then T is called an upper(resp. lower) semi- B-Fredhom operator. Moreover if T [n] is a Fredholm operator, then T is called a B-Fredholm operator. A semi-b-fredholm operator is an upper or a lower semi-b-fredholm operator. The index of a semi-b-fredholm operator T is the index of semi-fredholm operator T [d], where d is the degree of the stable iteration of T and defined as d = inf{n N; for all m N, m n (R(T n ) N(T )) (R(T m ) N(T ))}. T is called a B-Weyl operator if it is B-Fredholm of index 0. The B-Weyl spectrum of T is defined by σ BW (T ) = {λ C : T λi is not a B-Weyl operator}. The upper semi-b-weyl spectrum of T is defined by σ SBF (T ) = {λ C : T λi is not a upper semi- B-Weyl operator}.

3 Weyl s theorem and property (Saw) 435 Weyl s theorem holds for T [4] if T satisfies the equality σ(t ) σ w (T ) = π 00 (T ). a-weyl s theorem holds for T [7], if T satisfies the equality T satisfies property(w) if T satisfies property(saw) if σ a (T ) σ SF (T ) = πa 00(T ) σ a (T ) σ SF (T ) = π 00(T ) σ(t ) σ SBF (T ) = πa 00(T ) An operator T is said to have the single valued extension property(svep) at λ 0 C, if for every open neighborhood U of λ 0, the only analytic function f : U X which satisfies the equation (λi T )f(λ) = 0 for all λ U is the function f 0. An operator T is said to have SVEP, if T has SVEP at every point λ C. Aiena has proved the following result for operators whose adjoints have SVEP. Theorem 1.1 ([2], Lemma 2.15) If T L(X) and T has SVEP, then σ SF (T ) = σ w(t ) and σ(t ) = σ a (T ). 2 Property(Saw) and Weyl s Theorem The relationships between Weyl s and other Weyl type theorems are: Property(w) Weyl s theorem, a-weyl s theorem Weyl s theorem and Generalized Weyl s theorem Weyl s theorem. In this sequence, Sanabria[8] has shown that property(saw) Weyl s theorem The following example clearly shows that its converse is not true. Example 2.1 For the unilateral right shift operator T l 2 (N) defined as T (x 1, x 2, x 3,...) = (0, x 1, x 2,...), σ(t ) = D(0, 1), σ w (T ) = D(0, 1), σ SBF (T ) = Γ, π 00 (T ) = and π a 00(T ) = Hence weyl s theorem holds for T, but T does not satisfy property (Saw).

4 436 N. Jayanthi Example 2.2 For the projection operator T l 2 (N) defined as T (x 1, x 2, x 3,...) = (0, x 2, x 3,...) σ(t ) = {0, 1}, σ w (T ) = {1}, σ SBF (T ) =, π 00(T ) = {0} and π a 00(T ) = {0}. Hence weyl s theorem holds for T, but T does not satisfy property (Saw). The following theorem gives the sufficient condition for the converse to be true. Theorem 2.3 Let T L(X). If T satisfies weyl s theorem and σ w (T ) = (T ), then T satisfies property (Saw) σ SBF Proof: Assume that T satisfies weyl s theorem and σ w (T ) = σ SBF (T ). If λ σ(t ) σ SBF (T ), then λ σ(t ) σ w(t ) = π 00 (T ) π00(t a ). Hence σ(t ) σ SBF (T ) πa 00(T ). Conversely, let λ π00(t a ). Then λ iso σ a (T ) with 0 < α(λi T ) <, λ iso σ(t ) with 0 < α(λi T ) <. Hence λ π 00 (T ) = σ(t ) σ w (T ) = σ(t ) σ SBF (T ). Therefore π00(t a ) σ(t ) σ SBF (T ). Hence T satisfies property (Saw). we get the following results from Theorems 1.1 and 2.3 Corollary 2.4 If T L(X) and T has SVEP, then T satisfies property(saw) if and only if Weyl s theorem holds for T. Example 2.5 Let Q be defined for each x = (ξ i ) l 1 by Q(ξ 1, ξ 2, ξ 3,...ξ k...) = (0, α 1 ξ 1, α 2 ξ 2,...α k1 ξ k1...) where (α i ) is a sequence of complex numbers such that 0 < α i 1 and Σ i=1α i <.It follows that R(Q n ) R(Q n ), n = 1, 2,... Define the operator T on X = l 1 l 1 by T = Q 0. Then N(T ) = {0} l 1, R(T n ) = R(Q n ) {0} is not closed for any n N. Therefore σ(t ) = {0}, σ w (T ) = {0}, σ SBF (T ) = {0} = σ SF (T ) = {0}, σ a(t ) = {0}, π 00 (T ) =, π a 00(T ) =. Hence both weyl s theorem and property (Saw) hold for T. References [1] P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Speinger Kluwer Academic Publishers, Dordrecht, [2] P. Aiena and T. Biondi, Weyl Type theorems for Polaroid operators, Extracta Mathematicae, 23 (2008), no. 2,

5 Weyl s theorem and property (Saw) 437 [3] M. Berkani and J. Koliha, Weyl type therems for bounded linear operators, Acta Sci. Math. (Szeged), 69 (2003), [4] L.A. Coburn, Weyl s theorem for nonnormal operators, Michigan Math. J., 13 (1966), [5] H. Heuser, Functional Analysis, Marcel Dekker, New York, [6] V. Rakocevic, On a class of operators, Mat. Vesnik, 37 (1985), [7] V. Rakocevic, Operators obeying a-weyl s theorem, Rev. Roumaine Math. Pures Appl., 34 (1989), no. 10, [8] J. Sanabria, C. Carpintero, E. Rosas and O. Garcia, On Property(Saw) and other spectral properties type Weyl-Browder theorems, Revista Colombiana de Mathematicas, 51 (2017), Received: August 11, 2018; Published: September 5, 2018