Weyl Spectrum Research Articles

Weyl's theorem for the square of operator and perturbations

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

Weyl type theorems of [formula omitted] upper triangular operator matrices

Let MC=(AC0B) be an upper triangular operator matrix on the Hilbert space H⊕H, where H is a Hilbert space. In this paper, necessary and sufficient conditions for σ⁎(MC)=σ⁎(A)∪σ⁎(B) are obtained, where σ⁎ is the essential spectrum, the Weyl spectrum, the Browder spectrum, the essential approximate point spectrum and the Browder essential approximate point spectrum. For MC, we further investigate the relationship among Weyl type theorems such as Weyl's, a-Weyl's, Browder's and a-Browder's theorems.

Perturbation of Browder spectrum of upper triangular operator matrices

Let be a 2-by-2 upper triangular operator matrix acting on the Banach space or Hilbert space . For the most important spectra such as spectrum, essential spectrum and Weyl spectrum, the characterizations of the perturbations of on Banach space have been presented. However, the characterization for the Browder spectrum on the Banach space is still unknown. The goal of this paper is to present some necessary and sufficient conditions for to be Browder for some using an alternative approach based on matrix representation of operators and the ghost index theorem. Moreover, in the Hilbert space case, the characterizations of the Fredholm and invertible perturbations of Browder spectrum are also given.

NEW BROWDER AND WEYL TYPE THEOREMS

In this paper we introduce and study the new properties (W �), (UWa), (UW E) and (UW �). The main goal of this paper is to study relationship between these new properties and other Weyl type theorems. Moreover, we reconsider several earlier results obtained respec- tively in (11), (18), (14), (1) and (13) for which we give stronger versions. by σa(T) the approximate point spectrum of T. If the range R(T) of T is closed and α(T) < ∞ (resp. β(T) < ∞), then T is called an upper semi-Fredholm (resp. a lower semi-Fredholm) operator. If T ∈ L(X) is either upper or lower semi Fredholm, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T) = α(T) − β(T). If both of α(T) and β(T) are finite, then T is called a Fredholm operator. An operator T ∈ L(X) is called a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum σW(T) of T is defined by σW(T) = {λ ∈ C | T − λI is not a Weyl operator}. For a bounded linear operator T and a nonnegative 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 the range space R(T n ) is closed and T(n) is an upper (resp. a lower) semi-Fredholm operator, then T is called an upper (resp. a lower) semi-B-Fredholm operator. A semi-B-Fredholm operator T is an upper or a lower semi-B-Fredholm operator, and in this case the index of T is defined as the index of the semi-Fredholm operator T(n), see (12). Moreover if T(n) is a Fredholm operator, then T is called a B-Fredholm operator, see

THE SPECTRAL CONTINUITY OF ESSENTIALLY HYPONORMAL OPERATORS

If A is a unital Banach algebra, then the spectrum can be viewed as a function � : A ! S, mapping each T 2 A to its spectrum �(T), where S is the set, equipped with the Hausdorff metric, of all compact subsets of C. This paper is concerned with the continuity of the spectrumvia Browder's theorem. It is shown thatis continuous when � is restricted to the set of essentially hyponormal operators for which Browder's theorem holds, that is, the Weyl spectrum and the Browder spectrum coincide. Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of bounded linear operators acting on H. If T ∈ B(H) write σ(T) and σp(T) for the spectrum and the set of eigenvalues of T, respectively. Let S denote the set, equipped with the Hausdorff metric, of all compact subsets of C. If A is a unital Banach algebra, then the spectrum can be viewed as a function σ : A → S, mapping each T ∈ A to its spectrum σ(T). It is known that the function σ is upper semicontinuous and that in noncommutative algebras, σ does have points of discontinuity. J. Newburgh (17) gave the fundamental results on spectral continuity in general Banach algebras. J. Conway and B. Morrel (7) have undertaken a detailed study of spectral continuity in the case where the Banach algebra is B(H). It seems to be interesting and challenging to identify classes C of operators for which σ becomes continuous when restricted to C. The first result of this study is: σ is continuous on the set of normal operators. On the other hand, Newburgh's argument uses the fact that the inverses of normal resolvents are normaloid (cf. see Solution 105 of (10)) and this argument is extended to the set of hyponormal operators because the inverses of hyponormal resolvents are also hyponormal and hence normaloid.

Generalized Weyl Spectrum of Upper Triangular Operator Matrices

Let \({M_{C}}= \left (\begin{array}{ll} A \quad C \\ 0 \quad B \end{array}\right)\) be a 2 × 2 upper triangular operator matrix on Hilbert space \({\mathcal{H} \oplus\mathcal{K}}\) . For given operators \({A\in\mathcal{B}(\mathcal{H})}\) and \({B\in\mathcal{B}(\mathcal{K})}\) , sets \({\bigcap_{C\in\mathcal{B}(\mathcal{K}, \mathcal{H})} \sigma^{g}_{w}(M_{C})}\) and \({\bigcup_{C\in\mathcal{B}(\mathcal{K},\mathcal{H})} \sigma^{g}_{w}(M_{C})}\) are investigated, where \({\sigma^{g}_{w}(\cdot)}\) denotes the generalized Weyl spectrum.

Weyl’s theorem for algebraically quasi-paranormal operators

Let T or T? be an algebraically quasi-paranormal operator acting on Hilbert space. We prove : (i) Weyl?s theorem holds for f (T) for every f ? H(?(T)); (ii) a-Browder?s theorem holds for f (S) for every S ? T and f ? H(?(S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.

Subscalarity of operators for which T *T and T + T * commute

In this paper, we study operators for which T ∗ T and T + T ∗ commute. Let Θ denote the class of such operators in L(H). We show that every operator in Θ is subscalar of order 4. From this result, we give partial solutions to the invariant subspace problem. In addition, we examine some extensions of operators in Θ. Finally, we prove that if T belongs to Θ, then Weyl’s theorem holds for T and the spectral mapping theorem corresponding to the Weyl spectrum is satisfied for T . Let H be a complex separable Hilbert space and let L(H) denote the algebra of all bounded linear operators on H .I fT ∈L (H), then we write σ(T ), σp(T ), σap(T ), and σe(T ) for the spectrum, the point spectrum, the approximate point spectrum, and the essential spectrum of T , respectively. The notation r(T ) is used for the spectral radius of T . An operator S ∈L (H) is called scalar of order m (0 m ∞) if it possesses a spectral distribution of order m, that is, if there exists a continuous unital homomorphism of topological algebras Φ: C m 0 (C) −→ L (H)

Property $(aw)$ and perturbations

A bounded linear operator $T\in\mathbf{L}(\mathbb{X})$ acting on a Banach space satisfies property $(aw)$, a variant of Weyl's theorem, if the complement in the spectrum $\sigma(T)$ of the Weyl spectrum $\sigma_w(T)$ is the set of all isolated points of the approximate-point spectrum which are eigenvalues of finite multiplicity. In this article we consider the preservation of property $(aw)$ under a finite rank perturbation commuting with $T$, whenever $T$ is polaroid, or $T$ has analytical core $K(T-\lambda_0 I)=\{0\}$ for some $\lambda_0\in \mathbb{C}$. The preservation of property $(aw)$ is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations or under Riesz perturbations. The theory is exemplified in the case of some special classes of operators.

Spectrum of class absolute -*-k-paranormal operators for 0 ≤ k ≤ 1

In this paper, we shall introduce a new class absolute-*-k-paranormal operators given by a norm inequality and *-A(k) operator by operator inequality, we will discuss the inclusion relation of them. And we study spectral properties of class absolute-*-k-paranormal operators. We show that if T belongs to class absolute-*-k-paranormal operators, then its point spectrum and joint point spectrum are identical, its approximate point spectrum and joint approximate point spectrum are identical. Next as an application of them, for Weyl spectrum w(?) and essential approximate point spectrum ?ea, (?), we will show that if T or T*is absolute-*-k-paranormal for 0 ? k ? 1, then w(f (T)) = ? (w(T)), ?ea(? (T)) = ?(?ea(T)) for every ? ?? H(?(T)) where H(?(T)) denotes the set of all analytic functions on an open neighborhood of ?(T).

Weyl's Theorems and Extensions of Bounded Linear Operators

A bounded operator $T\in L(X)$, $X$ a Banach space, is said to satisfy Weyl's theorem if the set of all spectral points that do not belong to the Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues and having finite multiplicity. In this article we give sufficient conditions for which Weyl's theorem for an extension $\overline T$ of $T$ (respectively, for $T$) entails that Weyl's theorem holds for $T$ (respectively, for $\overline T$).

GENERALIZED WEYL'S THEOREM FOR ALGEBRAICALLY &lt;TEX&gt;$k$&lt;/TEX&gt;-QUASI-PARANORMAL OPERATORS

An operator <TEX>$T\;{\varepsilon}\;B(\mathcal{H})$</TEX> is said to be <TEX>$k$</TEX>-quasi-paranormal operator if <TEX>$||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$</TEX> for every <TEX>$x\;{\epsilon}\;\mathcal{H}$</TEX>, <TEX>$k$</TEX> is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of <TEX>$k$</TEX>-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically <TEX>$k$</TEX>-quasi-paranormal operator has Bishop's property (<TEX>$\beta$</TEX>), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for <TEX>$f(T)$</TEX> for every <TEX>$f\;{\epsilon}\;H({\sigma}(T))$</TEX>; (ii) generalized a - Browder's theorem holds for <TEX>$f(S)$</TEX> for every <TEX>$S\;{\prec}\;T$</TEX> and <TEX>$f\;{\epsilon}\;H({\sigma}(S))$</TEX>; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.

ON WEYL'S THEOREM FOR TENSOR PRODUCTS

AbstractLetAandBbe operators acting on infinite-dimensional spaces. In this paper we prove that ifAandBare isoloid, satisfy Weyl's theorem, and the Weyl spectrum identity holds, thenA⊗Bsatisfies Weyl's theorem.

A NOTE ON WEYL'S THEOREM FOR *-PARANORMAL OPERATORS

In this note we investigate Weyl's theorem for �-paranormal operators on a separable infinite dimensional Hilbert space. We prove that if T is a �-paranormal operator satisfying Property (E) - (TI)HT({�}) is closed for each � 2 C, where HT({�}) is a local spectral subspace T, then Weyl's theorem holds for T. Let H denote an infinite dimensional separable Hilbert space. Let B(H) and K(H) denote the algebra bounded linear operators and the ideal compact operators on H, respectively. If T 2 B(H) write N(T) and R(T) for the null space and range T; �(T) := dimN(T); �(T) := dimN(T � ); �(T) for the spectrum T; �ap(T) for the approximate point spectrum T; �0(T) for the set eigenvalues T. An operator T 2 B(H) is called Fredholm if it has closed range with finite dimensional null space and its range finite co-dimension. The index a Fredholm operator T 2 B(H) is given by ind(T) := �(T) − �(T). An operator T 2 B(H) is called Weyl if it is Fredholm index zero. An operator T 2 B(H) is called Browder if it is Fredholm of finite ascent and descent: equivalently ((11, Theorem 7.9.3)) if T is Fredholm and T − �I is invertible for sufficiently small� 6 0 in C. The essential spectrume(T), the Weyl spectrum !(T) and the Browder spectrumb(T) T 2 B(H) are defined by ((10), (11), (12))

Weyl's theorem for algebraically k-quasiclass A operators

If T or T is an algebraically k-quasiclass A operator acting on an infinite di- mensional separable Hilbert space and F is an operator commuting with T, and there exists a positive integer n such that F n has a finite rank, then we prove that Weyl's theorem holds for f(T)+F for every f2 H( (T)), where H( (T)) denotes the set of all analytic functions in a neighborhood of (T). Moreover, if T is an algebraically k-quasiclass A operator, then -Weyl's theorem holds for f(T). Also, we prove that if T or T is an algebraically k-quasiclass A operator then both the Weyl spectrum and the approximate point spectrum of T obey the spectral mapping theorem for every f2 H( (T)). We begin with some standard notation on Fredholm theory. Throughout this paper letH be a separable complex Hilbert space with inner producth ;i . Let B(H) and K(H) denote respectively, theC -algebra of all bounded linear operators and the ideal of compact operators acting on H. If T 2 B(H), we shall write kerT and ranT for the null space and the range of T respectively. Also let (T ) =dim kerT, (T ) =dim kerT and let (T ), a(T ) denote the spectrum, approximate point spectrum of T, respectively. Let p = p(T ) be the ascent of T; i.e., the smallest nonnegative integer p such that kerT p = kerT p+1 . If such an integer does not exist, we put p(T ) =1. Analogously, let q = q(T ) be the descent of T; i.e., the smallest nonnegative integer q such that ran T q =ranT q+1 , and if such an integer does not exist, we put q(T ) =1. It is well known that if p(T ) and q(T ) are both finite then p(T ) = q(T ). Moreover, 0 < p( T ) = q( T ) <1 precisely when is a pole of the resolvent of T, see Heuser (20, Proposition 50.2). An operator T 2 B(H) is called Fredholm if ranT is

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r &gt; 0 and q ≥ 1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.

Spectral Properties of n-perinormal operators

In this paper we study spectral properties of class (M,n) or n -perinormal operators. It is shown that if T belongs to class (M,n) , then its point spectrum and joint point spectrum are identical. As an application we show that the spectral mapping theorem holds for the essential approximate point spectrum and for Weyl spectrum. It is also shown that a -Browder’s theorem holds for n -perinormal operators. A general version of the famous Fuglede-Putnam’s theorem for n -perinormal operator is also presented. Mathematics subject classification (2010): Primary 47B47, 47A30, 47B20; Secondary 47B10.

Weyl’s Theorem for Algebraically Class A(s, t) Operators

In this study, we show that Weyl’s theorem holds for algebraically class A (s, t) operator acting on Hilbert space. We prove: (i) Weyl’s theorem holds for f (T) for every f belongs to holomorphic function of spectrum of T; (ii) generalized Weyl’s theorem holds for T; (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.

Almost α‐Hyponormal Operators with Weyl Spectrum of Area Zero

We define the class of almost α‐hyponormal operators and prove that for an operator T in this class, is trace‐class and its trace is zero when α ∈ (0, 1] and the area of the Weyl spectrum is zero.

On the a-Browder and a-Weyl spectra of tensor products

Given Banach space operators A ∈ B( Open image in new window ) and B ∈ B( Open image in new window ), let A⊗B ∈ B( Open image in new window ⊗ Open image in new window ) denote the tensor product of A and B. Let σa, σaw and σab denote the approximate point spectrum, the Weyl approximate point spectrum and the Browder approximate point spectrum, respectively. Then σaw(A⊗B) ⊆ σa(A)σaw(B) ⊂ σaw(A)σa(B) ⊆ σa(A)σab(B) ⊂ σab(A)σa(B) = σab(A⊗B), and a sufficient condition for the (a-Weyl spectrum) identity σaw(A⊗B) = σa(A)σaw(B) ⊂ σaw(A)σa(B) to hold is that σaw(A⊗B) = σab(A⊗B). Equivalent conditions are proved in Theorem 1, and the problem of the transference of a-Weyl’s theorem for a-isoloid operators A and B to their tensor product A⊗B is considered in Theorem 2. Necessary and sufficient conditions for the (plain) Weyl spectrum identity are revisited in Theorem 3.