Generalized Weyl’s Theorem for Some Classes of Operators

MECHERI, SALAH 대한수학회 2006 Kyungpook mathematical journal Vol.46 No.4

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set _(σBω)(A) of all λ ∈ C such that A-λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem _(σBω)(A) =σ(A) \ E(A), and the B-Weyl spectrum _(σBω)(A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalized Weyl's theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

WEYL'S THEOREM FOR ISOLOID AND REGULOID OPERATORS

Kim, An-Hyun,Yoo, Sung-Uk Korean Mathematical Society 1999 대한수학회논문집 Vol.14 No.1

In this paper we find some classes of operators for which Weyl`s theorem holds. The main result is as follows. If T$\in$L(\ulcorner) satisfies the following: (ⅰ) Either T or T\ulcorner is reduced by each of its eigenspaces; (ⅱ) Weyl`s theorem holds for T; (ⅲ) T is isoloid, then for every polynomial p, Weyl`s theorem holds for p(T).

Spectral Properties of k-quasi-class A(s, t) Operators

Mecheri, Salah,Braha, Naim Latif Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.3

In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator $T{\in}B(H)$ is said to be k-quasi-class A(s, t) if $$T^{*k}(({\mid}T^*{\mid}^t{\mid}T{\mid}^{2s}{\mid}T^*{\mid}^t)^{\frac{1}{t+s}}-{\mid}T^*{\mid}^{2t})T^k{\geq}0$$, where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop's property ${\beta}$ and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if $T^*$ is algebraically k-quasi-class A(s, t), then the generalized a-Weyl's theorem holds for T. Using these results we show that $T^*$ satisfies generalized the Weyl's theorem if and only if T satisfies the generalized Weyl's theorem if and only if T satisfies Weyl's theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators.

Weyl's theorem in several variables

Han, Y.M.,Kim, A.H. Academic Press 2010 Journal of mathematical analysis and applications Vol.370 No.2

In this note we consider Weyl's theorem and Browder's theorem in several variables. The main result is as follows. Let T be a doubly commuting n-tuple of hyponormal operators acting on a complex Hilbert space. If T has the quasitriangular property, i.e., the dimension of the left cohomology for the Koszul complex Λ(T-λ) is greater than or equal to the dimension of the right cohomology for Λ(T-λ) for all λ@?C<SUP>n</SUP>, then 'Weyl's theorem' holds for T, i.e., the complement in the Taylor spectrum of the Taylor Weyl spectrum coincides with the isolated joint eigenvalues of finite multiplicity.

WEYL SPECTRUM OF THE PRODUCTS OF OPERATORS

Cao, Xiaohong Korean Mathematical Society 2008 대한수학회지 Vol.45 No.3

Let $M_C=\(\array{A&C\\0&B}\)$ be a $2{\times}2$ upper triangular operator matrix acting on the Hilbert space $H{\bigoplus}K\;and\;let\;{\sigma}_w(\cdot)$ denote the Weyl spectrum. We give the necessary and sufficient conditions for operators A and B which ${\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w\(\array{A&C\\0&B}\)\;or\;{\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w(A){\cup}{\sigma}_w(B)$ holds for every $C{\in}B(K,\;H)$. We also study the Weyl's theorem for operator matrices.

WEYL@S THEOREMS FOR POSINORMAL OPERATORS

DUGGAL BHAGWATI PRASHAD,KUBRUSLY CARLOS Korean Mathematical Society 2005 대한수학회지 Vol.42 No.3

An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator $P{\in}B(H)$ such that $TT^*\;=\;T^*PT$. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to $T{\in}CTP(resp.,\;T{\in}TP)$, if to each complex number, $\lambda$ there corresponds a positive operator $P_\lambda$ such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P_{\lambda}^{\frac{1}{2}}(T-{\lambda}I)|^{2}$ (resp., if there exists a positive operator P such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P^{\frac{1}{2}}(T-{\lambda}I)|^{2}\;for\;all\;\lambda)$. This paper proves Weyl's theorem type results for TP and CTP operators. If $A\;{\in}\;TP$, if $B^*\;{\in}\;CTP$ is isoloid and if $d_{AB}\;{\in}\;B(B(H))$ denotes either of the elementary operators $\delta_{AB}(X)\;=\;AX\;-\;XB\;and\;\Delta_{AB}(X)\;=\;AXB\;-\;X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d^{\ast}_{AB}\;satisfies\;\alpha-Weyl's$ theorem.

Weyl Type Theorems for Unbounded Hyponormal Operators

GUPTA, ANURADHA,MAMTANI, KARUNA Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.3

If T is an unbounded hyponormal operator on an infinite dimensional complex Hilbert space H with ${\rho}(T){\neq}{\phi}$, then it is shown that T satisfies Weyl's theorem, generalized Weyl's theorem, Browder's theorem and generalized Browder's theorem. The equivalence of generalized Weyl's theorem with generalized Browder's theorem, property (gw) with property (gb) and property (w) with property (b) have also been established. It is also shown that a-Browder's theorem holds for T as well as its adjoint $T^*$.

Weyl spectrum of the products of operators

Xiaohong Cao 대한수학회 2008 대한수학회지 Vol.45 No.3

Let MC = (수식) be a 2×2 upper triangular operator matrix acting on the Hilbert space H K and let σw(ㆍ) denote theWeyl spectrum. We give the necessary and sufficient conditions for operators A and B which [수식] holds for every C ∈ B(K, H). We also study the Weyl’s theorem for operator matrices. Let MC = (수식) be a 2×2 upper triangular operator matrix acting on the Hilbert space H K and let σw(ㆍ) denote theWeyl spectrum. We give the necessary and sufficient conditions for operators A and B which [수식] holds for every C ∈ B(K, H). We also study the Weyl’s theorem for operator matrices.

ON WEYL'S THEOREM FOR QUASI-CLASS A OPERATORS

Duggal Bhagwati P.,Jeon, In-Ho,Kim, In-Hyoun Korean Mathematical Society 2006 대한수학회지 Vol.43 No.4

Let T be a bounded linear operator on a complex infinite dimensional Hilbert space $\scr{H}$. We say that T is a quasi-class A operator if $T^*\|T^2\|T{\geq}T^*\|T\|^2T$. In this paper we prove that if T is a quasi-class A operator and f is a function analytic on a neigh-borhood or the spectrum or T, then f(T) satisfies Weyl's theorem and f($T^*$) satisfies a-Weyl's theorem.

On Semiparallel and Weyl-semiparallel Hypersurfaces of Kaehler Manifolds

Ozgur, Cihan,Murathan, Cengizhan,Arslan, Kadri Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.1

We study on semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds. We prove that a (2n + 1)-dimensional Sasakian hypersurface M of a (2n+2)-dimensional Kaehler manifold $\widetilde{M}^{2n+2}$ is semiparallel if and only if it is totally umbilical with unit mean curvature, if dimM = 3 and $\widetilde{M}^4$ is a Calabi-Yau manifold, then $\widetilde{M}$ is flat at each point of M. We also prove that such a hypersurface M is Weyl-semiparallel if and only if it is either an ${\eta}$-Einstein manifold or semiparallel. We also investigate the extended classes of semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds.