scholarly journals WEYL TYPE THEOREMS FOR FUNCTIONS OF OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 493-505 ◽  
Author(s):  
SEN ZHU ◽  
CHUN GUANG LI ◽  
TING TING ZHOU
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Compact Perturbations ◽  
Weyl Type Theorems ◽  
Necessary And Sufficient

AbstractA-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

scholarly journals Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results

2003 ◽  
Vol 2003 (22) ◽  
pp. 1421-1431 ◽  
Author(s):  
Khalid Latrach ◽  
Abdelkader Dehici
Keyword(s):  
Banach Space ◽  
Fredholm Operator ◽  
Linear Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Compact Perturbations ◽  
Necessary And Sufficient ◽  
Fredholm Perturbations

Let(U(t))t≥0be aC0-semigroup of bounded linear operators on a Banach spaceX. In this paper, we establish that if, for somet0>0,U(t0)is a Fredholm (resp., semi-Fredholm) operator, then(U(t))t≥0is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that(U(t))t≥0can be imbedded in aC0-group onX. Also we study semigroups which are near the identity in the sense that there existst0>0such thatU(t0)−I∈𝒥(X), where𝒥(X)is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where𝒥(X)is replaced by some subsets of the set of polynomially compact perturbations.

Additive properties for the pseudo core inverse of morphisms in an additive category

2021 ◽  
Author(s):  
Jianlong Chen ◽  
Xiaofeng Chen ◽  
Dingguo Wang
Keyword(s):  
Sufficient Conditions ◽  
Linear Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Linear Operators ◽  
Additive Property ◽  
Bounded Linear ◽  
Additive Category ◽  
Core Inverse ◽  
Necessary And Sufficient

In this paper, given a morhism [Formula: see text] with its pseudo core inverse [Formula: see text] and a morphism [Formula: see text] such that [Formula: see text] is invertible, a necessary and sufficient condition and two sufficient conditions are presented under which the additive property, namely [Formula: see text] holds. Several interesting results about additive properties of core inverses of bounded linear operators presented in Huang et al. are generalized to the case of pseudo core inverse of morphism. Also, many results regarding additive properties of core-EP inverses of complex matrices studied by Ma and Stanimirović are extended to the cases of morphism.

scholarly journals Propertiesandfor Bounded Linear Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. H. M. Rashid
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Extension Property ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Single Valued Extension Property ◽  
Bounded Linear ◽  
Weyl Type Theorems

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

scholarly journals Generalized Weyl’s theorems for polaroid operators

2011 ◽  
Vol 27 (1) ◽  
pp. 24-33
Author(s):  
C. CARPINTERO ◽  
◽  
D. MUNOZ ◽  
E. ROSAS ◽  
O. GARCIA ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Generalized Weyl’S Theorem ◽  
Polaroid Operators ◽  
Weyl Theorem ◽  
Necessary And Sufficient

In this paper we establish necessary and sufficient conditions on bounded linear operators for which generalized Weyl’s theorem, or generalized a-Weyl theorem, holds. We also consider generalized Weyl’s theorems in the framework of polaroid operators and obtain improvements of some results recently established in [20] and [29].

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

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