scholarly journals Browder and Weyl spectra of upper triangular operator matrices

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 111-130 ◽  
Author(s):  
B.P. Duggal
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Weyl’S Theorem ◽  
Single Valued Extension Property ◽  
Browder’S Theorem ◽  
Subject Classifications ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Mathematics Subject Classifications ◽  
Necessary And Sufficient

Let MC = (A/0 C/B) ( B(X ( X ) be an upper triangulat Banach space operator. The relationship between the spectra of MC and M0, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. Operators MC and M0 satisfying Browder's, or a-Browder's, theorem are characterized, and we prove necessary and sufficient conditions for implications of the type 'M0 satisfies a-Browder's (or a-Weyl's) theorem ( MC satisfies a-Browder's (resp., a-Weyl's) theorem' to hold. 2010 Mathematics Subject Classifications. Primary 47B47, 47A10, 47A11. .

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

Generalized left and right Weyl spectra of upper triangular operator matrices

2017 ◽  
Vol 32 ◽  
pp. 41-50
Author(s):  
Guojun Hai ◽  
Dragana Cvetkovic-Ilic
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Operator ◽  
Operator Matrices ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Left And Right ◽  
Necessary And Sufficient

In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$ \bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) $$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a generalized left Weyl operator for each $C\in\B(\K,\H)$, are presented.

scholarly journals A method of constructing braided Hopf algebras

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 53-66
Author(s):  
Tianshui Ma ◽  
Shuanhong Wang ◽  
Shaoxian Xu
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Recent Work ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Braided Hopf Algebra ◽  
Subject Classifications ◽  
Mathematics Subject Classifications ◽  
Twisted Tensor Product ◽  
Necessary And Sufficient

Let A and B be two Hopf algebras and R ( Hom(B ( A, A ( B), the twisted tensor product Hopf algebra A#RB was introduced by S. Caenepeel et al in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for A#RB to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of A through a (?, R)-pair (A, B). Finally we give a method to construct Radford's biproduct directly by defining the module action and comodule action from the twisted tensor biproduct. 2010 Mathematics Subject Classifications. 16W30. .

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 AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 309-315 ◽  
Author(s):  
ROBERTO CONTI
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Cuntz Algebra ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Matrix Algebras ◽  
The Family ◽  
The Matrix ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

scholarly journals The point spectrum and residual spectrum of upper triangular operator matrices

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1759-1771
Author(s):  
Xiufeng Wu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Hilbert Spaces ◽  
Point Spectrum ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

The point and residual spectra of an operator are, respectively, split into 1,2-point spectrum and 1,2-residual spectrum, based on the denseness and closedness of its range. Let H,K be infinite dimensional complex separable Hilbert spaces and write MX = (AX0B) ? B(H?K). For given operators A ? B(H) and B ? B(K), the sets ? X?B(K,H) ?+,i(MX)(+ = p,r;i = 1,2), are characterized. Moreover, we obtain some necessary and sufficient condition such that ?*,i(MX) = ?*,i(A) ?*,i(B) (* = p,r;i = 1,2) for every X ? B(K,H).

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