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 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 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).

scholarly journals Characterizations of variational convergence of bifunctions defined on products of two sets

2020 ◽  
Vol 3 (SI3) ◽  
pp. First
Author(s):  
Diem Thi Hong Huynh
Keyword(s):  
Basic Property ◽  
Specific Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Variational Convergence ◽  
Product Spaces ◽  
The Third ◽  
Left And Right ◽  
Definition Of ◽  
Necessary And Sufficient

We present definitions of types of variational convergence of finite-valued bifunctions defined on rectangular domains and establish characterizations of these convergences. In the introduction, we present the origins of the research on variational convergence and then we lead to the specific problem of this paper. The content of the paper consists of 3 parts: variational convergance of fucntion; variational convergance of bifunction; and characterizations of variational convergence of bifunction, this part is the main results of this paper. In section 2, we presented the definition of epi convergence and presented a basic property problem that will be used to extend and develop the next two sections. In section 3, we start to present a new definition, the definition of convergence epi / hypo, minsup and maxinf. To clearly understand of these new definitions we have provided comments (remarks) and some examples which reader can check these definitions. The above contents serve the main result of this paper will apply in part 4. Now, we will explain more detail for this part as follows. Firstly, variational convergence of bifunctions is characterized by the epi- and hypo-convergence of related unifunctions, which are slices sup- and inf-projections. The second characterization expresses the equivalence of variational convergence of bifunctions and the same convergence of the so-called proper bifunctions defined on the whole product spaces. In the third one, the geometric reformulation, we establish explicitly the interval of all the limits by computing formulae of the left- and right-end limit bifunctions, and this is necessary and sufficient conditions of the sequence bifunctions to attain epi / hypo, minsup and maxinf convergence.

Solution of Some Weight Problems for the Riemann–Liouville and Weyl Operators

1998 ◽  
Vol 5 (6) ◽  
pp. 565-574
Author(s):  
A. Meskhi
Keyword(s):  
Weight Function ◽  
Liouville Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Operator ◽  
Weight Problems ◽  
Necessary And Sufficient

Abstract The necessary and sufficient conditions are found for the weight function 𝑣, which provide the boundedness and compactness of the Riemann–Liouville operator 𝑅 α from 𝐿𝑝 to . The criteria are also established for the weight function 𝑤, which guarantee the boundedness and compactness of the Weyl operator 𝑊 α from to 𝐿𝑞.

scholarly journals Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set

2018 ◽  
Vol 11 (3) ◽  
pp. 580-588
Author(s):  
Ronnason Chinram ◽  
Pattarawan Petchkaew ◽  
Samruam Baupradist
Keyword(s):  
Linear Transformation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Proper Subset ◽  
Transformation Semigroups ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Composition Of Functions

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Let X be a nonempty set and T(X) be the semigroup of all transformation from X into itself under the composition of functions. For a partition P = {X_α | α ∈ I} of the set X, let T(X,P) = {f ∈ T(X) | (X_α)f ⊆ X_α for all α ∈ I}. Then T(X,P) is a subsemigroup of T(X) and if P = {X}, T(X,P) = T(X). Our aim in this paper is to give necessary and sufficient conditions for elements in T(X,P) to be left or right magnifying. Moreover, we apply those conditions to give necessary and sufficient conditions for elements in some generalized linear transformation semigroups.

scholarly journals Spectra of 2 x 2 upper triangular operator matrices

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3587-3599
Author(s):  
Junjie Huang ◽  
Aichun Liu ◽  
Alatancang Chen
Keyword(s):  
Linear Operators ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Bounded Linear ◽  
Spectrum Of An Operator ◽  
Residual Spectrum ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Upper Triangular Operator Matrix ◽  
Necessary And Sufficient

The spectra of the 2 x 2 upper triangular operator matrix MC = (A C 0 B ) acting on a Hilbert space H1 ? H2 are investigated. We obtain a necessary and sufficient condition of ?(MC) = ?(A)??(B) for every C ? B(H2,H1), in terms of the spectral properties of two diagonal elements A and B of MC. Also, the analogues for the point spectrum, residual spectrum and continuous spectrum are further presented. Moveover, we construct some examples illustrating our main results. In particular, it is shown that the inclusion ?r(MC) ? ?r(A) ? ?r(B) for every C ? B(H2,H1) is not correct in general. Note that ?(T) (resp. ?r(T)) denotes the spectrum (resp. residual spectrum) of an operator T, and B(H2,H1) is the set of all bounded linear operators from H2 to H1.

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 Left and Right Magnifying Elements in the Semigroup of all Binary Relations

2020 ◽  
Vol 13 (4) ◽  
pp. 987-994
Author(s):  
Watchara Teparos ◽  
Soontorn Boonta ◽  
Thitiya Theparod
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Proper Subset ◽  
Binary Relations ◽  
Left And Right ◽  
Necessary And Sufficient

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = M a]. Let X be a nonempty set and BX the semigroup of binary relations on X. In this paper, we give necessary and sufficient conditions for elements in BX to be left or right magnifying.

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