scholarly journals Sums of A Pair of Orthogonal Frames

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 582
Author(s):  
Ghanshyam Bhatt
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Sufficient Condition ◽  
Simple Construction ◽  
Necessary And Sufficient Condition ◽  
Dual Frames ◽  
Bounded Linear ◽  
Frame Bounds ◽  
Necessary And Sufficient

Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.

scholarly journals The Sums of a Pair of Orthogonal Frames

2019 ◽  
Author(s):  
Ghanshyam Bhatt
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Linear ◽  
Frame Bounds ◽  
Necessary And Sufficient

A sum of different frames under the action of a bounded linear operator is studied with the help of analysis, synthesis and frame operators. In particular, it is shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. For a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate duals to be orthogonal.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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 On Convexity of Composition and Multiplication Operators on Weighted Hardy Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Karim Hedayatian ◽  
Lotfollah Karimi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Bounded Linear Operator ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Weighted Hardy Spaces ◽  
Bounded Linear ◽  
Necessary And Sufficient

A bounded linear operatorTon a Hilbert spaceℋ, satisfying‖T2h‖2+‖h‖2≥2‖Th‖2for everyh∈ℋ, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

scholarly journals Computation of eigenvectors corresponding to multiple eigenvalues

1971 ◽  
Vol 4 (3) ◽  
pp. 419-422 ◽  
Author(s):  
A.L. Andrew ◽  
G.C. Elton
Keyword(s):  
Numerical Methods ◽  
Linear Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Multiple Eigenvalues ◽  
Necessary And Sufficient

This paper examines a class of common numerical methods for computing eigenvectors of a compact linear operator. A necessary and sufficient condition is established for every element of an arbitrary given eigenspace to be the limit of a sequence of the approximate eigenvectors obtained by any given method in this class.

scholarly journals Continuous weaving fusion frames in Hilbert spaces

2020 ◽  
Vol 18 (05) ◽  
pp. 2050035
Author(s):  
Vahid Sadri ◽  
Reza Ahmadi ◽  
Gholamreza Rahimlou
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Sufficient Condition ◽  
Bounded Linear ◽  
Fusion Frames

In this paper, we first introduce the notation of weaving continuous fusion frames in separable Hilbert spaces. After reviewing the conditions for maintaining the weaving [Formula: see text]-fusion frames under the bounded linear operator and also, removing vectors from these frames, we will present a necessarily and sufficient condition about [Formula: see text]-woven and [Formula: see text]-fusion woven. Finally, perturbation of these frames will be introduced.

scholarly journals On Self-Adjoint Linear Relations

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 1-7
Author(s):  
Péter Berkics
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Classical Approach ◽  
Necessary And Sufficient Condition ◽  
Von Neumann ◽  
Necessary And Sufficient ◽  
Densely Defined ◽  
General Setup

A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if . In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

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 On the surjectivity of linear transformations

1996 ◽  
Vol 19 (3) ◽  
pp. 545-548
Author(s):  
M. Damlakhi ◽  
V. Anandam
Keyword(s):  
Reflexive Banach Space ◽  
Locally Convex Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Transformations ◽  
Bounded Linear ◽  
Smooth Coefficients ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Bounded Linear Transformation

LetBbe a reflexive Banach space,Xa locally convex space andT:B→X(not necessarily bounded) linear transformation. A necessary and sufficient condition is obtained so that for a givenv∈Xthere is a solution for the equationTu=v. This result is used to discuss the existence of anLp-weak solution ofDu=vwhereDis a differential operator with smooth coefficients andv∈Lp.

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