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 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 A New Inequality for Frames in Hilbert Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Bessel Sequences

We obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.

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.

(F,G)-operator frames for ℒ(ℋ,𝒦)

2020 ◽  
Vol 18 (05) ◽  
pp. 2050031
Author(s):  
J. Sedghi Moghaddam ◽  
A. Najati ◽  
F. Ghobadzadeh
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Atomic Systems ◽  
Pseudo Inverse

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

Normal operators on Banach spaces

1979 ◽  
Vol 20 (2) ◽  
pp. 163-168 ◽  
Author(s):  
Che-Kao Fong
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Normal Operators

A (bounded, linear) operator H on a Banach space is said to be hermitian if ∥exp(itH)∥ = 1 for all real t. An operator N on is said to be normal if N = H + iK, where H and K are commuting hermitian operators. These definitions generalize those familiar concepts of operators on Hilbert spaces. Also, the normal derivations defined in [1] are normal operators. For more details about hermitian operators and normal operators on general Banach spaces, see [4]. The main result concerning normal operators in the present paper is the following theorem.

scholarly journals Reverse inequalities for the numerical radius of linear operators in Hilbert spaces

2006 ◽  
Vol 73 (2) ◽  
pp. 255-262 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Upper Bounds ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear ◽  
The Difference ◽  
Reverse Inequalities

Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.

scholarly journals Class of bounded operators associated with an atomic system

2014 ◽  
Vol 46 (1) ◽  
pp. 85-90 ◽  
Author(s):  
P.Sam Johnson ◽  
G. Ramu
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Atomic System ◽  
Linear Operators ◽  
Bounded Operators ◽  
Frame Operator ◽  
Bessel Sequence ◽  
Bounded Linear ◽  
Atomic Systems

$K$-frames, more general than the ordinary frames, have been introduced by Laura G{\u{a}}vru{\c{t}}a in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Using the frame operator, we find a class of bounded linear operators in which a given Bessel sequence is an atomic system for every member in the class.

scholarly journals Additive perturbations and multiplicative perturbations for the core inverse of bounded linear operator in Hilbert space

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6131-6144
Author(s):  
Fapeng Du ◽  
Zuhair Nashed
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Upper Bound ◽  
Bounded Linear Operator ◽  
Upper Bounds ◽  
Bounded Linear ◽  
The Core ◽  
Core Inverse

In this paper, we present some characteristics and expressions of the core inverse A# of bounded linear operator A in Hilbert spaces. Additive perturbations of core inverse are investigated under the condition R( ?)?N(A#) = {0} and an upper bound of ||?#-A#|| is obtained. We also discuss the multiplicative perturbations. The expressions of core inverse of perturbed operator T = EAF and the upper bounds of ||T#-A#|| are obtained too.

scholarly journals Perturbations on K-fusion frames

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Animesh Bhandari ◽  
Saikat Mukherjee
Keyword(s):  
Hilbert Space ◽  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Large Data ◽  
Wireless Sensor ◽  
Bounded Linear ◽  
Fusion Frames

Abstract Fusion frames are widely studied for their applications in recovering signals from large data. These are proved to be very useful in many areas, for example, wireless sensor networks. In this paper, we discuss a generalization of fusion frames, K-fusion frames. K-fusion frames provide decompositions of a Hilbert space into atomic subspaces with respect to a bounded linear operator. This article studies various kinds of properties of K-fusion frames. Several perturbation results on K-fusion frames are formulated and analyzed.

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