scholarly journals On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1927-1931
Author(s):  
Saroj Malik ◽  
Néstor Thome
Keyword(s):  
Linear Operator ◽  
Least Squares ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Minimum Norm ◽  
Closed Range ◽  
Bounded Linear ◽  
Equivalent Conditions

For two given Hilbert spaces H and K and a given bounded linear operator A ? L(H,K) having closed range, it is well known that the Moore-Penrose inverse of A is a reflexive g-inverse G ? L(K,H) of A which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator G to be the Moore-Penrose inverse of A are investigated in terms of normal, EP, bi-normal, bi-EP, l-quasi-normal and r-quasi-normal and l-quasi-EP and r-quasi-EP operators.

(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].

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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.

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 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 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.

Steepest Descent and Least Squares Solvability

1974 ◽  
Vol 17 (2) ◽  
pp. 275-276 ◽  
Author(s):  
C. W. Groetsch
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Least Squares ◽  
Bounded Linear Operator ◽  
Steepest Descent ◽  
Normal Equation ◽  
Bounded Linear ◽  
Least Squares Solution ◽  
Image Position

Let T be a bounded linear operator defined on a Hilbert space H. An element z∈H is called a least squares solution of the equationif . It is easily shown that z is a least squares solution of (1) if and only if z satisfies the normal equation

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