Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator equations on Hilbert spaces

2010 ◽  
pp. 107-134 ◽  
Author(s):  
Saburou Saitoh
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Approximate Solutions ◽  
Reproducing Kernels ◽  
Operator Equations ◽  
Bounded Linear ◽  
Linear Operator Equations

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.

(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 The generalized condition numbers of bounded linear operators in Banach spaces

2004 ◽  
Vol 76 (2) ◽  
pp. 281-290 ◽  
Author(s):  
Guoliang Chen ◽  
Yimin Wei ◽  
Yifeng Xue
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Perturbation Analysis ◽  
Bounded Linear Operator ◽  
Linear Operators ◽  
Condition Numbers ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Ill Posed ◽  
Linear Operator Equations

AbstractFor any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

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.

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