scholarly journals Tractability of Tensor Product Linear Operators in Weighted Hilbert Spaces

2001 ◽  
Vol 8 (2) ◽  
pp. 415-426
Author(s):  
Henryk Woźniakowski
Keyword(s):  
Linear Operator ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Singular Values ◽  
Linear Operators ◽  
Initial Error ◽  
Continuous Linear Functional ◽  
Worst Case ◽  
Dimensional Version ◽  
The One

Abstract We study tractability in the worst case setting of tensor product linear operators defined over weighted tensor product Hilbert spaces. Tractability means that the minimal number of evaluations needed to reduce the initial error by a factor of ε in the d-dimensional case has a polynomial bound in both ε –1 and d. By one evaluation we mean the computation of an arbitrary continuous linear functional, and the initial error is the norm of the linear operator S d specifying the d-dimensional problem. We prove that nontrivial problems are tractable iff the dimension of the image under S 1 (the one-dimensional version of S d ) of the unweighted part of the Hilbert space is one, and the weights of the Hilbert spaces, as well as the singular values of the linear operator S 1, go to zero polynomially fast with their indices.

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 The Singular Value Expansion for Arbitrary Bounded Linear Operators

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1346
Author(s):  
Daniel K. Crane ◽  
Mark S. Gockenbach
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Singular Value ◽  
Linear Operators ◽  
Simple Analysis ◽  
Basic Tool ◽  
Bounded Linear ◽  
Orthonormal Bases ◽  
Value Decomposition ◽  
Well Posed

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

On Low Rank Approximation of Linear Operators in p-Norms and Some Algorithms

2016 ◽  
Vol 33 (04) ◽  
pp. 1650023
Author(s):  
Yang Liu
Keyword(s):  
Linear Operator ◽  
Sparse Matrix ◽  
Matrix Completion ◽  
Singular Values ◽  
Linear Operators ◽  
Low Rank ◽  
Minkowski Distance ◽  
Low Rank Approximation ◽  
Rank Approximation ◽  
Generalized Singular Values

In this paper, we study the optimal or best approximation of any linear operator by low rank linear operators, especially, any linear operator on the [Formula: see text]-space, [Formula: see text], under [Formula: see text] norm, or in Minkowski distance. Considering generalized singular values and using techniques from differential geometry, we extend the classical Schmidt–Mirsky theorem in the direction of the [Formula: see text]-norm of linear operators for some [Formula: see text] values. Also, we develop and provide algorithms for finding the solution to the low rank approximation problems in some nontrivial scenarios. The results can be applied to, in particular, matrix completion and sparse matrix recovery.

On eigenvalues and inverse singular values of compact linear operators in Hilbert space

1953 ◽  
Vol 49 (2) ◽  
pp. 201-212 ◽  
Author(s):  
J. P. O. Silberstein ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Linear Operator ◽  
Singular Values ◽  
Linear Operators ◽  
Complex Numbers ◽  
Identity Operator ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Image Position

1·1. In this paper we shall be concerned with the equationswhere K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.

scholarly journals Weighted Voronoi Diagrams in the Maximum Norm

2019 ◽  
Vol 29 (03) ◽  
pp. 239-250
Author(s):  
Günther Eder ◽  
Martin Held
Keyword(s):  
Simple Formula ◽  
Voronoi Diagrams ◽  
Incremental Algorithm ◽  
Maximum Norm ◽  
Worst Case ◽  
Line Segments ◽  
One Dimensional ◽  
Straight Line ◽  
Dimensional Version ◽  
The One

We consider multiplicatively weighted points, axis-aligned rectangular boxes and axis-aligned straight-line segments in the plane as input sites and study Voronoi diagrams of these sites in the maximum norm. For [Formula: see text] weighted input sites we establish a tight [Formula: see text] worst-case bound on the combinatorial complexity of their Voronoi diagram and introduce an incremental algorithm that allows its computation in [Formula: see text] time. Our approach also yields a truly simple [Formula: see text] algorithm for solving the one-dimensional version of this problem, where all weighted sites lie on a line.

scholarly journals Indicial equivalents of multiparameter definiteness conditions in finite dimensions

1984 ◽  
Vol 27 (3) ◽  
pp. 283-296 ◽  
Author(s):  
Paul Binding
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Column Vector ◽  
Image Position

Let Tm, Vmn be Hermitean linear operators on complex Hilbert spaces Hm, m=1…k. A nonzero column vector satisfyingwill be called an eigenvalue. This type of problem has been studied extensively by Atkinson [2] from the viewpoint of determinantal operators on the tensor product We shall connect his work with more recent investigations [5,7] of eigenvalue indices based on minimax principles for , which can be viewed as an operator on .

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

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