CONSTRUCTION OF DUAL g-FRAMES FOR CLOSED SUBSPACES

2011 ◽  
Vol 09 (06) ◽  
pp. 947-964 ◽  
Author(s):  
BAI-YUN YU ◽  
ZHI-BIAO SHU
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

In this paper, we introduce dual g-frames for a closed subspace in a separable Hilbert space and also give a characterization. Generally dual g-frames for a closed subspace are non-commutative. Therefore, we construct dual g-frames for closed subspaces from two aspects and give the corresponding formulas, respectively. Finally, we give a necessary and sufficient condition for commutative dual g-frame pairs for closed subspaces under certain conditions.

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

1981 ◽  
Vol 89 (3-4) ◽  
pp. 201-215 ◽  
Author(s):  
Richard Datko
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Exponential Stability ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Exponential Stability ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

Geometry on the Unit Ball of a Complex Hilbert Space

1978 ◽  
Vol 30 (01) ◽  
pp. 22-31 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Hyperbolic Geometry ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Open Unit Ball ◽  
Similar Role ◽  
Necessary And Sufficient

Furnishing the open unit ball of a complex Hilbert space with the Carathéodory-differential metric, we construct a model which plays a similar role as that of the Poincaré model for the hyperbolic geometry. In this note we study the question whether or not through a point in the model not lying on a given line there exists a unique perpendicular, and give a necessary and sufficient condition for the existence of a unique perpendicular. This enables us to divide a triangle into two right triangles. Many trigonometric identities in a general triangle are easy consequences of various identities which hold on a right triangle.

A two-parameter eigenvalue problem involving complex potentials

1990 ◽  
Vol 116 (1-2) ◽  
pp. 177-191
Author(s):  
M. Faierman
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Complex Potentials ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Parameter System ◽  
Complete Set ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe consider a two-parameter system of ordinary differential equations of the second order involving complex potentials and show that, unlike the case of real potentials, the eigenfunctions of the system do not necessarily form a complete set in the usual Hilbert space associated with the problem. We also give a necessary and sufficient condition for the eigenfunctions to be complete. Finally, we establish some results concerning the eigenvalues of the system.

scholarly journals On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

1997 ◽  
Vol 20 (3) ◽  
pp. 457-464
Author(s):  
A. A. El-Sabbagh
Keyword(s):  
Hilbert Space ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Matrix ◽  
Linear Relations ◽  
Symmetric Linear Relation ◽  
Necessary And Sufficient ◽  
Selfadjoint Extensions ◽  
Pick Problem

The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.

scholarly journals On the extension of positive linear functionals

2000 ◽  
Vol 23 (1) ◽  
pp. 31-35 ◽  
Author(s):  
Ergashboy Muhamadiev ◽  
Adel T. Diab
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Necessary And Sufficient Condition ◽  
Semi Group ◽  
Positive Linear Functionals ◽  
Necessary And Sufficient

A necessary and sufficient condition to extend a continuous linear real functionals which is positive with respect to a semi-group defined on a subspace of a linear space is discussed in this paper. The case of a closed subspace of a Banach space is also discussed.

scholarly journals Extensions of dissipative operators with closable imaginary part

2021 ◽  
Vol 41 (3) ◽  
pp. 381-393
Author(s):  
Christoph Fischbacher
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Operator ◽  
Dissipative Operators ◽  
First Order ◽  
Necessary And Sufficient ◽  
Symmetric Operators

Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

20.—Deficiency Indices of Polynomials in Symmetric Differential Expressions, II

1975 ◽  
Vol 73 ◽  
pp. 301-306 ◽  
Author(s):  
Anton Zettl
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Polynomial ◽  
Deficiency Indices ◽  
Linear Differential ◽  
Image Position ◽  
Necessary And Sufficient ◽  
The Relationship

SynopsisGiven a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

scholarly journals The moment spaces of normed linear spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 277-283
Author(s):  
Fowzi Ahmad Sejeeni ◽  
Matooq Ahmad Badri
Keyword(s):  
Normed Linear Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Independent Sequence ◽  
Linear Spaces ◽  
Linearly Independent ◽  
The Moment ◽  
Moment Spaces ◽  
Necessary And Sufficient

For a linearly independent sequence in a normed linear space the moment space is defined. Basic properties of moment spaces are discussed as well as a necessary and sufficient condition for the moment space to be a closed subspace of l∞.

scholarly journals The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Shifang Zhang ◽  
Huaijie Zhong ◽  
Long Long
Keyword(s):  
Separable Hilbert Space ◽  
Invertible Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Infinite Dimensional ◽  
Lower Semi ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

WhenA∈B(H)andB∈B(K)are given, we denote byMCthe operator acting on the infinite-dimensional separable Hilbert spaceH⊕Kof the formMC=(AC0B). In this paper, it is proved that there exists some operatorC∈B(K,H)such thatMCis upper semi-Browder if and only if there exists some left invertible operatorC∈B(K,H)such thatMCis upper semi-Browder. Moreover, a necessary and sufficient condition forMCto be upper semi-Browder for someC∈G(K,H)is given, whereG(K,H)denotes the subset of all of the invertible operators ofB(K,H).

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