Frames and Stable Bases for Shift-Invariant Subspaces of L2(ℝd)

1995 ◽  
Vol 47 (5) ◽  
pp. 1051-1094 ◽  
Author(s):  
Amos Ron ◽  
Zuowei Shen
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Invariant Subspaces ◽  
Fourier Domain ◽  
Main Theme ◽  
Bessel Sequence ◽  
Analogous Property ◽  
Finite Set ◽  
Image Position

AbstractLet X be a countable fundamental set in a Hilbert space H, and let T be the operator Whenever T is well-defined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L2(ℝd), and for sets X of the form with Φ either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT* and T* T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators.

A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

1991 ◽  
Vol 110 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
C Algebra ◽  
Alternative Characterization ◽  
Image Position ◽  
Algebra Of Operators

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such thatas i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

scholarly journals A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

A Generalization of Uniformly Rotund Banach Spaces

1979 ◽  
Vol 31 (3) ◽  
pp. 628-636 ◽  
Author(s):  
Francis Sullivan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Two Dimensional ◽  
Geometrical Characterization ◽  
Arbitrary Space ◽  
Image Position

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ XThus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for The space is said to be uniformly rotund if for each 0 we have .

Characterization of corneo-scleral geometry using Fourier domain profilometry in the healthy eye

2018 ◽  
Vol 41 ◽  
pp. S83-S84 ◽  
Author(s):  
David P. Piñero ◽  
Roberto Soto-Negro ◽  
Antonio Martínez-Abad ◽  
Pedro Ruiz-Fortes ◽  
Rafael J. Pérez-Cambrodí
Keyword(s):  
Fourier Domain

New characterizations of g-frames and g-Riesz bases

2018 ◽  
Vol 16 (06) ◽  
pp. 1850053 ◽  
Author(s):  
Dongwei Li ◽  
Jinsong Leng ◽  
Tingzhu Huang
Keyword(s):  
Riesz Basis ◽  
Riesz Bases ◽  
Gram Matrix ◽  
Necessary Condition ◽  
Topological Properties ◽  
Bessel Sequence ◽  
Bessel Sequences ◽  
Sufficient And Necessary Condition

In this paper, we give some new characterizations of g-frames, g-Bessel sequences and g-Riesz bases from their topological properties. By using the Gram matrix associated with the g-Bessel sequence, we present a sufficient and necessary condition under which the sequence is a g-Bessel sequence (or g-Riesz basis). Finally, we consider the excess of a g-frame and obtain some new results.

scholarly journals Characterization of Hilbert space manifolds revisited

1986 ◽  
Vol 24 (1-3) ◽  
pp. 53-69 ◽  
Author(s):  
Mladen Bestvina ◽  
Philip Bowers ◽  
Jerzy Mogilsky ◽  
John Walsh
Keyword(s):  
Hilbert Space
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