scholarly journals Gram matrix associated to controlled frames

2018 ◽  
Vol 16 (05) ◽  
pp. 1850035 ◽  
Author(s):  
E. Osgooei ◽  
A. Rahimi
Keyword(s):  
Hilbert Spaces ◽  
Diagnostic Tool ◽  
Riesz Basis ◽  
Trace Class ◽  
Trace Class Operator ◽  
Riesz Bases ◽  
Gram Matrix ◽  
Frame Operator ◽  
Class Operator ◽  
Interactive Algorithms

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every [Formula: see text]-controlled Riesz basis [Formula: see text] is in the form [Formula: see text], where [Formula: see text] is a bijective operator on [Formula: see text]. Furthermore, we propose an equivalent accessible condition to the sequence [Formula: see text] being a [Formula: see text]-controlled Riesz basis.

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.

E-integrability of trace-class operator-valued functions

Positivity ◽  
2020 ◽  
Author(s):  
Minoo Khosheghbal-Ghorabayi ◽  
Ghorban-Ali Bagheri-Bardi
Keyword(s):  
Trace Class ◽  
Trace Class Operator ◽  
Class Operator

scholarly journals Fredholm Determinant of an Integral Operator Driven by a Diffusion Process

2008 ◽  
Vol 2008 ◽  
pp. 1-17
Author(s):  
Adrian P. C. Lim
Keyword(s):  
Integral Operator ◽  
Diffusion Process ◽  
Fredholm Determinant ◽  
Trace Class ◽  
Trace Class Operator ◽  
Class Operator ◽  
Locally Lipschitz ◽  
Positive Matrices ◽  
The Law

This article aims to give a formula for differentiating, with respect to , an expression of the form , where and is a diffusion process starting from , taking values in a manifold, and the expectation is taken with respect to the law of this process. is a trace class operator defined by , where , are locally Lipschitz, positive matrices.

g-FRAMES AND MODULAR RIESZ BASES IN HILBERT C*-MODULES

2012 ◽  
Vol 10 (02) ◽  
pp. 1250013 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Hilbert Spaces ◽  
Riesz Basis ◽  
Riesz Bases ◽  
Parseval Frame ◽  
C Algebra ◽  
Perturbation Result ◽  
Natural Way

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces. We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

scholarly journals On zero-trace commutators

1986 ◽  
Vol 34 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Adjoint Operator ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Trace Class Operator ◽  
Class Operator ◽  
Closed Disc ◽  
Schmidt Operator

We present some results concerning the trace of certain trace class commutators of operators acting on a separable, complex Hilbert space. It is shown, among other things, that if X is a Hilbert-Schmidt operator and A is an operator such that AX − XA is a trace class operator, then tr (AX − XA) = 0 provided one of the following conditions holds: (a) A is subnormal and A*A − AA* is a trace class operator, (b) A is a hyponormal contraction and 1 − AA* is a trace class operator, (c) A2 is normal and A*A − AA* is a trace class operator, (d) A2 and A3 are normal. It is also shown that if A is a self - adjoint operator, if f is a function that is analytic on some neighbourhood of the closed disc{z: |z| ≥ ||A||}, and if X is a compact operator such that f (A) X − Xf (A) is a trace class operator, then tr (f (A) X − Xf (A))=0.

On the Trace of a Trace Class Operator

1974 ◽  
Vol 6 (1) ◽  
pp. 47-50 ◽  
Author(s):  
J. A. Erdös
Keyword(s):  
Trace Class ◽  
Trace Class Operator ◽  
Class Operator
Sign in / Sign up

Export Citation Format

Share Document