trace class operator
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Positivity ◽  
2020 ◽  
Author(s):  
Minoo Khosheghbal-Ghorabayi ◽  
Ghorban-Ali Bagheri-Bardi


Author(s):  
E. Osgooei ◽  
A. Rahimi

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.





2008 ◽  
Vol 2008 ◽  
pp. 1-17
Author(s):  
Adrian P. C. Lim

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.





1986 ◽  
Vol 34 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Fuad Kittaneh

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.



1974 ◽  
Vol 6 (1) ◽  
pp. 47-50 ◽  
Author(s):  
J. A. Erdös


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