Let A be a m × m complex matrix with zero trace. Then there are m ×m matrices B and C such that A = [B,C] and ‖B‖‖C‖2 ≤ (logm + O(1))‖A‖2 where ‖D‖ is the norm of D as an operator on `2 and ‖D‖2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely there is a zero trace m × m matrix A such that whenever A = [B,C], ‖B‖‖C‖2 ≥ | logm−O(1)|‖A‖2 for some abso...

We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. give a short, elementary proof that finitely generated nilpotent are HS-stable. investigate the permanence HS-stability under central quotients by showing is preserved finite quotients, but not in general. characterization for semidirect products G⋊γZ with G abelian. use it to construct first ex...

abstract. certain facts about frames and generalized frames (g- frames) are extended for the g-frames for hilbert c*-modules. it is shown that g-frames for hilbert c*-modules share several useful properties with those for hilbert spaces. the paper also character- izes the operators which preserve the class of g-frames for hilbert c*-modules. moreover, a necessary and suffcient condition is ob- ...

Notions of guardedness serve to delineate the admissibility of cycles, e.g. in recursion, corecursion, iteration, or tracing. We introduce an abstract notion of guardedness structure on a symmetric monoidal category, along with a corresponding notion of guarded traces, which are defined only if the cycles they induce are guarded. We relate structural guardedness, determined by propagating guard...

In this work we will investigate how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases as an extension of the known method of matrix representation by ONBs. We will give basic definitions of the functions connecting infinite matrices defining bounded operators on l and operators onH. We will show some structural results and give some...

Controlled frames in Hilbert spaces have been recently introduced by P. Balazs and etc. for improving the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper we develop a theory based on g-fusion frames on Hilbert spaces, which provides exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In part...

In this   paper,  we   present  some  refinements  of the   famous Young  type  inequality.   As  application  of   our   result, we  obtain  some  matrix inequalities   for   the  Hilbert-Schmidt norm  and   the  trace   norm. The results    obtained   in  this  paper  can  be   viewed   as  refinement  of  the   derived  results   by  H.  Kai  [Young  type  inequalities  for matrices,  J.  Ea...

In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form as dX(t) + (∫ t 0 b(t− s)AX(s) ds ) dt = dWQ (t), t ∈ (0, T ]; X(0) = X0 ∈ H, where WQ is a Q-Wiener process on the Hilbert space H and where...

In this letter we discuss a new entanglement measure. It is based on the Hilbert-Schmidt norm of operators. We give an explicit formula for calculating the entanglement of a large set of states on C 2 C 2. Furthermore we nd some relations between the entanglement of relative entropy and the Hilbert-Schmidt entanglement. A rigorous deenition of partial transposition is given in the appendix.