In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases. Then we construct Bessel sequences frames and Riesz bases for the class of Hilbert Schmidt operators using the tensor product of such sequences in the original...

Abstract We give a new class of equivalent norms for modulation spaces by replacing the window short-time Fourier transform Hilbert–Schmidt operator. The main result is applied to Cohen’s time-frequency distributions, Weyl operators and localization operators. In particular, any positive distribution with Schwartz kernel can be used an norm spaces. also obtain description as Wiener amalgam oper...

Elaborating on our joint work with Abramsky in [2,3] we further unravel the linear structure of Hilbert spaces into several constituents which play different roles in quantum mechanics. Some prove to be very crucial for particular features of the theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases. In particular do we pass f...

The idea that it should be easier to learn several tasks if they are related in some way is quite intuitive and has been found to work in many practical settings. There has been some interest in obtaining theoretical results to better understand this phenomenon (e.g. [3, 4]). Maurer [4] considers the case when the “relatedness” of the tasks is captured by requiring that all tasks share a common...

Unstable dynamical systems can be viewed from a variety of perspectives. We discuss the potential of an inputoutput map associated with an unstable system to represent a bounded map from L2(R) to itself and then develop criteria for optimal reduced order approximations to the original (unstable) system with respect to an L2-induced Hilbert-Schmidt norm. Our optimality criteria extend the Meier-...

We describe a method for causal inference that measures the strength of statistical dependence by the Hilbert-Schmidt norm of kernelbased conditional cross-covariance operators. We consider the increase of the dependence of two variables X and Y by conditioning on a third variable Z as a hint for Z being a common effect of X and Y . Based on this assumption, we collect “votes” for hypothetical ...

Bužek and Hillery proposed a universal quantum-copying machine (UQCM) (i.e., transformation) to analyze the possibility of cloning arbitrary states. The UQCM copies quantum-mechanical states with the quality of its output does not depend on the input. We propose a slightly different transformation to analyze a restricted set of input states. We impose the conditions (I) the density matrices of ...