The moments of operator-valued semicircular distribution are calculated and a new relation between random variables which is called semi-independence is introduced. The asymp- totically free matrix models of operator-valued semicircular distribution are given and a method is found to determine the freeness of some semicircular variables.
The free Fisher information of an operator random matrix is studied. When the covariance of a random matrix is a conditional expectation, the free Fisher information of such a matrix is the double of this conditional expectation’s Watatani index.
In this paper, the sum of standard generalized frames of Hilbert W-module is studied intensively by using operator-theoretic-methods, and some conditions are given to assure that the sum of two or more standard generalized frames is a standard generalized frame.
Operator-valued frames are natural generalization of frames that have been used in many applied areas such as quantum computing, packets encoding and sensor networks. We focus on developing the theory about operator-valued frame generators for projective unitary representations of finite or countable groups which can be viewed as the theory of quantum channels with group structures. We present new results for operator-valued frames concerning (general and structured) dilation property, orthogonal frames, frame representation and dual frames. Our results are complementary to some of the recent work of Kaftal, Larson and Zhang, and in some cases our treatment is more elementary and transparent.
