Infinite Dimensional Analysis, Quantum Probability and Related Topics

We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.

A canonical definition of the joint semi-quantum operators of a finite family of random variables, having finite moments of all orders, is given first in terms of an existence and uniqueness theorem. Then two characterizations, one for the polynomially symmetric, and another for the polynomially factorizable probability measures, having finite moments of all orders, are presented.

In this work, we establish Titchmarsh–Weyl theory for singular $q$-Dirac systems. Thus, we extend classical Titchmarsh–Weyl theory for Dirac systems to $q$-analogue of this system. We show that it does not occur for the limit-circle case for the $q$-Dirac system.

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

In this work, we study the spectral statistics for Anderson model on ${\ell }^2(\mathbb{N})$ with decaying randomness whose single-site distribution has unbounded support. Here, we consider the operator $H^{\omega }$ given by ${(H^{\omega }u)}_n=u_{n+1}+u_{n-1}+a_n{\omega }_n{u}_n$, $a_n\sim n^{-\alpha }$ and $\{{\omega }_n\}$ are real i.i.d random variables following symmetric distribution $\mu$ with fat tail, i.e. $\mu ({(-R,R)}^c)<\frac{C}{R^{\delta }}$ for $R\gg 1$, for some constant $C$. In case of $\alpha -\frac{1}{\delta }>\frac{1}{2}$, we are able to show that the eigenvalue process in $(-2,2)$ is the clock process.

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

In this paper, we study compound bi-free Poisson distributions for two-faced families of random variables. We prove a Poisson limit theorem for compound bi-free Poisson distributions. Furthermore, a bi-free infinitely divisible distribution for a two-faced family of self-adjoint random variables can be realized as the limit of a sequence of compound bi-free Poisson distributions of two-faced families of self-adjoint random variables. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a two-faced family of finitely many random variables, which has an almost sure random matrix model, and the left random variables commute with the right random variables in the two-faced family, then we can construct a random bi-matrix model for the compound bi-free Poisson distribution. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a commutative pair of random variables, we can construct an asymptotic bi-matrix model with entries of creation and annihilation operators for the compound bi-free Poisson distribution.

We study the analytic characterization of $S$-transform in a general setting of white noise functionals. Then, the measurability condition of the norms generating the underlining locally convex space is a necessary and sufficient condition for the analytic characterization of the $S$-transform in terms of analytic and growth conditions.