Armando Sánchez-Nungaray

Commutative Algebras of Toeplitz Operators on a Siegel Domain Associated with the Nilpotent Group of Its Biholomorphisms

We describe the commutative Banach (and even C *-) algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group of byholomorphisms of the unit ball in \(\mathbb{C}^n\). The key role in the study is played by the direct integral decomposition of the isomorphic image of the Bergman space (Theorem 2.2), by the symbols \(a\in L_\infty(\mathbb{R}^{l} \times\mathbb{R}_{+})\) of the form (3.1), and by Theorem 3.1 that describes the action of Toeplitz operators with such symbols as a direct integral of certain scalar multiplication operators. Theorem 3.3 describes then the Toeplitz operators with symbols \(b\in L_\infty(\mathbb{C}^{k}\) in the form of a direct integral of Toeplitz operators on the weighted Fock spaces.

Toeplitz Operators with Horizontal Symbols Acting on the Poly-Fock Spaces

We describe the C∗-algebra generated by the Toeplitz operators acting on each poly-Fock space of the complex plane C with the Gaussian measure, where the symbols are bounded functions depending only on x = Re z and have limit values at y = −∞ and y = ∞.TheC∗ algebra generated with this kind of symbols is isomorphic to theC∗-algebra functions on extended reals with values on the matrices of dimension n × n, and the limits at y = −∞ and y = ∞ are scalar multiples of the identity matrix.

Multiresolution Analysis Applied to the Monge-Kantorovich Problem

We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.

Separately radial and radial Toeplitz operators on the projective space and representation theory

We consider separately radial (with corresponding group Tn) and radial (with corresponding group U(n)) symbols on the projective space Pn(C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between Tn and U(n).

Commutative -Algebras of Toeplitz Operators via the Moment Map on the Polydisk

We found that in the polydisk there exist different classes of commutative -algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that the -algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutative -algebra of Toeplitz operators.