Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ${ℂ}^n$

Abstract

We introduce the so-called extended Lagrangian symbols, and we prove that the C ∗ -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂ n is isomorphic and isometric to the C ∗ -algebra of matrix-valued functions on a certain compactification of ℝ n obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.

Radial operators on polyanalytic weighted Bergman spaces

“…Recently, various authors investigated the behavior of vertical and angular Toeplitz operators acting on the polyanalytic Bergman or Fock spaces, supposing that the generating symbols are bounded and have limits at the boundary of the domain [2,23,31,32,34]. In particular, Ramírez-Ortega and Sánchez-Nungaray proved [31] that the C*-algebra generated by such vertical Toeplitz operators in the n-analytic Bergman space over the upper half-plane is isomerically isomorphic to the C*-algebra of continuous matrix functions on (0, +∞) having scalar limits at 0 and +∞.…”

“…Recall that L ∞ lim ([0, 1)) is defined by (2). In this paper, we consider the Toeplitz operators T n,α, a , where a ∈ L ∞ lim ([0, 1)).…”