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

Abstract

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=∞. The C⁎ algebra generated with this kind of symbols is isomorphic to the C⁎-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.

Radial operators on polyanalytic Bargmann-Segal-Fock spaces

“…3. In [23,26,29], all matrices have the same order n, whereas in our Theorem 1.1 the matrices have orders 1, 2, . .…”

“…Comparing our Theorem 1.1 with the main results of [23,26,29], we would like to point out three differences.…”

“…Hutník, Hutníková, Ramírez Ortega, Sánchez-Nungaray, Loaiza, and other authors [18,19,23,26,29] studied vertical and angular Toeplitz operators in polyanalytic and truepolyanalytic spaces, Bergman and Fock. In particular, vertical Toeplitz operators in the n-analytic Bergman space over the upper half-plane are represented in [26] as n×n matrices whose entries are continuous functions on (0, +∞), with some additional properties at 0 and +∞.…”

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

“…They considered an orthonormal basis of normalized complex Hermite polynomials to prove that the radial operators are diagonal. In [4], the authors studied Toeplitz operators acting on the one-dimensional poly-Fock space with horizontal symbols such that the limit values at x = ∞ and x = −∞ exist. They proved that the C * -algebra generated with this class of symbols is isomorphic to the C * -algebra of functions on ℝ with values on the m × m matrices, whose limit value at x = ∞ and x = −∞ are equal to some scalar multiples of the identity matrix.…”

Toeplitz Operators Acting on True-Poly-Bergman Type Spaces of the Two-Dimensional Siegel Domain: Nilpotent Symbols

“…On the other hand, in [17,18], the authors studied Toeplitz operators on the Fock space F 2 1 ðℂÞ with radial and bounded horizontal symbols; they found that spectral functions are uniformly continuous with respect to an adequate metric. Taking horizontal symbols having one-side limits at x = ±∞, in [19,20], the authors studied Toeplitz operators acting on poly-Fock spaces F 2 k ðℂÞ; they found the spectrum of the C * -algebra generated by such Toeplitz operators. Even though the authors described the C * -algebras generated by all spectral functions, the spectrum of the algebras is not fully understood in some cases; for this reason, additional conditions on the symbols are imposed.…”