1 Banach Spaces.- 2 Banach Algebras.- 3 Geometry of Hilbert Space.- 4 Operators on Hilbert Space and C*-Algebras.- 5 Compact Operators, Fredholm Operators, and Index Theory.- 6 The Hardy Spaces.- 7 Toeplitz Operators.- References.

On certain Toeplitz operators in two variables.

Banach Algebra Techniques in Operator Theory

The interplay between the theory of Toeplitz operators on the circle and the theory of pseudodifferential operators on the line (i. e. Wiener-Hopf operators) is by now well-known and well-understood. In this article we show that there is a parallel situation in higher dimensions. To begin with, by using pseudodifferential multipliers, one can simplify the composition rules for Toeplitz operators, (§ 3), and describe precisely how Toeplitz operators of “Bergmann type” are related to Toeplitz operators of “Szego type” (§ 9). Furthermore, it turns out that the ring of pseudodifferential operators on a compact manifold, M, is isomorphic with the ring of Toeplitz operators on an appropriate Grauert tube about M (§ § 4–6), and the ring of Weyl operators on ℝn is isomorphic with the ring of Toeplitz operators on the complex ball in ℂn (§ § 7–10).

TOEPLITZ OPERATORS IN n-DIMENSIONS

https://www.ams.org/tran/1971-158-01/S0002-9947-1971-0288591-1/S0002-9947-1971-0288591-1.pdf

On the *-algebra of Toeplitz operators on the quarterplane

/pdf/finite-linear-groups-of-degree-seven-ii-23ghtfgzrd.pdf

Finite linear groups of degree seven. II.

/pdf/toeplitz-operators-in-a-quarter-plane-5dhgm57tj3.pdf

Toeplitz operators in a quarter-plane

/pdf/on-certain-toeplitz-operators-in-two-variables-50o52u6rr6.pdf

/pdf/inversion-of-toeplitz-matrices-ii-31r27zv3pq.pdf

Inversion of Toeplitz matrices II

Strang has discussed stability of difference equations whose solutions satisfy the special boundary condition u=O on and outside of the boundaries. We shall use Toeplitz matrices to generalize this theory to systems of equations in one space variable with arbitrary homogeneous boundary conditions. The discussion will be confined to problems in the quarter plane with constant coefficients. The results can be easily generalized to variable coefficient and two point boundary value problems, using Kreiss' method [1] and/or Strang's in [2] and [3]. We shall derive Kreiss' sufficient conditions for stability of dissipative hyperbolic systems with constant coefficients as a corollary to a more general result. In particular, the condition of dissipativity is replaced by a weaker condition. We treat the explicit case in the main part of this work and add the implicit case as an appendix in part 7. The main results are stated in XIX and XXVIII. Kreiss' Theorem is derived in XXII. We give nondissipative examples in XXIII and XXIX. We hope to extend this technique to include problems in several space variables in the near future.

/pdf/systems-of-difference-equations-with-general-homogeneous-3rq89o73ti.pdf

On certain Toeplitz operators in two variables.

Citations

Banach Algebra Techniques in Operator Theory

TOEPLITZ OPERATORS IN n-DIMENSIONS

On the *-algebra of Toeplitz operators on the quarterplane

Finite linear groups of degree seven. II.

Toeplitz operators in a quarter-plane

References

Inversion of Toeplitz matrices II

Systems of difference equations with general homogeneous boundary conditions

Related Papers (5)

On Toeplitz operators

Systems of Toeplitz operators on

Local Toeplitz Operators

On the local theory of Toeplitz operators

A note on toeplitz operators