Spectral approximation for Segal-Bargmann space Toeplitz operators

Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk ${\displaystyle D^N}$ or on the Segal-Bargmann space over ${\displaystyle {ℂ}^N}$. Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large "finite section", that is, by the compression ${\displaystyle A_n}$ of A to the linear span of the monomials ${\displaystyle \{z_1^{k_1}...z_N^{k_N}:0\le k_j\le n\}}$. Unfortunately, in general the spectrum of ${\displaystyle A_n}$ does not mimic the spectrum of A as n goes to infinity. However, in the same way as in numerical analysis the question "Is A invertible?" is replaced by the question "What is ${\displaystyle \parallel A^{-1}\parallel }$?", it turns out that the mysteries of Λ(A_{n}) for large n may be much better understood by considering the pseudospectrum of ${\displaystyle A_n}$ rather than the usual spectrum. For ε > 0, the ε-pseudospectrum of an operator T is defined as the set ${\displaystyle {\Lambda }_{\epsilon }\left(T\right)=\{\lambda \in \{ℂ\}:\parallel {(T-\lambda I)}^{-1}\parallel \ge \frac{1}{\epsilon }\}}$. Our central result says that the limit ${\displaystyle \underset{n\to \infty }{lim}\parallel A_n^{-1}\parallel }$ exists and is equal to the maximum of ${\displaystyle \parallel A^{-1}\parallel }$ and the norms of the inverses of ${\displaystyle 2^N-1}$ other operators associated with A. This result implies that for each ε > 0 the ε-pseudospectrum of ${\displaystyle A_n}$ approaches the union of the ε-pseudospectra of A and the ${\displaystyle 2^N-1}$ operators associated with A. If in particular N = 1, it follows that Λ(A) = lim_{ε → 0} lim_{n → ∞} Λ_{ε}(A_{n}), whereas, as already said, the equality ${\displaystyle \Lambda \left(A\right)=\underset{n\to \infty }{lim}\underset{\epsilon \to 0}{lim}{\Lambda }_{\epsilon }\left(A_n\right)(=\underset{n\to \infty }{lim}\Lambda \left(A_n\right))}$ is in general not true. The paper does not aim at completeness, its purpose is rather to outline the ideas behind the theory, and especially, to illustrate the power of C*-algebra techniques for tackling the problem of spectral approximation. We therefore focus our attention on Segal-Bargmann space Toeplitz operators. Our main theorems include Fredholm criteria for such operators, results on the norms of the inverses of their large truncations, as well as the foundation of several approximation methods for solving equations with a Segal-Bargmann space Toeplitz operator.