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Spectral approximation for Segal-Bargmann space Toeplitz operators. (English) Zbl 0891.47015

Janas, Jan (ed.) et al., Linear operators. Proceedings of the semester organized at the Stefan Banach International Mathematical Center, Warsaw, Poland, February 7–May 15, 1994. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 38, 25-48 (1997).
Summary: Let \(A\) stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk \(\mathbb{D}^N\) or on the Segal-Bargmann space over \(\mathbb{C}^N\). Even in the case \(N=1\), the spectrum \(\Lambda(A)\) 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 \(A_n\) of \(A\) to the linear span of the monomials \(\{z^{k_1}_1\dots z^{k_N}_N: 0\leq k_j\leq n\}\). Unfortunately, in general the spectrum of \(A_n\) 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 \(|A^{-1}|\)?”, it turns out that the mysteries of \(\Lambda(A_n)\) for large \(n\) may be much better understood by considering the pseudospectrum of \(A_n\) rather than the usual spectrum. For \(\varepsilon>0\), the \(\varepsilon\)-pseudospectrum of an operator \(T\) is defined as the set \(\Lambda_\varepsilon(T)= \{\lambda\in \mathbb{C}:|(T-\lambda I)^{-1}|\geq 1/\varepsilon\}\). Our central result says that the limit \(\lim_{n\to\infty}|A^{-1}_n|\) exists and is equal to the maximum of \(|A^{-1}|\) and the norms of the inverses of \(2^N- 1\) other operators associated with \(A\). This result implies that for each \(\varepsilon>0\) the \(\varepsilon\)-pseudospectrum of \(A_n\) approaches the union of the \(\varepsilon\)-pseudospectra of \(A\) and the \(2^N- 1\) operators associated with \(A\). If in particular \(N=1\), it follows that \[ \Lambda(A)= \lim_{\varepsilon\to 0} \lim_{n\to \infty} \Lambda_\varepsilon(A_n), \] whereas, as already said, the equality \(\Lambda(A)= \lim_{n\to\infty} \lim_{\varepsilon\to 0} \Lambda_\varepsilon(A_n)\) (\(=\lim_{n\to\infty} \Lambda(A_n)\)) 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 trackling 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.
For the entire collection see [Zbl 0863.00036].

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A53 (Semi-) Fredholm operators; index theories
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
30E10 Approximation in the complex plane
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
45H05 Integral equations with miscellaneous special kernels
47N40 Applications of operator theory in numerical analysis
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