Complete spectral area estimates and self-commutators (Q1122116)

For a bounded linear operator \(T\) on a Hilbert space a set \(X\) in the complex plane including the spectrum \(\sigma(T)\) is called a complete spectral set if \(\| F(T)\| \leq \sup \{\| F(\lambda)\|:\lambda \in X\}\) for any \(k\times k\) matrix with elements in the algebra \(\mathrm{rat}(X)\) of all rational functions with poles off \(X\) and all \(k\geq 1\). In this case the estimate \(\mathrm{dist}(T^*,\mathcal A)\leq \{\mathrm{Area}(X)/\pi \}^{1/2}\) is shown, where \(\mathcal A\) denotes the norm closure of \(\{f(T): f\in\mathrm{rat}(X)\}\). If \(K\) is another bounded linear operator commuting with \(T\) then \(\| T^*K-KT^*\| \leq 2\{\mathrm{Area}(X)/\pi \}^{1/2}\| K\|\). For subnormal and hyponormal operators, respectively, these results can be improved in that \(X\) may be replaced by \(\sigma(T)\). A main tool for the proof is Alexander's estimate [\textit{H. Alexander}, J. Funct. Anal. 13, 13--19 (1973; Zbl 0256.32009)], for which a new proof is included.