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Anderson’s theorem and \(A\)-spectral radius bounds for semi-Hilbertian space operators. (English) Zbl 1517.47007

Summary: Let \(\mathcal{H}\) be a complex Hilbert space and let \(A\) be a positive bounded linear operator on \(\mathcal{H} \). Let \(T\) be an \(A\)-bounded operator on \(\mathcal{H} \). For \(\mathrm{rank}(A) = n < \infty \), we show that if \(W_A(T) \subseteq \overline{\mathbb{D}}( = \{\lambda \in \mathbb{C} : | \lambda | \leq 1 \})\) and \(W_A(T)\) intersects \(\partial \mathbb{D}( = \{\lambda \in \mathbb{C} : | \lambda | = 1 \})\) at more than \(n\) points, then \(W_A(T) = \overline{\mathbb{D}} \). In particular, when \(A\) is the identity operator on \(\mathbb{C}^n\), then this leads to Anderson’s theorem in the complex Hilbert space \(\mathbb{C}^n\). We introduce the notion of \(A\)-compact operators to study analogous result when the space \(\mathcal{H}\) is infinite dimensional. Further, we develop an upper bound for the \(\mathbb{A} \)-spectral radius of \(n \times n\) operator matrices with entries are commuting \(A\)-bounded operators, where \(\mathbb{A} = \operatorname{diag}(A, A, \dots, A)\) is an \(n \times n\) diagonal operator matrix. Several inequalities involving \(A\)-spectral radius of \(A\)-bounded operators are also given.

MSC:

47A12 Numerical range, numerical radius
47A63 Linear operator inequalities
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