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Some results related with statistical convergence and Berezin symbols. (English) Zbl 1055.40001

A functional Hilbert space is a Hilbert space \(\mathcal{H}\) of complex-valued functions on some set \(\Omega\) such that the evaluation functional \(f \to f(\lambda)\) is continuous for each \(\lambda \in \Omega\). Moreover, for each \(\lambda \in \Omega\) there exists a unique function \(k_{\lambda} \in \mathcal{H}\) such that \(f(\lambda) = (f, k_{\lambda})\) for all \(f \in \mathcal{H}\). For a bounded linear operator \(A\) on \(\mathcal{H}\), the Berezin symbol of \(A\) is the function \(\widetilde{A}\) defined by \(\widetilde{A}(\lambda) = (A\frac{k_{\lambda}}{\| k_{\lambda}\| }, \frac{k_{\lambda}}{\| k_{\lambda}\| })\), \(\lambda \in \Omega\). Let \(\{a_n\}\) be a bounded sequence of complex numbers. The authors show that \(\sum_{n = 0}^{\infty}a_n\) is discretely statistically Abel convergent if and only if the statistical limit of \(\mathcal{\widetilde{D}}_a(\sqrt{t_m})/(1 - t_m)\) is finite, where \(\mathcal{\widetilde{D}}_a\) is the diagonal operator with respect to the standard bases in \(\mathcal{H}\), and \(\{t_m\}\) is a sequence in \((0, 1)\) that statistically converges to 1. They obtain necessary and sufficient conditions, in terms of \(\mathcal{\widetilde{D}}\) for a compact operator to be in the Schatten class \(\mathfrak{T}_p\). Let \(A\) be the weak statistical limit of a sequence of bounded linear operators \(\{A_n\}\). Then they show that \(A\) is compact and \(\sum_{j = 1}^ns_j(A) \leq \sum_{j = 1}^n\sup_ms_j(A_m)\) \((n = 1, 2, \ldots)\), where \(s_i(A)\) denotes the largest eigenvalue of \((A^*A)^{1/2}\).

MSC:

40A05 Convergence and divergence of series and sequences
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
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