In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T (λ) a acting on weighted Bergman spaces A 2 λ (D) over the unit disk D, in dependence on λ, and compare their limit behavior under λ → +∞ with corresponding properties of the initial symbol a. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47A25, 47B10, 46E22.

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After 2000, an interest in the Hausdorff operators grew, first of all in the sense of a diversity of spaces on which these operators were considered. We try to introduce a 'correct' definition of the Hausdorff operator on Euclidean spaces. This is supplemented by a variety of known and possible applications.