

Functional analysis and its applications
April 19, 2018 10:30






Individual ergodic theorem in symmetric ideals of compact operators
A. N. Azizov^{}, V. I. Chilin^{} ^{} National University of Uzbekistan named after M. Ulugbek, Tashkent

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Abstract:
Let $H$ be a complex separable infinitedimensional Hilbert space, let $E$ be a fully symmetric sequence space, and let $\mathcal{C}_E$ be a symmetric ideal of compact operators in $H$ associated with $E$. It is proved that the averages $A_n(T) =\frac1{n + 1}\sum\limits_{k = 0}^n T^k $ for any positive DunfordSchwartz operator $T: \mathcal{C}_E \to \mathcal{C}_E$ converge in $\mathcal{C}_E$ with respect to the uniform norm. In addition, we show that for every noncompact bounded linear operator $x$ acting in $H$ there exists a positive DunfordSchwartz operator $T$ such that the averages $A_n(T)$ do not convergence with respect to the uniform norm.
Language: English

