Convergence of Iterates of Normal Operators in L^2 Spaces

Year 2024, Volume: 14 Issue: 2, 181 - 188, 31.07.2024

Heybetkulu Mustafayev

Abstract

Let (Ω, Σ, m) be a measure space with m being an σ-finite positive measure
and let N be a normal operator on L2(Ω, Σ, m). In this note, we study strong and almost
everywhere convergences of the sequences {ϕ (N)nf}n∈N in L2(Ω, Σ, m) spaces, where
ϕ is a continuous function on the spectrum of N.

Keywords

L 2 -space, normal operator, norm convergence, almost everywhere convergence.

References

• [1] A. Bellow, R. Jones, J. Rosenblatt, Almost everywhere convergence of convolutions powers, Erg. Theory and Dynam. Systems, 14, 1994, 415-432.
• [2] G. Cohen, Ch. Cuny, M. Lin, Almost everywhere convergence of powers of some positive Lp−contractions, J. Math. Anal. Appl., 420, 2014, 1129-1153.
• [3] J.B. Conway, A Course in Functional Analysis, Grad. Texts in Math., Springer-Verlag, 1985.
• [4] J-P Conze, M. Lin, Almost everywhere convergence of convolutions powers on compact Abelian groups, Ann. I’nstitut Henri Poincar´e, 49, 2013, 550-568.
• 5] R. Jones, J. Rosenblatt, A. Tempelman, Ergodic theorems for convolutions of a measure on a group, Illinois J. Math., 38, 1994, 521-553.
• [6] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, New York, 1985.
• [7] R. Larsen, Banach Algebras, Marcel-Dekker Inc., New York, 1973.
• [8] M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc., 43, 1974,337-340.
• [9] H. Mustafayev, Convergence of iterates of convolution operators in Lp spaces, Bull. Sci. Math., 152, 2019, 61-92.
• [10] H. Mustafayev, On the convergence of iterates of convolution operators in Banach spaces, Math. Scand., 126, 2020, 339-366.