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1997 | 66 | 1 | 137-153
Tytuł artykułu

Convergence of orthogonal series of projections in Banach spaces

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Abstrakty
EN
For a sequence $(A_j)$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums $S_n = ∑^n_{j=1} A_j$ in a "strong" sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of $S_n$ (i.e. $S_{n}f → Af$ μ-a.e. for all f ∈ 𝓓(A)).
Twórcy
  • Institute of Mathematics, Łódź University, S. Banacha 22, 90-238 Łódź, Poland
  • Institute of Mathematics, Łódź University, S. Banacha 22, 90-238 Łódź, Poland
Bibliografia
  • [1] E. Berkson and T. A. Gillespie, Stečkin's theorem, transference and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170.
  • [2] J. L. Ciach, R. Jajte and A. Paszkiewicz, On the almost sure approximation and convergence of linear operators in L₂-spaces, Probab. Math. Statist. 15 (1995), 215-225.
  • [3] H. R. Downson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, New York, 1978.
  • [4] N. Dunford and J. T. Schwartz, Linear Operators Part I: General Theory, Interscience, New York, 1958.
  • [5] N. Dunford and J. T. Schwartz, Linear Operators Part III : Spectral Operators, Interscience, New York, 1971.
  • [6] A. M. Garsia, Topics in Almost Everywhere Convergence, Lectures in Adv. Math. 4, Markham, Chicago, 1970.
  • [7] R. Jajte and A. Paszkiewicz, Almost sure approximation of unbounded operators in L₂(X,𝒜,μ), to appear.
  • [8] R. Jajte and A. Paszkiewicz, Topics in almost sure approximation of unbounded operators in L₂-spaces, in: Interaction between Functional Analysis, Harmonic Analysis, and Probability, Columbia, 1994, Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, 219-228.
  • [9] S. Kaczmarz, Sur la convergence et la sommabilité des développements orthogonaux, Studia Math. 1 (1929), 87-121.
  • [10] J. Marcinkiewicz, Sur la convergence des séries orthogonales, Studia Math. 6 (1933), 39-45.
  • [11] I. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1960), 334-343.
  • [12] I. R. Ringrose, On well-bounded operators, II, Proc. London Math. Soc. 13 (1963), 613-630.
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Bibliografia
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