Convergence of orthogonal series of projections in Banach spaces
Ryszard Jajte ; Adam Paszkiewicz
Annales Polonici Mathematici, Tome 66 (1997), p. 137-153 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

For a sequence (Aj) of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums Sn=∑j=1nAj 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 Sn (i.e. Snf→Af μ-a.e. for all f ∈ (A)).

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:269982 
@article{bwmeta1.element.bwnjournal-article-apmv66z1p137bwm,
     author = {Ryszard Jajte and Adam Paszkiewicz},
     title = {Convergence of orthogonal series of projections in Banach spaces},
     journal = {Annales Polonici Mathematici},
     volume = {66},
     year = {1997},
     pages = {137-153},
     zbl = {0886.47010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p137bwm}
}

Ryszard Jajte; Adam Paszkiewicz. Convergence of orthogonal series of projections in Banach spaces. Annales Polonici Mathematici, Tome 66 (1997) pp. 137-153. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p137bwm/

[000] [1] E. Berkson and T. A. Gillespie, Stečkin's theorem, transference and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. | Zbl 0607.47027

[001] [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. | Zbl 0859.60027

[002] [3] H. R. Downson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, New York, 1978.

[003] [4] N. Dunford and J. T. Schwartz, Linear Operators Part I: General Theory, Interscience, New York, 1958. | Zbl 0084.10402

[004] [5] N. Dunford and J. T. Schwartz, Linear Operators Part III : Spectral Operators, Interscience, New York, 1971. | Zbl 0243.47001

[005] [6] A. M. Garsia, Topics in Almost Everywhere Convergence, Lectures in Adv. Math. 4, Markham, Chicago, 1970. | Zbl 0198.38401

[006] [7] R. Jajte and A. Paszkiewicz, Almost sure approximation of unbounded operators in L₂(X,𝒜,μ), to appear. | Zbl 0905.47011

[007] [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. | Zbl 0874.47008

[008] [9] S. Kaczmarz, Sur la convergence et la sommabilité des développements orthogonaux, Studia Math. 1 (1929), 87-121. | Zbl 55.0165.01

[009] [10] J. Marcinkiewicz, Sur la convergence des séries orthogonales, Studia Math. 6 (1933), 39-45.

[010] [11] I. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1960), 334-343. | Zbl 0104.08902

[011] [12] I. R. Ringrose, On well-bounded operators, II, Proc. London Math. Soc. 13 (1963), 613-630. | Zbl 0121.33403