The search session has expired. Please query the service again.

Convergence of orthogonal series of projections in Banach spaces

Ryszard Jajte; Adam Paszkiewicz

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 137-153
  • ISSN: 0066-2216

Abstract

top
For a sequence ( A j ) of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums S n = ∑ j = 1 n 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 → A f μ-a.e. for all f ∈ (A)).

How to cite

top

Ryszard Jajte, and Adam Paszkiewicz. "Convergence of orthogonal series of projections in Banach spaces." Annales Polonici Mathematici 66.1 (1997): 137-153. <http://eudml.org/doc/269982>.

@article{RyszardJajte1997,
abstract = {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 ∈ (A)).},
author = {Ryszard Jajte, Adam Paszkiewicz},
journal = {Annales Polonici Mathematici},
keywords = {idempotent; mutually orthogonal projections; L₂-space; convergence almost everywhere; -space; idempotent operators},
language = {eng},
number = {1},
pages = {137-153},
title = {Convergence of orthogonal series of projections in Banach spaces},
url = {http://eudml.org/doc/269982},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Ryszard Jajte
AU - Adam Paszkiewicz
TI - Convergence of orthogonal series of projections in Banach spaces
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 137
EP - 153
AB - 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 ∈ (A)).
LA - eng
KW - idempotent; mutually orthogonal projections; L₂-space; convergence almost everywhere; -space; idempotent operators
UR - http://eudml.org/doc/269982
ER -

References

top
  1. [1] E. Berkson and T. A. Gillespie, Stečkin's theorem, transference and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. Zbl0607.47027
  2. [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. Zbl0859.60027
  3. [3] H. R. Downson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, New York, 1978. 
  4. [4] N. Dunford and J. T. Schwartz, Linear Operators Part I: General Theory, Interscience, New York, 1958. Zbl0084.10402
  5. [5] N. Dunford and J. T. Schwartz, Linear Operators Part III : Spectral Operators, Interscience, New York, 1971. Zbl0243.47001
  6. [6] A. M. Garsia, Topics in Almost Everywhere Convergence, Lectures in Adv. Math. 4, Markham, Chicago, 1970. Zbl0198.38401
  7. [7] R. Jajte and A. Paszkiewicz, Almost sure approximation of unbounded operators in L₂(X,𝒜,μ), to appear. Zbl0905.47011
  8. [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. Zbl0874.47008
  9. [9] S. Kaczmarz, Sur la convergence et la sommabilité des développements orthogonaux, Studia Math. 1 (1929), 87-121. Zbl55.0165.01
  10. [10] J. Marcinkiewicz, Sur la convergence des séries orthogonales, Studia Math. 6 (1933), 39-45. 
  11. [11] I. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1960), 334-343. Zbl0104.08902
  12. [12] I. R. Ringrose, On well-bounded operators, II, Proc. London Math. Soc. 13 (1963), 613-630. Zbl0121.33403

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.