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1
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Borel methods of summability and ergodic theorems

100%
Jajte R.
Annales Polonici Mathematici
|
2002
|
tom 78
|
nr 2
175-180
EN
Passing from Cesàro means to Borel-type methods of summability we prove some ergodic theorem for operators (acting in a Banach space) with spectrum contained in ℂ∖(1,∞).
2
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Automorphisms of the algebra of operators in $𝕃^{p}$ preserving conditioning

100%
Jajte R.
Colloquium Mathematicum
|
2010
|
tom 120
|
nr 2
263-266
EN
Let α be an isometric automorphism of the algebra $𝔹_{p}$ of bounded linear operators in $𝕃^{p}[0, 1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].
3
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Pointwise limit theorem for a class of unbounded operators in $𝕃^{r}$-spaces

100%
Jajte R.
Studia Mathematica
|
2007
|
tom 179
|
nr 1
49-61
EN
We distinguish a class of unbounded operators in $𝕃^{r}$, r ≥ 1, related to the self-adjoint operators in 𝕃². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in $𝕃^{r}$-spaces are applied.
4
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Almost sure approximation of unbounded operators in $L_2 (X,A,μ)$

64%
Jajte R., Paszkiewicz A.
Studia Mathematica
|
1998
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tom 128
|
nr 2
103-120
EN
The possibilities of almost sure approximation of unbounded operators in $L_2(X,A,μ)$ by multiples of projections or unitary operators are examined.
5
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Convergence of orthogonal series of projections in Banach spaces

64%
Jajte R., Paszkiewicz A.
Annales Polonici Mathematici
|
1997
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tom 66
|
nr 1
137-153
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)).
6
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The bundle convergence in von Neumann algebras and their $L_2$-spaces

51%
Hensz E., Jajte R., Paszkiewicz A.
Studia Mathematica
|
1996
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tom 120
|
nr 1
23-46
EN
A stronger version of almost uniform convergence in von Neumann algebras is introduced. This "bundle convergence" is additive and the limit is unique. Some extensions of classical limit theorems are obtained.
7
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Dilation theorems for completely positive maps and map-valued measures

51%
Hensz-Chądzyńska E., Jajte R., Paszkiewicz A.
Banach Center Publications
|
1998
|
tom 43
|
nr 1
231-239
EN
The Stinespring theorem is reformulated in terms of conditional expectations in a von Neumann algebra. A generalisation for map-valued measures is obtained.
8
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The unconditional pointwise convergence of orthogonal seriesin $L_2$ over a von Neumann algebra

51%
Hensz E., Jajte R., Paszkiewicz A.
Colloquium Mathematicum
|
1996
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tom 69
|
nr 2
167-178
EN
The paper is devoted to some problems concerning a convergence of pointwise type in the $L_2$-space over a von Neumann algebra M with a faithful normal state Φ [3]. Here $L_2 = L_2(M,Φ)$ is the completion of M under the norm $x → |x|^2 = Φ(x*x)^{1/2}$.
9
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Gleason measures and some inner products on the algebra of bounded operators on a Hilbert space

45%
Jajte R., Loebl R. I.
Studia Mathematica
|
1981
|
tom 70
|
nr 2
139-145
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