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1
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Some monotonicity and limit results for the regularised incomplete gamma function

100%
Chojnacki W.
Annales Polonici Mathematici
|
2008
|
tom 94
|
nr 3
283-291
EN
Letting P(u,x) denote the regularised incomplete gamma function, it is shown that for each α ≥ 0, P(x,x+α) decreases as x increases on the positive real semi-axis, and P(x,x+α) converges to 1/2 as x tends to infinity. The statistical significance of these results is explored.
2
Content available remote

On operator-valued cosine sequences on UMD spaces

100%
Chojnacki W.
Studia Mathematica
|
2010
|
tom 199
|
nr 3
267-278
EN
A two-sided sequence $(cₙ)_{n∈ℤ}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m} + c_{n-m} = 2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence $(cₙ)_{n∈ℤ}$ is bounded if $sup_{n∈ℤ} ||cₙ|| < ∞$. A (bounded) group decomposition for a cosine sequence $c = (cₙ)_{n∈ℤ}$ is a representation of c as $cₙ = (bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $sup_{n∈ℤ} ||bⁿ|| < ∞$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the so-called standard group decomposition. Here it is shown that if X is a complex UMD Banach space and, with 𝓛(X) denoting the algebra of all bounded linear operators on X, if c is an 𝓛(X)-valued bounded cosine sequence, then the standard group decomposition of c is bounded.
3
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Erratum à l'article "Sur un théorème de Day, un théorème de Mazur-Orlicz et une généralisation de quelques théorèmes de Silverman" (Colloquium Mathematicum 50 (1986), 257-262)

100%
Chojnacki W.
Colloquium Mathematicum
|
1992
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tom 63
|
nr 1
139
4
Content available remote

On group decompositions of bounded cosine sequences

100%
Chojnacki W.
Studia Mathematica
|
2007
|
tom 181
|
nr 1
61-85
EN
A two-sided sequence $(cₙ)_{n∈ℤ}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m} + c_{n-m} = 2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence $(cₙ)_{n∈ℤ}$ is bounded if $sup_{n∈ℤ} ||cₙ|| < ∞$. A (bounded) group decomposition for a cosine sequence $c = (cₙ)_{n∈ℤ}$ is a representation of c as $cₙ = (bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $sup_{n ∈ ℤ} ||bⁿ|| < ∞$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred to as a standard group decomposition. The present paper reveals various classes of bounded operator-valued cosine sequences for which the standard group decomposition is bounded. One such class consists of all bounded ℒ(X)-valued cosine sequences $(cₙ)_{n∈ℤ}$, with X a complex Banach space and ℒ(X) the algebra of all bounded linear operators on X, for which c₁ is scalar-type prespectral. Every bounded ℒ(H)-valued cosine sequence, where H is a complex Hilbert space, falls into this class. A different class of bounded cosine sequences with bounded standard group decomposition is formed by certain ℒ(X)-valued cosine sequences $(cₙ)_{n∈ℤ}$, with X a reflexive Banach space, for which c₁ is not scalar-type spectral-in fact, not even spectral. The isolation of this class uncovers a novel family of non-prespectral operators. Examples are also given of bounded ℒ(H)-valued cosine sequences, with H a complex Hilbert space, that admit an unbounded group decomposition, this being different from the standard group decomposition which in this case is necessarily bounded.
5
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Sur un théorème de Day, un théorème de Mazur-Orlicz et une généralisation de quelques théorèmes de Silverman

75%
Chojnacki W.
Colloquium Mathematicum
|
1985-1986
|
tom 50
|
nr 2
257-262
6
Content available remote

Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space

64%
Bobrowski A., Chojnacki W.
Studia Mathematica
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2013
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tom 217
|
nr 3
219-241
EN
We show that if the set of all bounded strongly continuous cosine families on a Banach space X is treated as a metric space under the metric of the uniform convergence associated with the operator norm on the space 𝓛(X) of all bounded linear operators on X, then the isolated points of this set are precisely the scalar cosine families. By definition, a scalar cosine family is a cosine family whose members are all scalar multiples of the identity operator. We also show that if the sets of all bounded cosine families and of all bounded strongly continuous cosine families on an infinite-dimensional separable Banach space X are viewed as topological spaces under the topology of the uniform convergence associated with the strong operator topology on 𝓛(X), then these sets have no isolated points. We present counterparts of all the above results for semigroups and groups of operators, relating to both the norm and strong operator topologies.
7
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On some functional equation generalizing Cauchy's and d'Alembert's functional equations

63%
Chojnacki W.
Colloquium Mathematicum
|
1988
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tom 55
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nr 1
169-178
8
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Sur certains éléments ergodiques dans $l^∞(G)$

44%
Chojnacki W.
Colloquium Mathematicum
|
1985-1986
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tom 50
|
nr 1
121-124
9
Content available remote

An analogue of the argument theorem of Bohr and its application

44%
Chojnacki W.
Studia Mathematica
|
1984
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tom 79
|
nr 1
51-59
10
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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On some distal functions

32%
Chojnacki W.
Colloquium Mathematicum
|
1989-1990
|
tom 58
|
nr 2
301-304
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