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Some theorems of Korovkin type

100%

Hachiro T., Okayasu T.

Studia Mathematica

2003

tom 155

nr 2

131-143

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}(X)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}(Y)$), and $ϕ_{∞}$ a linear isometry from M into C(Y) (resp., $C_{ℝ}(Y)$). We show, under the assumption that $Π_{N} ⊂ Π_{T}$, where $Π_{N}$ is the Choquet boundary for $N = Span(⋃_{1≤n≤∞}Nₙ)$, Nₙ = ϕₙ(M) (n = 1,2,..., ∞), and $Π_{T}$ the Choquet boundary for $T = ϕ_{∞}(S)$, that {ϕₙ(f)} converges pointwise to $ϕ_{∞}(f)$ for any f ∈ M provided {ϕₙ(f)} converges pointwise to ${ϕ_{∞}(f)}$ for any f ∈ S; that {ϕₙ(f)} converges uniformly on any compact subset of $Π_{N}$ to $ϕ_{∞}(f)$ for any f ∈ M provided {ϕₙ(f)} converges uniformly to $ϕ_{∞}(f)$ for any f ∈ S; and that, in the case where S is a function algebra, {ϕₙ} norm converges to $ϕ_{∞}$ on M provided {ϕₙ(f)} norm converges to $ϕ_{∞}$ on S. The proofs are in the spirit of the original one for the theorem of Korovkin.

On a binary relation between normal operators

81%

Okayasu T., Stochel J., Ueda Y.

Studia Mathematica

2011

tom 204

nr 3

247-264

The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that $E_{A}(Δ) ≤ T*E_{B}(Δ)T$ for all Borel subset Δ of the complex plane ℂ, where $E_{A}$ and $E_{B}$ are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is a partial order modulo unitary equivalence.

Ograniczanie wyników

2 Studia Mathematica

2 Okayasu T.

1 Hachiro T.

1 Stochel J.

1 Ueda Y.

1 2011

1 2003

Wyniki wyszukiwania

Some theorems of Korovkin type

On a binary relation between normal operators