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Spectral radius of weighted composition operators in $L^{p}$-spaces

100%

Zajkowski K.

Studia Mathematica

2010

tom 198

nr 3

301-307

We prove that for the spectral radius of a weighted composition operator $aT_{α}$, acting in the space $L^{p}(X,𝓑,μ)$, the following variational principle holds: $ln r (aT_{α}) = max_{ν ∈ M¹_{α,e}} ∫_{X} ln|a|dν$, where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, $M¹_{α,e}$ is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and $𝓑_{∞}$-measurable function, where $𝓑_{∞}= ⋂_{n=0}^{∞} α^{-n}(𝓑)$. This considerably extends the range of validity of the above formula, which was previously known in the case when α is a homeomorphism.

Spectral radius of operators associated with dynamical systems in the spaces C(X)

100%

Zajkowski K.

Banach Center Publications

2005

tom 67

nr 1

397-403

We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au(x) = (∑_{k=1}^{N} e^{φ_k}T_{α_k})u(x) = ∑_{k=1}^{N} e^{φ_k(x)}u(α_k(x))$, u ∈ C(X), where $φ_k ∈ C(X)$ and $α_k: X → X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = (φ_k)_{k=1}^{N}$. We prove that $ln(r(A)) = λ(φ) = max_{ν∈Mes} {∑_{k=1}^{N} ∫_{X} φ_{k}dν_{k} - λ*(ν)}$, where Mes is the set of all probability vectors of measures $ν = (ν_k)_{k=1}^{N}$ on X × {1,..., N} and λ* is some convex lower-semicontinuous functional on $(C^N(X))*$. In other words λ* is the Legendre conjugate to λ.

Ograniczanie wyników

1 Banach Center Publications

1 Studia Mathematica

2 Zajkowski K.

1 2010

1 2005

Wyniki wyszukiwania

Spectral radius of weighted composition operators in $L^{p}$-spaces

Spectral radius of operators associated with dynamical systems in the spaces C(X)