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Spectral radius of weighted composition operators in L p -spaces

Krzysztof Zajkowski — 2010

Studia Mathematica

We prove that for the spectral radius of a weighted composition operator a T α , acting in the space L p ( X , , μ ) , the following variational principle holds: l n r ( a T α ) = m a x ν M ¹ α , e X l n | a | d ν , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, M ¹ α , e is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where = n = 0 α - n ( ) . This considerably extends the range of validity of the above formula, which was previously known in the case...

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

Krzysztof Zajkowski — 2005

Banach Center Publications

We consider operators acting in the space C(X) (X is a compact topological space) of the form A u ( 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 l n ( r ( A ) ) = λ ( φ ) = m a x ν M e s k = 1 N X φ k d ν 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...

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