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Numerical range of operators acting on Banach spaces

Khadijeh Jahedi, Bahmann Yousefi (2012)

Czechoslovak Mathematical Journal

The aim of the paper is to propose a definition of numerical range of an operator on reflexive Banach spaces. Under this definition the numerical range will possess the basic properties of a canonical numerical range. We will determine necessary and sufficient conditions under which the numerical range of a composition operator on a weighted Hardy space is closed. We will also give some necessary conditions to show that when the closure of the numerical range of a composition operator on a small...

Numerical ranges of some composition operators.

Catherine Finet (2006)

RACSAM

This paper is a short survey on the numerical range of some composition operators. The first part is devoted to composition operators on the Hilbert Hardy space H2 on the unit disk. The results are due to P. Bourdon, J. Shapiro and V. Matache.In the second part we study the numerical range of composition operators on the Hilbert space H2 of Dirichlet series. These results are due to H. Queffélec and the author.The third part is devoted to compactness connected with fixed points in the setting of...

On a binary relation between normal operators

Takateru Okayasu, Jan Stochel, Yasunori Ueda (2011)

Studia Mathematica

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...

On a function that realizes the maximal spectral type

Krzysztof Frączek (1997)

Studia Mathematica

We show that for a unitary operator U on L 2 ( X , μ ) , where X is a compact manifold of class C r , r ∈ ℕ ∪ ∞ , ω , and μ is a finite Borel measure on X, there exists a C r function that realizes the maximal spectral type of U.

On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space

Driss Drissi (1998)

Studia Mathematica

Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.

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