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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 , 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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